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2
7/19/2019 10:46:21Prague Set Theory SeminarWednesday seminarDear all,

The seminar meets on Wednesday July 24th at 11:00 in the Institute of
Mathematics CAS, Zitna 25, seminar room, 3rd floor, front building.

Program: Jindřich Zapletal -- Chromatic numbers of locally countable
hypergraphs

I will introduce several variants of chromatic numbers of Borel locally
countable hypergraphs and show how they can be separated in choiceless
set theory.


Best,
David
3
7/19/2019 0:00:50Toronto Set Theory SeminarNo set theory seminar this weekHi everyone,

There will not be a set theory seminar on
19 July.

If someone would like to give a talk in the near future,
please let me know.

Osvald Guzman
4
7/15/2019 22:14:59NUS Logic SeminarLogic Seminar Talk at NUS todayInvitation to the Logic Seminar at the National University of Singapore

Date: Tuesday, 16 July 2019, 16:00 hrs

Room: S17#04-06, Department of Mathematics, NUS

Speaker: Andre Nies

Title: Random sequences of quantum bits

URL: http://www.comp.nus.edu.sg/~fstephan/logicseminar.html

Abstract:
Martin-Loef formalised in 1966 the intuitive notion of randomness
of infinite sequences of bits via algorithmic tests. In this talk,
we investigate what happens if we replace classical bits by
quantum bits.

We first provide a framework to formalise infinite sequences
of quantum bits as states of a suitable C* algebra.
Thereafter we introduce an analog of Martin-Loef's notion. We show
that for classical bit sequences the two notions coincide. We also
discuss quantum Kolmogorov complexity for finite sequences of quantum
bits and its relationship to quantum Martin-Loef randomness. Finally,
we consider an effective version of the Shannan-McMillan-Breiman theorem
in the quantum setting.

This is joint work with Volkher Scholz. The paper is available at
http://arxiv.org/abs/1709.08422.

5
7/11/2019 7:23:14Prague Set Theory SeminarWednesday seminarDear all,

The seminar meets on Wednesday July 17th at 11:00 in the Institute of
Mathematics CAS, Zitna 25, seminar room, 3rd floor, front building.

Program: Viera Šottová -- Ideals on natural numbers and combinatorial
characterization of sigma-ideal N

Historically, there is a lot of interesting results about Cichoń's
diagram as well as corresponded objects. Particularly, Bartoszyński
adduced several nice combinatorial characterizations of notions which we
are interested in.
We mainly study an ideal of Lebesgue measure zero sets, denoted N and we
briefly modify its characterization with respect to ideals on natural
numbers. We obtain a family N_J which is sigma-ideal as well.
Additionally, it is a subset of N. On the other hand, N_J expresses a
connection between ideals on natural numbers and ideals on real line. We
deal with common cardinal invariants of such families and their relation
to original notion.
Joint work with D. A. Mejia


Best,
David
6
7/4/2019 6:06:46Prague Set Theory SeminarWednesday seminarDear all,

The seminar meets on Wednesday July 10th.
THERE IS A TIME CHANGE, WE WILL MEET AT 12:00 (instead of 11) in the
Institute of Mathematics CAS, Zitna 25, seminar room, 3rd floor, front
building.

Program: Michael Hrušák -- Countably compact groups without convergent
sequences

In a joint work with J. van Mill, U.A. Ramos Garcia and S. Shelah we
solve a longstanding problem of E. van Douwen by constructing, in ZFC, a
countably compact topological group without non-trivial convergent
sequences.


Best,
David
7
7/2/2019 19:45:51Toronto Set Theory SeminarMaxim Burke: Comonotone approximation and interpolation by entire functionsPlace: Fields Institute (Room 210)
Date: July 5, 2019 (13:30-15:00)
Speaker: Maxim Burke
Title: Comonotone approximation and interpolation by entire functions
Abstract: We discuss some theorems on approximation of a real function f
whose derivatives up to order n are piecewise monotone by an entire
function g whose derivatives up to order n are comonotone with those of f
(increasing and decreasing on the same intervals) with interpolation on a
closed discrete set. One of the theorems depends on a conjecture regarding
the nature of the set of (n+1)-tuples (f(1),f'(1),f''(1),...) of final
values of C^n functions f on [0,1] whose derivatives at the origin are zero
and whose nth derivative is increasing but not constant. The work has its
origins in the problem of finding entire order-isomorphisms of everywhere
nonmeager subsets of R.
8
6/27/2019 21:33:50Toronto Set Theory SeminarNo set theory seminar this weekHi everyone, there will not be a set theoryseminar this week.

Osvaldo Guzman
9
6/21/2019 1:07:10Toronto Set Theory SeminarClovis Hamel: Definability, Topology of Function Spaces, and Continuous Logics.Place: Fields Institute (Room 210)
Date: 21 June, 2019 (13:00-13:50)
Speaker: Clovis Hamel
Title: Definability, Topology of Function Spaces, and Continuous Logics.
Abstract: In first-order logic, the notion of stability has been a driving
force of Model Theory in the last decades since Shelah introduced it. A
most relevant connection occurs in first-order logic: stability and
definability are equivalent. The classical definition of stability
involves the computation of cardinalities of spaces of types. However,
there are several equivalent definitions, most notably “no formula has the
order property”. We will present another approach to stability using
double limit conditions which is more suitable for continuous logics.
Using results from C${}_{p}$-theory, , i.e. the topology of real-valued
function spaces, we will show connections among double limit conditions,
stability and definability in various continuous logics. As an
application, we will expand some work of Casazza and Iovino concerning
Gower’s problem on the definability of pathological Banach spaces not
including isomorphic copies of $l^p$ or $c_0$ in compact logics to
stablish similar undefinability results for (continuous)
${\mathcal{L}}_{{\omega }_1,\omega }$. We will also discuss further lines
on research in this direction.
Clovis Hamel
10
6/19/2019 10:16:52Prague Set Theory SeminarWednesday seminarDear all,

There will be no seminars during the next two weeks (most of the regular
participants are away).

The seminar should meet again on Wednesday July 10th for a talk by
Michael Hrušák.

Best,
David
11
6/13/2019 4:59:55Prague Set Theory SeminarWednesday seminarDear all,

The seminar meets on Wednesday June 19th at 11:00 in the Institute of
Mathematics CAS, Zitna 25, seminar room, 3rd floor, front building.

Program: Jindřich Zapletal -- Matroids and the Axiom of Choice

Abstract: I will produce a model of ZF+DC with a Hamel basis or
transcendence basis for the reals over the rationals in an optimal way.
The two tasks are completely different, reflecting the model theoretic
distinctions between vectors spaces and fields. I will spend some time
outlining the basics of matroid theory, which is highly relevant for
this problem.



Best,
David
Jindřich Zapletal
12
6/13/2019 2:26:28Toronto Set Theory SeminarPiotr Szewczak: Products of gamma-setsPlace: Fields Institute (Room 210)
Date: 14 June, 2019 (13:30-15:00)
Speaker:Piotr Szewczak
Title: Products of gamma-sets
Abstract: Let X be a set of reals and Cp(X) be the set of all continuous
real-valued functions on X with the pointwise convergence topology. By the
result of Gerlits and Nagy the space Cp(X) has the Frechet-Urysohn
property (a generalization of first-countability) if and only if the set X
is a gamma-set (i.e., has a combinatorial covering property). The
existence of uncountable gamma-sets of reals is independent of ZFC. Tsaban
proved that sets with some special combinatorial structure are gamma-sets.
We generalize this class of sets and prove that their products have the
property gamma. We also show that for every set X from our class and every
gamma set Y, the product space X x Y have a strong property weaker than
gamma. These investigations are motivated by the result of Miller, Tsaban
and Zdomskyy that under CH,
there are two gamma-sets whose product space is not even Menger (in
particular it is not gamma). This is a joint work with Magdalena Włudecka.

Piotr Szewczak
13
6/9/2019 8:32:50IMPAN Working Group in Applications of Set TheoryPiotr Koszmider; Lifting uncountable combinatorics to the Banach space level: Banach spaces C(K) with few operators. Part 4.Seminar: Working group in applications of set theory, IMPAN

Thursday, 13.06.2019, 10:15, room 105, IMPAN

Speaker: Piotr Koszmider (IMPAN)

Title: "Lifting uncountable combinatorics to the Banach space level: Banach spaces C(K) with few operators. Part 4."

Abstact: "This series of 4 talks will be a minicourse on Banach spaces of continuous functions which have few operators, projections, injections etc. In particular they can be indecomposable and nonisomorphic with their hyperplanes. To obtain this linear operator level rigidity one needs to construct compact Ks which are not only rigid in the usual sense, i.e., in terms of continuous mappings on K. One needs to deal with weak* continuous functions from K into the space M(K) of Radon measures on K, so the combinatorics of the constructions needs stronger conditions than for endo-rigid Boolean algebras or strongly rigid compact spaces. We will present main arguments leading to C(K)s with the required properties but the proofs of many lemmas will be omitted. The talks should be accessible to everyone with general analytic and topological background".

This is the last regular meeting of the seminar this academic year. We may have some extraordinary meetings during the summer which will be announced here.

Visit our seminar page which may include some future talks at https://www.impan.pl/~set_theory/Seminar/
Piotr Koszmider
14
6/6/2019 23:07:26Toronto Set Theory SeminarNo seminar this weekHi everyone!

We will not have a seminar this week.
If you are iterested in giving a talk in the
near future, please let me know.

Osvaldo Guzman
15
6/6/2019 10:20:17Prague Set Theory SeminarWednesday seminarDear all,

The seminar meets on Wednesday June 12th at 11:00 in the Institute of
Mathematics CAS, Zitna 25, seminar room, 3rd floor, front building.

There is no fixed programme yet, walk in speakers are welcome.
The backup option is me talking about something, probably results of A.
Dow on P-points in the random model.

Best,
David
16
6/2/2019 13:01:06IMPAN Working Group in Applications of Set TheoryPiotr Koszmider; Lifting uncountable combinatorics to the Banach space level: Banach spaces C(K) with few operators. Part 3.Seminar: Working group in applications of set theory, IMPAN

NOTE: WE ARE BACK IN THE USUAL ROOM

Thursday, 06.06.2019, 10:15, room 105, IMPAN

Speaker: Piotr Koszmider (IMPAN)

Title: "Lifting uncountable combinatorics to the Banach space level: Banach spaces C(K) with few operators. Part 3."

Abstact: "This series of 4 talks will be a minicourse on Banach spaces of continuous functions which have few operators, projections, injections etc. In particular they can be indecomposable and nonisomorphic with their hyperplanes. To obtain this linear operator level rigidity one needs to construct compact Ks which are not only rigid in the usual sense, i.e., in terms of continuous mappings on K. One needs to deal with weak* continuous functions from K into the space M(K) of Radon measures on K, so the combinatorics of the constructions needs stronger conditions than for endo-rigid Boolean algebras or strongly rigid compact spaces. We will present main arguments leading to C(K)s with the required properties but the proofs of many lemmas will be omitted. The talks should be accessible to everyone with general analytic and topological background.".


Visit our seminar page which may include some future talks at https://www.impan.pl/~set_theory/Seminar/
Piotr Koszmider
17
5/31/2019 8:11:28Prague Set Theory SeminarWednesday seminarDear all,

The seminar meets on Wednesday June 5th at 11:00.

*** CHANGE OF LOCATION ***
Due to a scheduled interruption to the water supply in Zitna the seminar
will meet next week at an alternative location: room 119 (next to the
Department of Logic office), Faculty of Arts, Charles University,
Celetná 20, Praha 1.


Program: Šárka Stejskalová -- Stationary reflection and its variants

Abstract: If κ is a regular cardinal and S is a stationary subset of κ,
we say that S reflects at α of uncountable cofinality if S∩α is "large"
in a certain sense (for instance stationary or containing a club). We
will discuss several forms of stationary reflection and review an
argument for adding a club to ω2 which is contained in a stationary set
S concentrated on ordinals with countable cofinality plus all ordinals
of uncountable cofinality. This is for instance used by Harrington and
Shelah to show that from a Mahlo cardinal one can get a model where
every stationary subset of ω2 which concentrates on ordinal with
countable cofinality reflects at some α<ω2 with cofinality ω1 (i.e. S∩α
is stationary). We will indicate some original results in this direction.


Best,
David
Šárka Stejskalová
18
5/30/2019 0:41:51Toronto Set Theory SeminarDilip Raghavan: Order dimension of locally countable partial orders.Place: Fields Institute (Room 210)
Date: 31 May, 2019 (13:30-15:00)
Speaker: Dilip Raghavan
Title: Order dimension of locally countable partial orders
Abstract: I will present some recent results on order dimension, focusing
in the locally finite and locally countable orders. This is joint work
with several people.
Dilip Raghavan
19
5/29/2019 12:40:58Kurt Godel Research Center(KGRC) talk in the research seminar NEXT week, June 6The KGRC welcomes Neil Barton, Boaz Tsaban, Leandro Aurichi and Assaf
Shani as guests. Dr. Barton (host: Sy-David Friedman) will stay from June
3 to June 9. Professor Tsaban (host: Lyubomyr Zdomskyy) will stay from
June 16 to July 5 and give a talk for the European Set Theory Conference.
Professor Aurichi (host: Lyubomyr Zdomskyy) will stay from June 17 to July
3. Mr. Shani will stay from June 23 to July 31.

* * *

Please note that there will be no talk in the research seminar tomorrow,
May 30 (Ascension Day).

* * *

Research seminar
Kurt Göel Research Center
Thursday, June 6

"On convergent sequences of normalised measures on compact spaces"

Damian Sobota (KGRC)

The celebrated Josefson--Nissenzweig theorem---in a special case of a
Banach space $C(K)$ of continuous real-valued functions on an infinite
compact Hausdorff space $K$---asserts that there exists a sequence of
Radon measures $(\mu_n)$ on $K$ such that the total-variation of each
$\mu_n$ is $1$ and for every continuous function $f\in C(K)$ the sequence
of the integrals $\int_Kfd\mu_n$ converges to $0$. All the recent natural
proofs of the theorem start more or less as follows: "Assume there is not
such a sequence $(\mu_n)$ but with an additional property that each
$\mu_n$ is a finite linear combination of one-point measures (Dirac's
deltas). Then, ..." Although the proofs are correct, it appears that it is
not clear at all when this assumption is satisfied. During my talk I will
show when (and when not) it is the case that a compact space $K$ admits a
such a sequence of measures. As examples Efimov spaces, products of
compact spaces, Stone spaces of some funny Boolean algebras will appear.
This is a joint work with Lyubomyr Zdomskyy.

Time and Place

Tea at 3:30pm in the KGRC meeting room
Talk at 4:00pm in the KGRC lecture room
Damian Sobotahttps://youtu.be/BKAA64a6uB0
20
5/26/2019 13:53:42IMPAN Working Group in Applications of Set TheoryPiotr Koszmider; Lifting uncountable combinatorics to the Banach space level: Banach spaces C(K) with few operators. Part 2.Seminar: Working group in applications of set theory, IMPAN

NOTE CHANGE OF THE ROOM

Thursday, 30.05.2019, 10:15, ******ROOM 408******, IMPAN

Speaker: Piotr Koszmider (IMPAN)

Title: "Lifting uncountable combinatorics to the Banach space level: Banach spaces C(K) with few operators. Part 2."

Abstact: "This series of 4 talks will be a minicourse on Banach spaces of continuous functions which have few operators, projections, injections etc. In particular they can be indecomposable and nonisomorphic with their hyperplanes. To obtain this linear operator level rigidity one needs to construct compact Ks which are not only rigid in the usual sense, i.e., in terms of continuous mappings on K. One needs to deal with weak* continuous functions from K into the space M(K) of Radon measures on K, so the combinatorics of the constructions needs stronger conditions than for endo-rigid Boolean algebras or strongly rigid compact spaces. We will present main arguments leading to C(K)s with the required properties but the proofs of many lemmas will be omitted. The talks should be accessible to everyone with general analytic and topological background.".


Visit our seminar page which may include some future talks at https://www.impan.pl/~set_theory/Seminar/
Piotr Koszmider
21
5/23/2019 8:36:02Prague Set Theory SeminarWednesday seminarDear all,

The seminar meets on Wednesday May 29th at 11:00 in the Institute of
Mathematics CAS, Zitna 25, seminar room, 3rd floor, front building.

There is no fixed program yet, the backup option is me presenting
results of Raghavan on PID and weak squares.

Best,
David
22
5/21/2019
Kurt Godel Research Center
Asaf Karagila: Preservation theorems for symmetric extensions and Krivine-style results
Jean-Louis Krivine has used methods of realizability to prove several new independence results in ZF+DC. We show how to obtain some of these results using classical methods.

For the proof we also need theorems which lets us preserve some bits of choice in symmetric extensions. One of these theorems is an old folklore result, and the other is a new theorem.
Asaf Karagilahttp://www.logic.univie.ac.at/2019/Talk_05-23_a.html
23
5/20/2019This Week in Logic at CUNYThis Week in Logic at CUNY<div dir="ltr"><div style="font-family:arial,sans-serif;color:rgb(33,33,33)">Hi everyone,</div><div style="font-family:arial,sans-serif;color:rgb(33,33,33)"><br></div><div style="font-family:arial,sans-serif;color:rgb(33,33,33)">This will be our final newsletter for the Spring 2019 semester - regular mailings will resume at the end of August.  Special announcements may be sent in the interim as events warrant.</div><div style="font-family:arial,sans-serif;color:rgb(33,33,33)"><br></div><div style="font-family:arial,sans-serif;color:rgb(33,33,33)">Have a great summer,</div><div style="font-family:arial,sans-serif;color:rgb(33,33,33)">Jonas</div><div style="font-family:arial,sans-serif;color:rgb(33,33,33)"><br></div><div style="font-family:arial,sans-serif;color:rgb(33,33,33)"><br></div><div style="font-family:arial,sans-serif;color:rgb(33,33,33)"><br></div><div style="font-family:arial,sans-serif;color:rgb(33,33,33)">This<span style="color:rgb(0,0,0)"> Week in Logic at CUNY:</span></div><div><div><div><br></div><div><font color="#000000" face="arial, sans-serif">- - - - Tuesday, May 21, 2019 - - - -</font></div><div><font color="#000000" face="arial, sans-serif"><br></font></div><div><font color="#000000" face="arial, sans-serif"><br></font></div><div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><div><div>ASL 2019 Annual Meeting</div><div>May 20 — 23</div><div>CUNY Graduate Center</div><div><br></div><div>The Association for Symbolic Logic will have its 2019 North American Annual Meeting at the CUNY Graduate Center from May 20th through the 23rd. Additional information, including a schedule of speakers, is available on their website.</div><div><br></div><div><a href="https://asl2019.commons.gc.cuny.edu/" target="_blank">https://asl2019.commons.gc.cuny.edu/</a></div></div><div><br></div><br class="gmail-Apple-interchange-newline"></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><br></div></div><div><font color="#000000" face="arial, sans-serif"><br></font></div><div><font color="#000000" face="arial, sans-serif">- - - - Wednesday, May 22, 2019 - - - -</font></div><div><font color="#000000" face="arial, sans-serif"><br></font></div><div><div style="color:rgb(0,0,0);font-family:arial,sans-serif">ASL 2019 Annual Meeting</div><br class="gmail-m_-2657144423294426162gmail-Apple-interchange-newline"></div><div><font color="#000000" face="arial, sans-serif"><br></font></div><div><font color="#000000" face="arial, sans-serif">- - - - Thursday, May 23, 2019 - - - -</font></div><div><font color="#000000" face="arial, sans-serif"><br></font></div><div><div style="color:rgb(0,0,0);font-family:arial,sans-serif">ASL 2019 Annual Meeting</div><br class="gmail-m_-2657144423294426162gmail-Apple-interchange-newline"></div><div><font color="#000000" face="arial, sans-serif"><br></font></div><div><font color="#000000" face="arial, sans-serif">- - - - Friday, May 24, 2019 - - - -</font></div></div><div><font color="#000000" face="arial, sans-serif"><br></font></div><div><font color="#000000" face="arial, sans-serif"><br></font></div><div><font color="#000000" face="arial, sans-serif"><br></font></div><div><font color="#000000" face="arial, sans-serif"><br></font></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><font color="#212121">- - - - Other Logic News - - - -</font></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><br></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><b>Conference announcement:</b></div><div>JAF 2019 New York City</div><div>Journées sur les Arithmétiques Faibles</div><div>Weak Arithmetics Days</div><div>May 28-30 2019, The Graduate Center</div><div>City University of New York</div><div><br></div><div>Aim: Weak Arithmetics play a fundamental role in several areas of philosophy, mathematics, and computer science by studying the nature and properties of natural numbers from a logical point of view.  The aim of the conference is to provide a forum for researchers to present their results to members of communities who study or apply weak arithmetics in various fields and formalisms.<br></div><div><div><br></div><div>Topics: Proofs in arithmetic with restricted system of axioms.  Non-standard models of such systems.  Decidability, undecidability, and complexity of arithmetical theories.  Definability in arithmetic structures.  Machines, automata and words related to arithmetic.  Finite model theory, word structures.</div><div><br></div><div>Invited Speakers:  Gabriel Conant (Notre Dame), Damir Dzhafarov (University of Connecticut), Victoria Gitman (CUNY), Matt Kaufmann (University of Texas), Chris Miller (The Ohio State University), Russell Miller (CUNY), Arseniy Sheydvasser (CUNY).</div></div><div><br></div><div><a href="https://jaf2019nyc.com/" target="_blank">https://jaf2019nyc.com/</a><br></div><div><br></div><div><br></div><div><br></div><div><br></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><font color="#212121"><br></font></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><font color="#212121">- - - - Web Site - - - -</font></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><font color="#212121"><br></font></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><font color="#212121">Our new website is located at:  <a href="https://nylogic.github.io/" target="_blank" style="text-decoration-line:none">https://nylogic.github.io/</a></font></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><font color="#212121">Many thanks to Victoria Gitman for her development work!</font></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><font color="#212121"><br></font></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><font color="#212121">--------  ADMINISTRIVIA  --------</font></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><font color="#212121"><br></font></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><font color="#212121">To subscribe/unsubscribe to this list, please email your request to <a href="mailto:jreitz@nylogic.org" target="_blank" style="text-decoration-line:none">jreitz@nylogic.org</a>.</font></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><font color="#212121"><br></font></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><font color="#212121">If you have a logic-related event that you would like included in future mailings, please email <a href="mailto:jreitz@nylogic.org" target="_blank" style="text-decoration-line:none">jreitz@nylogic.org</a>. </font></div></div></div>
24
5/19/2019 15:47:16IMPAN Working Group in Applications of Set TheoryPiotr Koszmider; Lifting uncountable combinatorics to the Banach space level: Banach spaces C(K) with few operators. Part 1.Seminar: Working group in applications of set theory, IMPAN

Thursday, 23.05.2019, 10:15, room 105, IMPAN

Speaker: Piotr Koszmider (IMPAN)

Title: "Lifting uncountable combinatorics to the Banach space level: Banach spaces C(K) with few operators. Part 1."

Abstact: "This series of 4 talks will be a minicourse on Banach spaces of continuous functions which have few operators, projections, injections etc. In particular they can be indecomposable and nonisomorphic with their hyperplanes. To obtain this linear operator level rigidity one needs to construct compact Ks which are not only rigid in the usual sense, i.e., in terms of continuous mappings on K. One needs to deal with weak* continuous functions from K into the space M(K) of Radon measures on K, so the combinatorics of the constructions needs stronger conditions than for endo-rigid Boolean algebras or strongly rigid compact spaces. We will present main arguments leading to C(K)s with the required properties but the proofs of many lemmas will be omitted. The talks should be accessible to everyone with general analytic and topological background.".


Visit our seminar page which may include some future talks at https://www.impan.pl/~set_theory/Seminar/
25
5/15/2019 17:59:09Prague Set Theory SeminarWednesday seminarDear all,

The seminar meets on Wednesday May 22nd at 11:00 in the Institute of
Mathematics CAS, Zitna 25, seminar room, 3rd floor, front building.

Program: Jonathan Verner -- The RK ordering on P-points

Best,
David
26
5/12/2019 21:34:09This Week in Logic at CUNYThis Week in Logic at CUNY<div dir="ltr"><div dir="ltr"><div dir="ltr"><div dir="ltr"><div style="font-family:arial,sans-serif;color:rgb(33,33,33)">This Week in Logic at CUNY:<br></div><div><div><div style="color:rgb(33,33,33);font-family:arial,sans-serif"><div><br></div></div><div><font color="#000000"><div style="font-family:arial,sans-serif">- - - - Monday, May 13, 2019 - - - -</div><div style="font-family:arial,sans-serif"><br></div><div style="font-family:arial,sans-serif"><div>Logic and Metaphysics Workshop</div><div>Date: Monday, May 13th, 4.15-6.15  <br></div><div>Place: Room 7314, CUNY Graduate Center <br></div><div>Speaker: Martina Botti (Columbia)<br></div><div>Title: Composition as Identity: A New Approach<br></div><div><br></div><div>Abstract: I argue that the debate on composition as identity – the thesis that any composite object is identical to its parts – is deadlocked because both the defenders and the detractors of the claim have so far defended and criticized respectively something that is not composition as identity. After having made clear how composition as identity should properly be understood, I will set forth a new strategy to defend it. </div><div><br></div><div> </div></div><div style="font-family:arial,sans-serif"><br></div><div style="font-family:arial,sans-serif"><br></div><div style="font-family:arial,sans-serif">- - - - Tuesday, May 14, 2019 - - - -</div><div style="font-family:arial,sans-serif"><br></div><div style="font-family:arial,sans-serif"><br></div><div style="font-family:arial,sans-serif"><br></div><div style="font-family:arial,sans-serif">- - - - Wednesday, May 15, 2019 - - - -</div><div style="font-family:arial,sans-serif"><br></div><div style="font-family:arial,sans-serif"><div>The New York City Category Theory Seminar</div><div>Department of Computer Science</div><div>Department of Mathematics </div><div>The Graduate Center of The City University of New York </div></div><div><p class="MsoNormal"><font face="arial, helvetica, sans-serif">Speaker: <b>    Sergei Artemov, The Graduate Center, CUNY.</b><u></u><u></u></font></p><p class="MsoNormal"><font face="arial, helvetica, sans-serif">Date and Time: <b>    Wednesday May 15, 2019, 7:00 - 8:30 PM., Room 6417.</b><u></u><u></u></font></p><p class="MsoNormal"><font face="arial, helvetica, sans-serif">Title:<b>     On the Provability of Consistency.</b><u></u><u></u></font></p><p class="MsoNormal"><font face="arial, helvetica, sans-serif">Abstract: We revisit the foundational question concerning Peano arithmetic PA:<u></u><u></u></font></p><p class="MsoNormal" style="margin-left:30pt"><font face="arial, helvetica, sans-serif">(1) <i>can consistency of PA be established by means expressible in PA?</i><u></u><u></u></font></p><p class="MsoNormal"><font face="arial, helvetica, sans-serif">The usual answer to (1) is “No, by Gödel’s Second Incompleteness Theorem.” In that theorem (G2), Gödel used an arithmetization of contentual mathematical reasoning and established that the arithmetical formula representing PA-consistency is not derivable in PA. Applying G2 to (1), one makes use of the formalization thesis (FT):<u></u><u></u></font></p><p class="MsoNormal" style="margin-left:30pt"><font face="arial, helvetica, sans-serif">FT: <i>any proof by means expressible in PA admits Gödel’s arithmetization.</i><u></u><u></u></font></p><p class="MsoNormal"><font face="arial, helvetica, sans-serif">Historically, there has been no consensus on FT; Gödel (1931) and Hilbert (1934) argued against an even weaker version of FT with respect to finitary proofs, whereas von Neumann accepted it. <br><br>Note that the aforementioned negative answer to (1) is unwarranted: here is a counter-example to FT. Let <i>Ind(F)</i> denote the induction statement for an arithmetical formula <i>F</i>. The claim <i>C</i>, “for each formula <i>F</i>, <i>Ind(F)</i>,” is directly provable by means of PA: given any <i>F</i>, argue by induction to establish <i>Ind(F)</i>. However, <i>C</i> is not supported by any arithmetization as a single formula since PA is not finitely axiomatizable. <br><br>We provide a positive answer to (1). We offer a mathematical proof of PA-consistency,<u></u><u></u></font></p><p class="MsoNormal" style="margin-left:30pt"><font face="arial, helvetica, sans-serif"><i>No finite sequence of formulas is a PA-proof of 0=1,</i><u></u><u></u></font></p><p class="MsoNormal" style="margin-bottom:12pt"><font face="arial, helvetica, sans-serif">by means expressible in PA, namely, by partial truth definitions. Naturally, this proof does not admit Gödel’s arithmetization either.<u></u><u></u></font></p><div class="MsoNormal" align="center" style="text-align:center"></div><br class="gmail-m_-245045616821462084gmail-Apple-interchange-newline"></div><div style="font-family:arial,sans-serif"><br></div><div style="font-family:arial,sans-serif"><br></div><div style="font-family:arial,sans-serif"><br></div><div style="font-family:arial,sans-serif">- - - - Thursday, May 16, 2019 - - - -</div><div style="font-family:arial,sans-serif"><br></div><div style="font-family:arial,sans-serif">*** CUNY Final Exams begin today ***</div><div style="font-family:arial,sans-serif"><br></div><div style="font-family:arial,sans-serif"><br></div><div style="font-family:arial,sans-serif"><br></div><div style="font-family:arial,sans-serif">- - - - Friday, May 17, 2019 - - - -</div><div style="font-family:arial,sans-serif"><br></div><div><div><font face="arial, sans-serif">Set Theory Seminar </font><span style="font-family:arial,sans-serif">(RESCHEDULED from April 12)</span></div><div><font face="arial, sans-serif">CUNY Graduate Center, Room 6417</font></div><div><font face="arial, sans-serif">Friday, May 17, 10:00-11:45am  </font></div><div><font face="arial, sans-serif">Jonas Reitz, CUNY</font></div><div><font face="arial, sans-serif">Generalized Cohen Iterations</font></div><div><font face="arial, sans-serif"><br></font></div><div><font face="arial, sans-serif">Adding Cohen subsets to each of a class of cardinals in turn is a common construction in set theory, and underlies many fundamental results.  The construction comes in two basic flavors, products (as in Easton’s Theorem on the powers of regular cardinals) and iterations (forcing the GCH).  These flavors are  apparently quite similar, forcing at stage kappa to add subsets via the Cohen partial order Add(kappa,lambda).  They differ only in the universe over which Add(kappa,lambda) is defined - in the case of products the ground model poset is used at each stage, whereas in typical iterations the poset is taken from the partial extension up to kappa.  In this talk I will consider an alternative, in which we allow Add(kappa,lambda) to be defined over an arbitrary inner model (lying between the ground model and the extension up to kappa) at each stage.   These generalized Cohen iterations are ZFC-preserving, although neither the proof for products nor for traditional iterations transfers directly.   They allow constructions such as class iterations of class products of Cohen forcing, with applications including new work with Kameryn Williams on iterating the Mantle.</font></div></div><div><br></div></font></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><br></div></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><br></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><br></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><br></div><div><div><font color="#000000" face="arial, sans-serif">Next Week in Logic at CUNY:</font></div><div><font color="#000000" face="arial, sans-serif"><br></font></div><div><font color="#000000" face="arial, sans-serif">- - - - Monday, May 20, 2019 - - - -</font></div><div><font color="#000000" face="arial, sans-serif"><br></font></div><div><font color="#000000" face="arial, sans-serif"><div>Logic and Metaphysics Workshop</div><div>Date: Monday, May 20th, 4.15-6.15  </div><div>Place: Room 7314, CUNY Graduate Center </div><div>Speaker: Vincent Peluce (CUNY)</div><div>Title: The Perception of Time in Intuitionistic Arithmetic</div><div><br></div><div>Abstract: In L.E.J. Brouwer’s first act of intuitionism, the subject’s perception of time is put forth as the foundation on which arithmetic will be built. According to Brouwer, proper intuitionistic arithmetic, as with the rest of intuitionistic mathematics, is not tied to any particular formal system. When we try to axiomatically approximate an intuitionistic arithmetical system, we are faced with the problem of incorporating the subject and their perception into the axiom system itself. We discuss some unsatisfactory responses to this problem and then offer a solution.</div><div><br></div><div> </div><div><br></div><div><div><br></div><div>ASL 2019 Annual Meeting</div><div>May 20 — 23</div><div>CUNY Graduate Center</div><div><br></div><div>The Association for Symbolic Logic will have its 2019 North American Annual Meeting at the CUNY Graduate Center from May 20th through the 23rd. Additional information, including a schedule of speakers, is available on their website.</div><div><br></div><div><a href="https://asl2019.commons.gc.cuny.edu/">https://asl2019.commons.gc.cuny.edu/</a></div></div><div><br></div><div><br></div></font></div><div><font color="#000000" face="arial, sans-serif"><br></font></div><div><font color="#000000" face="arial, sans-serif"><br></font></div><div><font color="#000000" face="arial, sans-serif">- - - - Tuesday, May 21, 2019 - - - -</font></div><div><font color="#000000" face="arial, sans-serif"><br></font></div><div><font color="#000000" face="arial, sans-serif"><br></font></div><div><div style="color:rgb(0,0,0);font-family:arial,sans-serif">ASL 2019 Annual Meeting</div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><br></div></div><div><font color="#000000" face="arial, sans-serif"><br></font></div><div><font color="#000000" face="arial, sans-serif">- - - - Wednesday, May 22, 2019 - - - -</font></div><div><font color="#000000" face="arial, sans-serif"><br></font></div><div><div style="color:rgb(0,0,0);font-family:arial,sans-serif">ASL 2019 Annual Meeting</div><br class="gmail-Apple-interchange-newline"></div><div><font color="#000000" face="arial, sans-serif"><br></font></div><div><font color="#000000" face="arial, sans-serif">- - - - Thursday, May 23, 2019 - - - -</font></div><div><font color="#000000" face="arial, sans-serif"><br></font></div><div><div style="color:rgb(0,0,0);font-family:arial,sans-serif">ASL 2019 Annual Meeting</div><br class="gmail-Apple-interchange-newline"></div><div><font color="#000000" face="arial, sans-serif"><br></font></div><div><font color="#000000" face="arial, sans-serif">- - - - Friday, May 24, 2019 - - - -</font></div></div><div><font color="#000000" face="arial, sans-serif"><br></font></div><div><font color="#000000" face="arial, sans-serif"><br></font></div><div><font color="#000000" face="arial, sans-serif"><br></font></div><div><font color="#000000" face="arial, sans-serif"><br></font></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><font color="#212121">- - - - Other Logic News - - - -</font></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><br></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif">Please take note of two upcoming conferences this month at CUNY&#39;s Graduate Center:</div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><br></div><div style="font-family:arial,sans-serif"><div style="color:rgb(0,0,0)"><b>Conference announcement:</b></div><div style="font-family:Arial,Helvetica,sans-serif"><span style="background-color:rgba(255,255,255,0)">ASL 2019 Annual Meeting</span></div><div style="font-family:Arial,Helvetica,sans-serif">May 20 — 23<span style="background-color:rgba(255,255,255,0)"><br></span></div><div style="font-family:Arial,Helvetica,sans-serif"><span style="background-color:rgba(255,255,255,0)"><a href="https://www.gc.cuny.edu/Home" target="_blank" style="margin:0px;padding:0px;border:0px;vertical-align:baseline;outline:none">CUNY Graduate Center</a></span></div><div style="font-family:Arial,Helvetica,sans-serif"><span style="background-color:rgba(255,255,255,0)"><br></span></div><div style="font-family:Arial,Helvetica,sans-serif"><span style="color:rgb(0,0,0);font-family:arial,sans-serif">The Association for Symbolic Logic will have its 2019 North American Annual Meeting at the CUNY Graduate Center from May 20th through the 23rd. If you would like to attend, you are asked to <b>register by May 6th</b>. Additional information, including a schedule of speakers, is available on their website.</span><span style="background-color:rgba(255,255,255,0)"><br></span></div><div style="font-family:Arial,Helvetica,sans-serif"><span style="color:rgb(0,0,0);font-family:arial,sans-serif"><br></span></div><div style="font-family:Arial,Helvetica,sans-serif"><p style="margin:0px 0px 1.71429rem;padding:0px;border:0px;vertical-align:baseline;line-height:1.71429;font-family:&quot;Open Sans&quot;,Helvetica,Arial,sans-serif"><font color="#444444">GÖDEL LECTURE: </font><a href="https://www.math.ucsd.edu/people/profiles/samuel-buss/" target="_blank" style="color:rgb(159,159,159);margin:0px;padding:0px;border:0px;vertical-align:baseline;outline:none">S. Buss</a><br><font color="#444444">INVITED SPEAKERS:<br></font><a href="http://www.math.cmu.edu/~dbartoso/" target="_blank" style="color:rgb(159,159,159);margin:0px;padding:0px;border:0px;vertical-align:baseline;outline:none">D. Bartosova</a><font color="#444444">, </font><a href="http://www.logic.univie.ac.at/~vfischer/" target="_blank" style="color:rgb(159,159,159);margin:0px;padding:0px;border:0px;vertical-align:baseline;outline:none">V. Fischer</a><font color="#444444">, </font><a href="https://www.scs.gatech.edu/people/lance-fortnow" target="_blank" style="color:rgb(159,159,159);margin:0px;padding:0px;border:0px;vertical-align:baseline;outline:none">L. Fortnow</a><font color="#444444">, </font><a href="https://ms.mcmaster.ca/~haskell/" target="_blank" style="color:rgb(159,159,159);margin:0px;padding:0px;border:0px;vertical-align:baseline;outline:none">D. Haskell</a><font color="#444444">, </font><a href="https://www.uu.nl/staff/RIemhoff" target="_blank" style="color:rgb(159,159,159);margin:0px;padding:0px;border:0px;vertical-align:baseline;outline:none">R. Iemhoff</a><font color="#444444">, </font><a href="https://en.philo.umontreal.ca/repertoire-departement/vue/marquis-jean-pierre/" target="_blank" style="color:rgb(159,159,159);margin:0px;padding:0px;border:0px;vertical-align:baseline;outline:none">J.-P. Marquis</a><font color="#444444">, R. Patel, and </font><a href="http://www.math.wisc.edu/~msoskova/" target="_blank" style="color:rgb(159,159,159);margin:0px;padding:0px;border:0px;vertical-align:baseline;outline:none">M. Soskova</a><font color="#444444">. A tutorial will be given by </font><a href="http://web.cs.iastate.edu/~lutz/" target="_blank" style="color:rgb(159,159,159);margin:0px;padding:0px;border:0px;vertical-align:baseline;outline:none">J. Lutz</a><font color="#444444">.</font></p><p style="color:rgb(68,68,68);margin:0px 0px 1.71429rem;padding:0px;border:0px;vertical-align:baseline;line-height:1.71429;font-family:&quot;Open Sans&quot;,Helvetica,Arial,sans-serif"><a href="https://asl2019.commons.gc.cuny.edu/" target="_blank" style="font-family:Arial,Helvetica,sans-serif">https://asl2019.commons.gc.cuny.edu/</a><br></p></div></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><b><br></b></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><b><br></b></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><b>Conference announcement:</b></div><div>JAF 2019 New York City</div><div>Journées sur les Arithmétiques Faibles</div><div>Weak Arithmetics Days</div><div>May 28-30 2019, The Graduate Center</div><div>City University of New York</div><div><br></div><div>Aim: Weak Arithmetics play a fundamental role in several areas of philosophy, mathematics, and computer science by studying the nature and properties of natural numbers from a logical point of view.  The aim of the conference is to provide a forum for researchers to present their results to members of communities who study or apply weak arithmetics in various fields and formalisms.<br></div><div><div><br></div><div>Topics: Proofs in arithmetic with restricted system of axioms.  Non-standard models of such systems.  Decidability, undecidability, and complexity of arithmetical theories.  Definability in arithmetic structures.  Machines, automata and words related to arithmetic.  Finite model theory, word structures.</div><div><br></div><div>Invited Speakers:  Gabriel Conant (Notre Dame), Damir Dzhafarov (University of Connecticut), Victoria Gitman (CUNY), Matt Kaufmann (University of Texas), Chris Miller (The Ohio State University), Russell Miller (CUNY), Arseniy Sheydvasser (CUNY).</div></div><div><br></div><div><a href="https://jaf2019nyc.com/" target="_blank">https://jaf2019nyc.com/</a><br></div><div><br></div><div><br></div><div><br></div><div><br></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><font color="#212121"><br></font></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><font color="#212121">- - - - Web Site - - - -</font></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><font color="#212121"><br></font></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><font color="#212121">Our new website is located at:  <a href="https://nylogic.github.io/" target="_blank" style="text-decoration-line:none">https://nylogic.github.io/</a></font></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><font color="#212121">Many thanks to Victoria Gitman for her development work!</font></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><font color="#212121"><br></font></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><font color="#212121">--------  ADMINISTRIVIA  --------</font></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><font color="#212121"><br></font></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><font color="#212121">To subscribe/unsubscribe to this list, please email your request to <a href="mailto:jreitz@nylogic.org" target="_blank" style="text-decoration-line:none">jreitz@nylogic.org</a>.</font></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><font color="#212121"><br></font></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><font color="#212121">If you have a logic-related event that you would like included in future mailings, please email <a href="mailto:jreitz@nylogic.org" target="_blank" style="text-decoration-line:none">jreitz@nylogic.org</a>. </font></div></div></div></div></div></div>
27
5/9/2019 16:09:24Conference
BLAST 2019, Boulder, May 20-24
conference:  BLAST 2019

date:             May 20-24, 2019

location:       University of Colorado, Boulder CO

web:              https://math.colorado.edu/blast/2019/

email:            blast2019@colorado.edu

                                                     

SCOPE

The BLAST conference series focuses on related areas within the

foundations of mathematics, specifically Boolean Algebras,

Lattice Theory, Algebraic Logic, Universal Algebra, Set Theory,

and Set-theoretic and Point-free Topology.

This year's installment of BLAST will take place at the University

of Colorado at Boulder. The scientific program will include invited

lectures, tutorial lectures and 20-minute contributed talks. The

central BLAST web page, with links to past meetings, can be found here:

http://math.colorado.edu/blast/



BLAST 2019 INVITED SPEAKERS:

Guram Bezhanishvili (New Mexico State University)

Will Brian (University of North Carolina at Charlotte)

Ronnie Chen (University of Illinois at Urbana-Champaign)
Miklos Maroti (University of Szeged)

Ralph McKenzie (Vanderbilt University)

Matt Moore (University of Kansas)

Tommaso Moraschini (Czech Academy of Sciences)

Adam Prenosil (Vanderbilt University)

Douglas Ulrich (University of California at Irvine)
Amanda Vidal (Czech Academy of Sciences)

Ross Willard (University of Waterloo)



REGISTRATION INFORMATION


Registration is open at


https://math.colorado.edu/blast/2019/registration.html


Early registration helps in our planning.


The fee will be $80, and is payable only at the conference, in cash.

SUPPORT REQUESTS

There will be limited funds to support the participation of graduate

students and recent PhD's. Instructions for submitting a funding

request can be found on the conference web site.



ABSTRACT SUBMISSION

In addition to the invited talks and tutorial presentations, there will

be 20-minute contributed presentations. Information on the submission

procedures for titles and abstracts can be found at the conference web site.



ACCOMMODATIONS

Please visit the conference website at

https://math.colorado.edu/blast/2019/



DATES

May 13, 2019: Abstract submission deadline

May 20-24, 2019: Conference

 

THANKS

BLAST 2019 is supported by NSF, the Research and Innovation Office

of the University of Colorado, and the Department of Mathematics

of the University of Colorado.



LOCAL ORGANIZERS

William DeMeo, Keith Kearnes, Peter Mayr, Agnes Szendrei

Guram Bezhanishvili,Will Brian,Ronnie Chen,
Miklos Maroti,Ralph McKenzie,Matt Moore,Tommaso Moraschini,Adam Prenosil,Douglas Ulrich,Amanda Vidal,Ross Willard
05/20/2019https://math.colorado.edu/blast/2019
28
5/8/2019 16:09:24Prague Set Theory Seminar
Mexwell Levine: Partitions of Reflecting Stationary Sets
Dear all,

The seminar meets on Wednesday May 15th at 11:00 in the Institute of
Mathematics CAS, Zitna 25, seminar room, 3rd floor, front building.

Program: Mexwell Levine -- Partitions of Reflecting Stationary Sets
Abstract attached.

Best,
David
Maxwell Levine
29
5/6/2019 10:32:29IMPAN Working Group in Applications of Set TheoryArturo Martínez-Celis; Choice vs DeterminacySeminar: Working group in applications of set theory, IMPAN

NOTE THE USUAL TIME AND PLACE:

Thursday, 09.05.2019, 10:15, room 105, IMPAN

Speaker: Arturo Martínez-Celis (IMPAN)

Title: "Choice vs Determinacy"

Abstact: "We will discuss the concept of infinite game and winning strategies and we will present some examples, theorems and applications to topology. In particular we will prove that every uncountable Borel set has a homeomorphic copy of the Cantor set. The axiom of determinacy (AD) states that for certain kind of games, the Gale-Stewart games, one player has always a winning strategy. The aim of this talk is to present the differences between the universes satisfying AD and the universes satisfying the axiom of choice".


Visit our seminar page which may include some future talks at https://www.impan.pl/~set_theory/Seminar/
Arturo Martínez-Celis
30
5/6/2019 8:06:01NUS Logic SeminarLogic Seminar 8 May 2019 17:00 hrs at NUSInvitation to the Logic Seminar at the National University of Singapore

Date: Wednesday, 8 May 2019, 17:00 hrs

Room: S17#04-05, Department of Mathematics, NUS

Speaker: Borisa Kuzeljevic

Title: Matrices of countable elementary submodels

URL: http://www.comp.nus.edu.sg/~fstephan/logicseminar.html

Abstract:
We present an application of the forcing notion of finite
matrices whose rows consist of isomorphic countable elementary
submodels of a given structure of the form H_theta.
We will explain how forcing with this poset adds a Kurepa tree. If a
minor modification of the poset is considered, then the tree added is
actually an almost Souslin Kurepa tree.

This is a joint work with Stevo Todorcevic.

Borisa Kuzeljevic
31
5/5/2019 23:08:00This Week in Logic at CUNYThis Week in Logic at CUNY<div dir="ltr"><div dir="ltr"><div dir="ltr"><div dir="ltr"><div dir="ltr"><div dir="ltr"><div dir="ltr"><div dir="ltr"><div dir="ltr"><div dir="ltr"><div dir="ltr"><div dir="ltr"><div dir="ltr"><div style="font-family:arial,sans-serif;color:rgb(33,33,33)">This Week in Logic at CUNY:<br></div><div><div><div style="color:rgb(33,33,33);font-family:arial,sans-serif"><div><br></div></div><div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><font color="#000000" face="arial, sans-serif">- - - - Monday, May 6, 2019 - - - -</font></div><div><font color="#000000" face="arial, sans-serif"><div><br></div><div>Logic and Metaphysics Workshop</div><div>Date: Monday, May 6th, 4.15-6.15  <br></div><div>Place: Room 7314, CUNY Graduate Center <br></div><div>Speaker: Daniel Durante (Natal)<br></div><div>Title: No Metaphysical Disagreement Without Logical Incompatibility<br></div><div><br></div><div>Abstract: The purpose of this talk is to defend the logical incompatibility of the opposing views as a criterion for characterizing disagreements as genuinely metaphysical. That is, I intend to argue that a specific dispute is a metaphysical disagreement only when the conflicting views are governed by different logics. If correct, this criterion would not only help to separate merely verbal from genuine metaphysical debates, but it also would ground an argument against deflationism, guaranteeing the substantiality and relevance of metaphysics. I intend to clarify the criterion, to present its basic foundations and commitments, to give some logical and metaphysical motivations for its adoption and some examples of its application.</div><div><br></div><div><br></div><div> </div></font></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><font color="#000000" face="arial, sans-serif"><br></font></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><font color="#000000" face="arial, sans-serif"><br></font></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><font color="#000000" face="arial, sans-serif">- - - - Tuesday, May 7, 2019 - - - -</font></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><font color="#000000" face="arial, sans-serif"><br></font></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><font color="#000000" face="arial, sans-serif"><br></font></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><font color="#000000" face="arial, sans-serif"><br></font></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><font color="#000000" face="arial, sans-serif">- - - - Wednesday, May 8, 2019 - - - -</font></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><font color="#000000" face="arial, sans-serif"><br></font></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><font color="#000000" face="arial, sans-serif"><div>The New York City Category Theory Seminar</div><div>Department of Computer Science</div><div>Department of Mathematics </div><div>The Graduate Center of The City University of New York </div><div><div>Speaker:     Jonathan Funk, Queensborough Community College.</div><div>Date and Time:     Wednesday May 8, 2019, 7:00 - 8:30 PM., Room 6417.</div><div>Title:     Isotropy theory meets Galois theory.</div><div><br></div><div>Abstract: Isotropy theory for toposes is about internal symmetry of a topos. A topos may have trivial isotropy, said to be anisotropic. For example, a localic topos is anisotropic. The isotropy of a topos may be cancelled to yield what we call the isotropy quotient of a topos, although the quotient may itself have isotropy, or what we call higher isotropy of the given topos. (By analogy, the quotient of a group by its center may itself have non-trivial center.) Let us say that a topos is locally anisotropic if it has an etale cover by an anisotropic topos. </div><div><br></div><div>THEOREM: A locally anisotropic topos has no higher isotropy. Equivalently, its isotropy quotient is anisotropic. Furthermore, a locally anisotropic topos is recovered as the topos of actions for a connected groupoid internal to its isotropy quotient. </div><div><br></div><div>COROLLARY: An etendue, or locally localic topos, has no higher isotropy. An etendue may be recovered as the topos of actions for a connected groupoid internal to its isotropy quotient. </div><div><br></div><div>Our argumentation of the theorem brings into focus how isotropy theory and Galois theory for toposes meet in a natural and evidently effective way. </div><div><br></div><div>Joint work with Pieter Hofstra.</div></div></font></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><font color="#000000" face="arial, sans-serif"><br></font></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><font color="#000000" face="arial, sans-serif"><br></font></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><font color="#000000" face="arial, sans-serif"><br></font></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><font color="#000000" face="arial, sans-serif"><br></font></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><font color="#000000" face="arial, sans-serif"><br></font></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><font color="#000000" face="arial, sans-serif">- - - - Thursday, May 9, 2019 - - - -</font></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><font color="#000000" face="arial, sans-serif"><br></font></div><div><font color="#000000" face="arial, sans-serif"><div>Computer Science Colloquium</div><div>THURSDAY, May 9TH, 2019, 4:15pm – 6:15pm</div><div>Room 9205</div><div>Larry Moss, Indiana University</div><div>Bridging the Gap Between Logic and Machine Learning in Natural Language Inference</div><div><br></div><div>Abstract: The field of natural language inference (NLI) has seen strong progress in recent years, especially after the advent of deep learning.  The basic goal is to see whether one natural language (NL) sentence “follows from” another, and to do this on a computer, for sentences “in nature”.  Reflecting my own background, I wondered if there was anything whatsoever which 2000+years of work in logic could contribute to NLI.</div><div><br></div><div>This talk details work on making a connection between logic and computational linguistics.  It will touch on topics such as: combinatory categorical grammar and its syntax-semantics interference; work on monotonicity pioneered by Johan van Benthem; the typed lambda calculus; natural logic, and algorithms from it; datasets like SICK, FraCaS, and SNLI; and the bidirectional language model BERT. </div><div><br></div><div>The goal is to see whether theory scales up to practice.  I won’t give away the end of the story here, but suffice it to say the we are finding this work to be both challenging and interesting. </div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><br></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><br></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><br></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><br></div></font></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><font color="#000000" face="arial, sans-serif"><br></font></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><font color="#000000" face="arial, sans-serif">- - - - Friday, May 10, 2019 - - - -</font></div></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><font color="#000000" face="arial, sans-serif"><br></font></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><div><font color="#000000" face="arial, sans-serif">Set Theory Seminar</font></div><div><font color="#000000" face="arial, sans-serif">CUNY Graduate Center, Room 6417</font></div><div><font color="#000000" face="arial, sans-serif">Friday, May 10, 10:00-11:45am</font></div></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><p style="font-family:&quot;Open Sans&quot;,&quot;Helvetica Neue&quot;,Helvetica,Arial,sans-serif;box-sizing:border-box;margin-bottom:1em;margin-top:0px;color:rgb(96,108,113)"><strong style="box-sizing:border-box"><a href="http://kamerynjw.net/" target="_blank" style="font-family:arial,sans-serif;color:rgb(30,107,184);text-decoration-line:none;box-sizing:border-box;background-color:transparent;margin-top:0px">Kameryn Williams</a></strong>, University of Hawai‘i at Mānoa<br style="box-sizing:border-box"><strong style="box-sizing:border-box">Transfinite Recursion from Gödel–Bernays to Kelley–Morse</strong><br style="box-sizing:border-box"></p><p style="font-family:&quot;Open Sans&quot;,&quot;Helvetica Neue&quot;,Helvetica,Arial,sans-serif;box-sizing:border-box;margin-bottom:1em;color:rgb(96,108,113)">Gödel–Bernays set theory GB and Kelley–Morse set theory KM are two formal theories for second-order set theory, allowing both sets and proper classes as objects. GB is the weaker of the two theories, being conservative over ZF, while KM is the stronger. Set theorists have used KM in applications where GB is not strong enough; for instance, Kunen formulated his celebrated inconsistency result in the context of KM, as KM has the resources to directly allow talk of elementary embeddings of the universe of sets. But weaker theories than KM suffice for many of these applications. Between GB and KM there is a hierarchy of intermediate theories based upon restricting the logical complexity allowed in the comprehension axiom.</p><p style="font-family:&quot;Open Sans&quot;,&quot;Helvetica Neue&quot;,Helvetica,Arial,sans-serif;box-sizing:border-box;margin-bottom:1em;color:rgb(96,108,113)">In this talk I will present a hierarchy of second-order set theories which refines the comprehension-based hierarchy. This hierarchy is based upon transfinite recursion principles, ordered first by the logical complexity of the properties allowed and second by the lengths of well-orders on which we may carry out the recursions. Theories in this hierarchy are separated in terms of consistency strength. The substantive new result to establish this hierarchy is the following: Let <span id="gmail-m_-5092454438650748386gmail-MathJax-Element-1-Frame" style="box-sizing:border-box;display:inline-block;line-height:0;word-spacing:normal;white-space:nowrap;float:none;direction:ltr;max-width:none;max-height:none;min-width:0px;min-height:0px;border:0px;margin:0px;padding:1px 0px"><span id="gmail-m_-5092454438650748386gmail-MJXc-Node-1" style="box-sizing:border-box;display:inline-block;border-collapse:separate;border-spacing:0px;margin-top:0px"><span id="gmail-m_-5092454438650748386gmail-MJXc-Node-2" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_-5092454438650748386gmail-MJXc-Node-3" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw;margin-top:0px;padding-top:0.472em;padding-bottom:0.284em;box-sizing:content-box">k</span></span></span></span><span style="box-sizing:border-box;padding:1px 0px 0px;border:0px;height:1px;width:1px;overflow:hidden;display:block">k</span></span> be a natural number. Suppose <span id="gmail-m_-5092454438650748386gmail-MathJax-Element-2-Frame" style="box-sizing:border-box;display:inline-block;line-height:0;word-spacing:normal;white-space:nowrap;float:none;direction:ltr;max-width:none;max-height:none;min-width:0px;min-height:0px;border:0px;margin:0px;padding:1px 0px"><span id="gmail-m_-5092454438650748386gmail-MJXc-Node-4" style="box-sizing:border-box;display:inline-block;border-collapse:separate;border-spacing:0px;margin-top:0px"><span id="gmail-m_-5092454438650748386gmail-MJXc-Node-5" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_-5092454438650748386gmail-MJXc-Node-6" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.472em;padding-bottom:0.613em;box-sizing:content-box">(</span></span><span id="gmail-m_-5092454438650748386gmail-MJXc-Node-7" style="display:inline-block;box-sizing:content-box"><span style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw;margin-top:0px;padding-top:0.472em;padding-bottom:0.284em;padding-right:0.081em;box-sizing:content-box">M</span></span><span id="gmail-m_-5092454438650748386gmail-MJXc-Node-8" style="display:inline-block;box-sizing:content-box"><span style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;padding-bottom:0.519em;box-sizing:content-box">,</span></span><span id="gmail-m_-5092454438650748386gmail-MJXc-Node-9" style="display:inline-block;margin-left:0.167em;box-sizing:content-box"><span id="gmail-m_-5092454438650748386gmail-MJXc-Node-10" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_-5092454438650748386gmail-MJXc-Node-11" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-cal-R,MJXc-TeX-cal-Rw;margin-top:0px;padding-top:0.425em;padding-bottom:0.331em;padding-right:0.094em;box-sizing:content-box">X</span></span></span></span><span id="gmail-m_-5092454438650748386gmail-MJXc-Node-12" style="display:inline-block;box-sizing:content-box"><span style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.472em;padding-bottom:0.613em;box-sizing:content-box">)</span></span></span></span><span style="box-sizing:border-box;padding:1px 0px 0px;border:0px;height:1px;width:1px;overflow:hidden;display:block">(M,X)</span></span> satisfies GB and that <span id="gmail-m_-5092454438650748386gmail-MathJax-Element-3-Frame" style="box-sizing:border-box;display:inline-block;line-height:0;word-spacing:normal;white-space:nowrap;float:none;direction:ltr;max-width:none;max-height:none;min-width:0px;min-height:0px;border:0px;margin:0px;padding:1px 0px"><span id="gmail-m_-5092454438650748386gmail-MJXc-Node-13" style="box-sizing:border-box;display:inline-block;border-collapse:separate;border-spacing:0px;margin-top:0px"><span id="gmail-m_-5092454438650748386gmail-MJXc-Node-14" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_-5092454438650748386gmail-MJXc-Node-15" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.378em;padding-bottom:0.331em;box-sizing:content-box">Γ</span></span><span id="gmail-m_-5092454438650748386gmail-MJXc-Node-16" style="display:inline-block;margin-left:0.278em;box-sizing:content-box"><span style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.237em;padding-bottom:0.378em;box-sizing:content-box">∈</span></span><span id="gmail-m_-5092454438650748386gmail-MJXc-Node-17" style="display:inline-block;margin-left:0.278em;box-sizing:content-box"><span id="gmail-m_-5092454438650748386gmail-MJXc-Node-18" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_-5092454438650748386gmail-MJXc-Node-19" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-cal-R,MJXc-TeX-cal-Rw;margin-top:0px;padding-top:0.425em;padding-bottom:0.331em;padding-right:0.094em;box-sizing:content-box">X</span></span></span></span></span></span><span style="box-sizing:border-box;padding:1px 0px 0px;border:0px;height:1px;width:1px;overflow:hidden;display:block">Γ∈X</span></span> is a class well-order which is closed under addition. In case <span id="gmail-m_-5092454438650748386gmail-MathJax-Element-4-Frame" style="box-sizing:border-box;display:inline-block;line-height:0;word-spacing:normal;white-space:nowrap;float:none;direction:ltr;max-width:none;max-height:none;min-width:0px;min-height:0px;border:0px;margin:0px;padding:1px 0px"><span id="gmail-m_-5092454438650748386gmail-MJXc-Node-20" style="box-sizing:border-box;display:inline-block;border-collapse:separate;border-spacing:0px;margin-top:0px"><span id="gmail-m_-5092454438650748386gmail-MJXc-Node-21" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_-5092454438650748386gmail-MJXc-Node-22" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw;margin-top:0px;padding-top:0.472em;padding-bottom:0.284em;box-sizing:content-box">k</span></span><span id="gmail-m_-5092454438650748386gmail-MJXc-Node-23" style="display:inline-block;margin-left:0.278em;box-sizing:content-box"><span style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.096em;padding-bottom:0.331em;box-sizing:content-box">=</span></span><span id="gmail-m_-5092454438650748386gmail-MJXc-Node-24" style="display:inline-block;margin-left:0.278em;box-sizing:content-box"><span style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.378em;padding-bottom:0.378em;box-sizing:content-box">0</span></span></span></span><span style="box-sizing:border-box;padding:1px 0px 0px;border:0px;height:1px;width:1px;overflow:hidden;display:block">k=0</span></span> further assume <span id="gmail-m_-5092454438650748386gmail-MathJax-Element-5-Frame" style="box-sizing:border-box;display:inline-block;line-height:0;word-spacing:normal;white-space:nowrap;float:none;direction:ltr;max-width:none;max-height:none;min-width:0px;min-height:0px;border:0px;margin:0px;padding:1px 0px"><span id="gmail-m_-5092454438650748386gmail-MJXc-Node-25" style="box-sizing:border-box;display:inline-block;border-collapse:separate;border-spacing:0px;margin-top:0px"><span id="gmail-m_-5092454438650748386gmail-MJXc-Node-26" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_-5092454438650748386gmail-MJXc-Node-27" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.378em;padding-bottom:0.331em;box-sizing:content-box">Γ</span></span><span id="gmail-m_-5092454438650748386gmail-MJXc-Node-28" style="display:inline-block;margin-left:0.278em;box-sizing:content-box"><span style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.331em;padding-bottom:0.472em;box-sizing:content-box">≥</span></span><span id="gmail-m_-5092454438650748386gmail-MJXc-Node-29" style="display:inline-block;margin-left:0.278em;box-sizing:content-box"><span style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_-5092454438650748386gmail-MJXc-Node-30" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw;margin-top:0px;padding-top:0.237em;padding-bottom:0.284em;box-sizing:content-box">ω</span></span></span><span style="display:inline-block;vertical-align:0.513em;padding-left:0px;padding-right:0.071em;box-sizing:content-box"><span id="gmail-m_-5092454438650748386gmail-MJXc-Node-31" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw;margin-top:0px;padding-top:0.237em;padding-bottom:0.284em;box-sizing:content-box">ω</span></span></span></span></span></span><span style="box-sizing:border-box;padding:1px 0px 0px;border:0px;height:1px;width:1px;overflow:hidden;display:block">Γ≥ωω</span></span>. Then, if <span id="gmail-m_-5092454438650748386gmail-MathJax-Element-6-Frame" style="box-sizing:border-box;display:inline-block;line-height:0;word-spacing:normal;white-space:nowrap;float:none;direction:ltr;max-width:none;max-height:none;min-width:0px;min-height:0px;border:0px;margin:0px;padding:1px 0px"><span id="gmail-m_-5092454438650748386gmail-MJXc-Node-32" style="box-sizing:border-box;display:inline-block;border-collapse:separate;border-spacing:0px;margin-top:0px"><span id="gmail-m_-5092454438650748386gmail-MJXc-Node-33" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_-5092454438650748386gmail-MJXc-Node-34" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.472em;padding-bottom:0.613em;box-sizing:content-box">(</span></span><span id="gmail-m_-5092454438650748386gmail-MJXc-Node-35" style="display:inline-block;box-sizing:content-box"><span style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw;margin-top:0px;padding-top:0.472em;padding-bottom:0.284em;padding-right:0.081em;box-sizing:content-box">M</span></span><span id="gmail-m_-5092454438650748386gmail-MJXc-Node-36" style="display:inline-block;box-sizing:content-box"><span style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;padding-bottom:0.519em;box-sizing:content-box">,</span></span><span id="gmail-m_-5092454438650748386gmail-MJXc-Node-37" style="display:inline-block;margin-left:0.167em;box-sizing:content-box"><span id="gmail-m_-5092454438650748386gmail-MJXc-Node-38" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_-5092454438650748386gmail-MJXc-Node-39" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-cal-R,MJXc-TeX-cal-Rw;margin-top:0px;padding-top:0.425em;padding-bottom:0.331em;padding-right:0.094em;box-sizing:content-box">X</span></span></span></span><span id="gmail-m_-5092454438650748386gmail-MJXc-Node-40" style="display:inline-block;box-sizing:content-box"><span style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.472em;padding-bottom:0.613em;box-sizing:content-box">)</span></span></span></span><span style="box-sizing:border-box;padding:1px 0px 0px;border:0px;height:1px;width:1px;overflow:hidden;display:block">(M,X)</span></span> satisfies <span id="gmail-m_-5092454438650748386gmail-MathJax-Element-7-Frame" style="box-sizing:border-box;display:inline-block;line-height:0;word-spacing:normal;white-space:nowrap;float:none;direction:ltr;max-width:none;max-height:none;min-width:0px;min-height:0px;border:0px;margin:0px;padding:1px 0px"><span id="gmail-m_-5092454438650748386gmail-MJXc-Node-41" style="box-sizing:border-box;display:inline-block;border-collapse:separate;border-spacing:0px;margin-top:0px"><span id="gmail-m_-5092454438650748386gmail-MJXc-Node-42" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_-5092454438650748386gmail-MJXc-Node-43" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_-5092454438650748386gmail-MJXc-Node-44" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.378em;padding-bottom:0.331em;box-sizing:content-box">Π</span></span></span><span style="display:inline-block;vertical-align:-0.335em;box-sizing:content-box"><span style="display:block;margin-top:0px;padding-bottom:0.255em;padding-left:0px;padding-right:0.071em;box-sizing:content-box"><span id="gmail-m_-5092454438650748386gmail-MJXc-Node-46" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.378em;padding-bottom:0.331em;box-sizing:content-box">1</span></span></span><span style="display:block;padding-right:0.071em;box-sizing:content-box"><span id="gmail-m_-5092454438650748386gmail-MJXc-Node-45" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw;margin-top:0px;padding-top:0.472em;padding-bottom:0.284em;box-sizing:content-box">k</span></span></span></span></span></span></span><span style="box-sizing:border-box;padding:1px 0px 0px;border:0px;height:1px;width:1px;overflow:hidden;display:block">Πk1</span></span>-Transfinite Recursion for recursions along <span id="gmail-m_-5092454438650748386gmail-MathJax-Element-8-Frame" style="box-sizing:border-box;display:inline-block;line-height:0;word-spacing:normal;white-space:nowrap;float:none;direction:ltr;max-width:none;max-height:none;min-width:0px;min-height:0px;border:0px;margin:0px;padding:1px 0px"><span id="gmail-m_-5092454438650748386gmail-MJXc-Node-47" style="box-sizing:border-box;display:inline-block;border-collapse:separate;border-spacing:0px;margin-top:0px"><span id="gmail-m_-5092454438650748386gmail-MJXc-Node-48" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_-5092454438650748386gmail-MJXc-Node-49" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.378em;padding-bottom:0.331em;box-sizing:content-box">Γ</span></span></span></span><span style="box-sizing:border-box;padding:1px 0px 0px;border:0px;height:1px;width:1px;overflow:hidden;display:block">Γ</span></span>, there is <span id="gmail-m_-5092454438650748386gmail-MathJax-Element-9-Frame" style="box-sizing:border-box;display:inline-block;line-height:0;word-spacing:normal;white-space:nowrap;float:none;direction:ltr;max-width:none;max-height:none;min-width:0px;min-height:0px;border:0px;margin:0px;padding:1px 0px"><span id="gmail-m_-5092454438650748386gmail-MJXc-Node-50" style="box-sizing:border-box;display:inline-block;border-collapse:separate;border-spacing:0px;margin-top:0px"><span id="gmail-m_-5092454438650748386gmail-MJXc-Node-51" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_-5092454438650748386gmail-MJXc-Node-52" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_-5092454438650748386gmail-MJXc-Node-53" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_-5092454438650748386gmail-MJXc-Node-54" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-cal-R,MJXc-TeX-cal-Rw;margin-top:0px;padding-top:0.425em;padding-bottom:0.472em;padding-right:0.046em;box-sizing:content-box">Y</span></span></span></span><span id="gmail-m_-5092454438650748386gmail-MJXc-Node-55" style="display:inline-block;margin-left:0.278em;box-sizing:content-box"><span style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.331em;padding-bottom:0.472em;box-sizing:content-box">⊆</span></span><span id="gmail-m_-5092454438650748386gmail-MJXc-Node-56" style="display:inline-block;margin-left:0.278em;box-sizing:content-box"><span id="gmail-m_-5092454438650748386gmail-MJXc-Node-57" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_-5092454438650748386gmail-MJXc-Node-58" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-cal-R,MJXc-TeX-cal-Rw;margin-top:0px;padding-top:0.425em;padding-bottom:0.331em;padding-right:0.094em;box-sizing:content-box">X</span></span></span></span></span></span><span style="box-sizing:border-box;padding:1px 0px 0px;border:0px;height:1px;width:1px;overflow:hidden;display:block">Y⊆X</span></span> coded in <span id="gmail-m_-5092454438650748386gmail-MathJax-Element-10-Frame" style="box-sizing:border-box;display:inline-block;line-height:0;word-spacing:normal;white-space:nowrap;float:none;direction:ltr;max-width:none;max-height:none;min-width:0px;min-height:0px;border:0px;margin:0px;padding:1px 0px"><span id="gmail-m_-5092454438650748386gmail-MJXc-Node-59" style="box-sizing:border-box;display:inline-block;border-collapse:separate;border-spacing:0px;margin-top:0px"><span id="gmail-m_-5092454438650748386gmail-MJXc-Node-60" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_-5092454438650748386gmail-MJXc-Node-61" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_-5092454438650748386gmail-MJXc-Node-62" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_-5092454438650748386gmail-MJXc-Node-63" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-cal-R,MJXc-TeX-cal-Rw;margin-top:0px;padding-top:0.425em;padding-bottom:0.331em;padding-right:0.094em;box-sizing:content-box">X</span></span></span></span></span></span><span style="box-sizing:border-box;padding:1px 0px 0px;border:0px;height:1px;width:1px;overflow:hidden;display:block">X</span></span> so that <span id="gmail-m_-5092454438650748386gmail-MathJax-Element-11-Frame" style="box-sizing:border-box;display:inline-block;line-height:0;word-spacing:normal;white-space:nowrap;float:none;direction:ltr;max-width:none;max-height:none;min-width:0px;min-height:0px;border:0px;margin:0px;padding:1px 0px"><span id="gmail-m_-5092454438650748386gmail-MJXc-Node-64" style="box-sizing:border-box;display:inline-block;border-collapse:separate;border-spacing:0px;margin-top:0px"><span id="gmail-m_-5092454438650748386gmail-MJXc-Node-65" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_-5092454438650748386gmail-MJXc-Node-66" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.472em;padding-bottom:0.613em;box-sizing:content-box">(</span></span><span id="gmail-m_-5092454438650748386gmail-MJXc-Node-67" style="display:inline-block;box-sizing:content-box"><span style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw;margin-top:0px;padding-top:0.472em;padding-bottom:0.284em;padding-right:0.081em;box-sizing:content-box">M</span></span><span id="gmail-m_-5092454438650748386gmail-MJXc-Node-68" style="display:inline-block;box-sizing:content-box"><span style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;padding-bottom:0.519em;box-sizing:content-box">,</span></span><span id="gmail-m_-5092454438650748386gmail-MJXc-Node-69" style="display:inline-block;margin-left:0.167em;box-sizing:content-box"><span id="gmail-m_-5092454438650748386gmail-MJXc-Node-70" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_-5092454438650748386gmail-MJXc-Node-71" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-cal-R,MJXc-TeX-cal-Rw;margin-top:0px;padding-top:0.425em;padding-bottom:0.472em;padding-right:0.046em;box-sizing:content-box">Y</span></span></span></span><span id="gmail-m_-5092454438650748386gmail-MJXc-Node-72" style="display:inline-block;box-sizing:content-box"><span style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.472em;padding-bottom:0.613em;box-sizing:content-box">)</span></span></span></span><span style="box-sizing:border-box;padding:1px 0px 0px;border:0px;height:1px;width:1px;overflow:hidden;display:block">(M,Y)</span></span> satisfies GB plus the principle of <span id="gmail-m_-5092454438650748386gmail-MathJax-Element-12-Frame" style="box-sizing:border-box;display:inline-block;line-height:0;word-spacing:normal;white-space:nowrap;float:none;direction:ltr;max-width:none;max-height:none;min-width:0px;min-height:0px;border:0px;margin:0px;padding:1px 0px"><span id="gmail-m_-5092454438650748386gmail-MJXc-Node-73" style="box-sizing:border-box;display:inline-block;border-collapse:separate;border-spacing:0px;margin-top:0px"><span id="gmail-m_-5092454438650748386gmail-MJXc-Node-74" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_-5092454438650748386gmail-MJXc-Node-75" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_-5092454438650748386gmail-MJXc-Node-76" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.378em;padding-bottom:0.331em;box-sizing:content-box">Π</span></span></span><span style="display:inline-block;vertical-align:-0.335em;box-sizing:content-box"><span style="display:block;margin-top:0px;padding-bottom:0.255em;padding-left:0px;padding-right:0.071em;box-sizing:content-box"><span id="gmail-m_-5092454438650748386gmail-MJXc-Node-78" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.378em;padding-bottom:0.331em;box-sizing:content-box">1</span></span></span><span style="display:block;padding-right:0.071em;box-sizing:content-box"><span id="gmail-m_-5092454438650748386gmail-MJXc-Node-77" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw;margin-top:0px;padding-top:0.472em;padding-bottom:0.284em;box-sizing:content-box">k</span></span></span></span></span></span></span><span style="box-sizing:border-box;padding:1px 0px 0px;border:0px;height:1px;width:1px;overflow:hidden;display:block">Πk1</span></span>-Transfinite Recursion for recursions along well-orders of length <span id="gmail-m_-5092454438650748386gmail-MathJax-Element-13-Frame" style="box-sizing:border-box;display:inline-block;line-height:0;word-spacing:normal;white-space:nowrap;float:none;direction:ltr;max-width:none;max-height:none;min-width:0px;min-height:0px;border:0px;margin:0px;padding:1px 0px"><span id="gmail-m_-5092454438650748386gmail-MJXc-Node-79" style="box-sizing:border-box;display:inline-block;border-collapse:separate;border-spacing:0px;margin-top:0px"><span id="gmail-m_-5092454438650748386gmail-MJXc-Node-80" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_-5092454438650748386gmail-MJXc-Node-81" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.237em;padding-bottom:0.378em;box-sizing:content-box">&lt;</span></span><span id="gmail-m_-5092454438650748386gmail-MJXc-Node-82" style="display:inline-block;margin-left:0.278em;box-sizing:content-box"><span style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.378em;padding-bottom:0.331em;box-sizing:content-box">Γ</span></span></span></span><span style="box-sizing:border-box;padding:1px 0px 0px;border:0px;height:1px;width:1px;overflow:hidden;display:block">&lt;Γ</span></span>.</p></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><font color="#000000" face="arial, sans-serif"><br></font></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><font color="#000000" face="arial, sans-serif"><br></font></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><font color="#000000" face="arial, sans-serif"><br></font></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><span id="gmail-docs-internal-guid-738eaa5c-7fff-7cd5-0382-d44e5fdc7ede"><p dir="ltr" style="line-height:1.38;margin-top:0pt;margin-bottom:0pt"><span style="font-family:Arial;color:rgb(68,68,68);font-variant-numeric:normal;font-variant-east-asian:normal;vertical-align:baseline;white-space:pre-wrap">Model Theory Seminar</span></p><p dir="ltr" style="line-height:1.38;margin-top:0pt;margin-bottom:0pt"><span style="font-family:Arial;color:rgb(68,68,68);font-variant-numeric:normal;font-variant-east-asian:normal;vertical-align:baseline;white-space:pre-wrap">CUNY Graduate Center, Room 6417</span></p><p dir="ltr" style="line-height:1.38;margin-top:0pt;margin-bottom:0pt"><span style="font-family:Arial;color:rgb(68,68,68);font-variant-numeric:normal;font-variant-east-asian:normal;vertical-align:baseline;white-space:pre-wrap">Friday, May 10, 12:30-2:00pm</span></p></span><div>Alexander Van Abel, CUNY</div><div>Asymptotic Classes of Finite Structures</div><div><br></div><div>A one-dimensional asymptotic class, as introduced by Macpherson and Steinhorn in 2008, is a collection of finite structures whose definable subsets in a single variable grow approximately linearly with respect to the size of the structure, in a definable and well-behaved fashion. The motivating example is the collection of finite fields, as proved by Chatzidakis, van den Dries and Macintyre in 1992. In this talk, we survey Steinhorn and Macpherson&#39;s foundational 2008 paper. We give examples and nonexamples of one-dimensional asymptotic classes, as well as more general notions such as N-dimensional and multidimensional classes. We show how infinite ultra-products of one-dimensional asymptotic classes are model-theoretically nice, with particular emphasis on the existence of a well-behaved dimension and measure on definable subsets and the consequences of such.</div></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><br></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><br></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><br></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><font color="#000000" face="arial, sans-serif"><br></font></div><div><font color="#000000"><div style="font-family:arial,sans-serif">Next Week in Logic at CUNY:</div><div style="font-family:arial,sans-serif"><br></div><div style="font-family:arial,sans-serif">- - - - Monday, May 13, 2019 - - - -</div><div style="font-family:arial,sans-serif"><br></div><div style="font-family:arial,sans-serif"><div>Logic and Metaphysics Workshop</div><div>Date: Monday, May 13th, 4.15-6.15  <br></div><div>Place: Room 7314, CUNY Graduate Center <br></div><div>Speaker: Martina Botti (Columbia)<br></div><div>Title: Composition as Identity: A New Approach<br></div><div><br></div><div>Abstract: I argue that the debate on composition as identity – the thesis that any composite object is identical to its parts – is deadlocked because both the defenders and the detractors of the claim have so far defended and criticized respectively something that is not composition as identity. After having made clear how composition as identity should properly be understood, I will set forth a new strategy to defend it. </div><div><br></div><div> </div></div><div style="font-family:arial,sans-serif"><br></div><div style="font-family:arial,sans-serif"><br></div><div style="font-family:arial,sans-serif">- - - - Tuesday, May 14, 2019 - - - -</div><div style="font-family:arial,sans-serif"><br></div><div style="font-family:arial,sans-serif"><br></div><div style="font-family:arial,sans-serif"><br></div><div style="font-family:arial,sans-serif">- - - - Wednesday, May 15, 2019 - - - -</div><div style="font-family:arial,sans-serif"><br></div><div style="font-family:arial,sans-serif"><div>The New York City Category Theory Seminar</div><div>Department of Computer Science</div><div>Department of Mathematics </div><div>The Graduate Center of The City University of New York </div></div><div><p class="MsoNormal"><font face="arial, helvetica, sans-serif">Speaker: <b>    Sergei Artemov, The Graduate Center, CUNY.</b><u></u><u></u></font></p><p class="MsoNormal"><font face="arial, helvetica, sans-serif">Date and Time: <b>    Wednesday May 15, 2019, 7:00 - 8:30 PM., Room 6417.</b><u></u><u></u></font></p><p class="MsoNormal"><font face="arial, helvetica, sans-serif">Title:<b>     On the Provability of Consistency.</b><u></u><u></u></font></p><p class="MsoNormal"><font face="arial, helvetica, sans-serif">Abstract: We revisit the foundational question concerning Peano arithmetic PA:<u></u><u></u></font></p><p class="MsoNormal" style="margin-left:30pt"><font face="arial, helvetica, sans-serif">(1) <i>can consistency of PA be established by means expressible in PA?</i><u></u><u></u></font></p><p class="MsoNormal"><font face="arial, helvetica, sans-serif">The usual answer to (1) is “No, by Gödel’s Second Incompleteness Theorem.” In that theorem (G2), Gödel used an arithmetization of contentual mathematical reasoning and established that the arithmetical formula representing PA-consistency is not derivable in PA. Applying G2 to (1), one makes use of the formalization thesis (FT):<u></u><u></u></font></p><p class="MsoNormal" style="margin-left:30pt"><font face="arial, helvetica, sans-serif">FT: <i>any proof by means expressible in PA admits Gödel’s arithmetization.</i><u></u><u></u></font></p><p class="MsoNormal"><font face="arial, helvetica, sans-serif">Historically, there has been no consensus on FT; Gödel (1931) and Hilbert (1934) argued against an even weaker version of FT with respect to finitary proofs, whereas von Neumann accepted it. <br><br>Note that the aforementioned negative answer to (1) is unwarranted: here is a counter-example to FT. Let <i>Ind(F)</i> denote the induction statement for an arithmetical formula <i>F</i>. The claim <i>C</i>, “for each formula <i>F</i>, <i>Ind(F)</i>,” is directly provable by means of PA: given any <i>F</i>, argue by induction to establish <i>Ind(F)</i>. However, <i>C</i> is not supported by any arithmetization as a single formula since PA is not finitely axiomatizable. <br><br>We provide a positive answer to (1). We offer a mathematical proof of PA-consistency,<u></u><u></u></font></p><p class="MsoNormal" style="margin-left:30pt"><font face="arial, helvetica, sans-serif"><i>No finite sequence of formulas is a PA-proof of 0=1,</i><u></u><u></u></font></p><p class="MsoNormal" style="margin-bottom:12pt"><font face="arial, helvetica, sans-serif">by means expressible in PA, namely, by partial truth definitions. Naturally, this proof does not admit Gödel’s arithmetization either.<u></u><u></u></font></p><div class="MsoNormal" align="center" style="text-align:center"></div><br class="gmail-Apple-interchange-newline"></div><div style="font-family:arial,sans-serif"><br></div><div style="font-family:arial,sans-serif"><br></div><div style="font-family:arial,sans-serif"><br></div><div style="font-family:arial,sans-serif">- - - - Thursday, May 16, 2019 - - - -</div><div style="font-family:arial,sans-serif"><br></div><div style="font-family:arial,sans-serif">*** CUNY Final Exams begin today ***</div><div style="font-family:arial,sans-serif"><br></div><div style="font-family:arial,sans-serif"><br></div><div style="font-family:arial,sans-serif"><br></div><div style="font-family:arial,sans-serif">- - - - Friday, May 17, 2019 - - - -</div><div style="font-family:arial,sans-serif"><br></div><div><div><font face="arial, sans-serif">Set Theory Seminar </font><span style="font-family:arial,sans-serif">(RESCHEDULED from April 12)</span></div><div><font face="arial, sans-serif">CUNY Graduate Center, Room 6417</font></div><div><font face="arial, sans-serif">Friday, May 17, 10:00-11:45am  </font></div><div><font face="arial, sans-serif">Jonas Reitz, CUNY</font></div><div><font face="arial, sans-serif">Generalized Cohen Iterations</font></div><div><font face="arial, sans-serif"><br></font></div><div><font face="arial, sans-serif">Adding Cohen subsets to each of a class of cardinals in turn is a common construction in set theory, and underlies many fundamental results.  The construction comes in two basic flavors, products (as in Easton’s Theorem on the powers of regular cardinals) and iterations (forcing the GCH).  These flavors are  apparently quite similar, forcing at stage kappa to add subsets via the Cohen partial order Add(kappa,lambda).  They differ only in the universe over which Add(kappa,lambda) is defined - in the case of products the ground model poset is used at each stage, whereas in typical iterations the poset is taken from the partial extension up to kappa.  In this talk I will consider an alternative, in which we allow Add(kappa,lambda) to be defined over an arbitrary inner model (lying between the ground model and the extension up to kappa) at each stage.   These generalized Cohen iterations are ZFC-preserving, although neither the proof for products nor for traditional iterations transfers directly.   They allow constructions such as class iterations of class products of Cohen forcing, with applications including new work with Kameryn Williams on iterating the Mantle.</font></div></div><div><br></div></font></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><br></div></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><br></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><br></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><font color="#212121">- - - - Other Logic News - - - -</font></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><br></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif">Please take note of two upcoming conferences this month at CUNY&#39;s Graduate Center:</div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><br></div><div style="font-family:arial,sans-serif"><div style="color:rgb(0,0,0)"><b>Conference announcement:</b></div><div style="color:rgb(34,34,34);font-family:Arial,Helvetica,sans-serif"><span style="background-color:rgba(255,255,255,0)">ASL 2019 Annual Meeting</span></div><div style="color:rgb(34,34,34);font-family:Arial,Helvetica,sans-serif">May 20 — 23<span style="background-color:rgba(255,255,255,0)"><br></span></div><div style="color:rgb(34,34,34);font-family:Arial,Helvetica,sans-serif"><span style="background-color:rgba(255,255,255,0)"><a href="https://www.gc.cuny.edu/Home" target="_blank" style="margin:0px;padding:0px;border:0px;vertical-align:baseline;outline:none">CUNY Graduate Center</a></span></div><div style="color:rgb(34,34,34);font-family:Arial,Helvetica,sans-serif"><span style="background-color:rgba(255,255,255,0)"><br></span></div><div style="color:rgb(34,34,34);font-family:Arial,Helvetica,sans-serif"><span style="color:rgb(0,0,0);font-family:arial,sans-serif">The Association for Symbolic Logic will have its 2019 North American Annual Meeting at the CUNY Graduate Center from May 20th through the 23rd. If you would like to attend, you are asked to <b>register by May 6th</b>. Additional information, including a schedule of speakers, is available on their website.</span><span style="background-color:rgba(255,255,255,0)"><br></span></div><div style="color:rgb(34,34,34);font-family:Arial,Helvetica,sans-serif"><span style="color:rgb(0,0,0);font-family:arial,sans-serif"><br></span></div><div style="font-family:Arial,Helvetica,sans-serif"><p style="margin:0px 0px 1.71429rem;padding:0px;border:0px;vertical-align:baseline;line-height:1.71429;font-family:&quot;Open Sans&quot;,Helvetica,Arial,sans-serif"><font color="#444444">GÖDEL LECTURE: </font><a href="https://www.math.ucsd.edu/people/profiles/samuel-buss/" style="color:rgb(159,159,159);margin:0px;padding:0px;border:0px;vertical-align:baseline;outline:none">S. Buss</a><br><font color="#444444">INVITED SPEAKERS:<br></font><a href="http://www.math.cmu.edu/~dbartoso/" style="color:rgb(159,159,159);margin:0px;padding:0px;border:0px;vertical-align:baseline;outline:none">D. Bartosova</a><font color="#444444">, </font><a href="http://www.logic.univie.ac.at/~vfischer/" style="color:rgb(159,159,159);margin:0px;padding:0px;border:0px;vertical-align:baseline;outline:none">V. Fischer</a><font color="#444444">, </font><a href="https://www.scs.gatech.edu/people/lance-fortnow" style="color:rgb(159,159,159);margin:0px;padding:0px;border:0px;vertical-align:baseline;outline:none">L. Fortnow</a><font color="#444444">, </font><a href="https://ms.mcmaster.ca/~haskell/" style="color:rgb(159,159,159);margin:0px;padding:0px;border:0px;vertical-align:baseline;outline:none">D. Haskell</a><font color="#444444">, </font><a href="https://www.uu.nl/staff/RIemhoff" style="color:rgb(159,159,159);margin:0px;padding:0px;border:0px;vertical-align:baseline;outline:none">R. Iemhoff</a><font color="#444444">, </font><a href="https://en.philo.umontreal.ca/repertoire-departement/vue/marquis-jean-pierre/" style="color:rgb(159,159,159);margin:0px;padding:0px;border:0px;vertical-align:baseline;outline:none">J.-P. Marquis</a><font color="#444444">, R. Patel, and </font><a href="http://www.math.wisc.edu/~msoskova/" style="color:rgb(159,159,159);margin:0px;padding:0px;border:0px;vertical-align:baseline;outline:none">M. Soskova</a><font color="#444444">. A tutorial will be given by </font><a href="http://web.cs.iastate.edu/~lutz/" style="color:rgb(159,159,159);margin:0px;padding:0px;border:0px;vertical-align:baseline;outline:none">J. Lutz</a><font color="#444444">.</font></p><p style="color:rgb(68,68,68);margin:0px 0px 1.71429rem;padding:0px;border:0px;vertical-align:baseline;line-height:1.71429;font-family:&quot;Open Sans&quot;,Helvetica,Arial,sans-serif"><a href="https://asl2019.commons.gc.cuny.edu/" target="_blank" style="font-family:Arial,Helvetica,sans-serif">https://asl2019.commons.gc.cuny.edu/</a><br></p></div></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><b><br></b></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><b><br></b></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><b>Conference announcement:</b></div><div>JAF 2019 New York City</div><div>Journées sur les Arithmétiques Faibles</div><div>Weak Arithmetics Days</div><div>May 28-30 2019, The Graduate Center</div><div>City University of New York</div><div><br></div><div>Aim: Weak Arithmetics play a fundamental role in several areas of philosophy, mathematics, and computer science by studying the nature and properties of natural numbers from a logical point of view.  The aim of the conference is to provide a forum for researchers to present their results to members of communities who study or apply weak arithmetics in various fields and formalisms.<br></div><div><div><br></div><div>Topics: Proofs in arithmetic with restricted system of axioms.  Non-standard models of such systems.  Decidability, undecidability, and complexity of arithmetical theories.  Definability in arithmetic structures.  Machines, automata and words related to arithmetic.  Finite model theory, word structures.</div><div><br></div><div>Invited Speakers:  Gabriel Conant (Notre Dame), Damir Dzhafarov (University of Connecticut), Victoria Gitman (CUNY), Matt Kaufmann (University of Texas), Chris Miller (The Ohio State University), Russell Miller (CUNY), Arseniy Sheydvasser (CUNY).</div></div><div><br></div><div><a href="https://jaf2019nyc.com/">https://jaf2019nyc.com/</a><br></div><div><br></div><div><br></div><div><br></div><div><br></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><font color="#212121"><br></font></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><font color="#212121">- - - - Web Site - - - -</font></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><font color="#212121"><br></font></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><font color="#212121">Our new website is located at:  <a href="https://nylogic.github.io/" target="_blank" style="text-decoration-line:none">https://nylogic.github.io/</a></font></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><font color="#212121">Many thanks to Victoria Gitman for her development work!</font></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><font color="#212121"><br></font></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><font color="#212121">--------  ADMINISTRIVIA  --------</font></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><font color="#212121"><br></font></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><font color="#212121">To subscribe/unsubscribe to this list, please email your request to <a href="mailto:jreitz@nylogic.org" target="_blank" style="text-decoration-line:none">jreitz@nylogic.org</a>.</font></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><font color="#212121"><br></font></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><font color="#212121">If you have a logic-related event that you would like included in future mailings, please email <a href="mailto:jreitz@nylogic.org" target="_blank" style="text-decoration-line:none">jreitz@nylogic.org</a>. </font></div></div></div></div></div></div></div></div></div></div></div></div></div></div></div>
32
5/3/2019 0:53:36Toronto Set Theory SeminarNo set theory seminar this weekHi everyone!

We will not have a seminar this week.

See you soon!

Osvaldo Guzman
33
04/30/2019Bristol Logic SeminarF_sigma Games of Length omega^21st May 2019, 4:00 pm – 5:00 pm
Howard House, 2nd Floor Seminar Room

Speaker: Juan Aguilera (Technische Universität Vienna)

Title: Open graphs and hypergraphs on definable subsets of generalized Baire spaces

Abstract:

We show, in ZFC, that the determinacy of F_sigma games of length omega^2 is equivalent to the existence of an admissible model of AD that contains R and reflects Pi_1 statements about the next admissible set.
Juan Aguilerahttps://www.bristolmathsresearch.org/seminar/juan-aguilera/
34
04/30/2019ConferenceSet Theory in the United Kingdom 2, May 8Set Theory in the United Kingdom is a joint research group in set theory funded by the London Mathematical Society (Scheme 3) with members at the Universities of Bristol, Cambridge, East Anglia, Leeds, Oxford, Warwick and University College London.

STUK 2 will take place on Wednesday, 8 May 2019, 11.00-18.00 at the University of Bristol.

11-12 Victoria Gitman: Set theory in second-order

13-14 Andrew Brooke-Taylor: Set theory and category theory

Location: 4th floor seminar room, School of Mathematics, University of Bristol, Howard house, Queen’s avenue, Bristol BS8 1SD.
Victoria Gitman,Andrew Brooke-Taylor05/08/2019https://philippschlicht.github.io/meetings/
35
04/30/2019Bristol Logic SeminarSet theory in second-order8th May 2019, 11:00 am – 12:00 pm
Howard House, 4th Floor Seminar Room

Speaker: Victoria Gitman (CUNY)

Title: Set theory in second-order

Abstract:

Classes, from class forcing notions to elementary embeddings of the universe to inner models, play a fundamental role in modern set theory. But within first-order set theory we are limited to studying only definable classes and we cannot even express properties that necessitate quantifying over classes. Second-order set theory is a formal framework in which a model consists both of a collection of sets and a collection of classes (which are themselves collections of sets). In second-order set theory, we can study classes such as truth predicates, which can never be definable over a model of ZFC, and properties that, for instance, quantify over all inner models. With this formal background we can develop a theory of class forcing that explains why and when class forcing behaves differently from set forcing. In this talk, I will discuss a hierarchy of second-order set theories, starting from the weak Gödel-Bernays set theory GBC and going beyond the relatively strong Kelley-Morse theory KM. I will give an overview of a number of interesting second-order set theoretic principles that arose out of recent work in this area, such as, class choice principles, transfinite recursion with classes, determinacy of games on the ordinals, and the class Fodor Principle. The study of where these principles fit in the hierarchy of second-order set theories should serve as the beginning of a reverse mathematics program that I hope this talk will encourage set theorists to take part in.
Victoria Gitmanhttps://www.bristolmathsresearch.org/seminar/victoria-gitman/
36
04/30/2019Bristol Logic SeminarSet theory and category theory8th May 2019, 11:00 am – 12:00 pm
Howard House, 4th Floor Seminar Room

Speaker: Andrew Brooke-Taylor (University of Leeds)

Title: Set theory and category theory

Abstract:

Classes, from class forcing notions to elementary embeddings of the universe to inner models, play a fundamental role in modern set theory. But within first-order set theory we are limited to studying only definable classes and we cannot even express properties that necessitate quantifying over classes. Second-order set theory is a formal framework in which a model consists both of a collection of sets and a collection of classes (which are themselves collections of sets). In second-order set theory, we can study classes such as truth predicates, which can never be definable over a model of ZFC, and properties that, for instance, quantify over all inner models. With this formal background we can develop a theory of class forcing that explains why and when class forcing behaves differently from set forcing. In this talk, I will discuss a hierarchy of second-order set theories, starting from the weak Gödel-Bernays set theory GBC and going beyond the relatively strong Kelley-Morse theory KM. I will give an overview of a number of interesting second-order set theoretic principles that arose out of recent work in this area, such as, class choice principles, transfinite recursion with classes, determinacy of games on the ordinals, and the class Fodor Principle. The study of where these principles fit in the hierarchy of second-order set theories should serve as the beginning of a reverse mathematics program that I hope this talk will encourage set theorists to take part in.
Andrew Brooke-Taylorhttps://www.bristolmathsresearch.org/seminar/andrew-brooke-taylor-2/
37
04/30/2019Bristol Logic SeminarAxioms of Infinity in Zermelo Set Theory13th May 2019, 2:30 pm – 3:30 pm
Howard House, 4th Floor Seminar Room

Speaker: Adam Epstein (University of Warwick )

Title: Axioms of Infinity in Zermelo Set Theory

Abstract:

We show that without Replacement one cannot infer the existence of a definite infinite set from the existence of an otherwise unspecified infinite set.
Adam Epsteinhttps://www.bristolmathsresearch.org/seminar/adam-epstein/
38
4/29/2019Kurt Godel Research Center
Philipp Lücke: Simple definitions of complicated sets
For many types of pathological sets of real numbers (i.e. sets of reals constructed with the help of the Axiom of Choice), it is possible to use results from descriptive set theory to show that these sets cannot be defined by simple formulas in second-order arithmetic.
In this talk, I want to present results dealing with the <em>set theoretic</em> definability of pathological objects, i.e. with the question whether objects usually obtained from the Axiom of Choice can be defined in the structure $\langle\mathrm{V},\in\rangle$ using simple formulas.
I will focus on the definability of well-orderings of the reals and bistationary subsets of uncountable regular cardinals.
Philipp Lückehttp://www.logic.univie.ac.at/2019/Talk_05-02_a.html
39
4/28/2019 20:51:58This Week in Logic at CUNYThis Week in Logic at CUNY<div dir="ltr"><div dir="ltr"><div dir="ltr"><div dir="ltr"><div style="font-family:arial,sans-serif;color:rgb(33,33,33)">This Week in Logic at CUNY:<br></div><div><div><div style="color:rgb(33,33,33)"><div><br></div><div>- - - - Monday, Apr 29, 2019 - - - -<br></div><div><br></div><div><div>Logic and Metaphysics Workshop</div><div>Date: Monday, April 29th, 4.15-6.15  <br></div><div>Place: Room 7314, CUNY Graduate Center <br></div><div>Speaker: Tommy Kivatinos (CUNY)<br></div><div>Title: A Mechanistic Conception of Metaphysical Grounding<br></div><div><br></div><div>Abstract: A dominant theoretical framework in philosophy of science employs the notion of mechanistic dependence to elucidate how higher-level, less fundamental phenomena depend upon and arise out of lower-level, more fundamental phenomena. To elucidate the same thing, literature in metaphysics employs the notion of grounding. As I argue, regardless of whether the notion of mechanistic dependence or the notion of grounding is used to theoretically portray how higher-level phenomena arise out of lower-level phenomena, what is captured by such portrayals is the same. Thus, these notions pick out the same features of the world. With this as my basis, I identify the notion of grounding with the notion of mechanistic dependence, and thus, construct a mechanistic conception of grounding. Since mechanistic dependence is understood in terms of mechanisms, my conception frames grounding in terms of mechanisms. Moreover, the contemporary notion of mechanisms is shaped by how mechanisms are represented via the mechanistic models and mechanistic explanations provided by science. Thus, because my conception grounding identifies grounding with mechanistic dependence and thereby frames grounding in terms of mechanisms, this conception suggests that the notion of grounding is to be tailored to and constrained by the mechanistic models and mechanistic explanations provided by science. This leads the mechanistic conception of grounding to reject a wide variety of conventional claims about grounding, and thus, to offer a treatment of grounding that is highly revisionary. To reinforce the plausibility of the mechanistic conception of grounding, I discuss how grounding and mechanistic dependence are associated with explanation. Whereas mechanistic dependence is associated with mechanistic explanation, grounding is associated with grounding explanation. For each kind of explanation, some higher-level phenomenon P is explained by appeal to some low-level phenomenon that Parises out of. As I argue, these forms of explanation can be plausibly identified. This greatly supports the mechanistic conception of grounding. For if grounding explanations employ the notion of grounding and mechanistic explanations employ the notion of mechanistic explanation, and these forms of explanation can be identified, this suggests that these explanations employ the same notion. And, just as the notions of grounding and mechanistic dependence capture the same connection between higher-level and lower-level phenomena, grounding explanation and mechanistic explanation do so as well. Finally, to argue that the mechanistic conception is to be preferred to standard conceptions, I argue that my conception offers a powerful defense of grounding from recent criticisms.</div></div><div><br></div><div><br></div><div><br></div><div><br></div><div>- - - - Tuesday, Apr 30, 2019 - - - -</div><div><br></div><div><br></div><div><br></div><div>- - - - Wednesday, May 1, 2019 - - - -</div><div><br></div><div><br></div><div><br></div><div>- - - - Thursday, May 2, 2019 - - - -</div><div><br></div><div><br></div><div><br></div><div>- - - - Friday, May 3, 2019 - - - -</div></div><div style="color:rgb(33,33,33)"><br></div><div><div><font color="#000000" face="arial, sans-serif">Set Theory Seminar</font></div><div><font color="#000000" face="arial, sans-serif">CUNY Graduate Center, Room 6417</font></div><div><font color="#000000" face="arial, sans-serif">Friday, May 3, 10:00-11:45am</font></div><div><p style="box-sizing:border-box;margin-bottom:1em;margin-top:0px;color:rgb(96,108,113);font-family:&quot;Open Sans&quot;,&quot;Helvetica Neue&quot;,Helvetica,Arial,sans-serif"><strong style="box-sizing:border-box"><a href="https://nylogic.github.io/set-theory-seminar/2019/05/03/target=%22_blank%22" target="_blank" style="color:rgb(30,107,184);box-sizing:border-box;background-color:transparent;text-decoration-line:none;margin-top:0px">Joseph Van Name</a></strong>, CUNY<br style="box-sizing:border-box"><strong style="box-sizing:border-box">Lower bounds on the cardinalities of quotient algebras of elementary embeddings</strong><br style="box-sizing:border-box"></p><p style="box-sizing:border-box;margin-bottom:1em;color:rgb(96,108,113);font-family:&quot;Open Sans&quot;,&quot;Helvetica Neue&quot;,Helvetica,Arial,sans-serif">From non-trivial elementary embeddings <span id="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MathJax-Element-1-Frame" class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-chtml gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MathJax_CHTML" style="box-sizing:border-box;display:inline-block;line-height:0;word-spacing:normal;white-space:nowrap;float:none;direction:ltr;max-width:none;max-height:none;min-width:0px;min-height:0px;border:0px;margin:0px;padding:1px 0px"><span id="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-Node-1" class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-math" style="box-sizing:border-box;display:inline-block;border-collapse:separate;border-spacing:0px;margin-top:0px"><span id="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-Node-2" class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-mrow" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-Node-3" class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-msubsup" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-base" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-Node-4" class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-mi" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-char gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-TeX-math-I" style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw;margin-top:0px;padding-top:0.425em;padding-bottom:0.472em;box-sizing:content-box">j</span></span></span><span class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-sub" style="display:inline-block;vertical-align:-0.212em;padding-right:0.071em;box-sizing:content-box"><span id="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-Node-5" class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-texatom" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-Node-6" class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-mrow" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-Node-7" class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-mn" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-char gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-TeX-main-R" style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.378em;padding-bottom:0.331em;box-sizing:content-box">1</span></span></span></span></span></span><span id="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-Node-8" class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-mo gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-space1" style="display:inline-block;margin-left:0.167em;box-sizing:content-box"><span class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-char gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-TeX-main-R" style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;padding-bottom:0.331em;box-sizing:content-box">…</span></span><span id="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-Node-9" class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-msubsup gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-space1" style="display:inline-block;margin-left:0.167em;box-sizing:content-box"><span class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-base" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-Node-10" class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-mi" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-char gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-TeX-math-I" style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw;margin-top:0px;padding-top:0.425em;padding-bottom:0.472em;box-sizing:content-box">j</span></span></span><span class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-sub" style="display:inline-block;vertical-align:-0.212em;padding-right:0.071em;box-sizing:content-box"><span id="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-Node-11" class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-texatom" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-Node-12" class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-mrow" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-Node-13" class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-mi" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-char gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-TeX-math-I" style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw;margin-top:0px;padding-top:0.237em;padding-bottom:0.284em;box-sizing:content-box">r</span></span></span></span></span></span><span id="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-Node-14" class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-mo gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-space3" style="display:inline-block;margin-left:0.278em;box-sizing:content-box"><span class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-char gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-TeX-main-R" style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.143em;padding-bottom:0.331em;box-sizing:content-box">:</span></span><span id="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-Node-15" class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-msubsup gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-space3" style="display:inline-block;margin-left:0.278em;box-sizing:content-box"><span class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-base" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-Node-16" class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-mi" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-char gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-TeX-math-I" style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw;margin-top:0px;padding-top:0.472em;padding-bottom:0.284em;padding-right:0.186em;box-sizing:content-box">V</span></span></span><span class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-sub" style="display:inline-block;vertical-align:-0.219em;padding-right:0.071em;box-sizing:content-box"><span id="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-Node-17" class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-texatom" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-Node-18" class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-mrow" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-Node-19" class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-mi" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-char gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-TeX-math-I" style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw;margin-top:0px;padding-top:0.472em;padding-bottom:0.284em;box-sizing:content-box">λ</span></span></span></span></span></span><span id="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-Node-20" class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-mo gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-space3" style="display:inline-block;margin-left:0.278em;box-sizing:content-box"><span class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-char gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-TeX-main-R" style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.237em;padding-bottom:0.331em;box-sizing:content-box">→</span></span><span id="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-Node-21" class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-msubsup gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-space3" style="display:inline-block;margin-left:0.278em;box-sizing:content-box"><span class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-base" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-Node-22" class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-mi" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-char gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-TeX-math-I" style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw;margin-top:0px;padding-top:0.472em;padding-bottom:0.284em;padding-right:0.186em;box-sizing:content-box">V</span></span></span><span class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-sub" style="display:inline-block;vertical-align:-0.219em;padding-right:0.071em;box-sizing:content-box"><span id="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-Node-23" class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-texatom" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-Node-24" class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-mrow" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-Node-25" class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-mi" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-char gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-TeX-math-I" style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw;margin-top:0px;padding-top:0.472em;padding-bottom:0.284em;box-sizing:content-box">λ</span></span></span></span></span></span></span></span><span class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJX_Assistive_MathML" style="box-sizing:border-box;padding:1px 0px 0px;border:0px;height:1px;width:1px;overflow:hidden;display:block">j<span class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJX-TeXAtom-ORD">1</span>…j<span class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJX-TeXAtom-ORD">r</span>:V<span class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJX-TeXAtom-ORD">λ</span>→V<span class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJX-TeXAtom-ORD">λ</span></span></span>, we obtain a sequence of polynomials <span id="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MathJax-Element-2-Frame" class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-chtml gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MathJax_CHTML" style="box-sizing:border-box;display:inline-block;line-height:0;word-spacing:normal;white-space:nowrap;float:none;direction:ltr;max-width:none;max-height:none;min-width:0px;min-height:0px;border:0px;margin:0px;padding:1px 0px"><span id="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-Node-26" class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-math" style="box-sizing:border-box;display:inline-block;border-collapse:separate;border-spacing:0px;margin-top:0px"><span id="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-Node-27" class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-mrow" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-Node-28" class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-mo" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-char gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-TeX-main-R" style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.472em;padding-bottom:0.613em;box-sizing:content-box">(</span></span><span id="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-Node-29" class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-msubsup" style="display:inline-block;box-sizing:content-box"><span class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-base" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-Node-30" class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-mi" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-char gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-TeX-math-I" style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw;margin-top:0px;padding-top:0.237em;padding-bottom:0.472em;box-sizing:content-box">p</span></span></span><span class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-sub" style="display:inline-block;vertical-align:-0.212em;padding-right:0.071em;box-sizing:content-box"><span id="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-Node-31" class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-texatom" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-Node-32" class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-mrow" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-Node-33" class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-mi" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-char gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-TeX-math-I" style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw;margin-top:0px;padding-top:0.237em;padding-bottom:0.284em;box-sizing:content-box">n</span></span></span></span></span></span><span id="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-Node-34" class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-mo" style="display:inline-block;box-sizing:content-box"><span class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-char gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-TeX-main-R" style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.472em;padding-bottom:0.613em;box-sizing:content-box">(</span></span><span id="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-Node-35" class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-msubsup" style="display:inline-block;box-sizing:content-box"><span class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-base" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-Node-36" class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-mi" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-char gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-TeX-math-I" style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw;margin-top:0px;padding-top:0.237em;padding-bottom:0.284em;box-sizing:content-box">x</span></span></span><span class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-sub" style="display:inline-block;vertical-align:-0.212em;padding-right:0.071em;box-sizing:content-box"><span id="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-Node-37" class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-texatom" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-Node-38" class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-mrow" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-Node-39" class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-mn" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-char gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-TeX-main-R" style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.378em;padding-bottom:0.331em;box-sizing:content-box">1</span></span></span></span></span></span><span id="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-Node-40" class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-mo" style="display:inline-block;box-sizing:content-box"><span class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-char gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-TeX-main-R" style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;padding-bottom:0.519em;box-sizing:content-box">,</span></span><span id="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-Node-41" class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-mo gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-space1" style="display:inline-block;margin-left:0.167em;box-sizing:content-box"><span class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-char gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-TeX-main-R" style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;padding-bottom:0.331em;box-sizing:content-box">…</span></span><span id="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-Node-42" class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-mo gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-space1" style="display:inline-block;margin-left:0.167em;box-sizing:content-box"><span class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-char gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-TeX-main-R" style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;padding-bottom:0.519em;box-sizing:content-box">,</span></span><span id="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-Node-43" class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-msubsup gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-space1" style="display:inline-block;margin-left:0.167em;box-sizing:content-box"><span class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-base" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-Node-44" class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-mi" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-char gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-TeX-math-I" style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw;margin-top:0px;padding-top:0.237em;padding-bottom:0.284em;box-sizing:content-box">x</span></span></span><span class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-sub" style="display:inline-block;vertical-align:-0.212em;padding-right:0.071em;box-sizing:content-box"><span id="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-Node-45" class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-texatom" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-Node-46" class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-mrow" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-Node-47" class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-mi" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-char gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-TeX-math-I" style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw;margin-top:0px;padding-top:0.237em;padding-bottom:0.284em;box-sizing:content-box">r</span></span></span></span></span></span><span id="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-Node-48" class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-mo" style="display:inline-block;box-sizing:content-box"><span class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-char gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-TeX-main-R" style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.472em;padding-bottom:0.613em;box-sizing:content-box">)</span></span><span id="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-Node-49" class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-msubsup" style="display:inline-block;box-sizing:content-box"><span class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-base" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-Node-50" class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-mo" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-char gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-TeX-main-R" style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.472em;padding-bottom:0.613em;box-sizing:content-box">)</span></span></span><span class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-sub" style="display:inline-block;vertical-align:-0.212em;padding-right:0.071em;box-sizing:content-box"><span id="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-Node-51" class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-texatom" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-Node-52" class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-mrow" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-Node-53" class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-mi" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-char gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-TeX-math-I" style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw;margin-top:0px;padding-top:0.237em;padding-bottom:0.284em;box-sizing:content-box">n</span></span><span id="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-Node-54" class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-mo" style="display:inline-block;box-sizing:content-box"><span class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-char gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-TeX-main-R" style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.237em;padding-bottom:0.378em;box-sizing:content-box">∈</span></span><span id="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-Node-55" class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-mi" style="display:inline-block;box-sizing:content-box"><span class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-char gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-TeX-math-I" style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw;margin-top:0px;padding-top:0.237em;padding-bottom:0.284em;box-sizing:content-box">ω</span></span></span></span></span></span></span></span><span class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJX_Assistive_MathML" style="box-sizing:border-box;padding:1px 0px 0px;border:0px;height:1px;width:1px;overflow:hidden;display:block">(p<span class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJX-TeXAtom-ORD">n</span>(x<span class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJX-TeXAtom-ORD">1</span>,…,x<span class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJX-TeXAtom-ORD">r</span>))<span class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJX-TeXAtom-ORD">n∈ω</span></span></span> that satisfies the infinite product<span class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-chtml gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-display" style="box-sizing:border-box;display:block;line-height:0;text-align:center;word-spacing:normal;white-space:nowrap;float:none;direction:ltr;max-width:none;max-height:none;min-width:0px;min-height:0px;border:0px;margin:1em 0px;padding:0px"><span id="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MathJax-Element-3-Frame" class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-chtml gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MathJax_CHTML" style="box-sizing:border-box;display:inline-block;line-height:0;word-spacing:normal;float:none;direction:ltr;max-width:none;max-height:none;min-width:0px;min-height:0px;border:0px;margin:0px;padding:1px 0px"><span id="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-Node-56" class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-math" style="box-sizing:border-box;display:inline-block;border-collapse:separate;border-spacing:0px;margin-top:0px"><span id="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-Node-57" class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-mrow" style="display:inline-block;text-align:left;margin-top:0px;box-sizing:content-box"><span id="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-Node-58" class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-munderover" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-itable" style="display:inline-table;width:auto;margin-top:0px;box-sizing:content-box"><span class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-row" style="display:table-row;margin-top:0px;box-sizing:content-box"><span class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-cell" style="display:table-cell;margin-top:0px;box-sizing:content-box"><span class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-stack" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-over" style="display:block;margin-top:0px;padding-bottom:0.247em;padding-top:0.141em;padding-left:0.404em;box-sizing:content-box"><span id="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-Node-65" class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-texatom" style="display:inline-block;margin-top:0px;box-sizing:content-box;padding-left:0px;padding-right:0px"><span id="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-Node-66" class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-mrow" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-Node-67" class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-mi" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-char gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-TeX-main-R" style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.143em;padding-bottom:0.331em;box-sizing:content-box">∞</span></span></span></span></span><span class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-op" style="display:block;box-sizing:content-box"><span id="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-Node-59" class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-mo" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-char gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-TeX-size2-R" style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-size2-R,MJXc-TeX-size2-Rw;margin-top:0px;padding-top:0.707em;padding-bottom:0.707em;box-sizing:content-box">∏</span></span></span></span></span></span><span class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-row" style="display:table-row;box-sizing:content-box"><span class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-under" style="display:table-cell;margin-top:0px;padding-top:0.236em;padding-bottom:0.141em;padding-left:0.004em;box-sizing:content-box"><span id="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-Node-60" class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-texatom" style="display:inline-block;margin-top:0px;box-sizing:content-box;padding-left:0px;padding-right:0px"><span id="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-Node-61" class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-mrow" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-Node-62" class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-mi" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-char gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-TeX-math-I" style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw;margin-top:0px;padding-top:0.472em;padding-bottom:0.284em;box-sizing:content-box">k</span></span><span id="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-Node-63" class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-mo" style="display:inline-block;box-sizing:content-box"><span class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-char gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-TeX-main-R" style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.096em;padding-bottom:0.331em;box-sizing:content-box">=</span></span><span id="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-Node-64" class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-mn" style="display:inline-block;box-sizing:content-box"><span class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-char gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-TeX-main-R" style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.378em;padding-bottom:0.378em;box-sizing:content-box">0</span></span></span></span></span></span></span></span><span id="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-Node-68" class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-msubsup gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-space1" style="display:inline-block;margin-left:0.167em;box-sizing:content-box"><span class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-base" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-Node-69" class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-mi" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-char gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-TeX-math-I" style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw;margin-top:0px;padding-top:0.237em;padding-bottom:0.472em;box-sizing:content-box">p</span></span></span><span class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-sub" style="display:inline-block;vertical-align:-0.219em;padding-right:0.071em;box-sizing:content-box"><span id="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-Node-70" class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-texatom" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-Node-71" class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-mrow" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-Node-72" class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-mi" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-char gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-TeX-math-I" style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw;margin-top:0px;padding-top:0.472em;padding-bottom:0.284em;box-sizing:content-box">k</span></span></span></span></span></span><span id="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-Node-73" class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-mo" style="display:inline-block;box-sizing:content-box"><span class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-char gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-TeX-main-R" 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gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-space2" style="display:inline-block;margin-left:0.222em;box-sizing:content-box"><span class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-base" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-Node-104" class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-mi" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-char gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-TeX-math-I" style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw;margin-top:0px;padding-top:0.237em;padding-bottom:0.284em;box-sizing:content-box">x</span></span></span><span class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-sub" style="display:inline-block;vertical-align:-0.212em;padding-right:0.071em;box-sizing:content-box"><span id="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-Node-105" class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-texatom" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-Node-106" class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-mrow" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-Node-107" class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-mi" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-char gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-TeX-math-I" style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw;margin-top:0px;padding-top:0.237em;padding-bottom:0.284em;box-sizing:content-box">r</span></span></span></span></span></span><span id="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-Node-108" class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-mo" style="display:inline-block;box-sizing:content-box"><span class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-char gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-TeX-main-R" style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.472em;padding-bottom:0.613em;box-sizing:content-box">)</span></span></span></span><span class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-line" style="display:block;height:0px;border-bottom:1.3px solid;width:8.205em;box-sizing:content-box"></span></span><span class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-vsize" style="display:inline-block;width:0px;height:2.375em;vertical-align:-1.007em;box-sizing:content-box"></span></span><span id="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-Node-109" class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-mo" style="display:inline-block;box-sizing:content-box"><span class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-char gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-TeX-main-R" style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;padding-bottom:0.331em;box-sizing:content-box">.</span></span></span></span><span class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJX_Assistive_MathML gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJX_Assistive_MathML_Block" style="box-sizing:border-box;width:370.25px;padding:1px 0px 0px;border:0px;height:1px;overflow:hidden;display:block">∏<span class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJX-TeXAtom-ORD">k=0</span><span class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJX-TeXAtom-ORD">∞</span>p<span class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJX-TeXAtom-ORD">k</span>(x<span class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJX-TeXAtom-ORD">1</span>,…,x<span class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJX-TeXAtom-ORD">r</span>)=11−(x<span class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJX-TeXAtom-ORD">1</span>+⋯+x<span class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJX-TeXAtom-ORD">r</span>).</span></span></span>From this infinite product, we deduce lower bounds of the cardinality of <span id="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MathJax-Element-4-Frame" class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-chtml gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MathJax_CHTML" style="box-sizing:border-box;display:inline-block;line-height:0;word-spacing:normal;white-space:nowrap;float:none;direction:ltr;max-width:none;max-height:none;min-width:0px;min-height:0px;border:0px;margin:0px;padding:1px 0px"><span id="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-Node-110" class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-math" style="box-sizing:border-box;display:inline-block;border-collapse:separate;border-spacing:0px;margin-top:0px"><span id="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-Node-111" class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-mrow" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-Node-112" class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-texatom" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-Node-113" class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-mrow" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-Node-114" class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-mo" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-char gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-TeX-main-R" style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.472em;padding-bottom:0.613em;box-sizing:content-box">|</span></span></span></span><span id="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-Node-115" class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-mo" style="display:inline-block;box-sizing:content-box"><span class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-char gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-TeX-main-R" style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.472em;padding-bottom:0.613em;box-sizing:content-box">⟨</span></span><span id="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-Node-116" class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-msubsup" style="display:inline-block;box-sizing:content-box"><span class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-base" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-Node-117" class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-mi" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-char gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-TeX-math-I" style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw;margin-top:0px;padding-top:0.425em;padding-bottom:0.472em;box-sizing:content-box">j</span></span></span><span class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-sub" style="display:inline-block;vertical-align:-0.212em;padding-right:0.071em;box-sizing:content-box"><span id="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-Node-118" class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-texatom" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-Node-119" class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-mrow" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-Node-120" class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-mn" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-char gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-TeX-main-R" style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.378em;padding-bottom:0.331em;box-sizing:content-box">1</span></span></span></span></span></span><span id="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-Node-121" class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-mo" style="display:inline-block;box-sizing:content-box"><span class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-char gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-TeX-main-R" style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;padding-bottom:0.519em;box-sizing:content-box">,</span></span><span id="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-Node-122" class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-mo gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-space1" style="display:inline-block;margin-left:0.167em;box-sizing:content-box"><span class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-char gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-TeX-main-R" style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;padding-bottom:0.331em;box-sizing:content-box">.</span></span><span id="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-Node-123" class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-mo gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-space1" style="display:inline-block;margin-left:0.167em;box-sizing:content-box"><span class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-char gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-TeX-main-R" style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;padding-bottom:0.331em;box-sizing:content-box">.</span></span><span id="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-Node-124" class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-mo gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-space1" style="display:inline-block;margin-left:0.167em;box-sizing:content-box"><span class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-char gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-TeX-main-R" style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;padding-bottom:0.331em;box-sizing:content-box">.</span></span><span id="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-Node-125" class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-mo gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-space1" style="display:inline-block;margin-left:0.167em;box-sizing:content-box"><span class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-char gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-TeX-main-R" style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;padding-bottom:0.519em;box-sizing:content-box">,</span></span><span id="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-Node-126" class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-msubsup gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-space1" style="display:inline-block;margin-left:0.167em;box-sizing:content-box"><span class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-base" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-Node-127" class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-mi" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-char gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-TeX-math-I" style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw;margin-top:0px;padding-top:0.425em;padding-bottom:0.472em;box-sizing:content-box">j</span></span></span><span class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-sub" style="display:inline-block;vertical-align:-0.212em;padding-right:0.071em;box-sizing:content-box"><span id="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-Node-128" class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-texatom" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-Node-129" class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-mrow" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-Node-130" class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-mi" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-char gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-TeX-math-I" style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw;margin-top:0px;padding-top:0.237em;padding-bottom:0.284em;box-sizing:content-box">r</span></span></span></span></span></span><span id="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-Node-131" class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-mo" style="display:inline-block;box-sizing:content-box"><span class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-char gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-TeX-main-R" style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.472em;padding-bottom:0.613em;box-sizing:content-box">⟩</span></span><span id="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-Node-132" class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-texatom" style="display:inline-block;box-sizing:content-box"><span id="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-Node-133" class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-mrow" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-Node-134" class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-mo" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-char gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-TeX-main-R" style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.472em;padding-bottom:0.613em;box-sizing:content-box">/</span></span></span></span><span id="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-Node-135" class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-msubsup gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-space3" style="display:inline-block;margin-left:0.278em;box-sizing:content-box"><span class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-base" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-Node-136" class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-mo" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-char gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-TeX-main-R" style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.19em;padding-bottom:0.331em;box-sizing:content-box">≡</span></span></span><span class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-sup" style="display:inline-block;vertical-align:0.513em;padding-left:0px;padding-right:0.071em;box-sizing:content-box"><span id="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-Node-137" class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-texatom" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-Node-138" class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-mrow" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-Node-139" class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-mi" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-char gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-TeX-math-I" style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw;margin-top:0px;padding-top:0.237em;padding-bottom:0.284em;box-sizing:content-box">α</span></span></span></span></span></span><span id="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-Node-140" class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-texatom gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-space3" style="display:inline-block;margin-left:0.278em;box-sizing:content-box"><span id="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-Node-141" class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-mrow" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-Node-142" class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-mo" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-char gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-TeX-main-R" style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.472em;padding-bottom:0.613em;box-sizing:content-box">|</span></span></span></span></span></span><span class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJX_Assistive_MathML" style="box-sizing:border-box;padding:1px 0px 0px;border:0px;height:1px;width:1px;overflow:hidden;display:block"><span class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJX-TeXAtom-ORD">|</span>⟨j<span class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJX-TeXAtom-ORD">1</span>,...,j<span class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJX-TeXAtom-ORD">r</span>⟩<span class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJX-TeXAtom-ORD">/</span>≡<span class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJX-TeXAtom-ORD">α</span><span class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJX-TeXAtom-ORD">|</span></span></span> using analysis and analytic number theoretic techniques. Computer calculations that search for Laver-like algebras give some empirical evidence that these lower bounds cannot be greatly improved.</p><p style="box-sizing:border-box;margin-bottom:1em;color:rgb(96,108,113);font-family:&quot;Open Sans&quot;,&quot;Helvetica Neue&quot;,Helvetica,Arial,sans-serif"></p><br class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-Apple-interchange-newline"></div><div><font color="#000000" face="arial, sans-serif"><br></font></div><div><font color="#000000" face="arial, sans-serif"><br></font></div><div><font color="#000000" face="arial, sans-serif">Model Theory Seminar</font></div><div><font color="#000000" face="arial, sans-serif">CUNY Graduate Center, Room 6417</font></div><div><font color="#000000" face="arial, sans-serif">Friday, </font><span style="color:rgb(0,0,0);font-family:arial,sans-serif">May 3</span><font color="#000000" face="arial, sans-serif">, 12:30-2:00pm</font></div><div><font color="#000000" face="arial, sans-serif"><div>Artem Chernikov, UCLA</div><div>TBA</div></font></div><div><font color="#000000" face="arial, sans-serif"><br></font></div><div><br></div><div><br></div><div><br></div><div><font color="#000000" face="arial, sans-serif">Logic Workshop</font></div><div><font color="#000000" face="arial, sans-serif">CUNY Graduate Center, Room 6417</font></div><div><font color="#000000" face="arial, sans-serif">Friday, </font><span style="color:rgb(0,0,0);font-family:arial,sans-serif">May 3</span><font color="#000000" face="arial, sans-serif">, 2:00-3:30pm</font></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><p style="box-sizing:border-box;margin-bottom:1em;margin-top:0px;color:rgb(96,108,113);font-family:&quot;Open Sans&quot;,&quot;Helvetica Neue&quot;,Helvetica,Arial,sans-serif"><strong style="box-sizing:border-box"><a href="http://sites.math.rutgers.edu/~cherlin/" target="_blank" style="color:rgb(30,107,184);box-sizing:border-box;background-color:transparent;text-decoration-line:none;margin-top:0px">Gregory Cherlin</a></strong>, Rutgers University<br style="box-sizing:border-box"><strong style="box-sizing:border-box">Countable universal graphs with forbidden subgraphs</strong></p><p style="box-sizing:border-box;margin-bottom:1em;margin-top:0px;color:rgb(96,108,113);font-family:&quot;Open Sans&quot;,&quot;Helvetica Neue&quot;,Helvetica,Arial,sans-serif"><strong style="box-sizing:border-box;margin-top:0px">Abstract</strong></p><p style="box-sizing:border-box;margin-bottom:1em;color:rgb(96,108,113);font-family:&quot;Open Sans&quot;,&quot;Helvetica Neue&quot;,Helvetica,Arial,sans-serif"><strong style="box-sizing:border-box;margin-top:0px">Definition</strong>: A collection <span id="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MathJax-Element-1-Frame" class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-chtml gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MathJax_CHTML" style="box-sizing:border-box;display:inline-block;line-height:0;word-spacing:normal;white-space:nowrap;float:none;direction:ltr;max-width:none;max-height:none;min-width:0px;min-height:0px;border:0px;margin:0px;padding:1px 0px"><span id="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-Node-1" class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-math" style="box-sizing:border-box;display:inline-block;border-collapse:separate;border-spacing:0px;margin-top:0px"><span id="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-Node-2" class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-mrow" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-Node-3" class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-texatom" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-Node-4" class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-mrow" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-Node-5" class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-mi" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-char gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-TeX-cal-R" style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-cal-R,MJXc-TeX-cal-Rw;margin-top:0px;padding-top:0.425em;padding-bottom:0.331em;padding-right:0.11em;box-sizing:content-box">F</span></span></span></span></span></span><span class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJX_Assistive_MathML" style="box-sizing:border-box;padding:1px 0px 0px;border:0px;height:1px;width:1px;overflow:hidden;display:block"><span class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJX-TeXAtom-ORD"><span class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJX-tex-caligraphic">F</span></span></span></span> of forbidden subgraphs is a <em style="box-sizing:border-box">Rado family</em> if the class of countable <span id="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MathJax-Element-2-Frame" class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-chtml gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MathJax_CHTML" style="box-sizing:border-box;display:inline-block;line-height:0;word-spacing:normal;white-space:nowrap;float:none;direction:ltr;max-width:none;max-height:none;min-width:0px;min-height:0px;border:0px;margin:0px;padding:1px 0px"><span id="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-Node-6" class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-math" style="box-sizing:border-box;display:inline-block;border-collapse:separate;border-spacing:0px;margin-top:0px"><span id="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-Node-7" class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-mrow" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-Node-8" class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-texatom" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-Node-9" class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-mrow" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-Node-10" class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-mi" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-char gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-TeX-cal-R" style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-cal-R,MJXc-TeX-cal-Rw;margin-top:0px;padding-top:0.425em;padding-bottom:0.331em;padding-right:0.11em;box-sizing:content-box">F</span></span></span></span></span></span><span class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJX_Assistive_MathML" style="box-sizing:border-box;padding:1px 0px 0px;border:0px;height:1px;width:1px;overflow:hidden;display:block"><span class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJX-TeXAtom-ORD"><span class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJX-tex-caligraphic">F</span></span></span></span>-free graphs contains a universal structure with respect to embeddings as induced subgraphs. (By &#39;forbidden&#39; I mean forbidden as subgraphs; usage varies in the literature.)</p><p style="box-sizing:border-box;margin-bottom:1em;color:rgb(96,108,113);font-family:&quot;Open Sans&quot;,&quot;Helvetica Neue&quot;,Helvetica,Arial,sans-serif">We consider the following decision problem for Rado families.</p><p style="box-sizing:border-box;margin-bottom:1em;color:rgb(96,108,113);font-family:&quot;Open Sans&quot;,&quot;Helvetica Neue&quot;,Helvetica,Arial,sans-serif"><strong style="box-sizing:border-box;margin-top:0px">Problem</strong>: Given a finite set <span id="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MathJax-Element-3-Frame" class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-chtml gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MathJax_CHTML" style="box-sizing:border-box;display:inline-block;line-height:0;word-spacing:normal;white-space:nowrap;float:none;direction:ltr;max-width:none;max-height:none;min-width:0px;min-height:0px;border:0px;margin:0px;padding:1px 0px"><span id="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-Node-11" class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-math" style="box-sizing:border-box;display:inline-block;border-collapse:separate;border-spacing:0px;margin-top:0px"><span id="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-Node-12" class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-mrow" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-Node-13" class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-texatom" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-Node-14" class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-mrow" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-Node-15" class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-mi" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-char gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-TeX-cal-R" style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-cal-R,MJXc-TeX-cal-Rw;margin-top:0px;padding-top:0.425em;padding-bottom:0.331em;padding-right:0.11em;box-sizing:content-box">F</span></span></span></span></span></span><span class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJX_Assistive_MathML" style="box-sizing:border-box;padding:1px 0px 0px;border:0px;height:1px;width:1px;overflow:hidden;display:block"><span class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJX-TeXAtom-ORD"><span class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJX-tex-caligraphic">F</span></span></span></span> of finite, connected graphs, determine whether or not it is a Rado family.</p><p style="box-sizing:border-box;margin-bottom:1em;color:rgb(96,108,113);font-family:&quot;Open Sans&quot;,&quot;Helvetica Neue&quot;,Helvetica,Arial,sans-serif"><strong style="box-sizing:border-box;margin-top:0px">Question</strong>: Is this a decidable problem?</p><p style="box-sizing:border-box;margin-bottom:1em;color:rgb(96,108,113);font-family:&quot;Open Sans&quot;,&quot;Helvetica Neue&quot;,Helvetica,Arial,sans-serif">If we specialize to the sake of a single constraint then we speak instead of a <em style="box-sizing:border-box;margin-top:0px">Rado constraint.</em> Much is known, and much more conjectured, about the case of a <em style="box-sizing:border-box">single</em> constraint.</p><p style="box-sizing:border-box;margin-bottom:1em;color:rgb(96,108,113);font-family:&quot;Open Sans&quot;,&quot;Helvetica Neue&quot;,Helvetica,Arial,sans-serif"><strong style="box-sizing:border-box;margin-top:0px">Conjecture</strong>: The decision problem for the case of a single constraint is decidable.</p><p style="box-sizing:border-box;margin-bottom:1em;color:rgb(96,108,113);font-family:&quot;Open Sans&quot;,&quot;Helvetica Neue&quot;,Helvetica,Arial,sans-serif"><strong style="box-sizing:border-box;margin-top:0px">Discussion</strong></p><p style="box-sizing:border-box;margin-bottom:1em;color:rgb(96,108,113);font-family:&quot;Open Sans&quot;,&quot;Helvetica Neue&quot;,Helvetica,Arial,sans-serif">From the model theoretic side, a finite collection of forbidden subgraphs specifies a universal theory with some special properties: notably, there is a <em style="box-sizing:border-box;margin-top:0px">model companion</em>. When we restrict to families of connected constraints, the theory has joint embedding and the model companion is complete. In model theoretic terms the question becomes whether the model companion has a countable universal model with respect to elementary embedding, for which a purely type theoretic criterion is known.</p><p style="box-sizing:border-box;margin-bottom:1em;color:rgb(96,108,113);font-family:&quot;Open Sans&quot;,&quot;Helvetica Neue&quot;,Helvetica,Arial,sans-serif">Decision problems for other model theoretic properties of the model companion are natural. We have focused on this one because the question came to us from graph theory. However, even in that form, two variants are relevant.</p><ul style="box-sizing:border-box;margin-top:0px;color:rgb(96,108,113);font-family:&quot;Open Sans&quot;,&quot;Helvetica Neue&quot;,Helvetica,Arial,sans-serif"><li style="margin-left:15px;box-sizing:border-box;margin-top:0px">Can we decide whether the model companion is <span id="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MathJax-Element-4-Frame" class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-chtml gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MathJax_CHTML" style="box-sizing:border-box;display:inline-block;line-height:0;word-spacing:normal;white-space:nowrap;float:none;direction:ltr;max-width:none;max-height:none;min-width:0px;min-height:0px;border:0px;margin:0px;padding:1px 0px"><span id="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-Node-16" class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-math" style="box-sizing:border-box;display:inline-block;border-collapse:separate;border-spacing:0px;margin-top:0px"><span id="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-Node-17" class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-mrow" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-Node-18" class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-msubsup" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-base" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-Node-19" class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-mi" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-char gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-TeX-main-R" style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.425em;padding-bottom:0.331em;box-sizing:content-box">ℵ</span></span></span><span class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-sub" style="display:inline-block;vertical-align:-0.212em;padding-right:0.071em;box-sizing:content-box"><span id="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-Node-20" class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-mn" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-char gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-TeX-main-R" style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.378em;padding-bottom:0.378em;box-sizing:content-box">0</span></span></span></span></span></span><span class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJX_Assistive_MathML" style="box-sizing:border-box;padding:1px 0px 0px;border:0px;height:1px;width:1px;overflow:hidden;display:block">ℵ0</span></span>-categorical?</li><li style="margin-left:15px;box-sizing:border-box">Dropping the connectedness hypothesis on the constraints, can we decide whether the model companion is complete?</li></ul><p style="box-sizing:border-box;margin-bottom:1em;color:rgb(96,108,113);font-family:&quot;Open Sans&quot;,&quot;Helvetica Neue&quot;,Helvetica,Arial,sans-serif"></p><p style="box-sizing:border-box;margin-bottom:1em;color:rgb(96,108,113);font-family:&quot;Open Sans&quot;,&quot;Helvetica Neue&quot;,Helvetica,Arial,sans-serif">The main source of positive answers to the universality problem appears to be the <span id="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MathJax-Element-5-Frame" class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-chtml gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MathJax_CHTML" style="box-sizing:border-box;display:inline-block;line-height:0;word-spacing:normal;white-space:nowrap;float:none;direction:ltr;max-width:none;max-height:none;min-width:0px;min-height:0px;border:0px;margin:0px;padding:1px 0px"><span id="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-Node-21" class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-math" style="box-sizing:border-box;display:inline-block;border-collapse:separate;border-spacing:0px;margin-top:0px"><span id="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-Node-22" class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-mrow" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-Node-23" class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-msubsup" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-base" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-Node-24" class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-mi" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-char gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-TeX-main-R" style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.425em;padding-bottom:0.331em;box-sizing:content-box">ℵ</span></span></span><span class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-sub" style="display:inline-block;vertical-align:-0.212em;padding-right:0.071em;box-sizing:content-box"><span id="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-Node-25" class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-mn" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-char gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-TeX-main-R" style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.378em;padding-bottom:0.378em;box-sizing:content-box">0</span></span></span></span></span></span><span class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJX_Assistive_MathML" style="box-sizing:border-box;padding:1px 0px 0px;border:0px;height:1px;width:1px;overflow:hidden;display:block">ℵ0</span></span>-categorical case, and this has some theoretical justification. The most direct route to understanding the original graph theoretical problem appears to take the detour through <span id="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MathJax-Element-6-Frame" class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-chtml gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MathJax_CHTML" style="box-sizing:border-box;display:inline-block;line-height:0;word-spacing:normal;white-space:nowrap;float:none;direction:ltr;max-width:none;max-height:none;min-width:0px;min-height:0px;border:0px;margin:0px;padding:1px 0px"><span id="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-Node-26" class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-math" style="box-sizing:border-box;display:inline-block;border-collapse:separate;border-spacing:0px;margin-top:0px"><span id="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-Node-27" class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-mrow" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-Node-28" class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-msubsup" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-base" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-Node-29" class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-mi" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-char gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-TeX-main-R" style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.425em;padding-bottom:0.331em;box-sizing:content-box">ℵ</span></span></span><span class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-sub" style="display:inline-block;vertical-align:-0.212em;padding-right:0.071em;box-sizing:content-box"><span id="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-Node-30" class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-mn" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-mjx-char gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJXc-TeX-main-R" style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.378em;padding-bottom:0.378em;box-sizing:content-box">0</span></span></span></span></span></span><span class="gmail-m_439515022332123512gmail-m_5751542264780799491gmail-MJX_Assistive_MathML" style="box-sizing:border-box;padding:1px 0px 0px;border:0px;height:1px;width:1px;overflow:hidden;display:block">ℵ0</span></span>-categoricity.</p><p style="box-sizing:border-box;margin-bottom:1em;color:rgb(96,108,113);font-family:&quot;Open Sans&quot;,&quot;Helvetica Neue&quot;,Helvetica,Arial,sans-serif">As time allows, I would like to discuss the following three points.</p><ul style="box-sizing:border-box;margin-top:0px;color:rgb(96,108,113);font-family:&quot;Open Sans&quot;,&quot;Helvetica Neue&quot;,Helvetica,Arial,sans-serif"><li style="margin-left:15px;box-sizing:border-box;margin-top:0px">What we know, and what we expect, in the case of a single constraint;</li><li style="margin-left:15px;box-sizing:border-box">Methods of proof (<em style="box-sizing:border-box;margin-top:0px">algebraic closure; induction by pruning</em>);</li><li style="margin-left:15px;box-sizing:border-box">The status of the decision problem for j.e.p. and its analog in the theory of permutation pattern classes (work of Braunfeld).</li></ul><p style="box-sizing:border-box;margin-bottom:1em;color:rgb(96,108,113);font-family:&quot;Open Sans&quot;,&quot;Helvetica Neue&quot;,Helvetica,Arial,sans-serif"></p><p style="box-sizing:border-box;margin-bottom:1em;color:rgb(96,108,113);font-family:&quot;Open Sans&quot;,&quot;Helvetica Neue&quot;,Helvetica,Arial,sans-serif">This is joint work with Shelah, e.g. [Sh689].</p></div></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><br></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><br></div><div><div><font color="#000000" face="arial, sans-serif">Next Week in Logic at CUNY:</font></div><div><font color="#000000" face="arial, sans-serif"><br></font></div><div><font color="#000000" face="arial, sans-serif">- - - - Monday, May 6, 2019 - - - -</font></div><div><font color="#000000" face="arial, sans-serif"><br></font></div><div><font color="#000000" face="arial, sans-serif"><br></font></div><div><font color="#000000" face="arial, sans-serif"><br></font></div><div><font color="#000000" face="arial, sans-serif">- - - - Tuesday, May 7, 2019 - - - -</font></div><div><font color="#000000" face="arial, sans-serif"><br></font></div><div><font color="#000000" face="arial, sans-serif"><br></font></div><div><font color="#000000" face="arial, sans-serif"><br></font></div><div><font color="#000000" face="arial, sans-serif">- - - - Wednesday, May 8, 2019 - - - -</font></div><div><font color="#000000" face="arial, sans-serif"><br></font></div><div><font color="#000000" face="arial, sans-serif"><div>The New York City Category Theory Seminar</div><div>Department of Computer Science</div><div>Department of Mathematics </div><div>The Graduate Center of The City University of New York </div><div><div>Speaker:     Jonathan Funk, Queensborough Community College.</div><div>Date and Time:     Wednesday May 8, 2019, 7:00 - 8:30 PM., Room 6417.</div><div>Title:     Isotropy theory meets Galois theory.</div><div><br></div><div>Abstract: Isotropy theory for toposes is about internal symmetry of a topos. A topos may have trivial isotropy, said to be anisotropic. For example, a localic topos is anisotropic. The isotropy of a topos may be cancelled to yield what we call the isotropy quotient of a topos, although the quotient may itself have isotropy, or what we call higher isotropy of the given topos. (By analogy, the quotient of a group by its center may itself have non-trivial center.) Let us say that a topos is locally anisotropic if it has an etale cover by an anisotropic topos. </div><div><br></div><div>THEOREM: A locally anisotropic topos has no higher isotropy. Equivalently, its isotropy quotient is anisotropic. Furthermore, a locally anisotropic topos is recovered as the topos of actions for a connected groupoid internal to its isotropy quotient. </div><div><br></div><div>COROLLARY: An etendue, or locally localic topos, has no higher isotropy. An etendue may be recovered as the topos of actions for a connected groupoid internal to its isotropy quotient. </div><div><br></div><div>Our argumentation of the theorem brings into focus how isotropy theory and Galois theory for toposes meet in a natural and evidently effective way. </div><div><br></div><div>Joint work with Pieter Hofstra.</div></div></font></div><div><font color="#000000" face="arial, sans-serif"><br></font></div><div><font color="#000000" face="arial, sans-serif"><br></font></div><div><font color="#000000" face="arial, sans-serif"><br></font></div><div><font color="#000000" face="arial, sans-serif"><br></font></div><div><font color="#000000" face="arial, sans-serif"><br></font></div><div><font color="#000000" face="arial, sans-serif">- - - - Thursday, May 9, 2019 - - - -</font></div><div><font color="#000000" face="arial, sans-serif"><br></font></div><div><font color="#000000" face="arial, sans-serif"><br></font></div><div><font color="#000000" face="arial, sans-serif"><br></font></div><div><font color="#000000" face="arial, sans-serif">- - - - Friday, May 10, 2019 - - - -</font></div></div><div><font color="#000000" face="arial, sans-serif"><br></font></div><div><div><font color="#000000" face="arial, sans-serif">Set Theory Seminar</font></div><div><font color="#000000" face="arial, sans-serif">CUNY Graduate Center, Room 6417</font></div><div><font color="#000000" face="arial, sans-serif">Friday, May 10, 10:00-11:45am</font></div></div><div><p style="box-sizing:border-box;margin-bottom:1em;margin-top:0px;color:rgb(96,108,113);font-family:&quot;Open Sans&quot;,&quot;Helvetica Neue&quot;,Helvetica,Arial,sans-serif"><strong style="box-sizing:border-box"><a href="http://kamerynjw.net/" target="_blank" style="box-sizing:border-box;background-color:transparent;color:rgb(30,107,184);text-decoration-line:none;margin-top:0px">Kameryn Williams</a></strong>, University of Hawai‘i at Mānoa<br style="box-sizing:border-box"><strong style="box-sizing:border-box">Transfinite Recursion from Gödel–Bernays to Kelley–Morse</strong><br style="box-sizing:border-box"></p><p style="box-sizing:border-box;margin-bottom:1em;color:rgb(96,108,113);font-family:&quot;Open Sans&quot;,&quot;Helvetica Neue&quot;,Helvetica,Arial,sans-serif">Gödel–Bernays set theory GB and Kelley–Morse set theory KM are two formal theories for second-order set theory, allowing both sets and proper classes as objects. GB is the weaker of the two theories, being conservative over ZF, while KM is the stronger. Set theorists have used KM in applications where GB is not strong enough; for instance, Kunen formulated his celebrated inconsistency result in the context of KM, as KM has the resources to directly allow talk of elementary embeddings of the universe of sets. But weaker theories than KM suffice for many of these applications. Between GB and KM there is a hierarchy of intermediate theories based upon restricting the logical complexity allowed in the comprehension axiom.</p><p style="box-sizing:border-box;margin-bottom:1em;color:rgb(96,108,113);font-family:&quot;Open Sans&quot;,&quot;Helvetica Neue&quot;,Helvetica,Arial,sans-serif">In this talk I will present a hierarchy of second-order set theories which refines the comprehension-based hierarchy. This hierarchy is based upon transfinite recursion principles, ordered first by the logical complexity of the properties allowed and second by the lengths of well-orders on which we may carry out the recursions. Theories in this hierarchy are separated in terms of consistency strength. The substantive new result to establish this hierarchy is the following: Let <span id="gmail-MathJax-Element-1-Frame" class="gmail-mjx-chtml gmail-MathJax_CHTML" tabindex="0" style="box-sizing:border-box;display:inline-block;line-height:0;word-spacing:normal;white-space:nowrap;float:none;direction:ltr;max-width:none;max-height:none;min-width:0px;min-height:0px;border:0px;margin:0px;padding:1px 0px"><span id="gmail-MJXc-Node-1" class="gmail-mjx-math" style="box-sizing:border-box;display:inline-block;border-collapse:separate;border-spacing:0px;margin-top:0px"><span id="gmail-MJXc-Node-2" class="gmail-mjx-mrow" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-MJXc-Node-3" class="gmail-mjx-mi" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-math-I" style="display:block;white-space:pre;font-family:MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw;margin-top:0px;padding-top:0.472em;padding-bottom:0.284em;box-sizing:content-box">k</span></span></span></span><span class="gmail-MJX_Assistive_MathML" style="box-sizing:border-box;padding:1px 0px 0px;border:0px;height:1px;width:1px;overflow:hidden;display:block">k</span></span> be a natural number. Suppose <span id="gmail-MathJax-Element-2-Frame" class="gmail-mjx-chtml gmail-MathJax_CHTML" tabindex="0" style="box-sizing:border-box;display:inline-block;line-height:0;word-spacing:normal;white-space:nowrap;float:none;direction:ltr;max-width:none;max-height:none;min-width:0px;min-height:0px;border:0px;margin:0px;padding:1px 0px"><span id="gmail-MJXc-Node-4" class="gmail-mjx-math" style="box-sizing:border-box;display:inline-block;border-collapse:separate;border-spacing:0px;margin-top:0px"><span id="gmail-MJXc-Node-5" class="gmail-mjx-mrow" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-MJXc-Node-6" class="gmail-mjx-mo" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-main-R" style="display:block;white-space:pre;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.472em;padding-bottom:0.613em;box-sizing:content-box">(</span></span><span id="gmail-MJXc-Node-7" class="gmail-mjx-mi" style="display:inline-block;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-math-I" style="display:block;white-space:pre;font-family:MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw;margin-top:0px;padding-top:0.472em;padding-bottom:0.284em;padding-right:0.081em;box-sizing:content-box">M</span></span><span id="gmail-MJXc-Node-8" class="gmail-mjx-mo" style="display:inline-block;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-main-R" style="display:block;white-space:pre;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;padding-bottom:0.519em;box-sizing:content-box">,</span></span><span id="gmail-MJXc-Node-9" class="gmail-mjx-texatom gmail-MJXc-space1" style="display:inline-block;margin-left:0.167em;box-sizing:content-box"><span id="gmail-MJXc-Node-10" class="gmail-mjx-mrow" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-MJXc-Node-11" class="gmail-mjx-mi" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-cal-R" style="display:block;white-space:pre;font-family:MJXc-TeX-cal-R,MJXc-TeX-cal-Rw;margin-top:0px;padding-top:0.425em;padding-bottom:0.331em;padding-right:0.094em;box-sizing:content-box">X</span></span></span></span><span id="gmail-MJXc-Node-12" class="gmail-mjx-mo" style="display:inline-block;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-main-R" style="display:block;white-space:pre;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.472em;padding-bottom:0.613em;box-sizing:content-box">)</span></span></span></span><span class="gmail-MJX_Assistive_MathML" style="box-sizing:border-box;padding:1px 0px 0px;border:0px;height:1px;width:1px;overflow:hidden;display:block">(M,<span class="gmail-MJX-TeXAtom-ORD"><span class="gmail-MJX-tex-caligraphic">X</span></span>)</span></span> satisfies GB and that <span id="gmail-MathJax-Element-3-Frame" class="gmail-mjx-chtml gmail-MathJax_CHTML" tabindex="0" style="box-sizing:border-box;display:inline-block;line-height:0;word-spacing:normal;white-space:nowrap;float:none;direction:ltr;max-width:none;max-height:none;min-width:0px;min-height:0px;border:0px;margin:0px;padding:1px 0px"><span id="gmail-MJXc-Node-13" class="gmail-mjx-math" style="box-sizing:border-box;display:inline-block;border-collapse:separate;border-spacing:0px;margin-top:0px"><span id="gmail-MJXc-Node-14" class="gmail-mjx-mrow" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-MJXc-Node-15" class="gmail-mjx-mi" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-main-R" style="display:block;white-space:pre;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.378em;padding-bottom:0.331em;box-sizing:content-box">Γ</span></span><span id="gmail-MJXc-Node-16" class="gmail-mjx-mo gmail-MJXc-space3" style="display:inline-block;margin-left:0.278em;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-main-R" style="display:block;white-space:pre;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.237em;padding-bottom:0.378em;box-sizing:content-box">∈</span></span><span id="gmail-MJXc-Node-17" class="gmail-mjx-texatom gmail-MJXc-space3" style="display:inline-block;margin-left:0.278em;box-sizing:content-box"><span id="gmail-MJXc-Node-18" class="gmail-mjx-mrow" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-MJXc-Node-19" class="gmail-mjx-mi" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-cal-R" style="display:block;white-space:pre;font-family:MJXc-TeX-cal-R,MJXc-TeX-cal-Rw;margin-top:0px;padding-top:0.425em;padding-bottom:0.331em;padding-right:0.094em;box-sizing:content-box">X</span></span></span></span></span></span><span class="gmail-MJX_Assistive_MathML" style="box-sizing:border-box;padding:1px 0px 0px;border:0px;height:1px;width:1px;overflow:hidden;display:block">Γ∈<span class="gmail-MJX-TeXAtom-ORD"><span class="gmail-MJX-tex-caligraphic">X</span></span></span></span> is a class well-order which is closed under addition. In case <span id="gmail-MathJax-Element-4-Frame" class="gmail-mjx-chtml gmail-MathJax_CHTML" tabindex="0" style="box-sizing:border-box;display:inline-block;line-height:0;word-spacing:normal;white-space:nowrap;float:none;direction:ltr;max-width:none;max-height:none;min-width:0px;min-height:0px;border:0px;margin:0px;padding:1px 0px"><span id="gmail-MJXc-Node-20" class="gmail-mjx-math" style="box-sizing:border-box;display:inline-block;border-collapse:separate;border-spacing:0px;margin-top:0px"><span id="gmail-MJXc-Node-21" class="gmail-mjx-mrow" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-MJXc-Node-22" class="gmail-mjx-mi" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-math-I" style="display:block;white-space:pre;font-family:MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw;margin-top:0px;padding-top:0.472em;padding-bottom:0.284em;box-sizing:content-box">k</span></span><span id="gmail-MJXc-Node-23" class="gmail-mjx-mo gmail-MJXc-space3" style="display:inline-block;margin-left:0.278em;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-main-R" style="display:block;white-space:pre;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.096em;padding-bottom:0.331em;box-sizing:content-box">=</span></span><span id="gmail-MJXc-Node-24" class="gmail-mjx-mn gmail-MJXc-space3" style="display:inline-block;margin-left:0.278em;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-main-R" style="display:block;white-space:pre;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.378em;padding-bottom:0.378em;box-sizing:content-box">0</span></span></span></span><span class="gmail-MJX_Assistive_MathML" style="box-sizing:border-box;padding:1px 0px 0px;border:0px;height:1px;width:1px;overflow:hidden;display:block">k=0</span></span> further assume <span id="gmail-MathJax-Element-5-Frame" class="gmail-mjx-chtml gmail-MathJax_CHTML" tabindex="0" style="box-sizing:border-box;display:inline-block;line-height:0;word-spacing:normal;white-space:nowrap;float:none;direction:ltr;max-width:none;max-height:none;min-width:0px;min-height:0px;border:0px;margin:0px;padding:1px 0px"><span id="gmail-MJXc-Node-25" class="gmail-mjx-math" style="box-sizing:border-box;display:inline-block;border-collapse:separate;border-spacing:0px;margin-top:0px"><span id="gmail-MJXc-Node-26" class="gmail-mjx-mrow" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-MJXc-Node-27" class="gmail-mjx-mi" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-main-R" style="display:block;white-space:pre;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.378em;padding-bottom:0.331em;box-sizing:content-box">Γ</span></span><span id="gmail-MJXc-Node-28" class="gmail-mjx-mo gmail-MJXc-space3" style="display:inline-block;margin-left:0.278em;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-main-R" style="display:block;white-space:pre;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.331em;padding-bottom:0.472em;box-sizing:content-box">≥</span></span><span id="gmail-MJXc-Node-29" class="gmail-mjx-msubsup gmail-MJXc-space3" style="display:inline-block;margin-left:0.278em;box-sizing:content-box"><span class="gmail-mjx-base" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-MJXc-Node-30" class="gmail-mjx-mi" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-math-I" style="display:block;white-space:pre;font-family:MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw;margin-top:0px;padding-top:0.237em;padding-bottom:0.284em;box-sizing:content-box">ω</span></span></span><span class="gmail-mjx-sup" style="display:inline-block;vertical-align:0.513em;padding-left:0px;padding-right:0.071em;box-sizing:content-box"><span id="gmail-MJXc-Node-31" class="gmail-mjx-mi" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-math-I" style="display:block;white-space:pre;font-family:MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw;margin-top:0px;padding-top:0.237em;padding-bottom:0.284em;box-sizing:content-box">ω</span></span></span></span></span></span><span class="gmail-MJX_Assistive_MathML" style="box-sizing:border-box;padding:1px 0px 0px;border:0px;height:1px;width:1px;overflow:hidden;display:block">Γ≥ωω</span></span>. Then, if <span id="gmail-MathJax-Element-6-Frame" class="gmail-mjx-chtml gmail-MathJax_CHTML" tabindex="0" style="box-sizing:border-box;display:inline-block;line-height:0;word-spacing:normal;white-space:nowrap;float:none;direction:ltr;max-width:none;max-height:none;min-width:0px;min-height:0px;border:0px;margin:0px;padding:1px 0px"><span id="gmail-MJXc-Node-32" class="gmail-mjx-math" style="box-sizing:border-box;display:inline-block;border-collapse:separate;border-spacing:0px;margin-top:0px"><span id="gmail-MJXc-Node-33" class="gmail-mjx-mrow" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-MJXc-Node-34" class="gmail-mjx-mo" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-main-R" style="display:block;white-space:pre;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.472em;padding-bottom:0.613em;box-sizing:content-box">(</span></span><span id="gmail-MJXc-Node-35" class="gmail-mjx-mi" style="display:inline-block;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-math-I" style="display:block;white-space:pre;font-family:MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw;margin-top:0px;padding-top:0.472em;padding-bottom:0.284em;padding-right:0.081em;box-sizing:content-box">M</span></span><span id="gmail-MJXc-Node-36" class="gmail-mjx-mo" style="display:inline-block;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-main-R" style="display:block;white-space:pre;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;padding-bottom:0.519em;box-sizing:content-box">,</span></span><span id="gmail-MJXc-Node-37" class="gmail-mjx-texatom gmail-MJXc-space1" style="display:inline-block;margin-left:0.167em;box-sizing:content-box"><span id="gmail-MJXc-Node-38" class="gmail-mjx-mrow" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-MJXc-Node-39" class="gmail-mjx-mi" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-cal-R" style="display:block;white-space:pre;font-family:MJXc-TeX-cal-R,MJXc-TeX-cal-Rw;margin-top:0px;padding-top:0.425em;padding-bottom:0.331em;padding-right:0.094em;box-sizing:content-box">X</span></span></span></span><span id="gmail-MJXc-Node-40" class="gmail-mjx-mo" style="display:inline-block;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-main-R" style="display:block;white-space:pre;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.472em;padding-bottom:0.613em;box-sizing:content-box">)</span></span></span></span><span class="gmail-MJX_Assistive_MathML" style="box-sizing:border-box;padding:1px 0px 0px;border:0px;height:1px;width:1px;overflow:hidden;display:block">(M,<span class="gmail-MJX-TeXAtom-ORD"><span class="gmail-MJX-tex-caligraphic">X</span></span>)</span></span> satisfies <span id="gmail-MathJax-Element-7-Frame" class="gmail-mjx-chtml gmail-MathJax_CHTML" tabindex="0" style="box-sizing:border-box;display:inline-block;line-height:0;word-spacing:normal;white-space:nowrap;float:none;direction:ltr;max-width:none;max-height:none;min-width:0px;min-height:0px;border:0px;margin:0px;padding:1px 0px"><span id="gmail-MJXc-Node-41" class="gmail-mjx-math" style="box-sizing:border-box;display:inline-block;border-collapse:separate;border-spacing:0px;margin-top:0px"><span id="gmail-MJXc-Node-42" class="gmail-mjx-mrow" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-MJXc-Node-43" class="gmail-mjx-msubsup" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-mjx-base" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-MJXc-Node-44" class="gmail-mjx-mi" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-main-R" style="display:block;white-space:pre;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.378em;padding-bottom:0.331em;box-sizing:content-box">Π</span></span></span><span class="gmail-mjx-stack" style="display:inline-block;vertical-align:-0.335em;box-sizing:content-box"><span class="gmail-mjx-sup" style="display:block;margin-top:0px;padding-bottom:0.255em;padding-left:0px;padding-right:0.071em;box-sizing:content-box"><span id="gmail-MJXc-Node-46" class="gmail-mjx-mn" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-main-R" style="display:block;white-space:pre;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.378em;padding-bottom:0.331em;box-sizing:content-box">1</span></span></span><span class="gmail-mjx-sub" style="display:block;padding-right:0.071em;box-sizing:content-box"><span id="gmail-MJXc-Node-45" class="gmail-mjx-mi" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-math-I" style="display:block;white-space:pre;font-family:MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw;margin-top:0px;padding-top:0.472em;padding-bottom:0.284em;box-sizing:content-box">k</span></span></span></span></span></span></span><span class="gmail-MJX_Assistive_MathML" style="box-sizing:border-box;padding:1px 0px 0px;border:0px;height:1px;width:1px;overflow:hidden;display:block">Πk1</span></span>-Transfinite Recursion for recursions along <span id="gmail-MathJax-Element-8-Frame" class="gmail-mjx-chtml gmail-MathJax_CHTML" tabindex="0" style="box-sizing:border-box;display:inline-block;line-height:0;word-spacing:normal;white-space:nowrap;float:none;direction:ltr;max-width:none;max-height:none;min-width:0px;min-height:0px;border:0px;margin:0px;padding:1px 0px"><span id="gmail-MJXc-Node-47" class="gmail-mjx-math" style="box-sizing:border-box;display:inline-block;border-collapse:separate;border-spacing:0px;margin-top:0px"><span id="gmail-MJXc-Node-48" class="gmail-mjx-mrow" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-MJXc-Node-49" class="gmail-mjx-mi" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-main-R" style="display:block;white-space:pre;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.378em;padding-bottom:0.331em;box-sizing:content-box">Γ</span></span></span></span><span class="gmail-MJX_Assistive_MathML" style="box-sizing:border-box;padding:1px 0px 0px;border:0px;height:1px;width:1px;overflow:hidden;display:block">Γ</span></span>, there is <span id="gmail-MathJax-Element-9-Frame" class="gmail-mjx-chtml gmail-MathJax_CHTML" tabindex="0" style="box-sizing:border-box;display:inline-block;line-height:0;word-spacing:normal;white-space:nowrap;float:none;direction:ltr;max-width:none;max-height:none;min-width:0px;min-height:0px;border:0px;margin:0px;padding:1px 0px"><span id="gmail-MJXc-Node-50" class="gmail-mjx-math" style="box-sizing:border-box;display:inline-block;border-collapse:separate;border-spacing:0px;margin-top:0px"><span id="gmail-MJXc-Node-51" class="gmail-mjx-mrow" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-MJXc-Node-52" class="gmail-mjx-texatom" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-MJXc-Node-53" class="gmail-mjx-mrow" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-MJXc-Node-54" class="gmail-mjx-mi" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-cal-R" style="display:block;white-space:pre;font-family:MJXc-TeX-cal-R,MJXc-TeX-cal-Rw;margin-top:0px;padding-top:0.425em;padding-bottom:0.472em;padding-right:0.046em;box-sizing:content-box">Y</span></span></span></span><span id="gmail-MJXc-Node-55" class="gmail-mjx-mo gmail-MJXc-space3" style="display:inline-block;margin-left:0.278em;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-main-R" style="display:block;white-space:pre;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.331em;padding-bottom:0.472em;box-sizing:content-box">⊆</span></span><span id="gmail-MJXc-Node-56" class="gmail-mjx-texatom gmail-MJXc-space3" style="display:inline-block;margin-left:0.278em;box-sizing:content-box"><span id="gmail-MJXc-Node-57" class="gmail-mjx-mrow" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-MJXc-Node-58" class="gmail-mjx-mi" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-cal-R" style="display:block;white-space:pre;font-family:MJXc-TeX-cal-R,MJXc-TeX-cal-Rw;margin-top:0px;padding-top:0.425em;padding-bottom:0.331em;padding-right:0.094em;box-sizing:content-box">X</span></span></span></span></span></span><span class="gmail-MJX_Assistive_MathML" style="box-sizing:border-box;padding:1px 0px 0px;border:0px;height:1px;width:1px;overflow:hidden;display:block"><span class="gmail-MJX-TeXAtom-ORD"><span class="gmail-MJX-tex-caligraphic">Y</span></span>⊆<span class="gmail-MJX-TeXAtom-ORD"><span class="gmail-MJX-tex-caligraphic">X</span></span></span></span> coded in <span id="gmail-MathJax-Element-10-Frame" class="gmail-mjx-chtml gmail-MathJax_CHTML" tabindex="0" style="box-sizing:border-box;display:inline-block;line-height:0;word-spacing:normal;white-space:nowrap;float:none;direction:ltr;max-width:none;max-height:none;min-width:0px;min-height:0px;border:0px;margin:0px;padding:1px 0px"><span id="gmail-MJXc-Node-59" class="gmail-mjx-math" style="box-sizing:border-box;display:inline-block;border-collapse:separate;border-spacing:0px;margin-top:0px"><span id="gmail-MJXc-Node-60" class="gmail-mjx-mrow" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-MJXc-Node-61" class="gmail-mjx-texatom" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-MJXc-Node-62" class="gmail-mjx-mrow" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-MJXc-Node-63" class="gmail-mjx-mi" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-cal-R" style="display:block;white-space:pre;font-family:MJXc-TeX-cal-R,MJXc-TeX-cal-Rw;margin-top:0px;padding-top:0.425em;padding-bottom:0.331em;padding-right:0.094em;box-sizing:content-box">X</span></span></span></span></span></span><span class="gmail-MJX_Assistive_MathML" style="box-sizing:border-box;padding:1px 0px 0px;border:0px;height:1px;width:1px;overflow:hidden;display:block"><span class="gmail-MJX-TeXAtom-ORD"><span class="gmail-MJX-tex-caligraphic">X</span></span></span></span> so that <span id="gmail-MathJax-Element-11-Frame" class="gmail-mjx-chtml gmail-MathJax_CHTML" tabindex="0" style="box-sizing:border-box;display:inline-block;line-height:0;word-spacing:normal;white-space:nowrap;float:none;direction:ltr;max-width:none;max-height:none;min-width:0px;min-height:0px;border:0px;margin:0px;padding:1px 0px"><span id="gmail-MJXc-Node-64" class="gmail-mjx-math" style="box-sizing:border-box;display:inline-block;border-collapse:separate;border-spacing:0px;margin-top:0px"><span id="gmail-MJXc-Node-65" class="gmail-mjx-mrow" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-MJXc-Node-66" class="gmail-mjx-mo" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-main-R" style="display:block;white-space:pre;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.472em;padding-bottom:0.613em;box-sizing:content-box">(</span></span><span id="gmail-MJXc-Node-67" class="gmail-mjx-mi" style="display:inline-block;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-math-I" style="display:block;white-space:pre;font-family:MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw;margin-top:0px;padding-top:0.472em;padding-bottom:0.284em;padding-right:0.081em;box-sizing:content-box">M</span></span><span id="gmail-MJXc-Node-68" class="gmail-mjx-mo" style="display:inline-block;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-main-R" style="display:block;white-space:pre;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;padding-bottom:0.519em;box-sizing:content-box">,</span></span><span id="gmail-MJXc-Node-69" class="gmail-mjx-texatom gmail-MJXc-space1" style="display:inline-block;margin-left:0.167em;box-sizing:content-box"><span id="gmail-MJXc-Node-70" class="gmail-mjx-mrow" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-MJXc-Node-71" class="gmail-mjx-mi" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-cal-R" style="display:block;white-space:pre;font-family:MJXc-TeX-cal-R,MJXc-TeX-cal-Rw;margin-top:0px;padding-top:0.425em;padding-bottom:0.472em;padding-right:0.046em;box-sizing:content-box">Y</span></span></span></span><span id="gmail-MJXc-Node-72" class="gmail-mjx-mo" style="display:inline-block;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-main-R" style="display:block;white-space:pre;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.472em;padding-bottom:0.613em;box-sizing:content-box">)</span></span></span></span><span class="gmail-MJX_Assistive_MathML" style="box-sizing:border-box;padding:1px 0px 0px;border:0px;height:1px;width:1px;overflow:hidden;display:block">(M,<span class="gmail-MJX-TeXAtom-ORD"><span class="gmail-MJX-tex-caligraphic">Y</span></span>)</span></span> satisfies GB plus the principle of <span id="gmail-MathJax-Element-12-Frame" class="gmail-mjx-chtml gmail-MathJax_CHTML" tabindex="0" style="box-sizing:border-box;display:inline-block;line-height:0;word-spacing:normal;white-space:nowrap;float:none;direction:ltr;max-width:none;max-height:none;min-width:0px;min-height:0px;border:0px;margin:0px;padding:1px 0px"><span id="gmail-MJXc-Node-73" class="gmail-mjx-math" style="box-sizing:border-box;display:inline-block;border-collapse:separate;border-spacing:0px;margin-top:0px"><span id="gmail-MJXc-Node-74" class="gmail-mjx-mrow" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-MJXc-Node-75" class="gmail-mjx-msubsup" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-mjx-base" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-MJXc-Node-76" class="gmail-mjx-mi" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-main-R" style="display:block;white-space:pre;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.378em;padding-bottom:0.331em;box-sizing:content-box">Π</span></span></span><span class="gmail-mjx-stack" style="display:inline-block;vertical-align:-0.335em;box-sizing:content-box"><span class="gmail-mjx-sup" style="display:block;margin-top:0px;padding-bottom:0.255em;padding-left:0px;padding-right:0.071em;box-sizing:content-box"><span id="gmail-MJXc-Node-78" class="gmail-mjx-mn" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-main-R" style="display:block;white-space:pre;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.378em;padding-bottom:0.331em;box-sizing:content-box">1</span></span></span><span class="gmail-mjx-sub" style="display:block;padding-right:0.071em;box-sizing:content-box"><span id="gmail-MJXc-Node-77" class="gmail-mjx-mi" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-math-I" style="display:block;white-space:pre;font-family:MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw;margin-top:0px;padding-top:0.472em;padding-bottom:0.284em;box-sizing:content-box">k</span></span></span></span></span></span></span><span class="gmail-MJX_Assistive_MathML" style="box-sizing:border-box;padding:1px 0px 0px;border:0px;height:1px;width:1px;overflow:hidden;display:block">Πk1</span></span>-Transfinite Recursion for recursions along well-orders of length <span id="gmail-MathJax-Element-13-Frame" class="gmail-mjx-chtml gmail-MathJax_CHTML" tabindex="0" style="box-sizing:border-box;display:inline-block;line-height:0;word-spacing:normal;white-space:nowrap;float:none;direction:ltr;max-width:none;max-height:none;min-width:0px;min-height:0px;border:0px;margin:0px;padding:1px 0px"><span id="gmail-MJXc-Node-79" class="gmail-mjx-math" style="box-sizing:border-box;display:inline-block;border-collapse:separate;border-spacing:0px;margin-top:0px"><span id="gmail-MJXc-Node-80" class="gmail-mjx-mrow" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-MJXc-Node-81" class="gmail-mjx-mo" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-main-R" style="display:block;white-space:pre;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.237em;padding-bottom:0.378em;box-sizing:content-box">&lt;</span></span><span id="gmail-MJXc-Node-82" class="gmail-mjx-mi gmail-MJXc-space3" style="display:inline-block;margin-left:0.278em;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-main-R" style="display:block;white-space:pre;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.378em;padding-bottom:0.331em;box-sizing:content-box">Γ</span></span></span></span><span class="gmail-MJX_Assistive_MathML" style="box-sizing:border-box;padding:1px 0px 0px;border:0px;height:1px;width:1px;overflow:hidden;display:block">&lt;Γ</span></span>.</p></div><div><font color="#000000" face="arial, sans-serif"><br></font></div><div><font color="#000000" face="arial, sans-serif"><br></font></div><div><font color="#000000" face="arial, sans-serif"><br></font></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><br></div></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><font color="#212121">- - - - Other Logic News - - - -</font></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><br></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><b>Conference announcement:</b></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><span style="color:rgb(34,34,34);font-family:Arial,Helvetica,sans-serif">Model Theory and Mathematical Logic,</span><br style="color:rgb(34,34,34);font-family:Arial,Helvetica,sans-serif"><span style="color:rgb(34,34,34);font-family:Arial,Helvetica,sans-serif"> in honor of Chris Laskowski&#39;s 60th birthday,</span><br style="color:rgb(34,34,34);font-family:Arial,Helvetica,sans-serif"><span style="color:rgb(34,34,34);font-family:Arial,Helvetica,sans-serif"> June 21 - 23, 2019, at The University of Maryland, College Park</span><br style="color:rgb(34,34,34);font-family:Arial,Helvetica,sans-serif"><a href="http://www.umdlogic2019.com/" rel="noreferrer" target="_blank" style="font-family:Arial,Helvetica,sans-serif">http://www.umdlogic2019.com/</a><br></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><font color="#212121"><br></font></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><font color="#212121"><br></font></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><font color="#212121"><br></font></div><div><div><b style="color:rgb(0,0,0);font-family:arial,sans-serif">Conference announcement:</b><font color="#000000" face="arial, sans-serif"><b> <a href="mailto:scoskey@gmail.com" target="_blank">scoskey@gmail.com</a></b></font></div></div><div><font color="#000000" face="arial, sans-serif">The 2019 </font><a href="https://urldefense.proofpoint.com/v2/url?u=https-3A__math.boisestate.edu_best&amp;d=DwMFaQ&amp;c=pRW6ZPn_LDv0DnDIAK65Ad0CA4hBS-2mAmNa2_oHfF0&amp;r=rvtWDO9Zc-E9EhkvL8sfOAUVc4SuAmQdMXpD6PNOkGM&amp;m=wENt1oQTtXEQxL5XJzqTp39k1un9HMylVIRjM26QaC0&amp;s=zogVm_GCFVhwhg52vjaNoUo-ZDg_-qMcqBP74Fyv7qU&amp;e=" target="_blank" style="color:rgb(0,0,0);font-family:arial,sans-serif;text-decoration-line:none">Boise Extravaganza in Set Theory</a><font color="#000000" face="arial, sans-serif"> will take place in Ashland, Oregon, on the campus of Southern Oregon University, during June 19-21.</font><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><br></div><div><font color="#000000" face="arial, sans-serif">We are currently welcoming applications for </font><a href="https://urldefense.proofpoint.com/v2/url?u=https-3A__math.boisestate.edu_best_travel-2Dgrants_&amp;d=DwMFaQ&amp;c=pRW6ZPn_LDv0DnDIAK65Ad0CA4hBS-2mAmNa2_oHfF0&amp;r=rvtWDO9Zc-E9EhkvL8sfOAUVc4SuAmQdMXpD6PNOkGM&amp;m=wENt1oQTtXEQxL5XJzqTp39k1un9HMylVIRjM26QaC0&amp;s=G-M0o6olGB61CSXZnf_hF8qftpvGlv5561dy6PrlL7o&amp;e=" target="_blank" style="color:rgb(0,0,0);font-family:arial,sans-serif;text-decoration-line:none">travel grants</a><font color="#000000" face="arial, sans-serif"> from graduate students and postdocs (and other categories) in set theory </font><i style="color:rgb(0,0,0);font-family:arial,sans-serif">and related fields</i><font color="#000000" face="arial, sans-serif">. We will begin considering applications on </font><span style="color:rgb(0,0,0);font-family:arial,sans-serif">May 1</span><font color="#000000" face="arial, sans-serif">. We will continue accepting applications on a rolling basis until May 31.</font><br><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><br></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif">Plenary speakers:</div><font color="#000000" face="arial, sans-serif"> </font><u style="color:rgb(0,0,0);font-family:arial,sans-serif">Dana Bartosova</u><font color="#000000" face="arial, sans-serif"> (University of Florida)</font><br><font color="#000000" face="arial, sans-serif"> </font><u style="color:rgb(0,0,0);font-family:arial,sans-serif">Steve Jackson</u><font color="#000000" face="arial, sans-serif"> (University of North Texas)</font><br><font color="#000000" face="arial, sans-serif"> </font><u style="color:rgb(0,0,0);font-family:arial,sans-serif">Reese Johnston</u><font color="#000000" face="arial, sans-serif"> (University of Washington Robinson Center)</font><br><font color="#000000" face="arial, sans-serif"> </font><u style="color:rgb(0,0,0);font-family:arial,sans-serif">Assaf Shani</u><font color="#000000" face="arial, sans-serif"> (UCLA)</font><br><font color="#000000" face="arial, sans-serif"> </font><u style="color:rgb(0,0,0);font-family:arial,sans-serif">Piotr Szeczwak</u><font color="#000000" face="arial, sans-serif"> (Cardinal Stefan Wyszyński University, Warsaw)</font><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><br></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif">Contact:</div><div><font color="#000000" face="arial, sans-serif"><a href="mailto:scoskey@gmail.com" target="_blank">scoskey@gmail.com</a> or </font><a href="mailto:best@boisestate.edu" target="_blank" style="color:rgb(0,0,0);font-family:arial,sans-serif;text-decoration-line:none">best@boisestate.edu</a><br></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><br></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif">The BEST conference particularly aims to support the careers of young researchers in set theory (and related fields!), with travel support available for graduate students and postdocs. BEST features professional development opportunities and awards for student presentations. Please pass this invitation along to students, postdocs, and colleagues to submit an abstract and participate in BEST!<br><br>BEST is an international conference featuring talks on a broad range of recent advances in set theory research. The conference is organized by the Set Theory group at Boise State University and is structured as a symposium of the 100th annual meeting of the American Association for the Advancement of Science, Pacific Division (AAAS-PD). BEST is also supported by the NSF and Boise State University.<br><br>Organizers: Liljana Babinkostova (Boise State University), John Clemens (Boise State University), Samuel Coskey (Boise State University), Marion Scheepers (Boise State University). Scientific Committee: Natasha Dobrinen (University of Denver), Simon Thomas (Rutgers University)</div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><br></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><br></div></div></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><br></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><br></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><font color="#212121"><br></font></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><font color="#212121">- - - - Web Site - - - -</font></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><font color="#212121"><br></font></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><font color="#212121">Our new website is located at:  <a href="https://nylogic.github.io/" target="_blank" style="text-decoration-line:none">https://nylogic.github.io/</a></font></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><font color="#212121">Many thanks to Victoria Gitman for her development work!</font></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><font color="#212121"><br></font></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><font color="#212121">--------  ADMINISTRIVIA  --------</font></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><font color="#212121"><br></font></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><font color="#212121">To subscribe/unsubscribe to this list, please email your request to <a href="mailto:jreitz@nylogic.org" target="_blank" style="text-decoration-line:none">jreitz@nylogic.org</a>.</font></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><font color="#212121"><br></font></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><font color="#212121">If you have a logic-related event that you would like included in future mailings, please email <a href="mailto:jreitz@nylogic.org" target="_blank" style="text-decoration-line:none">jreitz@nylogic.org</a>. </font></div></div></div></div></div></div>
40
4/24/2019 21:10:29This Week in Logic at CUNYSPECIAL UPDATE: This Week in Logic at CUNY<div dir="ltr"><div dir="ltr"><div dir="ltr"><div style="font-family:arial,sans-serif;color:rgb(33,33,33)"><span style="color:rgb(0,0,0)">Hi everyone,</span></div><div style="font-family:arial,sans-serif;color:rgb(33,33,33)"><span style="color:rgb(0,0,0)"><br></span></div><div style="font-family:arial,sans-serif"><font color="#000000">Please note the addition of this Friday&#39;s talk in the Columbia Seminar on Logic Probability and Games.</font></div><div style="font-family:arial,sans-serif;color:rgb(33,33,33)"><span style="color:rgb(0,0,0)"><br></span></div><div style="font-family:arial,sans-serif;color:rgb(33,33,33)"><span style="color:rgb(0,0,0)">All the best,</span></div><div style="font-family:arial,sans-serif;color:rgb(33,33,33)"><span style="color:rgb(0,0,0)">Jonas</span></div><div style="font-family:arial,sans-serif;color:rgb(33,33,33)"><span style="color:rgb(0,0,0)"><br></span></div><div style="font-family:arial,sans-serif;color:rgb(33,33,33)"><span style="color:rgb(0,0,0)"><br></span></div><div style="font-family:arial,sans-serif;color:rgb(33,33,33)"><br></div><div style="font-family:arial,sans-serif;color:rgb(33,33,33)">This Week in Logic at CUNY:</div><div><div><div style="color:rgb(33,33,33)"><div><br></div><div><br></div><div><br></div><div><br></div><div>- - - - Thursday, Apr 25, 2019 - - - -</div><div><br></div><div><br></div><div><br></div><div>- - - - Friday, Apr 26, 2019 - - - -</div><div><br></div><div><p class="MsoNormal" style="color:rgb(34,34,34);text-align:justify">Message from Prof Haim Gaifman, Columbia University   <br>The Columbia Seminar Logic Probability and Games </p><p class="MsoNormal" style="color:rgb(34,34,34);text-align:justify">will meet this coming Friday April 26, 4:10-6:00, <br>at the Faculty House (for direction to the Faculty house click the link at the bottom). <br><br>Professor Simon Huttegger will give a talk entitled “Rethinking Convergence to the Truth”.  <br>In this talk he uses the framework of Robinson’s non-standard analysis, <br>which makes it possible to use infinitesimals as they have been used by <br>Leibniz (and other non-standard notions of the calculus and probability theory), <br>in order to throw light on problems arising in Bayesian probability theory,  and in the philosophy of science.  <br><br>Professor Huttegger, who teaches Logic and Philosophy of Science at UC Irvine,<br>is a central figure in this area.      The talk will be self-contained, <br>intended for an audience with different philosophical and technical interests.        <br><br>Note: We will be having dinner right after the meeting at the Faculty House. <br>Please let the rapporteur, Michael Nielsen (<a class="gmail-m_-1049225049523204082moz-txt-link-abbreviated" href="mailto:mn2683@columbia.edu" target="_blank">mn2683@columbia.edu</a>), know<br>if you would like to join us so that we can make the appropriate number of reservations <br>(please be advised that at this point the university agrees to cover the expenses of the <br>speaker and the rapporteur only and that the cost for all others is $30, payable by cash or check). </p><p class="MsoNormal" style="color:rgb(34,34,34);line-height:normal"><a href="http://universityseminars.columbia.edu/resources/directions-to-faculty-house/" target="_blank">Directions to Faculty House</a>.</p><p class="MsoNormal" style="color:rgb(34,34,34);line-height:normal"></p><p class="MsoNormal" style="color:rgb(34,34,34)"><br><br></p><p style="color:rgb(34,34,34)"></p><br class="gmail-Apple-interchange-newline"></div><div><br></div><div><br></div><div><br></div><div><br></div><div>Next Week in Logic at CUNY:</div><div><br></div><div>- - - - Monday, Apr 29, 2019 - - - -</div><div><br></div><div><div>Logic and Metaphysics Workshop</div><div>Date: Monday, April 29th, 4.15-6.15  <br></div><div>Place: Room 7314, CUNY Graduate Center <br></div><div>Speaker: Tommy Kivatinos (CUNY)<br></div><div>Title: A Mechanistic Conception of Metaphysical Grounding<br></div><div><br></div><div>Abstract: A dominant theoretical framework in philosophy of science employs the notion of mechanistic dependence to elucidate how higher-level, less fundamental phenomena depend upon and arise out of lower-level, more fundamental phenomena. To elucidate the same thing, literature in metaphysics employs the notion of grounding. As I argue, regardless of whether the notion of mechanistic dependence or the notion of grounding is used to theoretically portray how higher-level phenomena arise out of lower-level phenomena, what is captured by such portrayals is the same. Thus, these notions pick out the same features of the world. With this as my basis, I identify the notion of grounding with the notion of mechanistic dependence, and thus, construct a mechanistic conception of grounding. Since mechanistic dependence is understood in terms of mechanisms, my conception frames grounding in terms of mechanisms. Moreover, the contemporary notion of mechanisms is shaped by how mechanisms are represented via the mechanistic models and mechanistic explanations provided by science. Thus, because my conception grounding identifies grounding with mechanistic dependence and thereby frames grounding in terms of mechanisms, this conception suggests that the notion of grounding is to be tailored to and constrained by the mechanistic models and mechanistic explanations provided by science. This leads the mechanistic conception of grounding to reject a wide variety of conventional claims about grounding, and thus, to offer a treatment of grounding that is highly revisionary. To reinforce the plausibility of the mechanistic conception of grounding, I discuss how grounding and mechanistic dependence are associated with explanation. Whereas mechanistic dependence is associated with mechanistic explanation, grounding is associated with grounding explanation. For each kind of explanation, some higher-level phenomenon P is explained by appeal to some low-level phenomenon that Parises out of. As I argue, these forms of explanation can be plausibly identified. This greatly supports the mechanistic conception of grounding. For if grounding explanations employ the notion of grounding and mechanistic explanations employ the notion of mechanistic explanation, and these forms of explanation can be identified, this suggests that these explanations employ the same notion. And, just as the notions of grounding and mechanistic dependence capture the same connection between higher-level and lower-level phenomena, grounding explanation and mechanistic explanation do so as well. Finally, to argue that the mechanistic conception is to be preferred to standard conceptions, I argue that my conception offers a powerful defense of grounding from recent criticisms.</div></div><div><br></div><div><br></div><div><br></div><div><br></div><div>- - - - Tuesday, Apr 30, 2019 - - - -</div><div><br></div><div><br></div><div><br></div><div>- - - - Wednesday, May 1, 2019 - - - -</div><div><br></div><div><br></div><div><br></div><div>- - - - Thursday, May 2, 2019 - - - -</div><div><br></div><div><br></div><div><br></div><div>- - - - Friday, May 3, 2019 - - - -</div></div><div style="color:rgb(33,33,33)"><br></div><div><div><font color="#000000" face="arial, sans-serif">Set Theory Seminar</font></div><div><font color="#000000" face="arial, sans-serif">CUNY Graduate Center, Room 6417</font></div><div><font color="#000000" face="arial, sans-serif">Friday, May 3, 10:00-11:45am</font></div><div><p style="box-sizing:border-box;margin-bottom:1em;margin-top:0px;color:rgb(96,108,113);font-family:&quot;Open Sans&quot;,&quot;Helvetica Neue&quot;,Helvetica,Arial,sans-serif"><strong style="box-sizing:border-box"><a href="https://nylogic.github.io/set-theory-seminar/2019/05/03/target=%22_blank%22" target="_blank" style="color:rgb(30,107,184);box-sizing:border-box;background-color:transparent;text-decoration-line:none;margin-top:0px">Joseph Van Name</a></strong>, CUNY<br style="box-sizing:border-box"><strong style="box-sizing:border-box">Lower bounds on the cardinalities of quotient algebras of elementary embeddings</strong><br style="box-sizing:border-box"></p><p style="box-sizing:border-box;margin-bottom:1em;color:rgb(96,108,113);font-family:&quot;Open Sans&quot;,&quot;Helvetica Neue&quot;,Helvetica,Arial,sans-serif">From non-trivial elementary embeddings <span id="gmail-m_5751542264780799491gmail-MathJax-Element-1-Frame" class="gmail-m_5751542264780799491gmail-mjx-chtml gmail-m_5751542264780799491gmail-MathJax_CHTML" style="box-sizing:border-box;display:inline-block;line-height:0;word-spacing:normal;white-space:nowrap;float:none;direction:ltr;max-width:none;max-height:none;min-width:0px;min-height:0px;border:0px;margin:0px;padding:1px 0px"><span id="gmail-m_5751542264780799491gmail-MJXc-Node-1" class="gmail-m_5751542264780799491gmail-mjx-math" style="box-sizing:border-box;display:inline-block;border-collapse:separate;border-spacing:0px;margin-top:0px"><span id="gmail-m_5751542264780799491gmail-MJXc-Node-2" class="gmail-m_5751542264780799491gmail-mjx-mrow" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_5751542264780799491gmail-MJXc-Node-3" class="gmail-m_5751542264780799491gmail-mjx-msubsup" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-m_5751542264780799491gmail-mjx-base" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_5751542264780799491gmail-MJXc-Node-4" class="gmail-m_5751542264780799491gmail-mjx-mi" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-m_5751542264780799491gmail-mjx-char gmail-m_5751542264780799491gmail-MJXc-TeX-math-I" style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw;margin-top:0px;padding-top:0.425em;padding-bottom:0.472em;box-sizing:content-box">j</span></span></span><span class="gmail-m_5751542264780799491gmail-mjx-sub" style="display:inline-block;vertical-align:-0.212em;padding-right:0.071em;box-sizing:content-box"><span id="gmail-m_5751542264780799491gmail-MJXc-Node-5" class="gmail-m_5751542264780799491gmail-mjx-texatom" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_5751542264780799491gmail-MJXc-Node-6" class="gmail-m_5751542264780799491gmail-mjx-mrow" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_5751542264780799491gmail-MJXc-Node-7" class="gmail-m_5751542264780799491gmail-mjx-mn" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-m_5751542264780799491gmail-mjx-char gmail-m_5751542264780799491gmail-MJXc-TeX-main-R" style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.378em;padding-bottom:0.331em;box-sizing:content-box">1</span></span></span></span></span></span><span id="gmail-m_5751542264780799491gmail-MJXc-Node-8" class="gmail-m_5751542264780799491gmail-mjx-mo gmail-m_5751542264780799491gmail-MJXc-space1" style="display:inline-block;margin-left:0.167em;box-sizing:content-box"><span class="gmail-m_5751542264780799491gmail-mjx-char gmail-m_5751542264780799491gmail-MJXc-TeX-main-R" style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;padding-bottom:0.331em;box-sizing:content-box">…</span></span><span id="gmail-m_5751542264780799491gmail-MJXc-Node-9" class="gmail-m_5751542264780799491gmail-mjx-msubsup gmail-m_5751542264780799491gmail-MJXc-space1" style="display:inline-block;margin-left:0.167em;box-sizing:content-box"><span class="gmail-m_5751542264780799491gmail-mjx-base" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_5751542264780799491gmail-MJXc-Node-10" class="gmail-m_5751542264780799491gmail-mjx-mi" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-m_5751542264780799491gmail-mjx-char gmail-m_5751542264780799491gmail-MJXc-TeX-math-I" style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw;margin-top:0px;padding-top:0.425em;padding-bottom:0.472em;box-sizing:content-box">j</span></span></span><span class="gmail-m_5751542264780799491gmail-mjx-sub" style="display:inline-block;vertical-align:-0.212em;padding-right:0.071em;box-sizing:content-box"><span id="gmail-m_5751542264780799491gmail-MJXc-Node-11" class="gmail-m_5751542264780799491gmail-mjx-texatom" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_5751542264780799491gmail-MJXc-Node-12" class="gmail-m_5751542264780799491gmail-mjx-mrow" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_5751542264780799491gmail-MJXc-Node-13" class="gmail-m_5751542264780799491gmail-mjx-mi" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-m_5751542264780799491gmail-mjx-char gmail-m_5751542264780799491gmail-MJXc-TeX-math-I" style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw;margin-top:0px;padding-top:0.237em;padding-bottom:0.284em;box-sizing:content-box">r</span></span></span></span></span></span><span id="gmail-m_5751542264780799491gmail-MJXc-Node-14" class="gmail-m_5751542264780799491gmail-mjx-mo gmail-m_5751542264780799491gmail-MJXc-space3" style="display:inline-block;margin-left:0.278em;box-sizing:content-box"><span class="gmail-m_5751542264780799491gmail-mjx-char gmail-m_5751542264780799491gmail-MJXc-TeX-main-R" style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.143em;padding-bottom:0.331em;box-sizing:content-box">:</span></span><span id="gmail-m_5751542264780799491gmail-MJXc-Node-15" class="gmail-m_5751542264780799491gmail-mjx-msubsup gmail-m_5751542264780799491gmail-MJXc-space3" style="display:inline-block;margin-left:0.278em;box-sizing:content-box"><span class="gmail-m_5751542264780799491gmail-mjx-base" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_5751542264780799491gmail-MJXc-Node-16" class="gmail-m_5751542264780799491gmail-mjx-mi" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-m_5751542264780799491gmail-mjx-char gmail-m_5751542264780799491gmail-MJXc-TeX-math-I" style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw;margin-top:0px;padding-top:0.472em;padding-bottom:0.284em;padding-right:0.186em;box-sizing:content-box">V</span></span></span><span class="gmail-m_5751542264780799491gmail-mjx-sub" style="display:inline-block;vertical-align:-0.219em;padding-right:0.071em;box-sizing:content-box"><span id="gmail-m_5751542264780799491gmail-MJXc-Node-17" class="gmail-m_5751542264780799491gmail-mjx-texatom" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_5751542264780799491gmail-MJXc-Node-18" class="gmail-m_5751542264780799491gmail-mjx-mrow" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_5751542264780799491gmail-MJXc-Node-19" class="gmail-m_5751542264780799491gmail-mjx-mi" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-m_5751542264780799491gmail-mjx-char gmail-m_5751542264780799491gmail-MJXc-TeX-math-I" style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw;margin-top:0px;padding-top:0.472em;padding-bottom:0.284em;box-sizing:content-box">λ</span></span></span></span></span></span><span id="gmail-m_5751542264780799491gmail-MJXc-Node-20" class="gmail-m_5751542264780799491gmail-mjx-mo gmail-m_5751542264780799491gmail-MJXc-space3" style="display:inline-block;margin-left:0.278em;box-sizing:content-box"><span class="gmail-m_5751542264780799491gmail-mjx-char gmail-m_5751542264780799491gmail-MJXc-TeX-main-R" style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.237em;padding-bottom:0.331em;box-sizing:content-box">→</span></span><span id="gmail-m_5751542264780799491gmail-MJXc-Node-21" class="gmail-m_5751542264780799491gmail-mjx-msubsup gmail-m_5751542264780799491gmail-MJXc-space3" style="display:inline-block;margin-left:0.278em;box-sizing:content-box"><span class="gmail-m_5751542264780799491gmail-mjx-base" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_5751542264780799491gmail-MJXc-Node-22" class="gmail-m_5751542264780799491gmail-mjx-mi" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-m_5751542264780799491gmail-mjx-char gmail-m_5751542264780799491gmail-MJXc-TeX-math-I" style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw;margin-top:0px;padding-top:0.472em;padding-bottom:0.284em;padding-right:0.186em;box-sizing:content-box">V</span></span></span><span class="gmail-m_5751542264780799491gmail-mjx-sub" style="display:inline-block;vertical-align:-0.219em;padding-right:0.071em;box-sizing:content-box"><span id="gmail-m_5751542264780799491gmail-MJXc-Node-23" class="gmail-m_5751542264780799491gmail-mjx-texatom" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_5751542264780799491gmail-MJXc-Node-24" class="gmail-m_5751542264780799491gmail-mjx-mrow" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_5751542264780799491gmail-MJXc-Node-25" class="gmail-m_5751542264780799491gmail-mjx-mi" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-m_5751542264780799491gmail-mjx-char gmail-m_5751542264780799491gmail-MJXc-TeX-math-I" style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw;margin-top:0px;padding-top:0.472em;padding-bottom:0.284em;box-sizing:content-box">λ</span></span></span></span></span></span></span></span><span class="gmail-m_5751542264780799491gmail-MJX_Assistive_MathML" style="box-sizing:border-box;padding:1px 0px 0px;border:0px;height:1px;width:1px;overflow:hidden;display:block">j<span class="gmail-m_5751542264780799491gmail-MJX-TeXAtom-ORD">1</span>…j<span class="gmail-m_5751542264780799491gmail-MJX-TeXAtom-ORD">r</span>:V<span class="gmail-m_5751542264780799491gmail-MJX-TeXAtom-ORD">λ</span>→V<span class="gmail-m_5751542264780799491gmail-MJX-TeXAtom-ORD">λ</span></span></span>, we obtain a sequence of polynomials <span id="gmail-m_5751542264780799491gmail-MathJax-Element-2-Frame" class="gmail-m_5751542264780799491gmail-mjx-chtml gmail-m_5751542264780799491gmail-MathJax_CHTML" style="box-sizing:border-box;display:inline-block;line-height:0;word-spacing:normal;white-space:nowrap;float:none;direction:ltr;max-width:none;max-height:none;min-width:0px;min-height:0px;border:0px;margin:0px;padding:1px 0px"><span id="gmail-m_5751542264780799491gmail-MJXc-Node-26" class="gmail-m_5751542264780799491gmail-mjx-math" style="box-sizing:border-box;display:inline-block;border-collapse:separate;border-spacing:0px;margin-top:0px"><span id="gmail-m_5751542264780799491gmail-MJXc-Node-27" class="gmail-m_5751542264780799491gmail-mjx-mrow" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_5751542264780799491gmail-MJXc-Node-28" class="gmail-m_5751542264780799491gmail-mjx-mo" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-m_5751542264780799491gmail-mjx-char gmail-m_5751542264780799491gmail-MJXc-TeX-main-R" style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.472em;padding-bottom:0.613em;box-sizing:content-box">(</span></span><span id="gmail-m_5751542264780799491gmail-MJXc-Node-29" class="gmail-m_5751542264780799491gmail-mjx-msubsup" style="display:inline-block;box-sizing:content-box"><span class="gmail-m_5751542264780799491gmail-mjx-base" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_5751542264780799491gmail-MJXc-Node-30" class="gmail-m_5751542264780799491gmail-mjx-mi" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-m_5751542264780799491gmail-mjx-char gmail-m_5751542264780799491gmail-MJXc-TeX-math-I" style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw;margin-top:0px;padding-top:0.237em;padding-bottom:0.472em;box-sizing:content-box">p</span></span></span><span class="gmail-m_5751542264780799491gmail-mjx-sub" style="display:inline-block;vertical-align:-0.212em;padding-right:0.071em;box-sizing:content-box"><span id="gmail-m_5751542264780799491gmail-MJXc-Node-31" class="gmail-m_5751542264780799491gmail-mjx-texatom" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_5751542264780799491gmail-MJXc-Node-32" class="gmail-m_5751542264780799491gmail-mjx-mrow" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_5751542264780799491gmail-MJXc-Node-33" class="gmail-m_5751542264780799491gmail-mjx-mi" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-m_5751542264780799491gmail-mjx-char gmail-m_5751542264780799491gmail-MJXc-TeX-math-I" style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw;margin-top:0px;padding-top:0.237em;padding-bottom:0.284em;box-sizing:content-box">n</span></span></span></span></span></span><span id="gmail-m_5751542264780799491gmail-MJXc-Node-34" class="gmail-m_5751542264780799491gmail-mjx-mo" style="display:inline-block;box-sizing:content-box"><span class="gmail-m_5751542264780799491gmail-mjx-char gmail-m_5751542264780799491gmail-MJXc-TeX-main-R" style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.472em;padding-bottom:0.613em;box-sizing:content-box">(</span></span><span id="gmail-m_5751542264780799491gmail-MJXc-Node-35" class="gmail-m_5751542264780799491gmail-mjx-msubsup" style="display:inline-block;box-sizing:content-box"><span class="gmail-m_5751542264780799491gmail-mjx-base" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_5751542264780799491gmail-MJXc-Node-36" class="gmail-m_5751542264780799491gmail-mjx-mi" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-m_5751542264780799491gmail-mjx-char gmail-m_5751542264780799491gmail-MJXc-TeX-math-I" style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw;margin-top:0px;padding-top:0.237em;padding-bottom:0.284em;box-sizing:content-box">x</span></span></span><span class="gmail-m_5751542264780799491gmail-mjx-sub" style="display:inline-block;vertical-align:-0.212em;padding-right:0.071em;box-sizing:content-box"><span id="gmail-m_5751542264780799491gmail-MJXc-Node-37" class="gmail-m_5751542264780799491gmail-mjx-texatom" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_5751542264780799491gmail-MJXc-Node-38" class="gmail-m_5751542264780799491gmail-mjx-mrow" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_5751542264780799491gmail-MJXc-Node-39" class="gmail-m_5751542264780799491gmail-mjx-mn" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-m_5751542264780799491gmail-mjx-char gmail-m_5751542264780799491gmail-MJXc-TeX-main-R" style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.378em;padding-bottom:0.331em;box-sizing:content-box">1</span></span></span></span></span></span><span id="gmail-m_5751542264780799491gmail-MJXc-Node-40" class="gmail-m_5751542264780799491gmail-mjx-mo" style="display:inline-block;box-sizing:content-box"><span class="gmail-m_5751542264780799491gmail-mjx-char gmail-m_5751542264780799491gmail-MJXc-TeX-main-R" style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;padding-bottom:0.519em;box-sizing:content-box">,</span></span><span id="gmail-m_5751542264780799491gmail-MJXc-Node-41" class="gmail-m_5751542264780799491gmail-mjx-mo gmail-m_5751542264780799491gmail-MJXc-space1" style="display:inline-block;margin-left:0.167em;box-sizing:content-box"><span class="gmail-m_5751542264780799491gmail-mjx-char gmail-m_5751542264780799491gmail-MJXc-TeX-main-R" style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;padding-bottom:0.331em;box-sizing:content-box">…</span></span><span id="gmail-m_5751542264780799491gmail-MJXc-Node-42" class="gmail-m_5751542264780799491gmail-mjx-mo gmail-m_5751542264780799491gmail-MJXc-space1" style="display:inline-block;margin-left:0.167em;box-sizing:content-box"><span class="gmail-m_5751542264780799491gmail-mjx-char gmail-m_5751542264780799491gmail-MJXc-TeX-main-R" style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;padding-bottom:0.519em;box-sizing:content-box">,</span></span><span id="gmail-m_5751542264780799491gmail-MJXc-Node-43" class="gmail-m_5751542264780799491gmail-mjx-msubsup gmail-m_5751542264780799491gmail-MJXc-space1" style="display:inline-block;margin-left:0.167em;box-sizing:content-box"><span class="gmail-m_5751542264780799491gmail-mjx-base" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_5751542264780799491gmail-MJXc-Node-44" class="gmail-m_5751542264780799491gmail-mjx-mi" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-m_5751542264780799491gmail-mjx-char gmail-m_5751542264780799491gmail-MJXc-TeX-math-I" style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw;margin-top:0px;padding-top:0.237em;padding-bottom:0.284em;box-sizing:content-box">x</span></span></span><span class="gmail-m_5751542264780799491gmail-mjx-sub" style="display:inline-block;vertical-align:-0.212em;padding-right:0.071em;box-sizing:content-box"><span id="gmail-m_5751542264780799491gmail-MJXc-Node-45" class="gmail-m_5751542264780799491gmail-mjx-texatom" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_5751542264780799491gmail-MJXc-Node-46" class="gmail-m_5751542264780799491gmail-mjx-mrow" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_5751542264780799491gmail-MJXc-Node-47" class="gmail-m_5751542264780799491gmail-mjx-mi" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-m_5751542264780799491gmail-mjx-char gmail-m_5751542264780799491gmail-MJXc-TeX-math-I" style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw;margin-top:0px;padding-top:0.237em;padding-bottom:0.284em;box-sizing:content-box">r</span></span></span></span></span></span><span id="gmail-m_5751542264780799491gmail-MJXc-Node-48" class="gmail-m_5751542264780799491gmail-mjx-mo" style="display:inline-block;box-sizing:content-box"><span class="gmail-m_5751542264780799491gmail-mjx-char gmail-m_5751542264780799491gmail-MJXc-TeX-main-R" style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.472em;padding-bottom:0.613em;box-sizing:content-box">)</span></span><span id="gmail-m_5751542264780799491gmail-MJXc-Node-49" class="gmail-m_5751542264780799491gmail-mjx-msubsup" style="display:inline-block;box-sizing:content-box"><span class="gmail-m_5751542264780799491gmail-mjx-base" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_5751542264780799491gmail-MJXc-Node-50" class="gmail-m_5751542264780799491gmail-mjx-mo" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-m_5751542264780799491gmail-mjx-char gmail-m_5751542264780799491gmail-MJXc-TeX-main-R" style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.472em;padding-bottom:0.613em;box-sizing:content-box">)</span></span></span><span class="gmail-m_5751542264780799491gmail-mjx-sub" style="display:inline-block;vertical-align:-0.212em;padding-right:0.071em;box-sizing:content-box"><span id="gmail-m_5751542264780799491gmail-MJXc-Node-51" class="gmail-m_5751542264780799491gmail-mjx-texatom" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_5751542264780799491gmail-MJXc-Node-52" class="gmail-m_5751542264780799491gmail-mjx-mrow" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_5751542264780799491gmail-MJXc-Node-53" class="gmail-m_5751542264780799491gmail-mjx-mi" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-m_5751542264780799491gmail-mjx-char gmail-m_5751542264780799491gmail-MJXc-TeX-math-I" style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw;margin-top:0px;padding-top:0.237em;padding-bottom:0.284em;box-sizing:content-box">n</span></span><span id="gmail-m_5751542264780799491gmail-MJXc-Node-54" class="gmail-m_5751542264780799491gmail-mjx-mo" style="display:inline-block;box-sizing:content-box"><span class="gmail-m_5751542264780799491gmail-mjx-char gmail-m_5751542264780799491gmail-MJXc-TeX-main-R" style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.237em;padding-bottom:0.378em;box-sizing:content-box">∈</span></span><span id="gmail-m_5751542264780799491gmail-MJXc-Node-55" class="gmail-m_5751542264780799491gmail-mjx-mi" style="display:inline-block;box-sizing:content-box"><span class="gmail-m_5751542264780799491gmail-mjx-char gmail-m_5751542264780799491gmail-MJXc-TeX-math-I" style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw;margin-top:0px;padding-top:0.237em;padding-bottom:0.284em;box-sizing:content-box">ω</span></span></span></span></span></span></span></span><span class="gmail-m_5751542264780799491gmail-MJX_Assistive_MathML" style="box-sizing:border-box;padding:1px 0px 0px;border:0px;height:1px;width:1px;overflow:hidden;display:block">(p<span class="gmail-m_5751542264780799491gmail-MJX-TeXAtom-ORD">n</span>(x<span class="gmail-m_5751542264780799491gmail-MJX-TeXAtom-ORD">1</span>,…,x<span class="gmail-m_5751542264780799491gmail-MJX-TeXAtom-ORD">r</span>))<span class="gmail-m_5751542264780799491gmail-MJX-TeXAtom-ORD">n∈ω</span></span></span> that satisfies the infinite product<span class="gmail-m_5751542264780799491gmail-mjx-chtml gmail-m_5751542264780799491gmail-MJXc-display" style="box-sizing:border-box;display:block;line-height:0;text-align:center;word-spacing:normal;white-space:nowrap;float:none;direction:ltr;max-width:none;max-height:none;min-width:0px;min-height:0px;border:0px;margin:1em 0px;padding:0px"><span id="gmail-m_5751542264780799491gmail-MathJax-Element-3-Frame" class="gmail-m_5751542264780799491gmail-mjx-chtml gmail-m_5751542264780799491gmail-MathJax_CHTML" style="box-sizing:border-box;display:inline-block;line-height:0;word-spacing:normal;float:none;direction:ltr;max-width:none;max-height:none;min-width:0px;min-height:0px;border:0px;margin:0px;padding:1px 0px"><span id="gmail-m_5751542264780799491gmail-MJXc-Node-56" class="gmail-m_5751542264780799491gmail-mjx-math" style="box-sizing:border-box;display:inline-block;border-collapse:separate;border-spacing:0px;margin-top:0px"><span id="gmail-m_5751542264780799491gmail-MJXc-Node-57" class="gmail-m_5751542264780799491gmail-mjx-mrow" style="display:inline-block;text-align:left;margin-top:0px;box-sizing:content-box"><span id="gmail-m_5751542264780799491gmail-MJXc-Node-58" class="gmail-m_5751542264780799491gmail-mjx-munderover" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-m_5751542264780799491gmail-mjx-itable" style="display:inline-table;width:auto;margin-top:0px;box-sizing:content-box"><span class="gmail-m_5751542264780799491gmail-mjx-row" style="display:table-row;margin-top:0px;box-sizing:content-box"><span class="gmail-m_5751542264780799491gmail-mjx-cell" style="display:table-cell;margin-top:0px;box-sizing:content-box"><span class="gmail-m_5751542264780799491gmail-mjx-stack" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-m_5751542264780799491gmail-mjx-over" style="display:block;margin-top:0px;padding-bottom:0.247em;padding-top:0.141em;padding-left:0.404em;box-sizing:content-box"><span id="gmail-m_5751542264780799491gmail-MJXc-Node-65" class="gmail-m_5751542264780799491gmail-mjx-texatom" style="display:inline-block;margin-top:0px;box-sizing:content-box;padding-left:0px;padding-right:0px"><span id="gmail-m_5751542264780799491gmail-MJXc-Node-66" class="gmail-m_5751542264780799491gmail-mjx-mrow" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_5751542264780799491gmail-MJXc-Node-67" class="gmail-m_5751542264780799491gmail-mjx-mi" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-m_5751542264780799491gmail-mjx-char gmail-m_5751542264780799491gmail-MJXc-TeX-main-R" style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.143em;padding-bottom:0.331em;box-sizing:content-box">∞</span></span></span></span></span><span class="gmail-m_5751542264780799491gmail-mjx-op" style="display:block;box-sizing:content-box"><span id="gmail-m_5751542264780799491gmail-MJXc-Node-59" class="gmail-m_5751542264780799491gmail-mjx-mo" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-m_5751542264780799491gmail-mjx-char gmail-m_5751542264780799491gmail-MJXc-TeX-size2-R" style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-size2-R,MJXc-TeX-size2-Rw;margin-top:0px;padding-top:0.707em;padding-bottom:0.707em;box-sizing:content-box">∏</span></span></span></span></span></span><span class="gmail-m_5751542264780799491gmail-mjx-row" style="display:table-row;box-sizing:content-box"><span class="gmail-m_5751542264780799491gmail-mjx-under" style="display:table-cell;margin-top:0px;padding-top:0.236em;padding-bottom:0.141em;padding-left:0.004em;box-sizing:content-box"><span id="gmail-m_5751542264780799491gmail-MJXc-Node-60" class="gmail-m_5751542264780799491gmail-mjx-texatom" style="display:inline-block;margin-top:0px;box-sizing:content-box;padding-left:0px;padding-right:0px"><span id="gmail-m_5751542264780799491gmail-MJXc-Node-61" class="gmail-m_5751542264780799491gmail-mjx-mrow" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_5751542264780799491gmail-MJXc-Node-62" class="gmail-m_5751542264780799491gmail-mjx-mi" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-m_5751542264780799491gmail-mjx-char gmail-m_5751542264780799491gmail-MJXc-TeX-math-I" style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw;margin-top:0px;padding-top:0.472em;padding-bottom:0.284em;box-sizing:content-box">k</span></span><span id="gmail-m_5751542264780799491gmail-MJXc-Node-63" class="gmail-m_5751542264780799491gmail-mjx-mo" style="display:inline-block;box-sizing:content-box"><span class="gmail-m_5751542264780799491gmail-mjx-char gmail-m_5751542264780799491gmail-MJXc-TeX-main-R" style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.096em;padding-bottom:0.331em;box-sizing:content-box">=</span></span><span id="gmail-m_5751542264780799491gmail-MJXc-Node-64" class="gmail-m_5751542264780799491gmail-mjx-mn" style="display:inline-block;box-sizing:content-box"><span class="gmail-m_5751542264780799491gmail-mjx-char gmail-m_5751542264780799491gmail-MJXc-TeX-main-R" style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.378em;padding-bottom:0.378em;box-sizing:content-box">0</span></span></span></span></span></span></span></span><span id="gmail-m_5751542264780799491gmail-MJXc-Node-68" class="gmail-m_5751542264780799491gmail-mjx-msubsup gmail-m_5751542264780799491gmail-MJXc-space1" style="display:inline-block;margin-left:0.167em;box-sizing:content-box"><span class="gmail-m_5751542264780799491gmail-mjx-base" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_5751542264780799491gmail-MJXc-Node-69" class="gmail-m_5751542264780799491gmail-mjx-mi" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-m_5751542264780799491gmail-mjx-char gmail-m_5751542264780799491gmail-MJXc-TeX-math-I" style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw;margin-top:0px;padding-top:0.237em;padding-bottom:0.472em;box-sizing:content-box">p</span></span></span><span class="gmail-m_5751542264780799491gmail-mjx-sub" style="display:inline-block;vertical-align:-0.219em;padding-right:0.071em;box-sizing:content-box"><span id="gmail-m_5751542264780799491gmail-MJXc-Node-70" class="gmail-m_5751542264780799491gmail-mjx-texatom" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_5751542264780799491gmail-MJXc-Node-71" class="gmail-m_5751542264780799491gmail-mjx-mrow" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_5751542264780799491gmail-MJXc-Node-72" class="gmail-m_5751542264780799491gmail-mjx-mi" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-m_5751542264780799491gmail-mjx-char gmail-m_5751542264780799491gmail-MJXc-TeX-math-I" style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw;margin-top:0px;padding-top:0.472em;padding-bottom:0.284em;box-sizing:content-box">k</span></span></span></span></span></span><span id="gmail-m_5751542264780799491gmail-MJXc-Node-73" class="gmail-m_5751542264780799491gmail-mjx-mo" style="display:inline-block;box-sizing:content-box"><span class="gmail-m_5751542264780799491gmail-mjx-char gmail-m_5751542264780799491gmail-MJXc-TeX-main-R" style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.472em;padding-bottom:0.613em;box-sizing:content-box">(</span></span><span id="gmail-m_5751542264780799491gmail-MJXc-Node-74" class="gmail-m_5751542264780799491gmail-mjx-msubsup" style="display:inline-block;box-sizing:content-box"><span class="gmail-m_5751542264780799491gmail-mjx-base" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_5751542264780799491gmail-MJXc-Node-75" class="gmail-m_5751542264780799491gmail-mjx-mi" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-m_5751542264780799491gmail-mjx-char gmail-m_5751542264780799491gmail-MJXc-TeX-math-I" style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw;margin-top:0px;padding-top:0.237em;padding-bottom:0.284em;box-sizing:content-box">x</span></span></span><span class="gmail-m_5751542264780799491gmail-mjx-sub" style="display:inline-block;vertical-align:-0.212em;padding-right:0.071em;box-sizing:content-box"><span id="gmail-m_5751542264780799491gmail-MJXc-Node-76" class="gmail-m_5751542264780799491gmail-mjx-texatom" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_5751542264780799491gmail-MJXc-Node-77" class="gmail-m_5751542264780799491gmail-mjx-mrow" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_5751542264780799491gmail-MJXc-Node-78" class="gmail-m_5751542264780799491gmail-mjx-mn" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-m_5751542264780799491gmail-mjx-char gmail-m_5751542264780799491gmail-MJXc-TeX-main-R" style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.378em;padding-bottom:0.331em;box-sizing:content-box">1</span></span></span></span></span></span><span id="gmail-m_5751542264780799491gmail-MJXc-Node-79" class="gmail-m_5751542264780799491gmail-mjx-mo" style="display:inline-block;box-sizing:content-box"><span class="gmail-m_5751542264780799491gmail-mjx-char gmail-m_5751542264780799491gmail-MJXc-TeX-main-R" style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;padding-bottom:0.519em;box-sizing:content-box">,</span></span><span id="gmail-m_5751542264780799491gmail-MJXc-Node-80" class="gmail-m_5751542264780799491gmail-mjx-mo gmail-m_5751542264780799491gmail-MJXc-space1" style="display:inline-block;margin-left:0.167em;box-sizing:content-box"><span class="gmail-m_5751542264780799491gmail-mjx-char gmail-m_5751542264780799491gmail-MJXc-TeX-main-R" style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;padding-bottom:0.331em;box-sizing:content-box">…</span></span><span id="gmail-m_5751542264780799491gmail-MJXc-Node-81" class="gmail-m_5751542264780799491gmail-mjx-mo gmail-m_5751542264780799491gmail-MJXc-space1" style="display:inline-block;margin-left:0.167em;box-sizing:content-box"><span class="gmail-m_5751542264780799491gmail-mjx-char gmail-m_5751542264780799491gmail-MJXc-TeX-main-R" style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;padding-bottom:0.519em;box-sizing:content-box">,</span></span><span id="gmail-m_5751542264780799491gmail-MJXc-Node-82" class="gmail-m_5751542264780799491gmail-mjx-msubsup gmail-m_5751542264780799491gmail-MJXc-space1" style="display:inline-block;margin-left:0.167em;box-sizing:content-box"><span class="gmail-m_5751542264780799491gmail-mjx-base" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_5751542264780799491gmail-MJXc-Node-83" class="gmail-m_5751542264780799491gmail-mjx-mi" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-m_5751542264780799491gmail-mjx-char gmail-m_5751542264780799491gmail-MJXc-TeX-math-I" style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw;margin-top:0px;padding-top:0.237em;padding-bottom:0.284em;box-sizing:content-box">x</span></span></span><span class="gmail-m_5751542264780799491gmail-mjx-sub" style="display:inline-block;vertical-align:-0.212em;padding-right:0.071em;box-sizing:content-box"><span id="gmail-m_5751542264780799491gmail-MJXc-Node-84" class="gmail-m_5751542264780799491gmail-mjx-texatom" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_5751542264780799491gmail-MJXc-Node-85" class="gmail-m_5751542264780799491gmail-mjx-mrow" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_5751542264780799491gmail-MJXc-Node-86" class="gmail-m_5751542264780799491gmail-mjx-mi" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-m_5751542264780799491gmail-mjx-char gmail-m_5751542264780799491gmail-MJXc-TeX-math-I" style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw;margin-top:0px;padding-top:0.237em;padding-bottom:0.284em;box-sizing:content-box">r</span></span></span></span></span></span><span id="gmail-m_5751542264780799491gmail-MJXc-Node-87" class="gmail-m_5751542264780799491gmail-mjx-mo" style="display:inline-block;box-sizing:content-box"><span class="gmail-m_5751542264780799491gmail-mjx-char gmail-m_5751542264780799491gmail-MJXc-TeX-main-R" style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.472em;padding-bottom:0.613em;box-sizing:content-box">)</span></span><span id="gmail-m_5751542264780799491gmail-MJXc-Node-88" class="gmail-m_5751542264780799491gmail-mjx-mo gmail-m_5751542264780799491gmail-MJXc-space3" style="display:inline-block;margin-left:0.278em;box-sizing:content-box"><span class="gmail-m_5751542264780799491gmail-mjx-char gmail-m_5751542264780799491gmail-MJXc-TeX-main-R" style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.096em;padding-bottom:0.331em;box-sizing:content-box">=</span></span><span id="gmail-m_5751542264780799491gmail-MJXc-Node-89" class="gmail-m_5751542264780799491gmail-mjx-mfrac gmail-m_5751542264780799491gmail-MJXc-space3" style="display:inline-block;margin-left:0.278em;box-sizing:content-box"><span class="gmail-m_5751542264780799491gmail-mjx-box gmail-m_5751542264780799491gmail-MJXc-stacked" style="display:inline-block;height:0px;margin-top:0px;width:8.205em;padding:0px 0.12em;box-sizing:content-box"><span class="gmail-m_5751542264780799491gmail-mjx-numerator" style="display:block;text-align:center;margin-top:0px;width:8.205em;box-sizing:content-box"><span id="gmail-m_5751542264780799491gmail-MJXc-Node-90" class="gmail-m_5751542264780799491gmail-mjx-mn" style="display:inline-block;text-align:left;margin-top:0px;box-sizing:content-box"><span class="gmail-m_5751542264780799491gmail-mjx-char gmail-m_5751542264780799491gmail-MJXc-TeX-main-R" style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.378em;padding-bottom:0.331em;box-sizing:content-box">1</span></span></span><span class="gmail-m_5751542264780799491gmail-mjx-denominator" style="display:block;text-align:center;width:8.205em;box-sizing:content-box"><span id="gmail-m_5751542264780799491gmail-MJXc-Node-91" class="gmail-m_5751542264780799491gmail-mjx-mrow" style="display:inline-block;text-align:left;margin-top:0px;box-sizing:content-box"><span id="gmail-m_5751542264780799491gmail-MJXc-Node-92" class="gmail-m_5751542264780799491gmail-mjx-mn" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-m_5751542264780799491gmail-mjx-char gmail-m_5751542264780799491gmail-MJXc-TeX-main-R" style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.378em;padding-bottom:0.331em;box-sizing:content-box">1</span></span><span id="gmail-m_5751542264780799491gmail-MJXc-Node-93" class="gmail-m_5751542264780799491gmail-mjx-mo gmail-m_5751542264780799491gmail-MJXc-space2" style="display:inline-block;margin-left:0.222em;box-sizing:content-box"><span class="gmail-m_5751542264780799491gmail-mjx-char gmail-m_5751542264780799491gmail-MJXc-TeX-main-R" style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.284em;padding-bottom:0.425em;box-sizing:content-box">−</span></span><span id="gmail-m_5751542264780799491gmail-MJXc-Node-94" class="gmail-m_5751542264780799491gmail-mjx-mo gmail-m_5751542264780799491gmail-MJXc-space2" style="display:inline-block;margin-left:0.222em;box-sizing:content-box"><span class="gmail-m_5751542264780799491gmail-mjx-char gmail-m_5751542264780799491gmail-MJXc-TeX-main-R" style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.472em;padding-bottom:0.613em;box-sizing:content-box">(</span></span><span id="gmail-m_5751542264780799491gmail-MJXc-Node-95" class="gmail-m_5751542264780799491gmail-mjx-msubsup" style="display:inline-block;box-sizing:content-box"><span class="gmail-m_5751542264780799491gmail-mjx-base" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_5751542264780799491gmail-MJXc-Node-96" class="gmail-m_5751542264780799491gmail-mjx-mi" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-m_5751542264780799491gmail-mjx-char gmail-m_5751542264780799491gmail-MJXc-TeX-math-I" style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw;margin-top:0px;padding-top:0.237em;padding-bottom:0.284em;box-sizing:content-box">x</span></span></span><span class="gmail-m_5751542264780799491gmail-mjx-sub" style="display:inline-block;vertical-align:-0.212em;padding-right:0.071em;box-sizing:content-box"><span id="gmail-m_5751542264780799491gmail-MJXc-Node-97" class="gmail-m_5751542264780799491gmail-mjx-texatom" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_5751542264780799491gmail-MJXc-Node-98" class="gmail-m_5751542264780799491gmail-mjx-mrow" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_5751542264780799491gmail-MJXc-Node-99" class="gmail-m_5751542264780799491gmail-mjx-mn" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-m_5751542264780799491gmail-mjx-char gmail-m_5751542264780799491gmail-MJXc-TeX-main-R" style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.378em;padding-bottom:0.331em;box-sizing:content-box">1</span></span></span></span></span></span><span id="gmail-m_5751542264780799491gmail-MJXc-Node-100" class="gmail-m_5751542264780799491gmail-mjx-mo gmail-m_5751542264780799491gmail-MJXc-space2" style="display:inline-block;margin-left:0.222em;box-sizing:content-box"><span class="gmail-m_5751542264780799491gmail-mjx-char gmail-m_5751542264780799491gmail-MJXc-TeX-main-R" style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.284em;padding-bottom:0.425em;box-sizing:content-box">+</span></span><span id="gmail-m_5751542264780799491gmail-MJXc-Node-101" class="gmail-m_5751542264780799491gmail-mjx-mo gmail-m_5751542264780799491gmail-MJXc-space2" style="display:inline-block;margin-left:0.222em;box-sizing:content-box"><span class="gmail-m_5751542264780799491gmail-mjx-char gmail-m_5751542264780799491gmail-MJXc-TeX-main-R" style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.002em;padding-bottom:0.331em;box-sizing:content-box">⋯</span></span><span id="gmail-m_5751542264780799491gmail-MJXc-Node-102" class="gmail-m_5751542264780799491gmail-mjx-mo gmail-m_5751542264780799491gmail-MJXc-space2" style="display:inline-block;margin-left:0.222em;box-sizing:content-box"><span class="gmail-m_5751542264780799491gmail-mjx-char gmail-m_5751542264780799491gmail-MJXc-TeX-main-R" style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.284em;padding-bottom:0.425em;box-sizing:content-box">+</span></span><span id="gmail-m_5751542264780799491gmail-MJXc-Node-103" class="gmail-m_5751542264780799491gmail-mjx-msubsup gmail-m_5751542264780799491gmail-MJXc-space2" style="display:inline-block;margin-left:0.222em;box-sizing:content-box"><span class="gmail-m_5751542264780799491gmail-mjx-base" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_5751542264780799491gmail-MJXc-Node-104" class="gmail-m_5751542264780799491gmail-mjx-mi" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-m_5751542264780799491gmail-mjx-char gmail-m_5751542264780799491gmail-MJXc-TeX-math-I" style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw;margin-top:0px;padding-top:0.237em;padding-bottom:0.284em;box-sizing:content-box">x</span></span></span><span class="gmail-m_5751542264780799491gmail-mjx-sub" style="display:inline-block;vertical-align:-0.212em;padding-right:0.071em;box-sizing:content-box"><span id="gmail-m_5751542264780799491gmail-MJXc-Node-105" class="gmail-m_5751542264780799491gmail-mjx-texatom" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_5751542264780799491gmail-MJXc-Node-106" class="gmail-m_5751542264780799491gmail-mjx-mrow" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_5751542264780799491gmail-MJXc-Node-107" class="gmail-m_5751542264780799491gmail-mjx-mi" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-m_5751542264780799491gmail-mjx-char gmail-m_5751542264780799491gmail-MJXc-TeX-math-I" style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw;margin-top:0px;padding-top:0.237em;padding-bottom:0.284em;box-sizing:content-box">r</span></span></span></span></span></span><span id="gmail-m_5751542264780799491gmail-MJXc-Node-108" class="gmail-m_5751542264780799491gmail-mjx-mo" style="display:inline-block;box-sizing:content-box"><span class="gmail-m_5751542264780799491gmail-mjx-char gmail-m_5751542264780799491gmail-MJXc-TeX-main-R" style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.472em;padding-bottom:0.613em;box-sizing:content-box">)</span></span></span></span><span class="gmail-m_5751542264780799491gmail-mjx-line" style="display:block;height:0px;border-bottom:1.3px solid;width:8.205em;box-sizing:content-box"></span></span><span class="gmail-m_5751542264780799491gmail-mjx-vsize" style="display:inline-block;width:0px;height:2.375em;vertical-align:-1.007em;box-sizing:content-box"></span></span><span id="gmail-m_5751542264780799491gmail-MJXc-Node-109" class="gmail-m_5751542264780799491gmail-mjx-mo" style="display:inline-block;box-sizing:content-box"><span class="gmail-m_5751542264780799491gmail-mjx-char gmail-m_5751542264780799491gmail-MJXc-TeX-main-R" style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;padding-bottom:0.331em;box-sizing:content-box">.</span></span></span></span><span class="gmail-m_5751542264780799491gmail-MJX_Assistive_MathML gmail-m_5751542264780799491gmail-MJX_Assistive_MathML_Block" style="box-sizing:border-box;width:370.25px;padding:1px 0px 0px;border:0px;height:1px;overflow:hidden;display:block">∏<span class="gmail-m_5751542264780799491gmail-MJX-TeXAtom-ORD">k=0</span><span class="gmail-m_5751542264780799491gmail-MJX-TeXAtom-ORD">∞</span>p<span class="gmail-m_5751542264780799491gmail-MJX-TeXAtom-ORD">k</span>(x<span class="gmail-m_5751542264780799491gmail-MJX-TeXAtom-ORD">1</span>,…,x<span class="gmail-m_5751542264780799491gmail-MJX-TeXAtom-ORD">r</span>)=11−(x<span class="gmail-m_5751542264780799491gmail-MJX-TeXAtom-ORD">1</span>+⋯+x<span class="gmail-m_5751542264780799491gmail-MJX-TeXAtom-ORD">r</span>).</span></span></span>From this infinite product, we deduce lower bounds of the cardinality of <span id="gmail-m_5751542264780799491gmail-MathJax-Element-4-Frame" class="gmail-m_5751542264780799491gmail-mjx-chtml gmail-m_5751542264780799491gmail-MathJax_CHTML" style="box-sizing:border-box;display:inline-block;line-height:0;word-spacing:normal;white-space:nowrap;float:none;direction:ltr;max-width:none;max-height:none;min-width:0px;min-height:0px;border:0px;margin:0px;padding:1px 0px"><span id="gmail-m_5751542264780799491gmail-MJXc-Node-110" class="gmail-m_5751542264780799491gmail-mjx-math" style="box-sizing:border-box;display:inline-block;border-collapse:separate;border-spacing:0px;margin-top:0px"><span id="gmail-m_5751542264780799491gmail-MJXc-Node-111" class="gmail-m_5751542264780799491gmail-mjx-mrow" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_5751542264780799491gmail-MJXc-Node-112" class="gmail-m_5751542264780799491gmail-mjx-texatom" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_5751542264780799491gmail-MJXc-Node-113" class="gmail-m_5751542264780799491gmail-mjx-mrow" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_5751542264780799491gmail-MJXc-Node-114" class="gmail-m_5751542264780799491gmail-mjx-mo" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-m_5751542264780799491gmail-mjx-char gmail-m_5751542264780799491gmail-MJXc-TeX-main-R" style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.472em;padding-bottom:0.613em;box-sizing:content-box">|</span></span></span></span><span id="gmail-m_5751542264780799491gmail-MJXc-Node-115" class="gmail-m_5751542264780799491gmail-mjx-mo" style="display:inline-block;box-sizing:content-box"><span class="gmail-m_5751542264780799491gmail-mjx-char gmail-m_5751542264780799491gmail-MJXc-TeX-main-R" style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.472em;padding-bottom:0.613em;box-sizing:content-box">⟨</span></span><span id="gmail-m_5751542264780799491gmail-MJXc-Node-116" class="gmail-m_5751542264780799491gmail-mjx-msubsup" style="display:inline-block;box-sizing:content-box"><span class="gmail-m_5751542264780799491gmail-mjx-base" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_5751542264780799491gmail-MJXc-Node-117" class="gmail-m_5751542264780799491gmail-mjx-mi" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-m_5751542264780799491gmail-mjx-char gmail-m_5751542264780799491gmail-MJXc-TeX-math-I" style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw;margin-top:0px;padding-top:0.425em;padding-bottom:0.472em;box-sizing:content-box">j</span></span></span><span class="gmail-m_5751542264780799491gmail-mjx-sub" style="display:inline-block;vertical-align:-0.212em;padding-right:0.071em;box-sizing:content-box"><span id="gmail-m_5751542264780799491gmail-MJXc-Node-118" class="gmail-m_5751542264780799491gmail-mjx-texatom" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_5751542264780799491gmail-MJXc-Node-119" class="gmail-m_5751542264780799491gmail-mjx-mrow" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_5751542264780799491gmail-MJXc-Node-120" class="gmail-m_5751542264780799491gmail-mjx-mn" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-m_5751542264780799491gmail-mjx-char gmail-m_5751542264780799491gmail-MJXc-TeX-main-R" style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.378em;padding-bottom:0.331em;box-sizing:content-box">1</span></span></span></span></span></span><span id="gmail-m_5751542264780799491gmail-MJXc-Node-121" class="gmail-m_5751542264780799491gmail-mjx-mo" style="display:inline-block;box-sizing:content-box"><span class="gmail-m_5751542264780799491gmail-mjx-char gmail-m_5751542264780799491gmail-MJXc-TeX-main-R" style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;padding-bottom:0.519em;box-sizing:content-box">,</span></span><span id="gmail-m_5751542264780799491gmail-MJXc-Node-122" class="gmail-m_5751542264780799491gmail-mjx-mo gmail-m_5751542264780799491gmail-MJXc-space1" style="display:inline-block;margin-left:0.167em;box-sizing:content-box"><span class="gmail-m_5751542264780799491gmail-mjx-char gmail-m_5751542264780799491gmail-MJXc-TeX-main-R" style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;padding-bottom:0.331em;box-sizing:content-box">.</span></span><span id="gmail-m_5751542264780799491gmail-MJXc-Node-123" class="gmail-m_5751542264780799491gmail-mjx-mo gmail-m_5751542264780799491gmail-MJXc-space1" style="display:inline-block;margin-left:0.167em;box-sizing:content-box"><span class="gmail-m_5751542264780799491gmail-mjx-char gmail-m_5751542264780799491gmail-MJXc-TeX-main-R" style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;padding-bottom:0.331em;box-sizing:content-box">.</span></span><span id="gmail-m_5751542264780799491gmail-MJXc-Node-124" class="gmail-m_5751542264780799491gmail-mjx-mo gmail-m_5751542264780799491gmail-MJXc-space1" style="display:inline-block;margin-left:0.167em;box-sizing:content-box"><span class="gmail-m_5751542264780799491gmail-mjx-char gmail-m_5751542264780799491gmail-MJXc-TeX-main-R" style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;padding-bottom:0.331em;box-sizing:content-box">.</span></span><span id="gmail-m_5751542264780799491gmail-MJXc-Node-125" class="gmail-m_5751542264780799491gmail-mjx-mo gmail-m_5751542264780799491gmail-MJXc-space1" style="display:inline-block;margin-left:0.167em;box-sizing:content-box"><span class="gmail-m_5751542264780799491gmail-mjx-char gmail-m_5751542264780799491gmail-MJXc-TeX-main-R" style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;padding-bottom:0.519em;box-sizing:content-box">,</span></span><span id="gmail-m_5751542264780799491gmail-MJXc-Node-126" class="gmail-m_5751542264780799491gmail-mjx-msubsup gmail-m_5751542264780799491gmail-MJXc-space1" style="display:inline-block;margin-left:0.167em;box-sizing:content-box"><span class="gmail-m_5751542264780799491gmail-mjx-base" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_5751542264780799491gmail-MJXc-Node-127" class="gmail-m_5751542264780799491gmail-mjx-mi" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-m_5751542264780799491gmail-mjx-char gmail-m_5751542264780799491gmail-MJXc-TeX-math-I" style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw;margin-top:0px;padding-top:0.425em;padding-bottom:0.472em;box-sizing:content-box">j</span></span></span><span class="gmail-m_5751542264780799491gmail-mjx-sub" style="display:inline-block;vertical-align:-0.212em;padding-right:0.071em;box-sizing:content-box"><span id="gmail-m_5751542264780799491gmail-MJXc-Node-128" class="gmail-m_5751542264780799491gmail-mjx-texatom" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_5751542264780799491gmail-MJXc-Node-129" class="gmail-m_5751542264780799491gmail-mjx-mrow" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_5751542264780799491gmail-MJXc-Node-130" class="gmail-m_5751542264780799491gmail-mjx-mi" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-m_5751542264780799491gmail-mjx-char gmail-m_5751542264780799491gmail-MJXc-TeX-math-I" style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw;margin-top:0px;padding-top:0.237em;padding-bottom:0.284em;box-sizing:content-box">r</span></span></span></span></span></span><span id="gmail-m_5751542264780799491gmail-MJXc-Node-131" class="gmail-m_5751542264780799491gmail-mjx-mo" style="display:inline-block;box-sizing:content-box"><span class="gmail-m_5751542264780799491gmail-mjx-char gmail-m_5751542264780799491gmail-MJXc-TeX-main-R" style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.472em;padding-bottom:0.613em;box-sizing:content-box">⟩</span></span><span id="gmail-m_5751542264780799491gmail-MJXc-Node-132" class="gmail-m_5751542264780799491gmail-mjx-texatom" style="display:inline-block;box-sizing:content-box"><span id="gmail-m_5751542264780799491gmail-MJXc-Node-133" class="gmail-m_5751542264780799491gmail-mjx-mrow" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_5751542264780799491gmail-MJXc-Node-134" class="gmail-m_5751542264780799491gmail-mjx-mo" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-m_5751542264780799491gmail-mjx-char gmail-m_5751542264780799491gmail-MJXc-TeX-main-R" style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.472em;padding-bottom:0.613em;box-sizing:content-box">/</span></span></span></span><span id="gmail-m_5751542264780799491gmail-MJXc-Node-135" class="gmail-m_5751542264780799491gmail-mjx-msubsup gmail-m_5751542264780799491gmail-MJXc-space3" style="display:inline-block;margin-left:0.278em;box-sizing:content-box"><span class="gmail-m_5751542264780799491gmail-mjx-base" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_5751542264780799491gmail-MJXc-Node-136" class="gmail-m_5751542264780799491gmail-mjx-mo" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-m_5751542264780799491gmail-mjx-char gmail-m_5751542264780799491gmail-MJXc-TeX-main-R" style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.19em;padding-bottom:0.331em;box-sizing:content-box">≡</span></span></span><span class="gmail-m_5751542264780799491gmail-mjx-sup" style="display:inline-block;vertical-align:0.513em;padding-left:0px;padding-right:0.071em;box-sizing:content-box"><span id="gmail-m_5751542264780799491gmail-MJXc-Node-137" class="gmail-m_5751542264780799491gmail-mjx-texatom" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_5751542264780799491gmail-MJXc-Node-138" class="gmail-m_5751542264780799491gmail-mjx-mrow" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_5751542264780799491gmail-MJXc-Node-139" class="gmail-m_5751542264780799491gmail-mjx-mi" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-m_5751542264780799491gmail-mjx-char gmail-m_5751542264780799491gmail-MJXc-TeX-math-I" style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw;margin-top:0px;padding-top:0.237em;padding-bottom:0.284em;box-sizing:content-box">α</span></span></span></span></span></span><span id="gmail-m_5751542264780799491gmail-MJXc-Node-140" class="gmail-m_5751542264780799491gmail-mjx-texatom gmail-m_5751542264780799491gmail-MJXc-space3" style="display:inline-block;margin-left:0.278em;box-sizing:content-box"><span id="gmail-m_5751542264780799491gmail-MJXc-Node-141" class="gmail-m_5751542264780799491gmail-mjx-mrow" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_5751542264780799491gmail-MJXc-Node-142" class="gmail-m_5751542264780799491gmail-mjx-mo" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-m_5751542264780799491gmail-mjx-char gmail-m_5751542264780799491gmail-MJXc-TeX-main-R" style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.472em;padding-bottom:0.613em;box-sizing:content-box">|</span></span></span></span></span></span><span class="gmail-m_5751542264780799491gmail-MJX_Assistive_MathML" style="box-sizing:border-box;padding:1px 0px 0px;border:0px;height:1px;width:1px;overflow:hidden;display:block"><span class="gmail-m_5751542264780799491gmail-MJX-TeXAtom-ORD">|</span>⟨j<span class="gmail-m_5751542264780799491gmail-MJX-TeXAtom-ORD">1</span>,...,j<span class="gmail-m_5751542264780799491gmail-MJX-TeXAtom-ORD">r</span>⟩<span class="gmail-m_5751542264780799491gmail-MJX-TeXAtom-ORD">/</span>≡<span class="gmail-m_5751542264780799491gmail-MJX-TeXAtom-ORD">α</span><span class="gmail-m_5751542264780799491gmail-MJX-TeXAtom-ORD">|</span></span></span> using analysis and analytic number theoretic techniques. Computer calculations that search for Laver-like algebras give some empirical evidence that these lower bounds cannot be greatly improved.</p><p style="box-sizing:border-box;margin-bottom:1em;color:rgb(96,108,113);font-family:&quot;Open Sans&quot;,&quot;Helvetica Neue&quot;,Helvetica,Arial,sans-serif"></p><br class="gmail-m_5751542264780799491gmail-Apple-interchange-newline"></div><div><font color="#000000" face="arial, sans-serif"><br></font></div><div><font color="#000000" face="arial, sans-serif"><br></font></div><div><font color="#000000" face="arial, sans-serif">Model Theory Seminar</font></div><div><font color="#000000" face="arial, sans-serif">CUNY Graduate Center, Room 6417</font></div><div><font color="#000000" face="arial, sans-serif">Friday, </font><span style="color:rgb(0,0,0);font-family:arial,sans-serif">May 3</span><font color="#000000" face="arial, sans-serif">, 12:30-2:00pm</font></div><div><font color="#000000" face="arial, sans-serif"><div>Artem Chernikov, UCLA</div><div>TBA</div></font></div><div><font color="#000000" face="arial, sans-serif"><br></font></div><div><br></div><div><font color="#000000" face="arial, sans-serif">Logic Workshop</font></div><div><font color="#000000" face="arial, sans-serif">CUNY Graduate Center, Room 6417</font></div><div><font color="#000000" face="arial, sans-serif">Friday, </font><span style="color:rgb(0,0,0);font-family:arial,sans-serif">May 3</span><font color="#000000" face="arial, sans-serif">, 2:00-3:30pm</font></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><p style="box-sizing:border-box;margin-bottom:1em;margin-top:0px;color:rgb(96,108,113);font-family:&quot;Open Sans&quot;,&quot;Helvetica Neue&quot;,Helvetica,Arial,sans-serif"><strong style="box-sizing:border-box"><a href="http://sites.math.rutgers.edu/~cherlin/" target="_blank" style="color:rgb(30,107,184);box-sizing:border-box;background-color:transparent;text-decoration-line:none;margin-top:0px">Gregory Cherlin</a></strong>, Rutgers University<br style="box-sizing:border-box"><strong style="box-sizing:border-box">Countable universal graphs with forbidden subgraphs</strong><br style="box-sizing:border-box"></p><p align="center" style="box-sizing:border-box;margin-bottom:1em;color:rgb(96,108,113);font-family:&quot;Open Sans&quot;,&quot;Helvetica Neue&quot;,Helvetica,Arial,sans-serif"><strong style="box-sizing:border-box;margin-top:0px">Abstract</strong></p><p style="box-sizing:border-box;margin-bottom:1em;color:rgb(96,108,113);font-family:&quot;Open Sans&quot;,&quot;Helvetica Neue&quot;,Helvetica,Arial,sans-serif"><strong style="box-sizing:border-box;margin-top:0px">Definition</strong>: A collection <span id="gmail-m_5751542264780799491gmail-MathJax-Element-1-Frame" class="gmail-m_5751542264780799491gmail-mjx-chtml gmail-m_5751542264780799491gmail-MathJax_CHTML" style="box-sizing:border-box;display:inline-block;line-height:0;word-spacing:normal;white-space:nowrap;float:none;direction:ltr;max-width:none;max-height:none;min-width:0px;min-height:0px;border:0px;margin:0px;padding:1px 0px"><span id="gmail-m_5751542264780799491gmail-MJXc-Node-1" class="gmail-m_5751542264780799491gmail-mjx-math" style="box-sizing:border-box;display:inline-block;border-collapse:separate;border-spacing:0px;margin-top:0px"><span id="gmail-m_5751542264780799491gmail-MJXc-Node-2" class="gmail-m_5751542264780799491gmail-mjx-mrow" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_5751542264780799491gmail-MJXc-Node-3" class="gmail-m_5751542264780799491gmail-mjx-texatom" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_5751542264780799491gmail-MJXc-Node-4" class="gmail-m_5751542264780799491gmail-mjx-mrow" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_5751542264780799491gmail-MJXc-Node-5" class="gmail-m_5751542264780799491gmail-mjx-mi" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-m_5751542264780799491gmail-mjx-char gmail-m_5751542264780799491gmail-MJXc-TeX-cal-R" style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-cal-R,MJXc-TeX-cal-Rw;margin-top:0px;padding-top:0.425em;padding-bottom:0.331em;padding-right:0.11em;box-sizing:content-box">F</span></span></span></span></span></span><span class="gmail-m_5751542264780799491gmail-MJX_Assistive_MathML" style="box-sizing:border-box;padding:1px 0px 0px;border:0px;height:1px;width:1px;overflow:hidden;display:block"><span class="gmail-m_5751542264780799491gmail-MJX-TeXAtom-ORD"><span class="gmail-m_5751542264780799491gmail-MJX-tex-caligraphic">F</span></span></span></span> of forbidden subgraphs is a <em style="box-sizing:border-box">Rado family</em> if the class of countable <span id="gmail-m_5751542264780799491gmail-MathJax-Element-2-Frame" class="gmail-m_5751542264780799491gmail-mjx-chtml gmail-m_5751542264780799491gmail-MathJax_CHTML" style="box-sizing:border-box;display:inline-block;line-height:0;word-spacing:normal;white-space:nowrap;float:none;direction:ltr;max-width:none;max-height:none;min-width:0px;min-height:0px;border:0px;margin:0px;padding:1px 0px"><span id="gmail-m_5751542264780799491gmail-MJXc-Node-6" class="gmail-m_5751542264780799491gmail-mjx-math" style="box-sizing:border-box;display:inline-block;border-collapse:separate;border-spacing:0px;margin-top:0px"><span id="gmail-m_5751542264780799491gmail-MJXc-Node-7" class="gmail-m_5751542264780799491gmail-mjx-mrow" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_5751542264780799491gmail-MJXc-Node-8" class="gmail-m_5751542264780799491gmail-mjx-texatom" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_5751542264780799491gmail-MJXc-Node-9" class="gmail-m_5751542264780799491gmail-mjx-mrow" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_5751542264780799491gmail-MJXc-Node-10" class="gmail-m_5751542264780799491gmail-mjx-mi" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-m_5751542264780799491gmail-mjx-char gmail-m_5751542264780799491gmail-MJXc-TeX-cal-R" style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-cal-R,MJXc-TeX-cal-Rw;margin-top:0px;padding-top:0.425em;padding-bottom:0.331em;padding-right:0.11em;box-sizing:content-box">F</span></span></span></span></span></span><span class="gmail-m_5751542264780799491gmail-MJX_Assistive_MathML" style="box-sizing:border-box;padding:1px 0px 0px;border:0px;height:1px;width:1px;overflow:hidden;display:block"><span class="gmail-m_5751542264780799491gmail-MJX-TeXAtom-ORD"><span class="gmail-m_5751542264780799491gmail-MJX-tex-caligraphic">F</span></span></span></span>-free graphs contains a universal structure with respect to embeddings as induced subgraphs. (By &#39;forbidden&#39; I mean forbidden as subgraphs; usage varies in the literature.)</p><p style="box-sizing:border-box;margin-bottom:1em;color:rgb(96,108,113);font-family:&quot;Open Sans&quot;,&quot;Helvetica Neue&quot;,Helvetica,Arial,sans-serif">We consider the following decision problem for Rado families.</p><p style="box-sizing:border-box;margin-bottom:1em;color:rgb(96,108,113);font-family:&quot;Open Sans&quot;,&quot;Helvetica Neue&quot;,Helvetica,Arial,sans-serif"><strong style="box-sizing:border-box;margin-top:0px">Problem</strong>: Given a finite set <span id="gmail-m_5751542264780799491gmail-MathJax-Element-3-Frame" class="gmail-m_5751542264780799491gmail-mjx-chtml gmail-m_5751542264780799491gmail-MathJax_CHTML" style="box-sizing:border-box;display:inline-block;line-height:0;word-spacing:normal;white-space:nowrap;float:none;direction:ltr;max-width:none;max-height:none;min-width:0px;min-height:0px;border:0px;margin:0px;padding:1px 0px"><span id="gmail-m_5751542264780799491gmail-MJXc-Node-11" class="gmail-m_5751542264780799491gmail-mjx-math" style="box-sizing:border-box;display:inline-block;border-collapse:separate;border-spacing:0px;margin-top:0px"><span id="gmail-m_5751542264780799491gmail-MJXc-Node-12" class="gmail-m_5751542264780799491gmail-mjx-mrow" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_5751542264780799491gmail-MJXc-Node-13" class="gmail-m_5751542264780799491gmail-mjx-texatom" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_5751542264780799491gmail-MJXc-Node-14" class="gmail-m_5751542264780799491gmail-mjx-mrow" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_5751542264780799491gmail-MJXc-Node-15" class="gmail-m_5751542264780799491gmail-mjx-mi" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-m_5751542264780799491gmail-mjx-char gmail-m_5751542264780799491gmail-MJXc-TeX-cal-R" style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-cal-R,MJXc-TeX-cal-Rw;margin-top:0px;padding-top:0.425em;padding-bottom:0.331em;padding-right:0.11em;box-sizing:content-box">F</span></span></span></span></span></span><span class="gmail-m_5751542264780799491gmail-MJX_Assistive_MathML" style="box-sizing:border-box;padding:1px 0px 0px;border:0px;height:1px;width:1px;overflow:hidden;display:block"><span class="gmail-m_5751542264780799491gmail-MJX-TeXAtom-ORD"><span class="gmail-m_5751542264780799491gmail-MJX-tex-caligraphic">F</span></span></span></span> of finite, connected graphs, determine whether or not it is a Rado family.</p><p style="box-sizing:border-box;margin-bottom:1em;color:rgb(96,108,113);font-family:&quot;Open Sans&quot;,&quot;Helvetica Neue&quot;,Helvetica,Arial,sans-serif"><strong style="box-sizing:border-box;margin-top:0px">Question</strong>: Is this a decidable problem?</p><p style="box-sizing:border-box;margin-bottom:1em;color:rgb(96,108,113);font-family:&quot;Open Sans&quot;,&quot;Helvetica Neue&quot;,Helvetica,Arial,sans-serif">If we specialize to the sake of a single constraint then we speak instead of a <em style="box-sizing:border-box;margin-top:0px">Rado constraint.</em> Much is known, and much more conjectured, about the case of a <em style="box-sizing:border-box">single</em> constraint.</p><p style="box-sizing:border-box;margin-bottom:1em;color:rgb(96,108,113);font-family:&quot;Open Sans&quot;,&quot;Helvetica Neue&quot;,Helvetica,Arial,sans-serif"><strong style="box-sizing:border-box;margin-top:0px">Conjecture</strong>: The decision problem for the case of a single constraint is decidable.</p><p align="center" style="box-sizing:border-box;margin-bottom:1em;color:rgb(96,108,113);font-family:&quot;Open Sans&quot;,&quot;Helvetica Neue&quot;,Helvetica,Arial,sans-serif"><strong style="box-sizing:border-box;margin-top:0px">Discussion</strong></p><p style="box-sizing:border-box;margin-bottom:1em;color:rgb(96,108,113);font-family:&quot;Open Sans&quot;,&quot;Helvetica Neue&quot;,Helvetica,Arial,sans-serif">From the model theoretic side, a finite collection of forbidden subgraphs specifies a universal theory with some special properties: notably, there is a <em style="box-sizing:border-box;margin-top:0px">model companion</em>. When we restrict to families of connected constraints, the theory has joint embedding and the model companion is complete. In model theoretic terms the question becomes whether the model companion has a countable universal model with respect to elementary embedding, for which a purely type theoretic criterion is known.</p><p style="box-sizing:border-box;margin-bottom:1em;color:rgb(96,108,113);font-family:&quot;Open Sans&quot;,&quot;Helvetica Neue&quot;,Helvetica,Arial,sans-serif">Decision problems for other model theoretic properties of the model companion are natural. We have focused on this one because the question came to us from graph theory. However, even in that form, two variants are relevant.</p><ul style="box-sizing:border-box;margin-top:0px;color:rgb(96,108,113);font-family:&quot;Open Sans&quot;,&quot;Helvetica Neue&quot;,Helvetica,Arial,sans-serif"><li style="margin-left:15px;box-sizing:border-box;margin-top:0px">Can we decide whether the model companion is <span id="gmail-m_5751542264780799491gmail-MathJax-Element-4-Frame" class="gmail-m_5751542264780799491gmail-mjx-chtml gmail-m_5751542264780799491gmail-MathJax_CHTML" style="box-sizing:border-box;display:inline-block;line-height:0;word-spacing:normal;white-space:nowrap;float:none;direction:ltr;max-width:none;max-height:none;min-width:0px;min-height:0px;border:0px;margin:0px;padding:1px 0px"><span id="gmail-m_5751542264780799491gmail-MJXc-Node-16" class="gmail-m_5751542264780799491gmail-mjx-math" style="box-sizing:border-box;display:inline-block;border-collapse:separate;border-spacing:0px;margin-top:0px"><span id="gmail-m_5751542264780799491gmail-MJXc-Node-17" class="gmail-m_5751542264780799491gmail-mjx-mrow" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_5751542264780799491gmail-MJXc-Node-18" class="gmail-m_5751542264780799491gmail-mjx-msubsup" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-m_5751542264780799491gmail-mjx-base" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_5751542264780799491gmail-MJXc-Node-19" class="gmail-m_5751542264780799491gmail-mjx-mi" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-m_5751542264780799491gmail-mjx-char gmail-m_5751542264780799491gmail-MJXc-TeX-main-R" style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.425em;padding-bottom:0.331em;box-sizing:content-box">ℵ</span></span></span><span class="gmail-m_5751542264780799491gmail-mjx-sub" style="display:inline-block;vertical-align:-0.212em;padding-right:0.071em;box-sizing:content-box"><span id="gmail-m_5751542264780799491gmail-MJXc-Node-20" class="gmail-m_5751542264780799491gmail-mjx-mn" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-m_5751542264780799491gmail-mjx-char gmail-m_5751542264780799491gmail-MJXc-TeX-main-R" style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.378em;padding-bottom:0.378em;box-sizing:content-box">0</span></span></span></span></span></span><span class="gmail-m_5751542264780799491gmail-MJX_Assistive_MathML" style="box-sizing:border-box;padding:1px 0px 0px;border:0px;height:1px;width:1px;overflow:hidden;display:block">ℵ0</span></span>-categorical?</li><li style="margin-left:15px;box-sizing:border-box">Dropping the connectedness hypothesis on the constraints, can we decide whether the model companion is complete?</li></ul><p style="box-sizing:border-box;margin-bottom:1em;color:rgb(96,108,113);font-family:&quot;Open Sans&quot;,&quot;Helvetica Neue&quot;,Helvetica,Arial,sans-serif"></p><p style="box-sizing:border-box;margin-bottom:1em;color:rgb(96,108,113);font-family:&quot;Open Sans&quot;,&quot;Helvetica Neue&quot;,Helvetica,Arial,sans-serif">The main source of positive answers to the universality problem appears to be the <span id="gmail-m_5751542264780799491gmail-MathJax-Element-5-Frame" class="gmail-m_5751542264780799491gmail-mjx-chtml gmail-m_5751542264780799491gmail-MathJax_CHTML" style="box-sizing:border-box;display:inline-block;line-height:0;word-spacing:normal;white-space:nowrap;float:none;direction:ltr;max-width:none;max-height:none;min-width:0px;min-height:0px;border:0px;margin:0px;padding:1px 0px"><span id="gmail-m_5751542264780799491gmail-MJXc-Node-21" class="gmail-m_5751542264780799491gmail-mjx-math" style="box-sizing:border-box;display:inline-block;border-collapse:separate;border-spacing:0px;margin-top:0px"><span id="gmail-m_5751542264780799491gmail-MJXc-Node-22" class="gmail-m_5751542264780799491gmail-mjx-mrow" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_5751542264780799491gmail-MJXc-Node-23" class="gmail-m_5751542264780799491gmail-mjx-msubsup" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-m_5751542264780799491gmail-mjx-base" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_5751542264780799491gmail-MJXc-Node-24" class="gmail-m_5751542264780799491gmail-mjx-mi" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-m_5751542264780799491gmail-mjx-char gmail-m_5751542264780799491gmail-MJXc-TeX-main-R" style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.425em;padding-bottom:0.331em;box-sizing:content-box">ℵ</span></span></span><span class="gmail-m_5751542264780799491gmail-mjx-sub" style="display:inline-block;vertical-align:-0.212em;padding-right:0.071em;box-sizing:content-box"><span id="gmail-m_5751542264780799491gmail-MJXc-Node-25" class="gmail-m_5751542264780799491gmail-mjx-mn" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-m_5751542264780799491gmail-mjx-char gmail-m_5751542264780799491gmail-MJXc-TeX-main-R" style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.378em;padding-bottom:0.378em;box-sizing:content-box">0</span></span></span></span></span></span><span class="gmail-m_5751542264780799491gmail-MJX_Assistive_MathML" style="box-sizing:border-box;padding:1px 0px 0px;border:0px;height:1px;width:1px;overflow:hidden;display:block">ℵ0</span></span>-categorical case, and this has some theoretical justification. The most direct route to understanding the original graph theoretical problem appears to take the detour through <span id="gmail-m_5751542264780799491gmail-MathJax-Element-6-Frame" class="gmail-m_5751542264780799491gmail-mjx-chtml gmail-m_5751542264780799491gmail-MathJax_CHTML" style="box-sizing:border-box;display:inline-block;line-height:0;word-spacing:normal;white-space:nowrap;float:none;direction:ltr;max-width:none;max-height:none;min-width:0px;min-height:0px;border:0px;margin:0px;padding:1px 0px"><span id="gmail-m_5751542264780799491gmail-MJXc-Node-26" class="gmail-m_5751542264780799491gmail-mjx-math" style="box-sizing:border-box;display:inline-block;border-collapse:separate;border-spacing:0px;margin-top:0px"><span id="gmail-m_5751542264780799491gmail-MJXc-Node-27" class="gmail-m_5751542264780799491gmail-mjx-mrow" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_5751542264780799491gmail-MJXc-Node-28" class="gmail-m_5751542264780799491gmail-mjx-msubsup" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-m_5751542264780799491gmail-mjx-base" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_5751542264780799491gmail-MJXc-Node-29" class="gmail-m_5751542264780799491gmail-mjx-mi" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-m_5751542264780799491gmail-mjx-char gmail-m_5751542264780799491gmail-MJXc-TeX-main-R" style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.425em;padding-bottom:0.331em;box-sizing:content-box">ℵ</span></span></span><span class="gmail-m_5751542264780799491gmail-mjx-sub" style="display:inline-block;vertical-align:-0.212em;padding-right:0.071em;box-sizing:content-box"><span id="gmail-m_5751542264780799491gmail-MJXc-Node-30" class="gmail-m_5751542264780799491gmail-mjx-mn" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-m_5751542264780799491gmail-mjx-char gmail-m_5751542264780799491gmail-MJXc-TeX-main-R" style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.378em;padding-bottom:0.378em;box-sizing:content-box">0</span></span></span></span></span></span><span class="gmail-m_5751542264780799491gmail-MJX_Assistive_MathML" style="box-sizing:border-box;padding:1px 0px 0px;border:0px;height:1px;width:1px;overflow:hidden;display:block">ℵ0</span></span>-categoricity.</p><p style="box-sizing:border-box;margin-bottom:1em;color:rgb(96,108,113);font-family:&quot;Open Sans&quot;,&quot;Helvetica Neue&quot;,Helvetica,Arial,sans-serif">As time allows, I would like to discuss the following three points.</p><ul style="box-sizing:border-box;margin-top:0px;color:rgb(96,108,113);font-family:&quot;Open Sans&quot;,&quot;Helvetica Neue&quot;,Helvetica,Arial,sans-serif"><li style="margin-left:15px;box-sizing:border-box;margin-top:0px">What we know, and what we expect, in the case of a single constraint;</li><li style="margin-left:15px;box-sizing:border-box">Methods of proof (<em style="box-sizing:border-box;margin-top:0px">algebraic closure; induction by pruning</em>);</li><li style="margin-left:15px;box-sizing:border-box">The status of the decision problem for j.e.p. and its analog in the theory of permutation pattern classes (work of Braunfeld).</li></ul><p style="box-sizing:border-box;margin-bottom:1em;color:rgb(96,108,113);font-family:&quot;Open Sans&quot;,&quot;Helvetica Neue&quot;,Helvetica,Arial,sans-serif"></p><p style="box-sizing:border-box;margin-bottom:1em;color:rgb(96,108,113);font-family:&quot;Open Sans&quot;,&quot;Helvetica Neue&quot;,Helvetica,Arial,sans-serif">This is joint work with Shelah, e.g. [Sh689].</p></div></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><br></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><br></div></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><font color="#212121">- - - - Other Logic News - - - -</font></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><br></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><b>Conference announcement:</b></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><span style="color:rgb(34,34,34);font-family:Arial,Helvetica,sans-serif">Model Theory and Mathematical Logic,</span><br style="color:rgb(34,34,34);font-family:Arial,Helvetica,sans-serif"><span style="color:rgb(34,34,34);font-family:Arial,Helvetica,sans-serif"> in honor of Chris Laskowski&#39;s 60th birthday,</span><br style="color:rgb(34,34,34);font-family:Arial,Helvetica,sans-serif"><span style="color:rgb(34,34,34);font-family:Arial,Helvetica,sans-serif"> June 21 - 23, 2019, at The University of Maryland, College Park</span><br style="color:rgb(34,34,34);font-family:Arial,Helvetica,sans-serif"><a href="http://www.umdlogic2019.com/" rel="noreferrer" target="_blank" style="font-family:Arial,Helvetica,sans-serif">http://www.umdlogic2019.com/</a><br></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><font color="#212121"><br></font></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><font color="#212121"><br></font></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><font color="#212121"><br></font></div><div><div><b style="color:rgb(0,0,0);font-family:arial,sans-serif">Conference announcement:</b><font color="#000000" face="arial, sans-serif"><b> <a href="mailto:scoskey@gmail.com">scoskey@gmail.com</a></b></font></div></div><div><font color="#000000" face="arial, sans-serif">The 2019 </font><a href="https://urldefense.proofpoint.com/v2/url?u=https-3A__math.boisestate.edu_best&amp;d=DwMFaQ&amp;c=pRW6ZPn_LDv0DnDIAK65Ad0CA4hBS-2mAmNa2_oHfF0&amp;r=rvtWDO9Zc-E9EhkvL8sfOAUVc4SuAmQdMXpD6PNOkGM&amp;m=wENt1oQTtXEQxL5XJzqTp39k1un9HMylVIRjM26QaC0&amp;s=zogVm_GCFVhwhg52vjaNoUo-ZDg_-qMcqBP74Fyv7qU&amp;e=" target="_blank" style="color:rgb(0,0,0);font-family:arial,sans-serif;text-decoration-line:none">Boise Extravaganza in Set Theory</a><font color="#000000" face="arial, sans-serif"> will take place in Ashland, Oregon, on the campus of Southern Oregon University, during June 19-21.</font><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><br></div><div><font color="#000000" face="arial, sans-serif">We are currently welcoming applications for </font><a href="https://urldefense.proofpoint.com/v2/url?u=https-3A__math.boisestate.edu_best_travel-2Dgrants_&amp;d=DwMFaQ&amp;c=pRW6ZPn_LDv0DnDIAK65Ad0CA4hBS-2mAmNa2_oHfF0&amp;r=rvtWDO9Zc-E9EhkvL8sfOAUVc4SuAmQdMXpD6PNOkGM&amp;m=wENt1oQTtXEQxL5XJzqTp39k1un9HMylVIRjM26QaC0&amp;s=G-M0o6olGB61CSXZnf_hF8qftpvGlv5561dy6PrlL7o&amp;e=" target="_blank" style="color:rgb(0,0,0);font-family:arial,sans-serif;text-decoration-line:none">travel grants</a><font color="#000000" face="arial, sans-serif"> from graduate students and postdocs (and other categories) in set theory </font><i style="color:rgb(0,0,0);font-family:arial,sans-serif">and related fields</i><font color="#000000" face="arial, sans-serif">. We will begin considering applications on </font><span style="color:rgb(0,0,0);font-family:arial,sans-serif">May 1</span><font color="#000000" face="arial, sans-serif">. We will continue accepting applications on a rolling basis until May 31.</font><br><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><br></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif">Plenary speakers:</div><font color="#000000" face="arial, sans-serif"> </font><u style="color:rgb(0,0,0);font-family:arial,sans-serif">Dana Bartosova</u><font color="#000000" face="arial, sans-serif"> (University of Florida)</font><br><font color="#000000" face="arial, sans-serif"> </font><u style="color:rgb(0,0,0);font-family:arial,sans-serif">Steve Jackson</u><font color="#000000" face="arial, sans-serif"> (University of North Texas)</font><br><font color="#000000" face="arial, sans-serif"> </font><u style="color:rgb(0,0,0);font-family:arial,sans-serif">Reese Johnston</u><font color="#000000" face="arial, sans-serif"> (University of Washington Robinson Center)</font><br><font color="#000000" face="arial, sans-serif"> </font><u style="color:rgb(0,0,0);font-family:arial,sans-serif">Assaf Shani</u><font color="#000000" face="arial, sans-serif"> (UCLA)</font><br><font color="#000000" face="arial, sans-serif"> </font><u style="color:rgb(0,0,0);font-family:arial,sans-serif">Piotr Szeczwak</u><font color="#000000" face="arial, sans-serif"> (Cardinal Stefan Wyszyński University, Warsaw)</font><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><br></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif">Contact:</div><div><font color="#000000" face="arial, sans-serif"><a href="mailto:scoskey@gmail.com">scoskey@gmail.com</a> or </font><a href="mailto:best@boisestate.edu" target="_blank" style="color:rgb(0,0,0);font-family:arial,sans-serif;text-decoration-line:none">best@boisestate.edu</a><br></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><br></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif">The BEST conference particularly aims to support the careers of young researchers in set theory (and related fields!), with travel support available for graduate students and postdocs. BEST features professional development opportunities and awards for student presentations. Please pass this invitation along to students, postdocs, and colleagues to submit an abstract and participate in BEST!<br><br>BEST is an international conference featuring talks on a broad range of recent advances in set theory research. The conference is organized by the Set Theory group at Boise State University and is structured as a symposium of the 100th annual meeting of the American Association for the Advancement of Science, Pacific Division (AAAS-PD). BEST is also supported by the NSF and Boise State University.<br><br>Organizers: Liljana Babinkostova (Boise State University), John Clemens (Boise State University), Samuel Coskey (Boise State University), Marion Scheepers (Boise State University). Scientific Committee: Natasha Dobrinen (University of Denver), Simon Thomas (Rutgers University)</div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><br></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><br></div></div></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><br></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><br></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><font color="#212121"><br></font></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><font color="#212121">- - - - Web Site - - - -</font></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><font color="#212121"><br></font></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><font color="#212121">Our new website is located at:  <a href="https://nylogic.github.io/" target="_blank" style="text-decoration-line:none">https://nylogic.github.io/</a></font></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><font color="#212121">Many thanks to Victoria Gitman for her development work!</font></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><font color="#212121"><br></font></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><font color="#212121">--------  ADMINISTRIVIA  --------</font></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><font color="#212121"><br></font></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><font color="#212121">To subscribe/unsubscribe to this list, please email your request to <a href="mailto:jreitz@nylogic.org" target="_blank" style="text-decoration-line:none">jreitz@nylogic.org</a>.</font></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><font color="#212121"><br></font></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><font color="#212121">If you have a logic-related event that you would like included in future mailings, please email <a href="mailto:jreitz@nylogic.org" target="_blank" style="text-decoration-line:none">jreitz@nylogic.org</a>. </font></div></div></div></div></div>
41
4/24/2019 0:00:00Conference
Boise Extravaganza in Set Theory, second announcement
The 2019 Boise Extravaganza in Set Theory will take place in Ashland, Oregon, on the campus of Southern Oregon University, during June 19-21.

We are currently welcoming applications for travel grants from graduate students and postdocs (and other categories) in set theory and related fields. We will begin considering applications on May 1. We will continue accepting applications on a rolling basis until May 31.

Plenary speakers:
Dana Bartosova (University of Florida)
Steve Jackson (University of North Texas)
Reese Johnston (University of Washington Robinson Center)
Assaf Shani (UCLA)
Piotr Szeczwak (Cardinal Stefan Wyszyński University, Warsaw)

Contact:
myself or best@boisestate.edu

**

The BEST conference particularly aims to support the careers of young researchers in set theory (and related fields!), with travel support available for graduate students and postdocs. BEST features professional development opportunities and awards for student presentations. Please pass this invitation along to students, postdocs, and colleagues to submit an abstract and participate in BEST!

BEST is an international conference featuring talks on a broad range of recent advances in set theory research. The conference is organized by the Set Theory group at Boise State University and is structured as a symposium of the 100th annual meeting of the American Association for the Advancement of Science, Pacific Division (AAAS-PD). BEST is also supported by the NSF and Boise State University.

Organizers: Liljana Babinkostova (Boise State University), John Clemens (Boise State University), Samuel Coskey (Boise State University), Marion Scheepers (Boise State University). Scientific Committee: Natasha Dobrinen (University of Denver), Simon Thomas (Rutgers University)
Dana Bartošová,Steve Jackson,Reese Johnston,Assaf Shani,Piotr Szewczak
https://math.boisestate.edu/best
42
4/23/2019 7:10:38IMPAN Working Group in Applications of Set TheoryFulgencio Lopez; Compact extensions of first order logic; cont.Seminar: Working group in applications of set theory, IMPAN

NOTE UNUSUAL TIME AND PLACE:

Friday, 26.04.2019, 10:15, room 106, IMPAN

Speaker: Fulgencio Lopez (IM PAN)

Title: "Compact extensions of first order logic" continuation from 16.04.

Abstact: "We aim to provide an exhaustive proof of the results of Keisler and Magidor and Malitz about the compactness of certain extensions of first order logic. Keisler's Theorem refers to the extension L(Q) where Q is the quantifier "There is an uncountable subset with a one dimensional property", similarly we can define Q_n to be "There is an uncountable subset with an n-dimensional property". We will show that L(Q) is compact (Keisler's Theorem) and, assuming diamond, so is L(Q_n:n∈ N) (Magidor and Malitz). We will also discuss why this framework can sometimes be useful for constructions of set theoretical structures".

Until May 9 the meetings of the seminar will take place at unusual times due to holidays and to the scientific Council of the Institute.
We go back to the usual Thursday time on May 9.

Visit our seminar page which may include some future talks at https://www.impan.pl/~set_theory/Seminar/
Fulgencio Lopez
43
4/22/2019 22:52:21This Week in Logic at CUNYThis Week in Logic at CUNY<div dir="ltr"><div dir="ltr"><div dir="ltr"><div dir="ltr"><div dir="ltr"><div dir="ltr"><div style="font-family:arial,sans-serif;color:rgb(33,33,33)"><span style="color:rgb(0,0,0)">Hi everyone,</span></div><div style="font-family:arial,sans-serif;color:rgb(33,33,33)"><span style="color:rgb(0,0,0)"><br></span></div><div style="font-family:arial,sans-serif;color:rgb(33,33,33)"><span style="color:rgb(0,0,0)">I hope your Spring Break is going well.  I&#39;m sending out this special edition of &quot;This Week in Logic&quot; to announce this Wednesday&#39;s </span><span style="font-family:Arial,Helvetica,sans-serif">New York City Category Theory Seminar talk (and give a preview of next week&#39;s events).</span><br></div><div style="font-family:arial,sans-serif;color:rgb(33,33,33)"><span style="color:rgb(0,0,0)"><br></span></div><div style="font-family:arial,sans-serif;color:rgb(33,33,33)"><span style="color:rgb(0,0,0)">All the best,</span></div><div style="font-family:arial,sans-serif;color:rgb(33,33,33)"><span style="color:rgb(0,0,0)">Jonas</span></div><div style="font-family:arial,sans-serif;color:rgb(33,33,33)"><span style="color:rgb(0,0,0)"><br></span></div><div style="font-family:arial,sans-serif;color:rgb(33,33,33)"><span style="color:rgb(0,0,0)"><br></span></div><div style="font-family:arial,sans-serif;color:rgb(33,33,33)"><br></div><div style="font-family:arial,sans-serif;color:rgb(33,33,33)">This Week in Logic at CUNY:</div><div><div><div style="color:rgb(33,33,33)"><div><br></div><div>- - - - Monday, Apr 22, 2019 - - - -</div><div><br></div><div><div style="color:rgb(0,0,0);font-family:arial,sans-serif">*** NOTE: CUNY Spring Break April 19-28 ***</div><br class="gmail-Apple-interchange-newline"></div><div><br></div><div>- - - - Tuesday, Apr 23, 2019 - - - -</div><div><br></div><div><br></div><div><br></div><div>- - - - Wednesday, Apr 24, 2019 - - - -</div><div><br></div><div><div>The New York City Category Theory Seminar</div><div>Department of Computer Science</div><div>Department of Mathematics </div><div>The Graduate Center of The City University of New York </div><div>Speaker:     Andrei Rodin, Senior Researcher at the Institute of Philosophy of Russian Academy of Sciences.<br></div><div>Date and Time:     Wednesday April 24, 2019, 7:00 - 8:30 PM., Room 6417.<br></div><div>Title:     Directed Homotopy Type Theory and the (In)vertibility of Mathematics.<br></div><div><br></div><div>Abstract: Directed Homotopy Type theory (DHTT) is a generalization of Homotopy Type theory (HoTT) where fundamental groupoids of spaces are replaced by more general (higher) categories. Along with type formers for identity types which admit the standard HoTT interpretation in terms of invertible paths and their homotopies, DHTT comprises type formers for non-invertable homomorphisms of all levels which admit an interpreation in terms of non-invertable paths in appropriate spaces. The choice between DHTT and HoTT as foundational formal frameworks for building mathematical theories has an epistemological dimension, which concerns the epistemic significance of the invertibility condition. While HoTT and the related notion of Univalent Foundations support Mathematical Structuralism DHTT supports a more dynamic conception of Mathematics, which I shall outline in my talk. <br></div><div><br></div><div>Related papers: </div><div>Paige North, Towards a Directed Homotopy Type Theory, arXiv:1807.10566 <br></div><div>Andrei Rodin, Categories Without Structures, arXiv: 0907.5143 (published in Philosophia Mathematica 19/1 (2011), p. 20-46) <br></div><div>Michael Warren, Directed Type Theory (video of talk in IAS Princeton)<br></div></div><div><br></div><div><br></div><div><br></div><div>- - - - Thursday, Apr 25, 2019 - - - -</div><div><br></div><div><br></div><div><br></div><div>- - - - Friday, Apr 26, 2019 - - - -</div><div><br></div><div><br></div><div><br></div><div>Next Week in Logic at CUNY:</div><div><br></div><div>- - - - Monday, Apr 29, 2019 - - - -</div><div><br></div><div><div>Logic and Metaphysics Workshop</div><div>Date: Monday, April 29th, 4.15-6.15  <br></div><div>Place: Room 7314, CUNY Graduate Center <br></div><div>Speaker: Tommy Kivatinos (CUNY)<br></div><div>Title: A Mechanistic Conception of Metaphysical Grounding<br></div><div><br></div><div>Abstract: A dominant theoretical framework in philosophy of science employs the notion of mechanistic dependence to elucidate how higher-level, less fundamental phenomena depend upon and arise out of lower-level, more fundamental phenomena. To elucidate the same thing, literature in metaphysics employs the notion of grounding. As I argue, regardless of whether the notion of mechanistic dependence or the notion of grounding is used to theoretically portray how higher-level phenomena arise out of lower-level phenomena, what is captured by such portrayals is the same. Thus, these notions pick out the same features of the world. With this as my basis, I identify the notion of grounding with the notion of mechanistic dependence, and thus, construct a mechanistic conception of grounding. Since mechanistic dependence is understood in terms of mechanisms, my conception frames grounding in terms of mechanisms. Moreover, the contemporary notion of mechanisms is shaped by how mechanisms are represented via the mechanistic models and mechanistic explanations provided by science. Thus, because my conception grounding identifies grounding with mechanistic dependence and thereby frames grounding in terms of mechanisms, this conception suggests that the notion of grounding is to be tailored to and constrained by the mechanistic models and mechanistic explanations provided by science. This leads the mechanistic conception of grounding to reject a wide variety of conventional claims about grounding, and thus, to offer a treatment of grounding that is highly revisionary. To reinforce the plausibility of the mechanistic conception of grounding, I discuss how grounding and mechanistic dependence are associated with explanation. Whereas mechanistic dependence is associated with mechanistic explanation, grounding is associated with grounding explanation. For each kind of explanation, some higher-level phenomenon P is explained by appeal to some low-level phenomenon that Parises out of. As I argue, these forms of explanation can be plausibly identified. This greatly supports the mechanistic conception of grounding. For if grounding explanations employ the notion of grounding and mechanistic explanations employ the notion of mechanistic explanation, and these forms of explanation can be identified, this suggests that these explanations employ the same notion. And, just as the notions of grounding and mechanistic dependence capture the same connection between higher-level and lower-level phenomena, grounding explanation and mechanistic explanation do so as well. Finally, to argue that the mechanistic conception is to be preferred to standard conceptions, I argue that my conception offers a powerful defense of grounding from recent criticisms.</div></div><div><br></div><div><br></div><div><br></div><div><br></div><div>- - - - Tuesday, Apr 30, 2019 - - - -</div><div><br></div><div><br></div><div><br></div><div>- - - - Wednesday, May 1, 2019 - - - -</div><div><br></div><div><br></div><div><br></div><div>- - - - Thursday, May 2, 2019 - - - -</div><div><br></div><div><br></div><div><br></div><div>- - - - Friday, May 3, 2019 - - - -</div></div><div style="color:rgb(33,33,33)"><br></div><div><div><font color="#000000" face="arial, sans-serif">Set Theory Seminar</font></div><div><font color="#000000" face="arial, sans-serif">CUNY Graduate Center, Room 6417</font></div><div><font color="#000000" face="arial, sans-serif">Friday, May 3, 10:00-11:45am</font></div><div><p style="box-sizing:border-box;margin-bottom:1em;margin-top:0px;color:rgb(96,108,113);font-family:&quot;Open Sans&quot;,&quot;Helvetica Neue&quot;,Helvetica,Arial,sans-serif"><strong style="box-sizing:border-box"><a href="https://nylogic.github.io/set-theory-seminar/2019/05/03/target=%22_blank%22" style="box-sizing:border-box;background-color:transparent;color:rgb(30,107,184);text-decoration-line:none;margin-top:0px">Joseph Van Name</a></strong>, CUNY<br style="box-sizing:border-box"><strong style="box-sizing:border-box">Lower bounds on the cardinalities of quotient algebras of elementary embeddings</strong><br style="box-sizing:border-box"></p><p style="box-sizing:border-box;margin-bottom:1em;color:rgb(96,108,113);font-family:&quot;Open Sans&quot;,&quot;Helvetica Neue&quot;,Helvetica,Arial,sans-serif">From non-trivial elementary embeddings <span id="gmail-MathJax-Element-1-Frame" class="gmail-mjx-chtml gmail-MathJax_CHTML" tabindex="0" style="box-sizing:border-box;display:inline-block;line-height:0;word-spacing:normal;white-space:nowrap;float:none;direction:ltr;max-width:none;max-height:none;min-width:0px;min-height:0px;border:0px;margin:0px;padding:1px 0px"><span id="gmail-MJXc-Node-1" class="gmail-mjx-math" style="box-sizing:border-box;display:inline-block;border-collapse:separate;border-spacing:0px;margin-top:0px"><span id="gmail-MJXc-Node-2" class="gmail-mjx-mrow" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-MJXc-Node-3" class="gmail-mjx-msubsup" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-mjx-base" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-MJXc-Node-4" class="gmail-mjx-mi" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-math-I" style="display:block;white-space:pre;font-family:MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw;margin-top:0px;padding-top:0.425em;padding-bottom:0.472em;box-sizing:content-box">j</span></span></span><span class="gmail-mjx-sub" style="display:inline-block;vertical-align:-0.212em;padding-right:0.071em;box-sizing:content-box"><span id="gmail-MJXc-Node-5" class="gmail-mjx-texatom" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-MJXc-Node-6" class="gmail-mjx-mrow" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-MJXc-Node-7" class="gmail-mjx-mn" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-main-R" style="display:block;white-space:pre;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.378em;padding-bottom:0.331em;box-sizing:content-box">1</span></span></span></span></span></span><span id="gmail-MJXc-Node-8" class="gmail-mjx-mo gmail-MJXc-space1" style="display:inline-block;margin-left:0.167em;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-main-R" style="display:block;white-space:pre;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;padding-bottom:0.331em;box-sizing:content-box">…</span></span><span id="gmail-MJXc-Node-9" class="gmail-mjx-msubsup gmail-MJXc-space1" style="display:inline-block;margin-left:0.167em;box-sizing:content-box"><span class="gmail-mjx-base" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-MJXc-Node-10" class="gmail-mjx-mi" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-math-I" style="display:block;white-space:pre;font-family:MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw;margin-top:0px;padding-top:0.425em;padding-bottom:0.472em;box-sizing:content-box">j</span></span></span><span class="gmail-mjx-sub" style="display:inline-block;vertical-align:-0.212em;padding-right:0.071em;box-sizing:content-box"><span id="gmail-MJXc-Node-11" class="gmail-mjx-texatom" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-MJXc-Node-12" class="gmail-mjx-mrow" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-MJXc-Node-13" class="gmail-mjx-mi" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-math-I" style="display:block;white-space:pre;font-family:MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw;margin-top:0px;padding-top:0.237em;padding-bottom:0.284em;box-sizing:content-box">r</span></span></span></span></span></span><span id="gmail-MJXc-Node-14" class="gmail-mjx-mo gmail-MJXc-space3" style="display:inline-block;margin-left:0.278em;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-main-R" style="display:block;white-space:pre;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.143em;padding-bottom:0.331em;box-sizing:content-box">:</span></span><span id="gmail-MJXc-Node-15" class="gmail-mjx-msubsup gmail-MJXc-space3" style="display:inline-block;margin-left:0.278em;box-sizing:content-box"><span class="gmail-mjx-base" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-MJXc-Node-16" class="gmail-mjx-mi" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-math-I" style="display:block;white-space:pre;font-family:MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw;margin-top:0px;padding-top:0.472em;padding-bottom:0.284em;padding-right:0.186em;box-sizing:content-box">V</span></span></span><span class="gmail-mjx-sub" style="display:inline-block;vertical-align:-0.219em;padding-right:0.071em;box-sizing:content-box"><span id="gmail-MJXc-Node-17" class="gmail-mjx-texatom" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-MJXc-Node-18" class="gmail-mjx-mrow" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-MJXc-Node-19" class="gmail-mjx-mi" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-math-I" style="display:block;white-space:pre;font-family:MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw;margin-top:0px;padding-top:0.472em;padding-bottom:0.284em;box-sizing:content-box">λ</span></span></span></span></span></span><span id="gmail-MJXc-Node-20" class="gmail-mjx-mo gmail-MJXc-space3" style="display:inline-block;margin-left:0.278em;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-main-R" style="display:block;white-space:pre;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.237em;padding-bottom:0.331em;box-sizing:content-box">→</span></span><span id="gmail-MJXc-Node-21" class="gmail-mjx-msubsup gmail-MJXc-space3" style="display:inline-block;margin-left:0.278em;box-sizing:content-box"><span class="gmail-mjx-base" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-MJXc-Node-22" class="gmail-mjx-mi" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-math-I" style="display:block;white-space:pre;font-family:MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw;margin-top:0px;padding-top:0.472em;padding-bottom:0.284em;padding-right:0.186em;box-sizing:content-box">V</span></span></span><span class="gmail-mjx-sub" style="display:inline-block;vertical-align:-0.219em;padding-right:0.071em;box-sizing:content-box"><span id="gmail-MJXc-Node-23" class="gmail-mjx-texatom" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-MJXc-Node-24" class="gmail-mjx-mrow" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-MJXc-Node-25" class="gmail-mjx-mi" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-math-I" style="display:block;white-space:pre;font-family:MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw;margin-top:0px;padding-top:0.472em;padding-bottom:0.284em;box-sizing:content-box">λ</span></span></span></span></span></span></span></span><span class="gmail-MJX_Assistive_MathML" style="box-sizing:border-box;padding:1px 0px 0px;border:0px;height:1px;width:1px;overflow:hidden;display:block">j<span class="gmail-MJX-TeXAtom-ORD">1</span>…j<span class="gmail-MJX-TeXAtom-ORD">r</span>:V<span class="gmail-MJX-TeXAtom-ORD">λ</span>→V<span class="gmail-MJX-TeXAtom-ORD">λ</span></span></span>, we obtain a sequence of polynomials <span id="gmail-MathJax-Element-2-Frame" class="gmail-mjx-chtml gmail-MathJax_CHTML" tabindex="0" style="box-sizing:border-box;display:inline-block;line-height:0;word-spacing:normal;white-space:nowrap;float:none;direction:ltr;max-width:none;max-height:none;min-width:0px;min-height:0px;border:0px;margin:0px;padding:1px 0px"><span id="gmail-MJXc-Node-26" class="gmail-mjx-math" style="box-sizing:border-box;display:inline-block;border-collapse:separate;border-spacing:0px;margin-top:0px"><span id="gmail-MJXc-Node-27" class="gmail-mjx-mrow" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-MJXc-Node-28" class="gmail-mjx-mo" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-main-R" style="display:block;white-space:pre;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.472em;padding-bottom:0.613em;box-sizing:content-box">(</span></span><span id="gmail-MJXc-Node-29" class="gmail-mjx-msubsup" style="display:inline-block;box-sizing:content-box"><span class="gmail-mjx-base" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-MJXc-Node-30" class="gmail-mjx-mi" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-math-I" style="display:block;white-space:pre;font-family:MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw;margin-top:0px;padding-top:0.237em;padding-bottom:0.472em;box-sizing:content-box">p</span></span></span><span class="gmail-mjx-sub" style="display:inline-block;vertical-align:-0.212em;padding-right:0.071em;box-sizing:content-box"><span id="gmail-MJXc-Node-31" class="gmail-mjx-texatom" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-MJXc-Node-32" class="gmail-mjx-mrow" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-MJXc-Node-33" class="gmail-mjx-mi" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-math-I" style="display:block;white-space:pre;font-family:MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw;margin-top:0px;padding-top:0.237em;padding-bottom:0.284em;box-sizing:content-box">n</span></span></span></span></span></span><span id="gmail-MJXc-Node-34" class="gmail-mjx-mo" style="display:inline-block;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-main-R" style="display:block;white-space:pre;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.472em;padding-bottom:0.613em;box-sizing:content-box">(</span></span><span id="gmail-MJXc-Node-35" class="gmail-mjx-msubsup" style="display:inline-block;box-sizing:content-box"><span class="gmail-mjx-base" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-MJXc-Node-36" class="gmail-mjx-mi" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-math-I" style="display:block;white-space:pre;font-family:MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw;margin-top:0px;padding-top:0.237em;padding-bottom:0.284em;box-sizing:content-box">x</span></span></span><span class="gmail-mjx-sub" style="display:inline-block;vertical-align:-0.212em;padding-right:0.071em;box-sizing:content-box"><span id="gmail-MJXc-Node-37" class="gmail-mjx-texatom" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-MJXc-Node-38" class="gmail-mjx-mrow" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-MJXc-Node-39" class="gmail-mjx-mn" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-main-R" style="display:block;white-space:pre;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.378em;padding-bottom:0.331em;box-sizing:content-box">1</span></span></span></span></span></span><span id="gmail-MJXc-Node-40" class="gmail-mjx-mo" style="display:inline-block;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-main-R" style="display:block;white-space:pre;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;padding-bottom:0.519em;box-sizing:content-box">,</span></span><span id="gmail-MJXc-Node-41" class="gmail-mjx-mo gmail-MJXc-space1" style="display:inline-block;margin-left:0.167em;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-main-R" style="display:block;white-space:pre;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;padding-bottom:0.331em;box-sizing:content-box">…</span></span><span id="gmail-MJXc-Node-42" class="gmail-mjx-mo gmail-MJXc-space1" style="display:inline-block;margin-left:0.167em;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-main-R" style="display:block;white-space:pre;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;padding-bottom:0.519em;box-sizing:content-box">,</span></span><span id="gmail-MJXc-Node-43" class="gmail-mjx-msubsup gmail-MJXc-space1" style="display:inline-block;margin-left:0.167em;box-sizing:content-box"><span class="gmail-mjx-base" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-MJXc-Node-44" class="gmail-mjx-mi" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-math-I" style="display:block;white-space:pre;font-family:MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw;margin-top:0px;padding-top:0.237em;padding-bottom:0.284em;box-sizing:content-box">x</span></span></span><span class="gmail-mjx-sub" style="display:inline-block;vertical-align:-0.212em;padding-right:0.071em;box-sizing:content-box"><span id="gmail-MJXc-Node-45" class="gmail-mjx-texatom" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-MJXc-Node-46" class="gmail-mjx-mrow" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-MJXc-Node-47" class="gmail-mjx-mi" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-math-I" style="display:block;white-space:pre;font-family:MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw;margin-top:0px;padding-top:0.237em;padding-bottom:0.284em;box-sizing:content-box">r</span></span></span></span></span></span><span id="gmail-MJXc-Node-48" class="gmail-mjx-mo" style="display:inline-block;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-main-R" style="display:block;white-space:pre;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.472em;padding-bottom:0.613em;box-sizing:content-box">)</span></span><span id="gmail-MJXc-Node-49" class="gmail-mjx-msubsup" style="display:inline-block;box-sizing:content-box"><span class="gmail-mjx-base" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-MJXc-Node-50" class="gmail-mjx-mo" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-main-R" style="display:block;white-space:pre;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.472em;padding-bottom:0.613em;box-sizing:content-box">)</span></span></span><span class="gmail-mjx-sub" style="display:inline-block;vertical-align:-0.212em;padding-right:0.071em;box-sizing:content-box"><span id="gmail-MJXc-Node-51" class="gmail-mjx-texatom" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-MJXc-Node-52" class="gmail-mjx-mrow" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-MJXc-Node-53" class="gmail-mjx-mi" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-math-I" style="display:block;white-space:pre;font-family:MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw;margin-top:0px;padding-top:0.237em;padding-bottom:0.284em;box-sizing:content-box">n</span></span><span id="gmail-MJXc-Node-54" class="gmail-mjx-mo" style="display:inline-block;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-main-R" style="display:block;white-space:pre;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.237em;padding-bottom:0.378em;box-sizing:content-box">∈</span></span><span id="gmail-MJXc-Node-55" class="gmail-mjx-mi" style="display:inline-block;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-math-I" style="display:block;white-space:pre;font-family:MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw;margin-top:0px;padding-top:0.237em;padding-bottom:0.284em;box-sizing:content-box">ω</span></span></span></span></span></span></span></span><span class="gmail-MJX_Assistive_MathML" style="box-sizing:border-box;padding:1px 0px 0px;border:0px;height:1px;width:1px;overflow:hidden;display:block">(p<span class="gmail-MJX-TeXAtom-ORD">n</span>(x<span class="gmail-MJX-TeXAtom-ORD">1</span>,…,x<span class="gmail-MJX-TeXAtom-ORD">r</span>))<span class="gmail-MJX-TeXAtom-ORD">n∈ω</span></span></span> that satisfies the infinite product<span class="gmail-mjx-chtml gmail-MJXc-display" style="box-sizing:border-box;display:block;line-height:0;text-align:center;word-spacing:normal;white-space:nowrap;float:none;direction:ltr;max-width:none;max-height:none;min-width:0px;min-height:0px;border:0px;margin:1em 0px;padding:0px"><span id="gmail-MathJax-Element-3-Frame" class="gmail-mjx-chtml gmail-MathJax_CHTML" tabindex="0" style="box-sizing:border-box;display:inline-block;line-height:0;word-spacing:normal;float:none;direction:ltr;max-width:none;max-height:none;min-width:0px;min-height:0px;border:0px;margin:0px;padding:1px 0px"><span id="gmail-MJXc-Node-56" class="gmail-mjx-math" style="box-sizing:border-box;display:inline-block;border-collapse:separate;border-spacing:0px;margin-top:0px"><span id="gmail-MJXc-Node-57" class="gmail-mjx-mrow" style="display:inline-block;text-align:left;margin-top:0px;box-sizing:content-box"><span id="gmail-MJXc-Node-58" class="gmail-mjx-munderover" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-mjx-itable" style="display:inline-table;width:auto;margin-top:0px;box-sizing:content-box"><span class="gmail-mjx-row" style="display:table-row;margin-top:0px;box-sizing:content-box"><span class="gmail-mjx-cell" style="display:table-cell;margin-top:0px;box-sizing:content-box"><span class="gmail-mjx-stack" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-mjx-over" style="display:block;margin-top:0px;padding-bottom:0.247em;padding-top:0.141em;padding-left:0.404em;box-sizing:content-box"><span id="gmail-MJXc-Node-65" class="gmail-mjx-texatom" style="display:inline-block;margin-top:0px;box-sizing:content-box;padding-left:0px;padding-right:0px"><span id="gmail-MJXc-Node-66" class="gmail-mjx-mrow" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-MJXc-Node-67" class="gmail-mjx-mi" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-main-R" style="display:block;white-space:pre;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.143em;padding-bottom:0.331em;box-sizing:content-box">∞</span></span></span></span></span><span class="gmail-mjx-op" style="display:block;box-sizing:content-box"><span id="gmail-MJXc-Node-59" class="gmail-mjx-mo" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-size2-R" style="display:block;white-space:pre;font-family:MJXc-TeX-size2-R,MJXc-TeX-size2-Rw;margin-top:0px;padding-top:0.707em;padding-bottom:0.707em;box-sizing:content-box">∏</span></span></span></span></span></span><span class="gmail-mjx-row" style="display:table-row;box-sizing:content-box"><span class="gmail-mjx-under" style="display:table-cell;margin-top:0px;padding-top:0.236em;padding-bottom:0.141em;padding-left:0.004em;box-sizing:content-box"><span id="gmail-MJXc-Node-60" class="gmail-mjx-texatom" style="display:inline-block;margin-top:0px;box-sizing:content-box;padding-left:0px;padding-right:0px"><span id="gmail-MJXc-Node-61" class="gmail-mjx-mrow" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-MJXc-Node-62" class="gmail-mjx-mi" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-math-I" style="display:block;white-space:pre;font-family:MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw;margin-top:0px;padding-top:0.472em;padding-bottom:0.284em;box-sizing:content-box">k</span></span><span id="gmail-MJXc-Node-63" class="gmail-mjx-mo" style="display:inline-block;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-main-R" style="display:block;white-space:pre;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.096em;padding-bottom:0.331em;box-sizing:content-box">=</span></span><span id="gmail-MJXc-Node-64" class="gmail-mjx-mn" style="display:inline-block;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-main-R" style="display:block;white-space:pre;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.378em;padding-bottom:0.378em;box-sizing:content-box">0</span></span></span></span></span></span></span></span><span id="gmail-MJXc-Node-68" class="gmail-mjx-msubsup gmail-MJXc-space1" style="display:inline-block;margin-left:0.167em;box-sizing:content-box"><span class="gmail-mjx-base" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-MJXc-Node-69" class="gmail-mjx-mi" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-math-I" style="display:block;white-space:pre;font-family:MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw;margin-top:0px;padding-top:0.237em;padding-bottom:0.472em;box-sizing:content-box">p</span></span></span><span class="gmail-mjx-sub" style="display:inline-block;vertical-align:-0.219em;padding-right:0.071em;box-sizing:content-box"><span id="gmail-MJXc-Node-70" class="gmail-mjx-texatom" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-MJXc-Node-71" class="gmail-mjx-mrow" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-MJXc-Node-72" class="gmail-mjx-mi" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-math-I" style="display:block;white-space:pre;font-family:MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw;margin-top:0px;padding-top:0.472em;padding-bottom:0.284em;box-sizing:content-box">k</span></span></span></span></span></span><span id="gmail-MJXc-Node-73" class="gmail-mjx-mo" style="display:inline-block;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-main-R" style="display:block;white-space:pre;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.472em;padding-bottom:0.613em;box-sizing:content-box">(</span></span><span id="gmail-MJXc-Node-74" class="gmail-mjx-msubsup" style="display:inline-block;box-sizing:content-box"><span class="gmail-mjx-base" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-MJXc-Node-75" class="gmail-mjx-mi" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-math-I" style="display:block;white-space:pre;font-family:MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw;margin-top:0px;padding-top:0.237em;padding-bottom:0.284em;box-sizing:content-box">x</span></span></span><span class="gmail-mjx-sub" style="display:inline-block;vertical-align:-0.212em;padding-right:0.071em;box-sizing:content-box"><span id="gmail-MJXc-Node-76" class="gmail-mjx-texatom" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-MJXc-Node-77" class="gmail-mjx-mrow" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-MJXc-Node-78" class="gmail-mjx-mn" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-main-R" style="display:block;white-space:pre;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.378em;padding-bottom:0.331em;box-sizing:content-box">1</span></span></span></span></span></span><span id="gmail-MJXc-Node-79" class="gmail-mjx-mo" style="display:inline-block;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-main-R" style="display:block;white-space:pre;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;padding-bottom:0.519em;box-sizing:content-box">,</span></span><span id="gmail-MJXc-Node-80" class="gmail-mjx-mo gmail-MJXc-space1" style="display:inline-block;margin-left:0.167em;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-main-R" style="display:block;white-space:pre;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;padding-bottom:0.331em;box-sizing:content-box">…</span></span><span id="gmail-MJXc-Node-81" class="gmail-mjx-mo gmail-MJXc-space1" style="display:inline-block;margin-left:0.167em;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-main-R" style="display:block;white-space:pre;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;padding-bottom:0.519em;box-sizing:content-box">,</span></span><span id="gmail-MJXc-Node-82" class="gmail-mjx-msubsup gmail-MJXc-space1" style="display:inline-block;margin-left:0.167em;box-sizing:content-box"><span class="gmail-mjx-base" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-MJXc-Node-83" class="gmail-mjx-mi" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-math-I" style="display:block;white-space:pre;font-family:MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw;margin-top:0px;padding-top:0.237em;padding-bottom:0.284em;box-sizing:content-box">x</span></span></span><span class="gmail-mjx-sub" style="display:inline-block;vertical-align:-0.212em;padding-right:0.071em;box-sizing:content-box"><span id="gmail-MJXc-Node-84" class="gmail-mjx-texatom" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-MJXc-Node-85" class="gmail-mjx-mrow" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-MJXc-Node-86" class="gmail-mjx-mi" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-math-I" style="display:block;white-space:pre;font-family:MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw;margin-top:0px;padding-top:0.237em;padding-bottom:0.284em;box-sizing:content-box">r</span></span></span></span></span></span><span id="gmail-MJXc-Node-87" class="gmail-mjx-mo" style="display:inline-block;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-main-R" style="display:block;white-space:pre;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.472em;padding-bottom:0.613em;box-sizing:content-box">)</span></span><span id="gmail-MJXc-Node-88" class="gmail-mjx-mo gmail-MJXc-space3" style="display:inline-block;margin-left:0.278em;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-main-R" style="display:block;white-space:pre;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.096em;padding-bottom:0.331em;box-sizing:content-box">=</span></span><span id="gmail-MJXc-Node-89" class="gmail-mjx-mfrac gmail-MJXc-space3" style="display:inline-block;margin-left:0.278em;box-sizing:content-box"><span class="gmail-mjx-box gmail-MJXc-stacked" style="display:inline-block;height:0px;margin-top:0px;width:8.205em;padding:0px 0.12em;box-sizing:content-box"><span class="gmail-mjx-numerator" style="display:block;text-align:center;margin-top:0px;width:8.205em;box-sizing:content-box"><span id="gmail-MJXc-Node-90" class="gmail-mjx-mn" style="display:inline-block;text-align:left;margin-top:0px;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-main-R" style="display:block;white-space:pre;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.378em;padding-bottom:0.331em;box-sizing:content-box">1</span></span></span><span class="gmail-mjx-denominator" style="display:block;text-align:center;width:8.205em;box-sizing:content-box"><span id="gmail-MJXc-Node-91" class="gmail-mjx-mrow" style="display:inline-block;text-align:left;margin-top:0px;box-sizing:content-box"><span id="gmail-MJXc-Node-92" class="gmail-mjx-mn" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-main-R" style="display:block;white-space:pre;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.378em;padding-bottom:0.331em;box-sizing:content-box">1</span></span><span id="gmail-MJXc-Node-93" class="gmail-mjx-mo gmail-MJXc-space2" style="display:inline-block;margin-left:0.222em;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-main-R" style="display:block;white-space:pre;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.284em;padding-bottom:0.425em;box-sizing:content-box">−</span></span><span id="gmail-MJXc-Node-94" class="gmail-mjx-mo gmail-MJXc-space2" style="display:inline-block;margin-left:0.222em;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-main-R" style="display:block;white-space:pre;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.472em;padding-bottom:0.613em;box-sizing:content-box">(</span></span><span id="gmail-MJXc-Node-95" class="gmail-mjx-msubsup" style="display:inline-block;box-sizing:content-box"><span class="gmail-mjx-base" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-MJXc-Node-96" class="gmail-mjx-mi" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-math-I" style="display:block;white-space:pre;font-family:MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw;margin-top:0px;padding-top:0.237em;padding-bottom:0.284em;box-sizing:content-box">x</span></span></span><span class="gmail-mjx-sub" style="display:inline-block;vertical-align:-0.212em;padding-right:0.071em;box-sizing:content-box"><span id="gmail-MJXc-Node-97" class="gmail-mjx-texatom" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-MJXc-Node-98" class="gmail-mjx-mrow" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-MJXc-Node-99" class="gmail-mjx-mn" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-main-R" style="display:block;white-space:pre;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.378em;padding-bottom:0.331em;box-sizing:content-box">1</span></span></span></span></span></span><span id="gmail-MJXc-Node-100" class="gmail-mjx-mo gmail-MJXc-space2" style="display:inline-block;margin-left:0.222em;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-main-R" style="display:block;white-space:pre;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.284em;padding-bottom:0.425em;box-sizing:content-box">+</span></span><span id="gmail-MJXc-Node-101" class="gmail-mjx-mo gmail-MJXc-space2" style="display:inline-block;margin-left:0.222em;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-main-R" style="display:block;white-space:pre;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.002em;padding-bottom:0.331em;box-sizing:content-box">⋯</span></span><span id="gmail-MJXc-Node-102" class="gmail-mjx-mo gmail-MJXc-space2" style="display:inline-block;margin-left:0.222em;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-main-R" style="display:block;white-space:pre;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.284em;padding-bottom:0.425em;box-sizing:content-box">+</span></span><span id="gmail-MJXc-Node-103" class="gmail-mjx-msubsup gmail-MJXc-space2" style="display:inline-block;margin-left:0.222em;box-sizing:content-box"><span class="gmail-mjx-base" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-MJXc-Node-104" class="gmail-mjx-mi" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-math-I" style="display:block;white-space:pre;font-family:MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw;margin-top:0px;padding-top:0.237em;padding-bottom:0.284em;box-sizing:content-box">x</span></span></span><span class="gmail-mjx-sub" style="display:inline-block;vertical-align:-0.212em;padding-right:0.071em;box-sizing:content-box"><span id="gmail-MJXc-Node-105" class="gmail-mjx-texatom" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-MJXc-Node-106" class="gmail-mjx-mrow" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-MJXc-Node-107" class="gmail-mjx-mi" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-math-I" style="display:block;white-space:pre;font-family:MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw;margin-top:0px;padding-top:0.237em;padding-bottom:0.284em;box-sizing:content-box">r</span></span></span></span></span></span><span id="gmail-MJXc-Node-108" class="gmail-mjx-mo" style="display:inline-block;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-main-R" style="display:block;white-space:pre;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.472em;padding-bottom:0.613em;box-sizing:content-box">)</span></span></span></span><span class="gmail-mjx-line" style="display:block;height:0px;border-bottom:1.3px solid;width:8.205em;box-sizing:content-box"></span></span><span class="gmail-mjx-vsize" style="display:inline-block;width:0px;height:2.375em;vertical-align:-1.007em;box-sizing:content-box"></span></span><span id="gmail-MJXc-Node-109" class="gmail-mjx-mo" style="display:inline-block;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-main-R" style="display:block;white-space:pre;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;padding-bottom:0.331em;box-sizing:content-box">.</span></span></span></span><span class="gmail-MJX_Assistive_MathML gmail-MJX_Assistive_MathML_Block" style="box-sizing:border-box;width:370.25px;padding:1px 0px 0px;border:0px;height:1px;overflow:hidden;display:block">∏<span class="gmail-MJX-TeXAtom-ORD">k=0</span><span class="gmail-MJX-TeXAtom-ORD">∞</span>p<span class="gmail-MJX-TeXAtom-ORD">k</span>(x<span class="gmail-MJX-TeXAtom-ORD">1</span>,…,x<span class="gmail-MJX-TeXAtom-ORD">r</span>)=11−(x<span class="gmail-MJX-TeXAtom-ORD">1</span>+⋯+x<span class="gmail-MJX-TeXAtom-ORD">r</span>).</span></span></span>From this infinite product, we deduce lower bounds of the cardinality of <span id="gmail-MathJax-Element-4-Frame" class="gmail-mjx-chtml gmail-MathJax_CHTML" tabindex="0" style="box-sizing:border-box;display:inline-block;line-height:0;word-spacing:normal;white-space:nowrap;float:none;direction:ltr;max-width:none;max-height:none;min-width:0px;min-height:0px;border:0px;margin:0px;padding:1px 0px"><span id="gmail-MJXc-Node-110" class="gmail-mjx-math" style="box-sizing:border-box;display:inline-block;border-collapse:separate;border-spacing:0px;margin-top:0px"><span id="gmail-MJXc-Node-111" class="gmail-mjx-mrow" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-MJXc-Node-112" class="gmail-mjx-texatom" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-MJXc-Node-113" class="gmail-mjx-mrow" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-MJXc-Node-114" class="gmail-mjx-mo" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-main-R" style="display:block;white-space:pre;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.472em;padding-bottom:0.613em;box-sizing:content-box">|</span></span></span></span><span id="gmail-MJXc-Node-115" class="gmail-mjx-mo" style="display:inline-block;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-main-R" style="display:block;white-space:pre;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.472em;padding-bottom:0.613em;box-sizing:content-box">⟨</span></span><span id="gmail-MJXc-Node-116" class="gmail-mjx-msubsup" style="display:inline-block;box-sizing:content-box"><span class="gmail-mjx-base" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-MJXc-Node-117" class="gmail-mjx-mi" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-math-I" style="display:block;white-space:pre;font-family:MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw;margin-top:0px;padding-top:0.425em;padding-bottom:0.472em;box-sizing:content-box">j</span></span></span><span class="gmail-mjx-sub" style="display:inline-block;vertical-align:-0.212em;padding-right:0.071em;box-sizing:content-box"><span id="gmail-MJXc-Node-118" class="gmail-mjx-texatom" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-MJXc-Node-119" class="gmail-mjx-mrow" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-MJXc-Node-120" class="gmail-mjx-mn" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-main-R" style="display:block;white-space:pre;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.378em;padding-bottom:0.331em;box-sizing:content-box">1</span></span></span></span></span></span><span id="gmail-MJXc-Node-121" class="gmail-mjx-mo" style="display:inline-block;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-main-R" style="display:block;white-space:pre;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;padding-bottom:0.519em;box-sizing:content-box">,</span></span><span id="gmail-MJXc-Node-122" class="gmail-mjx-mo gmail-MJXc-space1" style="display:inline-block;margin-left:0.167em;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-main-R" style="display:block;white-space:pre;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;padding-bottom:0.331em;box-sizing:content-box">.</span></span><span id="gmail-MJXc-Node-123" class="gmail-mjx-mo gmail-MJXc-space1" style="display:inline-block;margin-left:0.167em;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-main-R" style="display:block;white-space:pre;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;padding-bottom:0.331em;box-sizing:content-box">.</span></span><span id="gmail-MJXc-Node-124" class="gmail-mjx-mo gmail-MJXc-space1" style="display:inline-block;margin-left:0.167em;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-main-R" style="display:block;white-space:pre;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;padding-bottom:0.331em;box-sizing:content-box">.</span></span><span id="gmail-MJXc-Node-125" class="gmail-mjx-mo gmail-MJXc-space1" style="display:inline-block;margin-left:0.167em;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-main-R" style="display:block;white-space:pre;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;padding-bottom:0.519em;box-sizing:content-box">,</span></span><span id="gmail-MJXc-Node-126" class="gmail-mjx-msubsup gmail-MJXc-space1" style="display:inline-block;margin-left:0.167em;box-sizing:content-box"><span class="gmail-mjx-base" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-MJXc-Node-127" class="gmail-mjx-mi" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-math-I" style="display:block;white-space:pre;font-family:MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw;margin-top:0px;padding-top:0.425em;padding-bottom:0.472em;box-sizing:content-box">j</span></span></span><span class="gmail-mjx-sub" style="display:inline-block;vertical-align:-0.212em;padding-right:0.071em;box-sizing:content-box"><span id="gmail-MJXc-Node-128" class="gmail-mjx-texatom" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-MJXc-Node-129" class="gmail-mjx-mrow" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-MJXc-Node-130" class="gmail-mjx-mi" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-math-I" style="display:block;white-space:pre;font-family:MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw;margin-top:0px;padding-top:0.237em;padding-bottom:0.284em;box-sizing:content-box">r</span></span></span></span></span></span><span id="gmail-MJXc-Node-131" class="gmail-mjx-mo" style="display:inline-block;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-main-R" style="display:block;white-space:pre;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.472em;padding-bottom:0.613em;box-sizing:content-box">⟩</span></span><span id="gmail-MJXc-Node-132" class="gmail-mjx-texatom" style="display:inline-block;box-sizing:content-box"><span id="gmail-MJXc-Node-133" class="gmail-mjx-mrow" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-MJXc-Node-134" class="gmail-mjx-mo" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-main-R" style="display:block;white-space:pre;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.472em;padding-bottom:0.613em;box-sizing:content-box">/</span></span></span></span><span id="gmail-MJXc-Node-135" class="gmail-mjx-msubsup gmail-MJXc-space3" style="display:inline-block;margin-left:0.278em;box-sizing:content-box"><span class="gmail-mjx-base" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-MJXc-Node-136" class="gmail-mjx-mo" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-main-R" style="display:block;white-space:pre;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.19em;padding-bottom:0.331em;box-sizing:content-box">≡</span></span></span><span class="gmail-mjx-sup" style="display:inline-block;vertical-align:0.513em;padding-left:0px;padding-right:0.071em;box-sizing:content-box"><span id="gmail-MJXc-Node-137" class="gmail-mjx-texatom" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-MJXc-Node-138" class="gmail-mjx-mrow" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-MJXc-Node-139" class="gmail-mjx-mi" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-math-I" style="display:block;white-space:pre;font-family:MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw;margin-top:0px;padding-top:0.237em;padding-bottom:0.284em;box-sizing:content-box">α</span></span></span></span></span></span><span id="gmail-MJXc-Node-140" class="gmail-mjx-texatom gmail-MJXc-space3" style="display:inline-block;margin-left:0.278em;box-sizing:content-box"><span id="gmail-MJXc-Node-141" class="gmail-mjx-mrow" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-MJXc-Node-142" class="gmail-mjx-mo" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-main-R" style="display:block;white-space:pre;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.472em;padding-bottom:0.613em;box-sizing:content-box">|</span></span></span></span></span></span><span class="gmail-MJX_Assistive_MathML" style="box-sizing:border-box;padding:1px 0px 0px;border:0px;height:1px;width:1px;overflow:hidden;display:block"><span class="gmail-MJX-TeXAtom-ORD">|</span>⟨j<span class="gmail-MJX-TeXAtom-ORD">1</span>,...,j<span class="gmail-MJX-TeXAtom-ORD">r</span>⟩<span class="gmail-MJX-TeXAtom-ORD">/</span>≡<span class="gmail-MJX-TeXAtom-ORD">α</span><span class="gmail-MJX-TeXAtom-ORD">|</span></span></span> using analysis and analytic number theoretic techniques. Computer calculations that search for Laver-like algebras give some empirical evidence that these lower bounds cannot be greatly improved.</p><p style="box-sizing:border-box;margin-bottom:1em;color:rgb(96,108,113);font-family:&quot;Open Sans&quot;,&quot;Helvetica Neue&quot;,Helvetica,Arial,sans-serif"></p><br class="gmail-Apple-interchange-newline"></div><div><font color="#000000" face="arial, sans-serif"><br></font></div><div><font color="#000000" face="arial, sans-serif"><br></font></div><div><font color="#000000" face="arial, sans-serif">Model Theory Seminar</font></div><div><font color="#000000" face="arial, sans-serif">CUNY Graduate Center, Room 6417</font></div><div><font color="#000000" face="arial, sans-serif">Friday, </font><span style="color:rgb(0,0,0);font-family:arial,sans-serif">May 3</span><font color="#000000" face="arial, sans-serif">, 12:30-2:00pm</font></div><div><font color="#000000" face="arial, sans-serif"><div>Artem Chernikov, UCLA</div><div>TBA</div></font></div><div><font color="#000000" face="arial, sans-serif"><br></font></div><div><br></div><div><font color="#000000" face="arial, sans-serif">Logic Workshop</font></div><div><font color="#000000" face="arial, sans-serif">CUNY Graduate Center, Room 6417</font></div><div><font color="#000000" face="arial, sans-serif">Friday, </font><span style="color:rgb(0,0,0);font-family:arial,sans-serif">May 3</span><font color="#000000" face="arial, sans-serif">, 2:00-3:30pm</font></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><p style="box-sizing:border-box;margin-bottom:1em;margin-top:0px;color:rgb(96,108,113);font-family:&quot;Open Sans&quot;,&quot;Helvetica Neue&quot;,Helvetica,Arial,sans-serif"><strong style="box-sizing:border-box"><a href="http://sites.math.rutgers.edu/~cherlin/" target="_blank" style="box-sizing:border-box;background-color:transparent;color:rgb(30,107,184);text-decoration-line:none;margin-top:0px">Gregory Cherlin</a></strong>, Rutgers University<br style="box-sizing:border-box"><strong style="box-sizing:border-box">Countable universal graphs with forbidden subgraphs</strong><br style="box-sizing:border-box"></p><p align="center" style="box-sizing:border-box;margin-bottom:1em;color:rgb(96,108,113);font-family:&quot;Open Sans&quot;,&quot;Helvetica Neue&quot;,Helvetica,Arial,sans-serif"><strong style="box-sizing:border-box;margin-top:0px">Abstract</strong></p><p style="box-sizing:border-box;margin-bottom:1em;color:rgb(96,108,113);font-family:&quot;Open Sans&quot;,&quot;Helvetica Neue&quot;,Helvetica,Arial,sans-serif"><strong style="box-sizing:border-box;margin-top:0px">Definition</strong>: A collection <span id="gmail-MathJax-Element-1-Frame" class="gmail-mjx-chtml gmail-MathJax_CHTML" tabindex="0" style="box-sizing:border-box;display:inline-block;line-height:0;word-spacing:normal;white-space:nowrap;float:none;direction:ltr;max-width:none;max-height:none;min-width:0px;min-height:0px;border:0px;margin:0px;padding:1px 0px"><span id="gmail-MJXc-Node-1" class="gmail-mjx-math" style="box-sizing:border-box;display:inline-block;border-collapse:separate;border-spacing:0px;margin-top:0px"><span id="gmail-MJXc-Node-2" class="gmail-mjx-mrow" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-MJXc-Node-3" class="gmail-mjx-texatom" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-MJXc-Node-4" class="gmail-mjx-mrow" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-MJXc-Node-5" class="gmail-mjx-mi" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-cal-R" style="display:block;white-space:pre;font-family:MJXc-TeX-cal-R,MJXc-TeX-cal-Rw;margin-top:0px;padding-top:0.425em;padding-bottom:0.331em;padding-right:0.11em;box-sizing:content-box">F</span></span></span></span></span></span><span class="gmail-MJX_Assistive_MathML" style="box-sizing:border-box;padding:1px 0px 0px;border:0px;height:1px;width:1px;overflow:hidden;display:block"><span class="gmail-MJX-TeXAtom-ORD"><span class="gmail-MJX-tex-caligraphic">F</span></span></span></span> of forbidden subgraphs is a <em style="box-sizing:border-box">Rado family</em> if the class of countable <span id="gmail-MathJax-Element-2-Frame" class="gmail-mjx-chtml gmail-MathJax_CHTML" tabindex="0" style="box-sizing:border-box;display:inline-block;line-height:0;word-spacing:normal;white-space:nowrap;float:none;direction:ltr;max-width:none;max-height:none;min-width:0px;min-height:0px;border:0px;margin:0px;padding:1px 0px"><span id="gmail-MJXc-Node-6" class="gmail-mjx-math" style="box-sizing:border-box;display:inline-block;border-collapse:separate;border-spacing:0px;margin-top:0px"><span id="gmail-MJXc-Node-7" class="gmail-mjx-mrow" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-MJXc-Node-8" class="gmail-mjx-texatom" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-MJXc-Node-9" class="gmail-mjx-mrow" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-MJXc-Node-10" class="gmail-mjx-mi" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-cal-R" style="display:block;white-space:pre;font-family:MJXc-TeX-cal-R,MJXc-TeX-cal-Rw;margin-top:0px;padding-top:0.425em;padding-bottom:0.331em;padding-right:0.11em;box-sizing:content-box">F</span></span></span></span></span></span><span class="gmail-MJX_Assistive_MathML" style="box-sizing:border-box;padding:1px 0px 0px;border:0px;height:1px;width:1px;overflow:hidden;display:block"><span class="gmail-MJX-TeXAtom-ORD"><span class="gmail-MJX-tex-caligraphic">F</span></span></span></span>-free graphs contains a universal structure with respect to embeddings as induced subgraphs. (By &#39;forbidden&#39; I mean forbidden as subgraphs; usage varies in the literature.)</p><p style="box-sizing:border-box;margin-bottom:1em;color:rgb(96,108,113);font-family:&quot;Open Sans&quot;,&quot;Helvetica Neue&quot;,Helvetica,Arial,sans-serif">We consider the following decision problem for Rado families.</p><p style="box-sizing:border-box;margin-bottom:1em;color:rgb(96,108,113);font-family:&quot;Open Sans&quot;,&quot;Helvetica Neue&quot;,Helvetica,Arial,sans-serif"><strong style="box-sizing:border-box;margin-top:0px">Problem</strong>: Given a finite set <span id="gmail-MathJax-Element-3-Frame" class="gmail-mjx-chtml gmail-MathJax_CHTML" tabindex="0" style="box-sizing:border-box;display:inline-block;line-height:0;word-spacing:normal;white-space:nowrap;float:none;direction:ltr;max-width:none;max-height:none;min-width:0px;min-height:0px;border:0px;margin:0px;padding:1px 0px"><span id="gmail-MJXc-Node-11" class="gmail-mjx-math" style="box-sizing:border-box;display:inline-block;border-collapse:separate;border-spacing:0px;margin-top:0px"><span id="gmail-MJXc-Node-12" class="gmail-mjx-mrow" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-MJXc-Node-13" class="gmail-mjx-texatom" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-MJXc-Node-14" class="gmail-mjx-mrow" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-MJXc-Node-15" class="gmail-mjx-mi" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-cal-R" style="display:block;white-space:pre;font-family:MJXc-TeX-cal-R,MJXc-TeX-cal-Rw;margin-top:0px;padding-top:0.425em;padding-bottom:0.331em;padding-right:0.11em;box-sizing:content-box">F</span></span></span></span></span></span><span class="gmail-MJX_Assistive_MathML" style="box-sizing:border-box;padding:1px 0px 0px;border:0px;height:1px;width:1px;overflow:hidden;display:block"><span class="gmail-MJX-TeXAtom-ORD"><span class="gmail-MJX-tex-caligraphic">F</span></span></span></span> of finite, connected graphs, determine whether or not it is a Rado family.</p><p style="box-sizing:border-box;margin-bottom:1em;color:rgb(96,108,113);font-family:&quot;Open Sans&quot;,&quot;Helvetica Neue&quot;,Helvetica,Arial,sans-serif"><strong style="box-sizing:border-box;margin-top:0px">Question</strong>: Is this a decidable problem?</p><p style="box-sizing:border-box;margin-bottom:1em;color:rgb(96,108,113);font-family:&quot;Open Sans&quot;,&quot;Helvetica Neue&quot;,Helvetica,Arial,sans-serif">If we specialize to the sake of a single constraint then we speak instead of a <em style="box-sizing:border-box;margin-top:0px">Rado constraint.</em> Much is known, and much more conjectured, about the case of a <em style="box-sizing:border-box">single</em> constraint.</p><p style="box-sizing:border-box;margin-bottom:1em;color:rgb(96,108,113);font-family:&quot;Open Sans&quot;,&quot;Helvetica Neue&quot;,Helvetica,Arial,sans-serif"><strong style="box-sizing:border-box;margin-top:0px">Conjecture</strong>: The decision problem for the case of a single constraint is decidable.</p><p align="center" style="box-sizing:border-box;margin-bottom:1em;color:rgb(96,108,113);font-family:&quot;Open Sans&quot;,&quot;Helvetica Neue&quot;,Helvetica,Arial,sans-serif"><strong style="box-sizing:border-box;margin-top:0px">Discussion</strong></p><p style="box-sizing:border-box;margin-bottom:1em;color:rgb(96,108,113);font-family:&quot;Open Sans&quot;,&quot;Helvetica Neue&quot;,Helvetica,Arial,sans-serif">From the model theoretic side, a finite collection of forbidden subgraphs specifies a universal theory with some special properties: notably, there is a <em style="box-sizing:border-box;margin-top:0px">model companion</em>. When we restrict to families of connected constraints, the theory has joint embedding and the model companion is complete. In model theoretic terms the question becomes whether the model companion has a countable universal model with respect to elementary embedding, for which a purely type theoretic criterion is known.</p><p style="box-sizing:border-box;margin-bottom:1em;color:rgb(96,108,113);font-family:&quot;Open Sans&quot;,&quot;Helvetica Neue&quot;,Helvetica,Arial,sans-serif">Decision problems for other model theoretic properties of the model companion are natural. We have focused on this one because the question came to us from graph theory. However, even in that form, two variants are relevant.</p><ul style="box-sizing:border-box;margin-top:0px;color:rgb(96,108,113);font-family:&quot;Open Sans&quot;,&quot;Helvetica Neue&quot;,Helvetica,Arial,sans-serif"><li style="box-sizing:border-box;margin-top:0px">Can we decide whether the model companion is <span id="gmail-MathJax-Element-4-Frame" class="gmail-mjx-chtml gmail-MathJax_CHTML" tabindex="0" style="box-sizing:border-box;display:inline-block;line-height:0;word-spacing:normal;white-space:nowrap;float:none;direction:ltr;max-width:none;max-height:none;min-width:0px;min-height:0px;border:0px;margin:0px;padding:1px 0px"><span id="gmail-MJXc-Node-16" class="gmail-mjx-math" style="box-sizing:border-box;display:inline-block;border-collapse:separate;border-spacing:0px;margin-top:0px"><span id="gmail-MJXc-Node-17" class="gmail-mjx-mrow" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-MJXc-Node-18" class="gmail-mjx-msubsup" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-mjx-base" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-MJXc-Node-19" class="gmail-mjx-mi" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-main-R" style="display:block;white-space:pre;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.425em;padding-bottom:0.331em;box-sizing:content-box">ℵ</span></span></span><span class="gmail-mjx-sub" style="display:inline-block;vertical-align:-0.212em;padding-right:0.071em;box-sizing:content-box"><span id="gmail-MJXc-Node-20" class="gmail-mjx-mn" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-main-R" style="display:block;white-space:pre;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.378em;padding-bottom:0.378em;box-sizing:content-box">0</span></span></span></span></span></span><span class="gmail-MJX_Assistive_MathML" style="box-sizing:border-box;padding:1px 0px 0px;border:0px;height:1px;width:1px;overflow:hidden;display:block">ℵ0</span></span>-categorical?</li><li style="box-sizing:border-box">Dropping the connectedness hypothesis on the constraints, can we decide whether the model companion is complete?</li></ul><p style="box-sizing:border-box;margin-bottom:1em;color:rgb(96,108,113);font-family:&quot;Open Sans&quot;,&quot;Helvetica Neue&quot;,Helvetica,Arial,sans-serif"></p><p style="box-sizing:border-box;margin-bottom:1em;color:rgb(96,108,113);font-family:&quot;Open Sans&quot;,&quot;Helvetica Neue&quot;,Helvetica,Arial,sans-serif">The main source of positive answers to the universality problem appears to be the <span id="gmail-MathJax-Element-5-Frame" class="gmail-mjx-chtml gmail-MathJax_CHTML" tabindex="0" style="box-sizing:border-box;display:inline-block;line-height:0;word-spacing:normal;white-space:nowrap;float:none;direction:ltr;max-width:none;max-height:none;min-width:0px;min-height:0px;border:0px;margin:0px;padding:1px 0px"><span id="gmail-MJXc-Node-21" class="gmail-mjx-math" style="box-sizing:border-box;display:inline-block;border-collapse:separate;border-spacing:0px;margin-top:0px"><span id="gmail-MJXc-Node-22" class="gmail-mjx-mrow" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-MJXc-Node-23" class="gmail-mjx-msubsup" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-mjx-base" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-MJXc-Node-24" class="gmail-mjx-mi" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-main-R" style="display:block;white-space:pre;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.425em;padding-bottom:0.331em;box-sizing:content-box">ℵ</span></span></span><span class="gmail-mjx-sub" style="display:inline-block;vertical-align:-0.212em;padding-right:0.071em;box-sizing:content-box"><span id="gmail-MJXc-Node-25" class="gmail-mjx-mn" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-main-R" style="display:block;white-space:pre;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.378em;padding-bottom:0.378em;box-sizing:content-box">0</span></span></span></span></span></span><span class="gmail-MJX_Assistive_MathML" style="box-sizing:border-box;padding:1px 0px 0px;border:0px;height:1px;width:1px;overflow:hidden;display:block">ℵ0</span></span>-categorical case, and this has some theoretical justification. The most direct route to understanding the original graph theoretical problem appears to take the detour through <span id="gmail-MathJax-Element-6-Frame" class="gmail-mjx-chtml gmail-MathJax_CHTML" tabindex="0" style="box-sizing:border-box;display:inline-block;line-height:0;word-spacing:normal;white-space:nowrap;float:none;direction:ltr;max-width:none;max-height:none;min-width:0px;min-height:0px;border:0px;margin:0px;padding:1px 0px"><span id="gmail-MJXc-Node-26" class="gmail-mjx-math" style="box-sizing:border-box;display:inline-block;border-collapse:separate;border-spacing:0px;margin-top:0px"><span id="gmail-MJXc-Node-27" class="gmail-mjx-mrow" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-MJXc-Node-28" class="gmail-mjx-msubsup" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-mjx-base" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-MJXc-Node-29" class="gmail-mjx-mi" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-main-R" style="display:block;white-space:pre;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.425em;padding-bottom:0.331em;box-sizing:content-box">ℵ</span></span></span><span class="gmail-mjx-sub" style="display:inline-block;vertical-align:-0.212em;padding-right:0.071em;box-sizing:content-box"><span id="gmail-MJXc-Node-30" class="gmail-mjx-mn" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-main-R" style="display:block;white-space:pre;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.378em;padding-bottom:0.378em;box-sizing:content-box">0</span></span></span></span></span></span><span class="gmail-MJX_Assistive_MathML" style="box-sizing:border-box;padding:1px 0px 0px;border:0px;height:1px;width:1px;overflow:hidden;display:block">ℵ0</span></span>-categoricity.</p><p style="box-sizing:border-box;margin-bottom:1em;color:rgb(96,108,113);font-family:&quot;Open Sans&quot;,&quot;Helvetica Neue&quot;,Helvetica,Arial,sans-serif">As time allows, I would like to discuss the following three points.</p><ul style="box-sizing:border-box;margin-top:0px;color:rgb(96,108,113);font-family:&quot;Open Sans&quot;,&quot;Helvetica Neue&quot;,Helvetica,Arial,sans-serif"><li style="box-sizing:border-box;margin-top:0px">What we know, and what we expect, in the case of a single constraint;</li><li style="box-sizing:border-box">Methods of proof (<em style="box-sizing:border-box;margin-top:0px">algebraic closure; induction by pruning</em>);</li><li style="box-sizing:border-box">The status of the decision problem for j.e.p. and its analog in the theory of permutation pattern classes (work of Braunfeld).</li></ul><p style="box-sizing:border-box;margin-bottom:1em;color:rgb(96,108,113);font-family:&quot;Open Sans&quot;,&quot;Helvetica Neue&quot;,Helvetica,Arial,sans-serif"></p><p style="box-sizing:border-box;margin-bottom:1em;color:rgb(96,108,113);font-family:&quot;Open Sans&quot;,&quot;Helvetica Neue&quot;,Helvetica,Arial,sans-serif">This is joint work with Shelah, e.g. [Sh689].</p></div></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><br></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><br></div></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><font color="#212121">- - - - Other Logic News - - - -</font></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><br></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><b>Conference announcement:</b></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><span style="color:rgb(34,34,34);font-family:Arial,Helvetica,sans-serif">Model Theory and Mathematical Logic,</span><br style="color:rgb(34,34,34);font-family:Arial,Helvetica,sans-serif"><span style="color:rgb(34,34,34);font-family:Arial,Helvetica,sans-serif"> in honor of Chris Laskowski&#39;s 60th birthday,</span><br style="color:rgb(34,34,34);font-family:Arial,Helvetica,sans-serif"><span style="color:rgb(34,34,34);font-family:Arial,Helvetica,sans-serif"> June 21 - 23, 2019, at The University of Maryland, College Park</span><br style="color:rgb(34,34,34);font-family:Arial,Helvetica,sans-serif"><a href="http://www.umdlogic2019.com/" rel="noreferrer" target="_blank" style="font-family:Arial,Helvetica,sans-serif">http://www.umdlogic2019.com/</a><br></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><font color="#212121"><br></font></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><font color="#212121"><br></font></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><font color="#212121">- - - - Web Site - - - -</font></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><font color="#212121"><br></font></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><font color="#212121">Our new website is located at:  <a href="https://nylogic.github.io/" target="_blank" style="text-decoration-line:none">https://nylogic.github.io/</a></font></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><font color="#212121">Many thanks to Victoria Gitman for her development work!</font></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><font color="#212121"><br></font></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><font color="#212121">--------  ADMINISTRIVIA  --------</font></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><font color="#212121"><br></font></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><font color="#212121">To subscribe/unsubscribe to this list, please email your request to <a href="mailto:jreitz@nylogic.org" target="_blank" style="text-decoration-line:none">jreitz@nylogic.org</a>.</font></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><font color="#212121"><br></font></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><font color="#212121">If you have a logic-related event that you would like included in future mailings, please email <a href="mailto:jreitz@nylogic.org" target="_blank" style="text-decoration-line:none">jreitz@nylogic.org</a>. </font></div></div></div></div></div></div></div></div>
44
4/18/2019 10:25:39Prague Set Theory SeminarWednesday seminarDear all,

The seminar meets on Wednesday April 24th at 11:00 in the Institute of
Mathematics CAS, Zitna 25, seminar room, 3rd floor, front building.

There is no fixed program yet. If nobody is willing to talk about
something, I may have a backup topic (probably reviewing some results
about forcing axioms and weak square principles).

Best,
David
45
4/17/2019 0:00:00ConferenceYoung Set Theory Workshop and European Set Theory Conference - early registration extendedDear All,

We have extended our early registration period till May 15 for both the Young Set Theory Workshop and the European Set Theory Conference. Please follow the links below and note that the registration is separate for the two meetings. There is limited financial support available for young researchers through the ASL and NSF; please see the websites.

Details: European Set Theory Conference

If you indicated to give a contributed lecture at the ESTC, please submit your abstract as soon as possible (you can use the code received at registration to edit your original registration and submit an abstract).

Details: Advanced Class 2019 (Young Set Theory Workshop)

Please upload a short research statement with your registration to the YSTW (or if you registered already, please add this statement using the code you received).

We are very much looking forward to seeing you in Vienna.
https://sites.google.com/view/estc2019/estc-2019/registration
46
4/14/2019 21:02:48This Week in Logic at CUNYThis Week in Logic at CUNY<div dir="ltr"><div style="font-family:arial,sans-serif;color:rgb(33,33,33)"><span style="color:rgb(0,0,0)">Hi everyone,</span></div><div style="font-family:arial,sans-serif;color:rgb(33,33,33)"><span style="color:rgb(0,0,0)"><br></span></div><div style="font-family:arial,sans-serif;color:rgb(33,33,33)"><span style="color:rgb(0,0,0)">Heads up, CUNY Spring Break takes place starting this Friday, April 19-28.  Regular mailings of &quot;This Week in Logic&quot; will resume on Sunday, April 28.</span><br></div><div style="font-family:arial,sans-serif;color:rgb(33,33,33)"><span style="color:rgb(0,0,0)"><br></span></div><div style="font-family:arial,sans-serif;color:rgb(33,33,33)"><span style="color:rgb(0,0,0)">All the best,</span></div><div style="font-family:arial,sans-serif;color:rgb(33,33,33)"><span style="color:rgb(0,0,0)">Jonas</span></div><div style="font-family:arial,sans-serif;color:rgb(33,33,33)"><span style="color:rgb(0,0,0)"><br></span></div><div style="font-family:arial,sans-serif;color:rgb(33,33,33)"><span style="color:rgb(0,0,0)"><br></span></div><div style="font-family:arial,sans-serif;color:rgb(33,33,33)"><br></div><div style="font-family:arial,sans-serif;color:rgb(33,33,33)">This Week in Logic at CUNY:</div><div style="font-family:arial,sans-serif;color:rgb(33,33,33)"><br></div><div><div><font color="#212121"><div><font face="arial, sans-serif">- - - - Monday, Apr 15, 2019 - - - -</font></div><div><font face="arial, sans-serif"><br></font></div><div><font face="arial, sans-serif"><div>Logic and Metaphysics Workshop</div><div>Date: Monday, April 15th, 4.15-6.15  <br></div><div>Place: Room 7314, CUNY Graduate Center <br></div><div>Speaker: Jenn McDonald (CUNY)<br></div><div>Title: Structural Counterfactuals and the Importation Problem<br></div><div><br></div><div>Abstract: Structural causal models lend themselves to an analysis of counterfactuals – a structural semantics of counterfactuals. The basic idea is that a causal model allows for the clear and precise evaluation of any counterfactual encoded by it. Many argue that a structural semantics is superior to a more traditional similarity semantics, in part due to the latter’s independence from any notion of similarity(Galles &amp; Pearl, 1998; Gallow, 2016; Hiddleston, 2005; Hitchcock, 2018; Pearl, 2000; Starr, 2019). I argue, though, that this is too quick. A similarity semantics employs the notion of similarity to answer what Priest (2018) calls the importation problem– the question of what information is to be held fixed in a counterfactual evaluation. I argue that where similarity semantics relies on an unarticulated notion of similarity, a structural semantics relies on an unarticulated notion of aptness. The superiority of structural semantics depends on its ability to deliver on a principled guide to determining which model(s) is apt. In this talk I go some way towards providing this guide.</div><div><br></div><div><br></div><div><br></div></font></div><div><font face="arial, sans-serif"><br></font></div><div><font face="arial, sans-serif">- - - - Tuesday, Apr 16, 2019 - - - -</font></div><div><font face="arial, sans-serif"><br></font></div><div><font face="arial, sans-serif"><br></font></div><div><font face="arial, sans-serif"><br></font></div><div><font face="arial, sans-serif">- - - - Wednesday, Apr 17, 2019 - - - -</font></div><div><font face="arial, sans-serif"><br></font></div><div><font face="arial, sans-serif"><div>The New York City Category Theory Seminar</div><div>Department of Computer Science</div><div>Department of Mathematics </div><div>The Graduate Center of The City University of New York </div><div>Speaker:     Tibor Beke, University of Massachusetts, Lowell.</div><div>Date and Time:     Wednesday April 17, 2019, 7:00 - 8:30 PM., Room 6417.<br></div><div>Title:    Schanuel functors and the Grothendieck (semi)ring of some theories.<br></div><div><br></div><div>Abstract: In a little known article, Schanuel defines a functor from semirings to idempotent semirings and a notion of dimension that is not linearly ordered. He uses it to give an elegant presentation of the Grothendieck semiring of semi-algebraic sets, from which the (much better known) structure of the Grothendieck ring of semi-algebraic sets easily follows. I will review his work and related results on the Grothendieck semiring of algebraically closed fields and similar geometric structures.</div><div><br></div><div><br></div><div><br></div></font></div><div><font face="arial, sans-serif"><br></font></div><div><font face="arial, sans-serif">- - - - Thursday, Apr 18, 2019 - - - -</font></div><div><font face="arial, sans-serif"><br></font></div><div><font face="arial, sans-serif"><br></font></div><div><font face="arial, sans-serif"><br></font></div><div><font face="arial, sans-serif">- - - - Friday, Apr 19, 2019 - - - -</font></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><br></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif">*** NOTE: CUNY Spring Break April 19-28 ***</div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><br></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><br></div></font></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><font color="#212121">- - - - Other Logic News - - - -</font></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><br></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><font color="#212121"><br></font></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><font color="#212121"><br></font></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><font color="#212121">- - - - Web Site - - - -</font></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><font color="#212121"><br></font></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><font color="#212121">Our new website is located at:  <a href="https://nylogic.github.io/" target="_blank" style="text-decoration-line:none">https://nylogic.github.io/</a></font></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><font color="#212121">Many thanks to Victoria Gitman for her development work!</font></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><font color="#212121"><br></font></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><font color="#212121">--------  ADMINISTRIVIA  --------</font></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><font color="#212121"><br></font></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><font color="#212121">To subscribe/unsubscribe to this list, please email your request to <a href="mailto:jreitz@nylogic.org" target="_blank" style="text-decoration-line:none">jreitz@nylogic.org</a>.</font></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><font color="#212121"><br></font></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><font color="#212121">If you have a logic-related event that you would like included in future mailings, please email <a href="mailto:jreitz@nylogic.org" target="_blank" style="text-decoration-line:none">jreitz@nylogic.org</a>. </font></div></div></div>
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4/14/2019 3:21:39Barcelona Set Theory SeminarRe: Set Theory Seminar<html><head><meta http-equiv="Content-Type" content="text/html charset=us-ascii"></head><body style="word-wrap: break-word; -webkit-nbsp-mode: space; -webkit-line-break: after-white-space;" class="">Dear Colleagues,<br class=""><div><div style="word-wrap: break-word; -webkit-nbsp-mode: space; -webkit-line-break: after-white-space;" class=""><div class=""><div class=""><div style="word-wrap: break-word; -webkit-nbsp-mode: space; -webkit-line-break: after-white-space;" class=""><div class=""><div class=""><div style="word-wrap: break-word; -webkit-nbsp-mode: space; -webkit-line-break: after-white-space;" class=""><div class=""><div class=""><div style="word-wrap: break-word; -webkit-nbsp-mode: space; -webkit-line-break: after-white-space;" class=""><div class=""><div style="word-wrap: break-word; -webkit-nbsp-mode: space; -webkit-line-break: after-white-space;" class=""><div class=""><div class=""><div style="word-wrap: break-word; -webkit-nbsp-mode: space; -webkit-line-break: after-white-space;" class=""><div class=""><div class=""><div style="word-wrap: break-word; -webkit-nbsp-mode: space; -webkit-line-break: after-white-space;" class=""><div class=""><div class=""><div style="word-wrap: break-word; -webkit-nbsp-mode: space; -webkit-line-break: after-white-space;" class=""><div class=""><div class=""><div style="word-wrap: break-word; -webkit-nbsp-mode: space; -webkit-line-break: after-white-space;" class=""><div class=""><div class=""><div style="word-wrap: break-word; -webkit-nbsp-mode: space; -webkit-line-break: after-white-space;" class=""><div class=""><div style="word-wrap: break-word; -webkit-nbsp-mode: space; -webkit-line-break: after-white-space;" class=""><div class=""><div style="word-wrap: break-word; -webkit-nbsp-mode: space; -webkit-line-break: after-white-space;" class=""><div class=""><div class=""><div style="word-wrap: break-word; -webkit-nbsp-mode: space; -webkit-line-break: after-white-space;" class=""><div class=""><div style="word-wrap: break-word; -webkit-nbsp-mode: space; -webkit-line-break: after-white-space;" class=""><div class=""><div class=""><div style="word-wrap: break-word; -webkit-nbsp-mode: space; -webkit-line-break: after-white-space;" class=""><div class=""><div class=""><div style="word-wrap: break-word; -webkit-nbsp-mode: space; -webkit-line-break: after-white-space;" class=""><div class=""><div class=""><div style="word-wrap: break-word; -webkit-nbsp-mode: space; -webkit-line-break: after-white-space;" class=""><div class="">Please find attached the announcement of the next session of the Barcelona Set Theory seminar.</div><div class="">You are all welcome to attend.&nbsp;</div><div class="">Best regards,</div><div class="">Joan</div><div class=""><br class=""></div><div class=""></div></div></div></div></div></div></div></div></div></div></div></div></div></div></div></div></div></div></div></div></div></div></div></div></div></div></div></div></div></div></div></div></div></div></div><div style="word-wrap: break-word; -webkit-nbsp-mode: space; -webkit-line-break: after-white-space;" class=""><div class=""><div style="letter-spacing: normal; orphans: auto; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; widows: auto; word-spacing: 0px; -webkit-text-stroke-width: 0px; word-wrap: break-word; -webkit-nbsp-mode: space; -webkit-line-break: after-white-space;" class=""><span class="Apple-style-span" style="border-collapse: separate; font-variant-ligatures: normal; font-variant-position: normal; font-variant-numeric: normal; font-variant-alternates: normal; font-variant-east-asian: normal; line-height: normal; border-spacing: 0px;"><div style="word-wrap: break-word; -webkit-nbsp-mode: space; -webkit-line-break: after-white-space;" class=""></div></span></div></div></div></div></div><div style="word-wrap: break-word; -webkit-nbsp-mode: space; -webkit-line-break: after-white-space;" class=""><br class=""></div><div style="word-wrap: break-word; -webkit-nbsp-mode: space; -webkit-line-break: after-white-space;" class=""></div></div></div></div></div></div></div><div style="word-wrap: break-word; -webkit-nbsp-mode: space; -webkit-line-break: after-white-space;" class=""><div class=""><div style="letter-spacing: normal; orphans: auto; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; widows: auto; word-spacing: 0px; -webkit-text-stroke-width: 0px; word-wrap: break-word; -webkit-nbsp-mode: space; -webkit-line-break: after-white-space;" class=""><span class="Apple-style-span" style="border-collapse: separate; font-family: Helvetica; font-style: normal; font-variant-ligatures: normal; font-variant-position: normal; font-variant-caps: normal; font-variant-numeric: normal; font-variant-alternates: normal; font-variant-east-asian: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: -webkit-auto; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; border-spacing: 0px; -webkit-text-decorations-in-effect: none; -webkit-text-stroke-width: 0px;"><div style="word-wrap: break-word; -webkit-nbsp-mode: space; -webkit-line-break: after-white-space;" class=""><span class="Apple-style-span" style="border-collapse: separate; font-family: Helvetica; font-style: normal; font-variant-ligatures: normal; font-variant-position: normal; font-variant-caps: normal; font-variant-numeric: normal; font-variant-alternates: normal; font-variant-east-asian: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: -webkit-auto; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; border-spacing: 0px; -webkit-text-decorations-in-effect: none; -webkit-text-stroke-width: 0px;"><div style="word-wrap: break-word; -webkit-nbsp-mode: space; -webkit-line-break: after-white-space;" class=""></div></span></div></span></div></div></div></div></div></div></div></body></html>
<html><head><meta http-equiv="Content-Type" content="text/html charset=utf-8"></head><body style="word-wrap: break-word; -webkit-nbsp-mode: space; -webkit-line-break: after-white-space;" class=""><div><br class=""><meta http-equiv="Content-Type" content="text/html charset=utf-8" class=""><div style="word-wrap: break-word; -webkit-nbsp-mode: space; -webkit-line-break: after-white-space;" class=""><div class=""><div class=""><div style="word-wrap: break-word; -webkit-nbsp-mode: space; -webkit-line-break: after-white-space;" class=""><div class=""><div style="letter-spacing: normal; orphans: auto; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; widows: auto; word-spacing: 0px; -webkit-text-stroke-width: 0px; word-wrap: break-word; -webkit-nbsp-mode: space; -webkit-line-break: after-white-space;" class=""><span class="Apple-style-span" style="border-collapse: separate; font-variant-ligatures: normal; font-variant-position: normal; font-variant-numeric: normal; font-variant-alternates: normal; font-variant-east-asian: normal; line-height: normal; border-spacing: 0px;"><div style="word-wrap: break-word; -webkit-nbsp-mode: space; -webkit-line-break: after-white-space;" class=""><span class="Apple-style-span" style="border-collapse: separate; font-family: Helvetica; font-style: normal; font-variant-ligatures: normal; font-variant-position: normal; font-variant-caps: normal; font-variant-numeric: normal; font-variant-alternates: normal; font-variant-east-asian: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: -webkit-auto; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; border-spacing: 0px; -webkit-text-decorations-in-effect: none; -webkit-text-stroke-width: 0px;"><div style="word-wrap: break-word; -webkit-nbsp-mode: space; -webkit-line-break: after-white-space;" class=""></div></span><br class="Apple-interchange-newline"></div></span><br class="Apple-interchange-newline"></div><br class="Apple-interchange-newline"><br class=""></div></div></div></div></div></div><br class=""><div class="">
<div style="color: rgb(0, 0, 0); letter-spacing: normal; orphans: auto; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; widows: auto; word-spacing: 0px; -webkit-text-stroke-width: 0px; word-wrap: break-word; -webkit-nbsp-mode: space; -webkit-line-break: after-white-space;" class=""><span class="Apple-style-span" style="border-collapse: separate; color: rgb(0, 0, 0); font-family: Helvetica; font-style: normal; font-variant-ligatures: normal; font-variant-position: normal; font-variant-caps: normal; font-variant-numeric: normal; font-variant-alternates: normal; font-variant-east-asian: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: -webkit-auto; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; border-spacing: 0px; -webkit-text-decorations-in-effect: none; -webkit-text-stroke-width: 0px;"><div style="word-wrap: break-word; -webkit-nbsp-mode: space; -webkit-line-break: after-white-space;" class=""><span class="Apple-style-span" style="border-collapse: separate; color: rgb(0, 0, 0); font-family: Helvetica; font-style: normal; font-variant-ligatures: normal; font-variant-position: normal; font-variant-caps: normal; font-variant-numeric: normal; font-variant-alternates: normal; font-variant-east-asian: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: -webkit-auto; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; border-spacing: 0px; -webkit-text-decorations-in-effect: none; -webkit-text-stroke-width: 0px;"><div style="word-wrap: break-word; -webkit-nbsp-mode: space; -webkit-line-break: after-white-space;" class=""><span class="Apple-style-span" style="border-collapse: separate; color: rgb(0, 0, 0); font-family: Helvetica; font-size: 12px; font-style: normal; font-variant-ligatures: normal; font-variant-position: normal; font-variant-caps: normal; font-variant-numeric: normal; font-variant-alternates: normal; font-variant-east-asian: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; border-spacing: 0px; -webkit-text-decorations-in-effect: none; -webkit-text-stroke-width: 0px;"><div style="word-wrap: break-word; -webkit-nbsp-mode: space; -webkit-line-break: after-white-space;" class=""><span class="Apple-style-span" style="border-collapse: separate; color: rgb(0, 0, 0); font-family: Helvetica; font-size: 12px; font-style: normal; font-variant-ligatures: normal; font-variant-position: normal; font-variant-caps: normal; font-variant-numeric: normal; font-variant-alternates: normal; font-variant-east-asian: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; border-spacing: 0px; -webkit-text-decorations-in-effect: none; -webkit-text-stroke-width: 0px;"><div style="word-wrap: break-word; -webkit-nbsp-mode: space; -webkit-line-break: after-white-space;" class=""><div class=""><div class="">Joan Bagaria&nbsp;🎗</div><div class="">ICREA Research Professor&nbsp;</div><div class="">Universitat de Barcelona</div><div class="">Departament de Matemàtiques i Informàtica</div><div class="">Gran Via de les Corts Catalanes 585</div><div class="">08007 Barcelona</div><div class="">Catalonia&nbsp;</div><div class=""><br class=""></div><div class="">Phone: +34 93 402 1609</div><div class=""><a href="mailto:joan.bagaria@icrea.cat" class="">joan.bagaria@icrea.cat</a></div><div class=""><a href="mailto:bagaria@ub.edu" class="">bagaria@ub.edu</a></div><div class=""><br class=""></div><div class=""><br class=""></div></div><br class=""></div></span><br class="Apple-interchange-newline"></div></span><br class="Apple-interchange-newline"></div></span><br class="Apple-interchange-newline"></div></span><br class="Apple-interchange-newline"></div><br class="Apple-interchange-newline"><br class="Apple-interchange-newline">
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4/12/2019 9:39:02IMPAN Working Group in Applications of Set TheoryFulgencio Lopez: Compact extensions of first order logicSeminar: Working group in applications of set theory, IMPAN

NOTE UNUSUAL TIME AND PLACE:

Tuesday, 16.04.2019, 10:15, room 106, IMPAN

Speaker: Fulgencio Lopez (IM PAN)

Title: "Compact extensions of first order logic"

Abstact: "We aim to provide an exhaustive proof of the results of Keisler and Magidor and Malitz about the compactness of certain extensions of first order logic. Keisler's Theorem refers to the extension L(Q) where Q is the quantifier "There is an uncountable subset with a one dimensional property", similarly we can define Q_n to be "There is an uncountable subset with an n-dimensional property". We will show that L(Q) is compact (Keisler's Theorem) and, assuming diamond, so is L(Q_n:n∈ N) (Magidor and Malitz). We will also discuss why this framework can sometimes be useful for constructions of set theoretical structures".

Until May 9 the meetings of the seminar will take place at unusual times due to holidays and to the scientific Council of the Institute

Visit our seminar page which may include some future talks at https://www.impan.pl/~set_theory/Seminar/
Fulgencio Lopez
49
4/12/2019 6:51:29This Week in Logic at CUNYUPDATE: This Week in Logic at CUNY<div dir="ltr"><div style="font-family:arial,sans-serif;color:rgb(33,33,33)">Hi everyone,</div><div style="font-family:arial,sans-serif;color:rgb(33,33,33)"><br></div><div style="font-family:arial,sans-serif;color:rgb(33,33,33)">This morning&#39;s Set Theory Seminar talk is cancelled, due to illness.</div><div style="font-family:arial,sans-serif;color:rgb(33,33,33)"><br></div><div style="font-family:arial,sans-serif;color:rgb(33,33,33)">Regrets,</div><div style="font-family:arial,sans-serif;color:rgb(33,33,33)">Jonas</div><div style="font-family:arial,sans-serif;color:rgb(33,33,33)"><br></div><div style="font-family:arial,sans-serif;color:rgb(33,33,33)"><br></div><div style="font-family:arial,sans-serif;color:rgb(33,33,33)">This Week in Logic at CUNY:</div><div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><div><br></div></div><div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><font color="#212121">- - - - Monday, Apr 8, 2019 - - - -</font></div><div><font color="#212121" face="arial, sans-serif"><div><br></div><div>Logic and Metaphysics Workshop</div><div>Date: Monday, April 8th, 4.15-6.15  </div><div>Place: Room 7314, CUNY Graduate Center </div><div>Speaker: Chris Scambler (NYU)</div><div>Title: Classical Logic and the Strict Tolerant Hierarchy</div><div><br></div><div>Abstract: In this talk I will do three things. First: I will present the central results from Barrio, Pailos and Szmuc’s recent paper “A hierarchy of classical and paraconsistent logics” (forthcoming in the JPL) along with some generalizations derived by observing certain symmetries; second, I will discuss the relation between the strict tolerant logics and classical logic, K3 and LP; third, I will try to convey the exact state of uncertainty about the philosophical significance of the foregoing I find myself in on the day.</div><div><br></div><div><br></div><div> </div></font></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><font color="#212121"><br></font></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><font color="#212121"><br></font></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><font color="#212121">- - - - Tuesday, Apr 9, 2019 - - - -</font></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><font color="#212121"><br></font></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><font color="#212121"><br></font></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><font color="#212121"><br></font></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><font color="#212121">- - - - Wednesday, Apr 10, 2019 - - - -</font></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><font color="#212121"><br></font></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><font color="#212121"><div style="color:rgb(0,0,0)">MOPA (Models of Peano Arithmetic)</div><div style="color:rgb(0,0,0)">CUNY Graduate Center, Room 4213.03 (Math Thesis Room)</div><div style="color:rgb(0,0,0)">Wednesday, April 10, 6:30-8:00pm</div><div style="color:rgb(0,0,0)">Erez Shochat, St. Francis College</div><div style="color:rgb(0,0,0)">Introduction to Loeb Measure</div><div style="color:rgb(0,0,0)"><br></div><div style="color:rgb(0,0,0)">In this talk we will outline results and facts from nonstandard analysis and introduce the concept of Loeb Measure.</div></font></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><font color="#212121"><br></font></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><font color="#212121"><br></font></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><font color="#212121"><br></font></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><font color="#212121"><br></font></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><font color="#212121">- - - - Thursday, Apr 11, 2019 - - - -</font></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><font color="#212121"><br></font></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><font color="#212121"><br></font></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><font color="#212121"><br></font></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><font color="#212121">- - - - Friday, Apr 12, 2019 - - - -</font></div></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><font color="#212121"><br></font></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><div style="color:rgb(33,33,33)">Set Theory Seminar</div><div style="color:rgb(33,33,33)">CUNY Graduate Center, Room 6417</div><div><font color="#000000">Friday, April 12, 10:00-11:45am</font></div><div><font color="#000000"><span style="font-family:&quot;Open Sans&quot;,&quot;Helvetica Neue&quot;,Helvetica,Arial,sans-serif">Jonas Reitz, CUNY</span><br style="box-sizing:border-box;font-family:&quot;Open Sans&quot;,&quot;Helvetica Neue&quot;,Helvetica,Arial,sans-serif"></font><div><font face="Open Sans, Helvetica Neue, Helvetica, Arial, sans-serif" color="#000000"><b>Generalized Cohen Iterations</b></font></div><div><font face="Open Sans, Helvetica Neue, Helvetica, Arial, sans-serif" color="#000000"><br></font></div><div><font face="Open Sans, Helvetica Neue, Helvetica, Arial, sans-serif" color="#000000">Adding Cohen subsets to each of a class of cardinals in turn is a common construction in set theory, and underlies many fundamental results.  The construction comes in two basic flavors, products (as in Easton’s Theorem on the powers of regular cardinals) and iterations (forcing the GCH).  These flavors are  apparently quite similar, forcing at stage kappa to add subsets via the Cohen partial order Add(kappa,lambda).  They differ only in the universe over which Add(kappa,lambda) is defined - in the case of products the ground model poset is used at each stage, whereas in typical iterations the poset is taken from the partial extension up to kappa.  In this talk I will consider an alternative, in which we allow Add(kappa,lambda) to be defined over an arbitrary inner model (lying between the ground model and the extension up to kappa) at each stage.   These generalized Cohen iterations are ZFC-preserving, although neither the proof for products nor for traditional iterations transfers directly.   They allow constructions such as class iterations of class products of Cohen forcing, with applications including new work with Kameryn Williams on iterating the Mantle.</font></div></div><div><font color="#000000"><br></font></div><div><font color="#000000"><br></font></div><div><font color="#000000"><br></font></div><div><font color="#000000">Model Theory Seminar</font></div><div style="color:rgb(33,33,33)">CUNY Graduate Center, Room 6417</div><div style="color:rgb(33,33,33)">Friday, April 12, 12:30-2:00pm</div><div style="color:rgb(33,33,33)"><strong style="box-sizing:border-box;color:rgb(96,108,113);font-family:&quot;Open Sans&quot;,&quot;Helvetica Neue&quot;,Helvetica,Arial,sans-serif"><a href="http://sites.nd.edu/rachaelalvir/" target="_blank" style="color:rgb(30,107,184);font-family:arial,sans-serif;text-decoration-line:none;box-sizing:border-box;background-color:transparent;margin-top:0px">Rachael Alvir</a></strong><span style="color:rgb(96,108,113);font-family:&quot;Open Sans&quot;,&quot;Helvetica Neue&quot;,Helvetica,Arial,sans-serif">, University of Notre Dame</span><br style="box-sizing:border-box;color:rgb(96,108,113);font-family:&quot;Open Sans&quot;,&quot;Helvetica Neue&quot;,Helvetica,Arial,sans-serif"><p style="box-sizing:border-box;margin-bottom:1em;margin-top:0px;color:rgb(96,108,113);font-family:&quot;Open Sans&quot;,&quot;Helvetica Neue&quot;,Helvetica,Arial,sans-serif"><strong style="box-sizing:border-box">Scott Sentences of Scattered Linear Orders</strong><br style="box-sizing:border-box"></p><p style="box-sizing:border-box;margin-bottom:1em;color:rgb(96,108,113);font-family:&quot;Open Sans&quot;,&quot;Helvetica Neue&quot;,Helvetica,Arial,sans-serif">The logic <span id="gmail-m_1984291059849704914gmail-MathJax-Element-1-Frame" class="gmail-m_1984291059849704914gmail-mjx-chtml gmail-m_1984291059849704914gmail-MathJax_CHTML" style="box-sizing:border-box;display:inline-block;line-height:0;word-spacing:normal;white-space:nowrap;float:none;direction:ltr;max-width:none;max-height:none;min-width:0px;min-height:0px;border:0px;margin:0px;padding:1px 0px"><span id="gmail-m_1984291059849704914gmail-MJXc-Node-1" class="gmail-m_1984291059849704914gmail-mjx-math" style="box-sizing:border-box;display:inline-block;border-collapse:separate;border-spacing:0px;margin-top:0px"><span id="gmail-m_1984291059849704914gmail-MJXc-Node-2" class="gmail-m_1984291059849704914gmail-mjx-mrow" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_1984291059849704914gmail-MJXc-Node-3" class="gmail-m_1984291059849704914gmail-mjx-msubsup" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-m_1984291059849704914gmail-mjx-base" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_1984291059849704914gmail-MJXc-Node-4" class="gmail-m_1984291059849704914gmail-mjx-mi" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-m_1984291059849704914gmail-mjx-char gmail-m_1984291059849704914gmail-MJXc-TeX-math-I" style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw;margin-top:0px;padding-top:0.472em;padding-bottom:0.284em;box-sizing:content-box">L</span></span></span><span class="gmail-m_1984291059849704914gmail-mjx-sub" style="display:inline-block;vertical-align:-0.212em;padding-right:0.071em;box-sizing:content-box"><span id="gmail-m_1984291059849704914gmail-MJXc-Node-5" class="gmail-m_1984291059849704914gmail-mjx-texatom" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_1984291059849704914gmail-MJXc-Node-6" class="gmail-m_1984291059849704914gmail-mjx-mrow" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_1984291059849704914gmail-MJXc-Node-7" class="gmail-m_1984291059849704914gmail-mjx-msubsup" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-m_1984291059849704914gmail-mjx-base" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_1984291059849704914gmail-MJXc-Node-8" class="gmail-m_1984291059849704914gmail-mjx-mi" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-m_1984291059849704914gmail-mjx-char gmail-m_1984291059849704914gmail-MJXc-TeX-math-I" style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw;margin-top:0px;padding-top:0.237em;padding-bottom:0.284em;box-sizing:content-box">ω</span></span></span><span class="gmail-m_1984291059849704914gmail-mjx-sub" style="display:inline-block;vertical-align:-0.212em;padding-right:0.071em;box-sizing:content-box"><span id="gmail-m_1984291059849704914gmail-MJXc-Node-9" class="gmail-m_1984291059849704914gmail-mjx-mn" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-m_1984291059849704914gmail-mjx-char gmail-m_1984291059849704914gmail-MJXc-TeX-main-R" style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.378em;padding-bottom:0.331em;box-sizing:content-box">1</span></span></span></span><span id="gmail-m_1984291059849704914gmail-MJXc-Node-10" class="gmail-m_1984291059849704914gmail-mjx-mi" style="display:inline-block;box-sizing:content-box"><span class="gmail-m_1984291059849704914gmail-mjx-char gmail-m_1984291059849704914gmail-MJXc-TeX-math-I" style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw;margin-top:0px;padding-top:0.237em;padding-bottom:0.284em;box-sizing:content-box">ω</span></span></span></span></span></span></span></span><span class="gmail-m_1984291059849704914gmail-MJX_Assistive_MathML" style="box-sizing:border-box;padding:1px 0px 0px;border:0px;height:1px;width:1px;overflow:hidden;display:block">L<span class="gmail-m_1984291059849704914gmail-MJX-TeXAtom-ORD">ω1ω</span></span></span> is obtained by closing finitary first-order logic under countable disjunction and conjunction. There is a kind of normal form for such sentences. For any structure <span id="gmail-m_1984291059849704914gmail-MathJax-Element-2-Frame" class="gmail-m_1984291059849704914gmail-mjx-chtml gmail-m_1984291059849704914gmail-MathJax_CHTML" style="box-sizing:border-box;display:inline-block;line-height:0;word-spacing:normal;white-space:nowrap;float:none;direction:ltr;max-width:none;max-height:none;min-width:0px;min-height:0px;border:0px;margin:0px;padding:1px 0px"><span id="gmail-m_1984291059849704914gmail-MJXc-Node-11" class="gmail-m_1984291059849704914gmail-mjx-math" style="box-sizing:border-box;display:inline-block;border-collapse:separate;border-spacing:0px;margin-top:0px"><span id="gmail-m_1984291059849704914gmail-MJXc-Node-12" class="gmail-m_1984291059849704914gmail-mjx-mrow" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_1984291059849704914gmail-MJXc-Node-13" class="gmail-m_1984291059849704914gmail-mjx-texatom" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_1984291059849704914gmail-MJXc-Node-14" class="gmail-m_1984291059849704914gmail-mjx-mrow" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_1984291059849704914gmail-MJXc-Node-15" class="gmail-m_1984291059849704914gmail-mjx-mi" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-m_1984291059849704914gmail-mjx-char gmail-m_1984291059849704914gmail-MJXc-TeX-cal-R" style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-cal-R,MJXc-TeX-cal-Rw;margin-top:0px;padding-top:0.472em;padding-bottom:0.378em;padding-right:0.021em;box-sizing:content-box">A</span></span></span></span></span></span><span class="gmail-m_1984291059849704914gmail-MJX_Assistive_MathML" style="box-sizing:border-box;padding:1px 0px 0px;border:0px;height:1px;width:1px;overflow:hidden;display:block"><span class="gmail-m_1984291059849704914gmail-MJX-TeXAtom-ORD"><span class="gmail-m_1984291059849704914gmail-MJX-tex-caligraphic">A</span></span></span></span> there is a sentence of <span id="gmail-m_1984291059849704914gmail-MathJax-Element-3-Frame" class="gmail-m_1984291059849704914gmail-mjx-chtml gmail-m_1984291059849704914gmail-MathJax_CHTML" style="box-sizing:border-box;display:inline-block;line-height:0;word-spacing:normal;white-space:nowrap;float:none;direction:ltr;max-width:none;max-height:none;min-width:0px;min-height:0px;border:0px;margin:0px;padding:1px 0px"><span id="gmail-m_1984291059849704914gmail-MJXc-Node-16" class="gmail-m_1984291059849704914gmail-mjx-math" style="box-sizing:border-box;display:inline-block;border-collapse:separate;border-spacing:0px;margin-top:0px"><span id="gmail-m_1984291059849704914gmail-MJXc-Node-17" class="gmail-m_1984291059849704914gmail-mjx-mrow" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_1984291059849704914gmail-MJXc-Node-18" class="gmail-m_1984291059849704914gmail-mjx-msubsup" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-m_1984291059849704914gmail-mjx-base" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_1984291059849704914gmail-MJXc-Node-19" class="gmail-m_1984291059849704914gmail-mjx-mi" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-m_1984291059849704914gmail-mjx-char gmail-m_1984291059849704914gmail-MJXc-TeX-math-I" style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw;margin-top:0px;padding-top:0.472em;padding-bottom:0.284em;box-sizing:content-box">L</span></span></span><span class="gmail-m_1984291059849704914gmail-mjx-sub" style="display:inline-block;vertical-align:-0.212em;padding-right:0.071em;box-sizing:content-box"><span id="gmail-m_1984291059849704914gmail-MJXc-Node-20" class="gmail-m_1984291059849704914gmail-mjx-texatom" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_1984291059849704914gmail-MJXc-Node-21" class="gmail-m_1984291059849704914gmail-mjx-mrow" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_1984291059849704914gmail-MJXc-Node-22" class="gmail-m_1984291059849704914gmail-mjx-msubsup" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-m_1984291059849704914gmail-mjx-base" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_1984291059849704914gmail-MJXc-Node-23" class="gmail-m_1984291059849704914gmail-mjx-mi" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-m_1984291059849704914gmail-mjx-char gmail-m_1984291059849704914gmail-MJXc-TeX-math-I" style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw;margin-top:0px;padding-top:0.237em;padding-bottom:0.284em;box-sizing:content-box">ω</span></span></span><span class="gmail-m_1984291059849704914gmail-mjx-sub" style="display:inline-block;vertical-align:-0.212em;padding-right:0.071em;box-sizing:content-box"><span id="gmail-m_1984291059849704914gmail-MJXc-Node-24" class="gmail-m_1984291059849704914gmail-mjx-mn" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-m_1984291059849704914gmail-mjx-char gmail-m_1984291059849704914gmail-MJXc-TeX-main-R" style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.378em;padding-bottom:0.331em;box-sizing:content-box">1</span></span></span></span><span id="gmail-m_1984291059849704914gmail-MJXc-Node-25" class="gmail-m_1984291059849704914gmail-mjx-mi" style="display:inline-block;box-sizing:content-box"><span class="gmail-m_1984291059849704914gmail-mjx-char gmail-m_1984291059849704914gmail-MJXc-TeX-math-I" style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw;margin-top:0px;padding-top:0.237em;padding-bottom:0.284em;box-sizing:content-box">ω</span></span></span></span></span></span></span></span><span class="gmail-m_1984291059849704914gmail-MJX_Assistive_MathML" style="box-sizing:border-box;padding:1px 0px 0px;border:0px;height:1px;width:1px;overflow:hidden;display:block">L<span class="gmail-m_1984291059849704914gmail-MJX-TeXAtom-ORD">ω1ω</span></span></span>, known as its <em style="box-sizing:border-box">Scott sentence</em>, which describes <span id="gmail-m_1984291059849704914gmail-MathJax-Element-4-Frame" class="gmail-m_1984291059849704914gmail-mjx-chtml gmail-m_1984291059849704914gmail-MathJax_CHTML" style="box-sizing:border-box;display:inline-block;line-height:0;word-spacing:normal;white-space:nowrap;float:none;direction:ltr;max-width:none;max-height:none;min-width:0px;min-height:0px;border:0px;margin:0px;padding:1px 0px"><span id="gmail-m_1984291059849704914gmail-MJXc-Node-26" class="gmail-m_1984291059849704914gmail-mjx-math" style="box-sizing:border-box;display:inline-block;border-collapse:separate;border-spacing:0px;margin-top:0px"><span id="gmail-m_1984291059849704914gmail-MJXc-Node-27" class="gmail-m_1984291059849704914gmail-mjx-mrow" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_1984291059849704914gmail-MJXc-Node-28" class="gmail-m_1984291059849704914gmail-mjx-texatom" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_1984291059849704914gmail-MJXc-Node-29" class="gmail-m_1984291059849704914gmail-mjx-mrow" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_1984291059849704914gmail-MJXc-Node-30" class="gmail-m_1984291059849704914gmail-mjx-mi" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-m_1984291059849704914gmail-mjx-char gmail-m_1984291059849704914gmail-MJXc-TeX-cal-R" style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-cal-R,MJXc-TeX-cal-Rw;margin-top:0px;padding-top:0.472em;padding-bottom:0.378em;padding-right:0.021em;box-sizing:content-box">A</span></span></span></span></span></span><span class="gmail-m_1984291059849704914gmail-MJX_Assistive_MathML" style="box-sizing:border-box;padding:1px 0px 0px;border:0px;height:1px;width:1px;overflow:hidden;display:block"><span class="gmail-m_1984291059849704914gmail-MJX-TeXAtom-ORD"><span class="gmail-m_1984291059849704914gmail-MJX-tex-caligraphic">A</span></span></span></span> up to isomorphism among countable structures. Given a countable scattered linear order <span id="gmail-m_1984291059849704914gmail-MathJax-Element-5-Frame" class="gmail-m_1984291059849704914gmail-mjx-chtml gmail-m_1984291059849704914gmail-MathJax_CHTML" style="box-sizing:border-box;display:inline-block;line-height:0;word-spacing:normal;white-space:nowrap;float:none;direction:ltr;max-width:none;max-height:none;min-width:0px;min-height:0px;border:0px;margin:0px;padding:1px 0px"><span id="gmail-m_1984291059849704914gmail-MJXc-Node-31" class="gmail-m_1984291059849704914gmail-mjx-math" style="box-sizing:border-box;display:inline-block;border-collapse:separate;border-spacing:0px;margin-top:0px"><span id="gmail-m_1984291059849704914gmail-MJXc-Node-32" class="gmail-m_1984291059849704914gmail-mjx-mrow" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_1984291059849704914gmail-MJXc-Node-33" class="gmail-m_1984291059849704914gmail-mjx-mi" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-m_1984291059849704914gmail-mjx-char gmail-m_1984291059849704914gmail-MJXc-TeX-math-I" style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw;margin-top:0px;padding-top:0.472em;padding-bottom:0.284em;box-sizing:content-box">L</span></span></span></span><span class="gmail-m_1984291059849704914gmail-MJX_Assistive_MathML" style="box-sizing:border-box;padding:1px 0px 0px;border:0px;height:1px;width:1px;overflow:hidden;display:block">L</span></span> of Hausdorff rank <span id="gmail-m_1984291059849704914gmail-MathJax-Element-6-Frame" class="gmail-m_1984291059849704914gmail-mjx-chtml gmail-m_1984291059849704914gmail-MathJax_CHTML" style="box-sizing:border-box;display:inline-block;line-height:0;word-spacing:normal;white-space:nowrap;float:none;direction:ltr;max-width:none;max-height:none;min-width:0px;min-height:0px;border:0px;margin:0px;padding:1px 0px"><span id="gmail-m_1984291059849704914gmail-MJXc-Node-34" class="gmail-m_1984291059849704914gmail-mjx-math" style="box-sizing:border-box;display:inline-block;border-collapse:separate;border-spacing:0px;margin-top:0px"><span id="gmail-m_1984291059849704914gmail-MJXc-Node-35" class="gmail-m_1984291059849704914gmail-mjx-mrow" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_1984291059849704914gmail-MJXc-Node-36" class="gmail-m_1984291059849704914gmail-mjx-mi" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-m_1984291059849704914gmail-mjx-char gmail-m_1984291059849704914gmail-MJXc-TeX-math-I" style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw;margin-top:0px;padding-top:0.237em;padding-bottom:0.284em;box-sizing:content-box">α</span></span><span id="gmail-m_1984291059849704914gmail-MJXc-Node-37" class="gmail-m_1984291059849704914gmail-mjx-mo gmail-m_1984291059849704914gmail-MJXc-space3" style="display:inline-block;margin-left:0.278em;box-sizing:content-box"><span class="gmail-m_1984291059849704914gmail-mjx-char gmail-m_1984291059849704914gmail-MJXc-TeX-main-R" style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.237em;padding-bottom:0.378em;box-sizing:content-box">&lt;</span></span><span id="gmail-m_1984291059849704914gmail-MJXc-Node-38" class="gmail-m_1984291059849704914gmail-mjx-msubsup gmail-m_1984291059849704914gmail-MJXc-space3" style="display:inline-block;margin-left:0.278em;box-sizing:content-box"><span class="gmail-m_1984291059849704914gmail-mjx-base" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_1984291059849704914gmail-MJXc-Node-39" class="gmail-m_1984291059849704914gmail-mjx-mi" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-m_1984291059849704914gmail-mjx-char gmail-m_1984291059849704914gmail-MJXc-TeX-math-I" style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw;margin-top:0px;padding-top:0.237em;padding-bottom:0.284em;box-sizing:content-box">ω</span></span></span><span class="gmail-m_1984291059849704914gmail-mjx-sub" style="display:inline-block;vertical-align:-0.212em;padding-right:0.071em;box-sizing:content-box"><span id="gmail-m_1984291059849704914gmail-MJXc-Node-40" class="gmail-m_1984291059849704914gmail-mjx-mn" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-m_1984291059849704914gmail-mjx-char gmail-m_1984291059849704914gmail-MJXc-TeX-main-R" style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.378em;padding-bottom:0.331em;box-sizing:content-box">1</span></span></span></span></span></span><span class="gmail-m_1984291059849704914gmail-MJX_Assistive_MathML" style="box-sizing:border-box;padding:1px 0px 0px;border:0px;height:1px;width:1px;overflow:hidden;display:block">α&lt;ω1</span></span>, we show that it has a <span id="gmail-m_1984291059849704914gmail-MathJax-Element-7-Frame" class="gmail-m_1984291059849704914gmail-mjx-chtml gmail-m_1984291059849704914gmail-MathJax_CHTML" style="box-sizing:border-box;display:inline-block;line-height:0;word-spacing:normal;white-space:nowrap;float:none;direction:ltr;max-width:none;max-height:none;min-width:0px;min-height:0px;border:0px;margin:0px;padding:1px 0px"><span id="gmail-m_1984291059849704914gmail-MJXc-Node-41" class="gmail-m_1984291059849704914gmail-mjx-math" style="box-sizing:border-box;display:inline-block;border-collapse:separate;border-spacing:0px;margin-top:0px"><span id="gmail-m_1984291059849704914gmail-MJXc-Node-42" class="gmail-m_1984291059849704914gmail-mjx-mrow" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_1984291059849704914gmail-MJXc-Node-43" class="gmail-m_1984291059849704914gmail-mjx-mi" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-m_1984291059849704914gmail-mjx-char gmail-m_1984291059849704914gmail-MJXc-TeX-math-I" style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw;margin-top:0px;padding-top:0.472em;padding-bottom:0.284em;padding-right:0.003em;box-sizing:content-box">d</span></span></span></span><span class="gmail-m_1984291059849704914gmail-MJX_Assistive_MathML" style="box-sizing:border-box;padding:1px 0px 0px;border:0px;height:1px;width:1px;overflow:hidden;display:block">d</span></span>-<span id="gmail-m_1984291059849704914gmail-MathJax-Element-8-Frame" class="gmail-m_1984291059849704914gmail-mjx-chtml gmail-m_1984291059849704914gmail-MathJax_CHTML" style="box-sizing:border-box;display:inline-block;line-height:0;word-spacing:normal;white-space:nowrap;float:none;direction:ltr;max-width:none;max-height:none;min-width:0px;min-height:0px;border:0px;margin:0px;padding:1px 0px"><span id="gmail-m_1984291059849704914gmail-MJXc-Node-44" class="gmail-m_1984291059849704914gmail-mjx-math" style="box-sizing:border-box;display:inline-block;border-collapse:separate;border-spacing:0px;margin-top:0px"><span id="gmail-m_1984291059849704914gmail-MJXc-Node-45" class="gmail-m_1984291059849704914gmail-mjx-mrow" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_1984291059849704914gmail-MJXc-Node-46" class="gmail-m_1984291059849704914gmail-mjx-msubsup" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-m_1984291059849704914gmail-mjx-base" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_1984291059849704914gmail-MJXc-Node-47" class="gmail-m_1984291059849704914gmail-mjx-mi" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-m_1984291059849704914gmail-mjx-char gmail-m_1984291059849704914gmail-MJXc-TeX-main-R" style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.378em;padding-bottom:0.331em;box-sizing:content-box">Σ</span></span></span><span class="gmail-m_1984291059849704914gmail-mjx-sub" style="display:inline-block;vertical-align:-0.212em;padding-right:0.071em;box-sizing:content-box"><span id="gmail-m_1984291059849704914gmail-MJXc-Node-48" class="gmail-m_1984291059849704914gmail-mjx-texatom" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_1984291059849704914gmail-MJXc-Node-49" class="gmail-m_1984291059849704914gmail-mjx-mrow" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_1984291059849704914gmail-MJXc-Node-50" class="gmail-m_1984291059849704914gmail-mjx-mn" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-m_1984291059849704914gmail-mjx-char gmail-m_1984291059849704914gmail-MJXc-TeX-main-R" style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.378em;padding-bottom:0.331em;box-sizing:content-box">2</span></span><span id="gmail-m_1984291059849704914gmail-MJXc-Node-51" class="gmail-m_1984291059849704914gmail-mjx-mi" style="display:inline-block;box-sizing:content-box"><span class="gmail-m_1984291059849704914gmail-mjx-char gmail-m_1984291059849704914gmail-MJXc-TeX-math-I" style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw;margin-top:0px;padding-top:0.237em;padding-bottom:0.284em;box-sizing:content-box">α</span></span><span id="gmail-m_1984291059849704914gmail-MJXc-Node-52" class="gmail-m_1984291059849704914gmail-mjx-mo" style="display:inline-block;box-sizing:content-box"><span class="gmail-m_1984291059849704914gmail-mjx-char gmail-m_1984291059849704914gmail-MJXc-TeX-main-R" style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.284em;padding-bottom:0.425em;box-sizing:content-box">+</span></span><span id="gmail-m_1984291059849704914gmail-MJXc-Node-53" class="gmail-m_1984291059849704914gmail-mjx-mn" style="display:inline-block;box-sizing:content-box"><span class="gmail-m_1984291059849704914gmail-mjx-char gmail-m_1984291059849704914gmail-MJXc-TeX-main-R" style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.378em;padding-bottom:0.331em;box-sizing:content-box">1</span></span></span></span></span></span></span></span><span class="gmail-m_1984291059849704914gmail-MJX_Assistive_MathML" style="box-sizing:border-box;padding:1px 0px 0px;border:0px;height:1px;width:1px;overflow:hidden;display:block">Σ<span class="gmail-m_1984291059849704914gmail-MJX-TeXAtom-ORD">2α+1</span></span></span> Scott sentence. From Ash&#39;s calculation of the back and forth relations for all countable well-orders, we obtain that this upper bound is tight, i.e., for every <span id="gmail-m_1984291059849704914gmail-MathJax-Element-9-Frame" class="gmail-m_1984291059849704914gmail-mjx-chtml gmail-m_1984291059849704914gmail-MathJax_CHTML" style="box-sizing:border-box;display:inline-block;line-height:0;word-spacing:normal;white-space:nowrap;float:none;direction:ltr;max-width:none;max-height:none;min-width:0px;min-height:0px;border:0px;margin:0px;padding:1px 0px"><span id="gmail-m_1984291059849704914gmail-MJXc-Node-54" class="gmail-m_1984291059849704914gmail-mjx-math" style="box-sizing:border-box;display:inline-block;border-collapse:separate;border-spacing:0px;margin-top:0px"><span id="gmail-m_1984291059849704914gmail-MJXc-Node-55" class="gmail-m_1984291059849704914gmail-mjx-mrow" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_1984291059849704914gmail-MJXc-Node-56" class="gmail-m_1984291059849704914gmail-mjx-mi" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-m_1984291059849704914gmail-mjx-char gmail-m_1984291059849704914gmail-MJXc-TeX-math-I" style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw;margin-top:0px;padding-top:0.237em;padding-bottom:0.284em;box-sizing:content-box">α</span></span><span id="gmail-m_1984291059849704914gmail-MJXc-Node-57" class="gmail-m_1984291059849704914gmail-mjx-mo gmail-m_1984291059849704914gmail-MJXc-space3" style="display:inline-block;margin-left:0.278em;box-sizing:content-box"><span class="gmail-m_1984291059849704914gmail-mjx-char gmail-m_1984291059849704914gmail-MJXc-TeX-main-R" style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.237em;padding-bottom:0.378em;box-sizing:content-box">&lt;</span></span><span id="gmail-m_1984291059849704914gmail-MJXc-Node-58" class="gmail-m_1984291059849704914gmail-mjx-msubsup gmail-m_1984291059849704914gmail-MJXc-space3" style="display:inline-block;margin-left:0.278em;box-sizing:content-box"><span class="gmail-m_1984291059849704914gmail-mjx-base" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_1984291059849704914gmail-MJXc-Node-59" class="gmail-m_1984291059849704914gmail-mjx-mi" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-m_1984291059849704914gmail-mjx-char gmail-m_1984291059849704914gmail-MJXc-TeX-math-I" style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw;margin-top:0px;padding-top:0.237em;padding-bottom:0.284em;box-sizing:content-box">ω</span></span></span><span class="gmail-m_1984291059849704914gmail-mjx-sub" style="display:inline-block;vertical-align:-0.212em;padding-right:0.071em;box-sizing:content-box"><span id="gmail-m_1984291059849704914gmail-MJXc-Node-60" class="gmail-m_1984291059849704914gmail-mjx-mn" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-m_1984291059849704914gmail-mjx-char gmail-m_1984291059849704914gmail-MJXc-TeX-main-R" style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.378em;padding-bottom:0.331em;box-sizing:content-box">1</span></span></span></span></span></span><span class="gmail-m_1984291059849704914gmail-MJX_Assistive_MathML" style="box-sizing:border-box;padding:1px 0px 0px;border:0px;height:1px;width:1px;overflow:hidden;display:block">α&lt;ω1</span></span> there is a linear order whose optimal Scott sentence has this complexity.</p></div><div style="color:rgb(33,33,33)"><br></div><div style="color:rgb(33,33,33)"><br></div><div style="color:rgb(33,33,33)"><br></div><div style="color:rgb(33,33,33)">Logic Workshop</div><div style="color:rgb(33,33,33)">CUNY Graduate Center, Room 6417</div><div style="color:rgb(33,33,33)">Friday, April 12, 2:00-3:30pm</div><div style="color:rgb(33,33,33)"><p style="font-family:&quot;Open Sans&quot;,&quot;Helvetica Neue&quot;,Helvetica,Arial,sans-serif;box-sizing:border-box;margin-bottom:1em;margin-top:0px;color:rgb(96,108,113)"><strong style="box-sizing:border-box"><a href="https://math.nd.edu/people/faculty/julia-f-knight/" target="_blank" style="color:rgb(30,107,184);font-family:arial,sans-serif;text-decoration-line:none;box-sizing:border-box;background-color:transparent;margin-top:0px">Julia Knight</a></strong>, University of Notre Dame<br style="box-sizing:border-box"><strong style="box-sizing:border-box">Coding structures</strong><br style="box-sizing:border-box"></p><p style="font-family:&quot;Open Sans&quot;,&quot;Helvetica Neue&quot;,Helvetica,Arial,sans-serif;box-sizing:border-box;margin-bottom:1em;color:rgb(96,108,113)">A <em style="box-sizing:border-box;margin-top:0px">Turing computable embedding</em> is a Turing operator that maps one class of structures to another so as to preserve isomorphism. The embedding codes the input structure in the output structure. It is interesting when there is an effective decoding. It is also interesting when the decoding is very difficult. Recently, Harrison-Trainor, Melnikov, R. Miller, and Montalbán have defined very general notions of interpretation, in which the interpreting formulas have no fixed arity. Uniformly defined interpretations give us decoding. I will discuss some known Turing computable embeddings, looking for uniform interpretations that yield effective, or Borel, decoding.</p><ol style="box-sizing:border-box;margin-top:0px;color:rgb(96,108,113);font-family:&quot;Open Sans&quot;,&quot;Helvetica Neue&quot;,Helvetica,Arial,sans-serif"><li style="margin-left:15px;box-sizing:border-box;margin-top:0px">Marker&#39;s embedding of directed graphs in undirected graphs,</li><li style="margin-left:15px;box-sizing:border-box">Mal&#39;tsev&#39;s embedding of fields in groups,</li><li style="margin-left:15px;box-sizing:border-box">Friedman and Stanley&#39;s embedding of graphs in linear orderings.</li></ol><span style="color:rgb(96,108,113);font-family:&quot;Open Sans&quot;,&quot;Helvetica Neue&quot;,Helvetica,Arial,sans-serif">The first two embeddings come with uniform &#39;effective&#39; interpretations, which give uniform effective decoding. For the third, we do not even have uniform interpretation via </span><span id="gmail-m_1984291059849704914gmail-m_2794932661880136167gmail-MathJax-Element-1-Frame" style="box-sizing:border-box;display:inline-block;line-height:0;white-space:nowrap;float:none;direction:ltr;max-width:none;max-height:none;min-width:0px;min-height:0px;border:0px;margin:0px;padding:1px 0px;color:rgb(96,108,113);font-family:&quot;Open Sans&quot;,&quot;Helvetica Neue&quot;,Helvetica,Arial,sans-serif"><span id="gmail-m_1984291059849704914gmail-m_2794932661880136167gmail-MJXc-Node-1" style="box-sizing:border-box;display:inline-block;border-collapse:separate;border-spacing:0px;margin-top:0px"><span id="gmail-m_1984291059849704914gmail-m_2794932661880136167gmail-MJXc-Node-2" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_1984291059849704914gmail-m_2794932661880136167gmail-MJXc-Node-3" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_1984291059849704914gmail-m_2794932661880136167gmail-MJXc-Node-4" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw;margin-top:0px;padding-top:0.472em;padding-bottom:0.284em;box-sizing:content-box">L</span></span></span><span style="display:inline-block;vertical-align:-0.212em;padding-right:0.071em;box-sizing:content-box"><span id="gmail-m_1984291059849704914gmail-m_2794932661880136167gmail-MJXc-Node-5" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_1984291059849704914gmail-m_2794932661880136167gmail-MJXc-Node-6" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_1984291059849704914gmail-m_2794932661880136167gmail-MJXc-Node-7" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_1984291059849704914gmail-m_2794932661880136167gmail-MJXc-Node-8" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw;margin-top:0px;padding-top:0.237em;padding-bottom:0.284em;box-sizing:content-box">ω</span></span></span><span style="display:inline-block;vertical-align:-0.212em;padding-right:0.071em;box-sizing:content-box"><span id="gmail-m_1984291059849704914gmail-m_2794932661880136167gmail-MJXc-Node-9" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.378em;padding-bottom:0.331em;box-sizing:content-box">1</span></span></span></span><span id="gmail-m_1984291059849704914gmail-m_2794932661880136167gmail-MJXc-Node-10" style="display:inline-block;box-sizing:content-box"><span style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw;margin-top:0px;padding-top:0.237em;padding-bottom:0.284em;box-sizing:content-box">ω</span></span></span></span></span></span></span></span><span style="box-sizing:border-box;padding:1px 0px 0px;border:0px;height:1px;width:1px;overflow:hidden;display:block">Lω1ω</span></span><span style="color:rgb(96,108,113);font-family:&quot;Open Sans&quot;,&quot;Helvetica Neue&quot;,Helvetica,Arial,sans-serif"> formulas.</span><br></div></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><font color="#212121"><br></font></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><font color="#212121"><br></font></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><font color="#212121"><br></font></div><div><font color="#212121"><div><font face="arial, sans-serif">Next Week in Logic at CUNY:</font></div><div><font face="arial, sans-serif"><br></font></div><div><font face="arial, sans-serif">- - - - Monday, Apr 15, 2019 - - - -</font></div><div><font face="arial, sans-serif"><br></font></div><div><font face="arial, sans-serif"><div>Logic and Metaphysics Workshop</div><div>Date: Monday, April 15th, 4.15-6.15  <br></div><div>Place: Room 7314, CUNY Graduate Center <br></div><div>Speaker: Jenn McDonald (CUNY)<br></div><div>Title: Structural Counterfactuals and the Importation Problem<br></div><div><br></div><div>Abstract: Structural causal models lend themselves to an analysis of counterfactuals – a structural semantics of counterfactuals. The basic idea is that a causal model allows for the clear and precise evaluation of any counterfactual encoded by it. Many argue that a structural semantics is superior to a more traditional similarity semantics, in part due to the latter’s independence from any notion of similarity(Galles &amp; Pearl, 1998; Gallow, 2016; Hiddleston, 2005; Hitchcock, 2018; Pearl, 2000; Starr, 2019). I argue, though, that this is too quick. A similarity semantics employs the notion of similarity to answer what Priest (2018) calls the importation problem– the question of what information is to be held fixed in a counterfactual evaluation. I argue that where similarity semantics relies on an unarticulated notion of similarity, a structural semantics relies on an unarticulated notion of aptness. The superiority of structural semantics depends on its ability to deliver on a principled guide to determining which model(s) is apt. In this talk I go some way towards providing this guide.</div><div><br></div><div><br></div><div><br></div></font></div><div><font face="arial, sans-serif"><br></font></div><div><font face="arial, sans-serif">- - - - Tuesday, Apr 16, 2019 - - - -</font></div><div><font face="arial, sans-serif"><br></font></div><div><font face="arial, sans-serif"><br></font></div><div><font face="arial, sans-serif"><br></font></div><div><font face="arial, sans-serif">- - - - Wednesday, Apr 17, 2019 - - - -</font></div><div><font face="arial, sans-serif"><br></font></div><div><font face="arial, sans-serif"><div>The New York City Category Theory Seminar</div><div>Department of Computer Science</div><div>Department of Mathematics </div><div>The Graduate Center of The City University of New York </div><div>Speaker:     Tibor Beke, University of Massachusetts, Lowell.</div><div>Date and Time:     Wednesday April 17, 2019, 7:00 - 8:30 PM., Room 6417.<br></div><div>Title:    Schanuel functors and the Grothendieck (semi)ring of some theories.<br></div><div><br></div><div>Abstract: In a little known article, Schanuel defines a functor from semirings to idempotent semirings and a notion of dimension that is not linearly ordered. He uses it to give an elegant presentation of the Grothendieck semiring of semi-algebraic sets, from which the (much better known) structure of the Grothendieck ring of semi-algebraic sets easily follows. I will review his work and related results on the Grothendieck semiring of algebraically closed fields and similar geometric structures.</div><div><br></div><div><br></div><div><br></div></font></div><div><font face="arial, sans-serif"><br></font></div><div><font face="arial, sans-serif">- - - - Thursday, Apr 18, 2019 - - - -</font></div><div><font face="arial, sans-serif"><br></font></div><div><font face="arial, sans-serif"><br></font></div><div><font face="arial, sans-serif"><br></font></div><div><font face="arial, sans-serif">- - - - Friday, Apr 19, 2019 - - - -</font></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><br></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif">*** CUNY Spring Break April 19-28 ***</div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><br></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><br></div></font></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><font color="#212121">- - - - Other Logic News - - - -</font></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><br></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><font color="#212121"><br></font></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><font color="#212121"><br></font></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><font color="#212121">- - - - Web Site - - - -</font></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><font color="#212121"><br></font></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><font color="#212121">Our new website is located at:  <a href="https://nylogic.github.io/" target="_blank" style="text-decoration-line:none">https://nylogic.github.io/</a></font></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><font color="#212121">Many thanks to Victoria Gitman for her development work!</font></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><font color="#212121"><br></font></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><font color="#212121">--------  ADMINISTRIVIA  --------</font></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><font color="#212121"><br></font></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><font color="#212121">To subscribe/unsubscribe to this list, please email your request to <a href="mailto:jreitz@nylogic.org" target="_blank" style="text-decoration-line:none">jreitz@nylogic.org</a>.</font></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><font color="#212121"><br></font></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><font color="#212121">If you have a logic-related event that you would like included in future mailings, please email <a href="mailto:jreitz@nylogic.org" target="_blank" style="text-decoration-line:none">jreitz@nylogic.org</a>. </font></div></div></div>
50
4/11/2019 0:34:58NUS Logic SeminarLogic Seminar 17 April 2019 17:00 hrs at NUSInvitation to the Logic Seminar at the National University of Singapore

Date: Wednesday, 17 April 2019, 17:00 hrs

Room: S17#04-06, Department of Mathematics, NUS

Speaker: Liu Tianyu

Title: Arithmetic algorithm of surreal numbers

URL: http://www.comp.nus.edu.sg/~fstephan/logicseminar.html

Abstract:
The surreal number system is a totally ordered proper class
containing the real numbers as well as infinite and infinitesimal
numbers. A surreal number is sometimes defined as a function from an
initial segment of the ordinals into the set $(+,-)$, usually leads to
an infinite sign sequence. It can also be expressed uniquely in a
normal form, as $\sum_{i<\alpha} \omega^{a_i}r_i$.

In this talk I will present some algorithms for addition,
multiplication and division on the real field. Besides, we will cover
the normal form and sign sequence of surreal numbers, which will be
crucial for arithmetic algorithm of surreal numbers of length $>\omega$.

51
4/10/2019 9:00:00University of Vienna Mathematics ColloquiumIlijas Farah: Some necessary uses of set theory in mathematicsMathematisches Kolloquium | 10.04.2019 16:15 - 19:00
Ilijas Farah (York University, Canada)

Abstract:

Every now and then, a difficult mathematical problem turns out to be difficult for a particularly objective reason: Provably, it cannot be solved by using `conventional' means. Some classical examples are proving the Continuum Hypothesis, duplicating the cube, and solving the quintic equation in radicals. I’ll discuss more recent examples of such problems, giving emphasis to those arising from the study of operator algebras.
Ilijas Farahhttps://mathematik.univie.ac.at/newsevents/nachrichtenvolldarstellung/news/some-necessary-uses-of-set-theory-in-mathematics/
52
4/10/2019 2:16:40Toronto Set Theory SeminarWilliam Chen: A Frechet space defined from a square principlePlace: Fields Institute (Room 210)
Date: Friday 12, 2019 (13:30-15:00)
Speaker: William Chen
Title: A Frechet space defined from a square principle
Abstract: From $\Box(\kappa)$, we construct a Frechet $\alpha_1$-space
whose tightness in the topology obtained by declaring the $G_\delta$ sets
to be open is equal to $\kappa$. This complements the fact that under PFA,
every Frechet $\alpha_1$-space has tightness at most $\omega_1$ in the
$G_\delta$ modification. This space also satisfies several other
interesting properties from the theory of Frechet spaces.
(joint with P. Szeptycki)
William Chen
53
4/8/2019 11:39:32Prague Set Theory SeminarRalf Schindler: Variants of the extender algebra and their applicationsDear all,

There is no seminar this Wednesday.

The seminar meets again on Wednesday April 17th at 11:00 in the
Institute of Mathematics CAS, Zitna 25, seminar room, 3rd floor, front
building.

Program: Ralf Schindler -- Variants of the extender algebra and their
applications

Abstract. In the 1970’ies, Bukovský identified a beautiful and handy
criterion for when V is a forcing extension of a given inner model,
which proved very useful recently in set theoretical geology. In the
1990’ies, Woodin isolated his extender algebra which makes use of a
large cardinal, a Woodin cardinal. It turns out that Bukovský’s theorem
and Woodin’s extender algebra may be presented in a uniform fashion –
one proof and one forcing gives both results. We will discuss the proof
and applications in set theoretic and inner model theoretic geology and
then address (and answer) a question of T. Usuba on the \kappa-mantle.
In part, this is joint work with G. Sargsyan, F. Schlutzenberg, and my
student A. Lietz.



Best,
David
Ralf Schindler
54
4/8/2019 11:36:42Barcelona Set Theory SeminarRe: Set Theory Seminar<html><head><meta http-equiv="Content-Type" content="text/html charset=us-ascii"></head><body style="word-wrap: break-word; -webkit-nbsp-mode: space; -webkit-line-break: after-white-space;" class=""><div><div class="">Dear Colleagues,</div><div class=""><div style="word-wrap: break-word; -webkit-nbsp-mode: space; -webkit-line-break: after-white-space;" class=""><div class=""><div class=""><div style="word-wrap: break-word; -webkit-nbsp-mode: space; -webkit-line-break: after-white-space;" class=""><div class=""><div class=""><div style="word-wrap: break-word; -webkit-nbsp-mode: space; -webkit-line-break: after-white-space;" class=""><div class=""><div style="word-wrap: break-word; -webkit-nbsp-mode: space; -webkit-line-break: after-white-space;" class=""><div class=""><div class=""><div style="word-wrap: break-word; -webkit-nbsp-mode: space; -webkit-line-break: after-white-space;" class=""><div class=""><div class=""><div style="word-wrap: break-word; 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-webkit-nbsp-mode: space; -webkit-line-break: after-white-space;" class=""><div class=""><div class=""><div style="word-wrap: break-word; -webkit-nbsp-mode: space; -webkit-line-break: after-white-space;" class=""><div class=""><div class=""><div style="word-wrap: break-word; -webkit-nbsp-mode: space; -webkit-line-break: after-white-space;" class=""><div class="">Please find attached the announcement of the next session of the Barcelona Set Theory seminar.</div><div class="">You are all welcome to attend.&nbsp;</div><div class="">Best regards,</div><div class="">Joan</div><div class=""><br class=""></div><div class=""></div></div></div></div></div></div></div></div></div></div></div></div></div></div></div></div></div></div></div></div></div></div></div></div></div></div></div></div></div></div></div></div></div></div></div><div style="word-wrap: break-word; -webkit-nbsp-mode: space; -webkit-line-break: after-white-space;" class=""><div class=""><div style="letter-spacing: normal; orphans: auto; 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<html><head><meta http-equiv="Content-Type" content="text/html charset=utf-8"></head><body style="word-wrap: break-word; -webkit-nbsp-mode: space; -webkit-line-break: after-white-space;" class=""><div><div class=""><div style="word-wrap: break-word; -webkit-nbsp-mode: space; -webkit-line-break: after-white-space;" class=""><div class=""><div style="letter-spacing: normal; orphans: auto; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; widows: auto; word-spacing: 0px; -webkit-text-stroke-width: 0px; word-wrap: break-word; -webkit-nbsp-mode: space; -webkit-line-break: after-white-space;" class=""><span class="Apple-style-span" style="border-collapse: separate; font-family: Helvetica; font-style: normal; font-variant-ligatures: normal; font-variant-position: normal; font-variant-caps: normal; font-variant-numeric: normal; font-variant-alternates: normal; font-variant-east-asian: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: -webkit-auto; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; border-spacing: 0px; -webkit-text-decorations-in-effect: none; -webkit-text-stroke-width: 0px;"><div style="word-wrap: break-word; -webkit-nbsp-mode: space; -webkit-line-break: after-white-space;" class=""><span class="Apple-style-span" style="border-collapse: separate; font-family: Helvetica; font-style: normal; font-variant-ligatures: normal; font-variant-position: normal; font-variant-caps: normal; font-variant-numeric: normal; font-variant-alternates: normal; font-variant-east-asian: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: -webkit-auto; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; border-spacing: 0px; -webkit-text-decorations-in-effect: none; -webkit-text-stroke-width: 0px;"><div style="word-wrap: break-word; -webkit-nbsp-mode: space; -webkit-line-break: after-white-space;" class=""></div></span><br class="Apple-interchange-newline"></div></span><br class="Apple-interchange-newline"></div><br class="Apple-interchange-newline"><br class="Apple-interchange-newline">
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55
4/7/2019 20:49:07This Week in Logic at CUNYThis Week in Logic at CUNY<div dir="ltr"><div dir="ltr"><div dir="ltr"><div dir="ltr"><div dir="ltr"><div dir="ltr"><div dir="ltr"><div style="font-family:arial,sans-serif;color:rgb(33,33,33)">This Week in Logic at CUNY:</div><div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><div><br></div></div><div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><font color="#212121">- - - - Monday, Apr 8, 2019 - - - -</font></div><div><font color="#212121" face="arial, sans-serif"><div><br></div><div>Logic and Metaphysics Workshop</div><div>Date: Monday, April 8th, 4.15-6.15  </div><div>Place: Room 7314, CUNY Graduate Center </div><div>Speaker: Chris Scambler (NYU)</div><div>Title: Classical Logic and the Strict Tolerant Hierarchy</div><div><br></div><div>Abstract: In this talk I will do three things. First: I will present the central results from Barrio, Pailos and Szmuc’s recent paper “A hierarchy of classical and paraconsistent logics” (forthcoming in the JPL) along with some generalizations derived by observing certain symmetries; second, I will discuss the relation between the strict tolerant logics and classical logic, K3 and LP; third, I will try to convey the exact state of uncertainty about the philosophical significance of the foregoing I find myself in on the day.</div><div><br></div><div><br></div><div> </div></font></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><font color="#212121"><br></font></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><font color="#212121"><br></font></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><font color="#212121">- - - - Tuesday, Apr 9, 2019 - - - -</font></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><font color="#212121"><br></font></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><font color="#212121"><br></font></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><font color="#212121"><br></font></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><font color="#212121">- - - - Wednesday, Apr 10, 2019 - - - -</font></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><font color="#212121"><br></font></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><font color="#212121"><div style="color:rgb(0,0,0)">MOPA (Models of Peano Arithmetic)</div><div style="color:rgb(0,0,0)">CUNY Graduate Center, Room 4213.03 (Math Thesis Room)</div><div style="color:rgb(0,0,0)">Wednesday, April 10, 6:30-8:00pm</div><div style="color:rgb(0,0,0)">Erez Shochat, St. Francis College</div><div style="color:rgb(0,0,0)">Introduction to Loeb Measure</div><div style="color:rgb(0,0,0)"><br></div><div style="color:rgb(0,0,0)">In this talk we will outline results and facts from nonstandard analysis and introduce the concept of Loeb Measure.</div></font></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><font color="#212121"><br></font></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><font color="#212121"><br></font></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><font color="#212121"><br></font></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><font color="#212121"><br></font></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><font color="#212121">- - - - Thursday, Apr 11, 2019 - - - -</font></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><font color="#212121"><br></font></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><font color="#212121"><br></font></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><font color="#212121"><br></font></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><font color="#212121">- - - - Friday, Apr 12, 2019 - - - -</font></div></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><font color="#212121"><br></font></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><div style="color:rgb(33,33,33)">Set Theory Seminar</div><div style="color:rgb(33,33,33)">CUNY Graduate Center, Room 6417</div><div><font color="#000000">Friday, April 12, 10:00-11:45am</font></div><div><font color="#000000"><span style="font-family:&quot;Open Sans&quot;,&quot;Helvetica Neue&quot;,Helvetica,Arial,sans-serif">Jonas Reitz, CUNY</span><br style="box-sizing:border-box;font-family:&quot;Open Sans&quot;,&quot;Helvetica Neue&quot;,Helvetica,Arial,sans-serif"></font><div><font face="Open Sans, Helvetica Neue, Helvetica, Arial, sans-serif" color="#000000"><b>Generalized Cohen Iterations</b></font></div><div><font face="Open Sans, Helvetica Neue, Helvetica, Arial, sans-serif" color="#000000"><br></font></div><div><font face="Open Sans, Helvetica Neue, Helvetica, Arial, sans-serif" color="#000000">Adding Cohen subsets to each of a class of cardinals in turn is a common construction in set theory, and underlies many fundamental results.  The construction comes in two basic flavors, products (as in Easton’s Theorem on the powers of regular cardinals) and iterations (forcing the GCH).  These flavors are  apparently quite similar, forcing at stage kappa to add subsets via the Cohen partial order Add(kappa,lambda).  They differ only in the universe over which Add(kappa,lambda) is defined - in the case of products the ground model poset is used at each stage, whereas in typical iterations the poset is taken from the partial extension up to kappa.  In this talk I will consider an alternative, in which we allow Add(kappa,lambda) to be defined over an arbitrary inner model (lying between the ground model and the extension up to kappa) at each stage.   These generalized Cohen iterations are ZFC-preserving, although neither the proof for products nor for traditional iterations transfers directly.   They allow constructions such as class iterations of class products of Cohen forcing, with applications including new work with Kameryn Williams on iterating the Mantle.</font></div></div><div><font color="#000000"><br></font></div><div><font color="#000000"><br></font></div><div><font color="#000000"><br></font></div><div><font color="#000000">Model Theory Seminar</font></div><div style="color:rgb(33,33,33)">CUNY Graduate Center, Room 6417</div><div style="color:rgb(33,33,33)">Friday, April 12, 12:30-2:00pm</div><div style="color:rgb(33,33,33)"><strong style="box-sizing:border-box;color:rgb(96,108,113);font-family:&quot;Open Sans&quot;,&quot;Helvetica Neue&quot;,Helvetica,Arial,sans-serif"><a href="http://sites.nd.edu/rachaelalvir/" target="_blank" style="font-family:arial,sans-serif;color:rgb(30,107,184);text-decoration-line:none;box-sizing:border-box;background-color:transparent;margin-top:0px">Rachael Alvir</a></strong><span style="color:rgb(96,108,113);font-family:&quot;Open Sans&quot;,&quot;Helvetica Neue&quot;,Helvetica,Arial,sans-serif">, University of Notre Dame</span><br style="box-sizing:border-box;color:rgb(96,108,113);font-family:&quot;Open Sans&quot;,&quot;Helvetica Neue&quot;,Helvetica,Arial,sans-serif"><p style="box-sizing:border-box;margin-bottom:1em;margin-top:0px;color:rgb(96,108,113);font-family:&quot;Open Sans&quot;,&quot;Helvetica Neue&quot;,Helvetica,Arial,sans-serif"><strong style="box-sizing:border-box">Scott Sentences of Scattered Linear Orders</strong><br style="box-sizing:border-box"></p><p style="box-sizing:border-box;margin-bottom:1em;color:rgb(96,108,113);font-family:&quot;Open Sans&quot;,&quot;Helvetica Neue&quot;,Helvetica,Arial,sans-serif">The logic <span id="gmail-MathJax-Element-1-Frame" class="gmail-mjx-chtml gmail-MathJax_CHTML" tabindex="0" style="box-sizing:border-box;display:inline-block;line-height:0;word-spacing:normal;white-space:nowrap;float:none;direction:ltr;max-width:none;max-height:none;min-width:0px;min-height:0px;border:0px;margin:0px;padding:1px 0px"><span id="gmail-MJXc-Node-1" class="gmail-mjx-math" style="box-sizing:border-box;display:inline-block;border-collapse:separate;border-spacing:0px;margin-top:0px"><span id="gmail-MJXc-Node-2" class="gmail-mjx-mrow" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-MJXc-Node-3" class="gmail-mjx-msubsup" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-mjx-base" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-MJXc-Node-4" class="gmail-mjx-mi" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-math-I" style="display:block;white-space:pre;font-family:MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw;margin-top:0px;padding-top:0.472em;padding-bottom:0.284em;box-sizing:content-box">L</span></span></span><span class="gmail-mjx-sub" style="display:inline-block;vertical-align:-0.212em;padding-right:0.071em;box-sizing:content-box"><span id="gmail-MJXc-Node-5" class="gmail-mjx-texatom" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-MJXc-Node-6" class="gmail-mjx-mrow" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-MJXc-Node-7" class="gmail-mjx-msubsup" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-mjx-base" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-MJXc-Node-8" class="gmail-mjx-mi" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-math-I" style="display:block;white-space:pre;font-family:MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw;margin-top:0px;padding-top:0.237em;padding-bottom:0.284em;box-sizing:content-box">ω</span></span></span><span class="gmail-mjx-sub" style="display:inline-block;vertical-align:-0.212em;padding-right:0.071em;box-sizing:content-box"><span id="gmail-MJXc-Node-9" class="gmail-mjx-mn" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-main-R" style="display:block;white-space:pre;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.378em;padding-bottom:0.331em;box-sizing:content-box">1</span></span></span></span><span id="gmail-MJXc-Node-10" class="gmail-mjx-mi" style="display:inline-block;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-math-I" style="display:block;white-space:pre;font-family:MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw;margin-top:0px;padding-top:0.237em;padding-bottom:0.284em;box-sizing:content-box">ω</span></span></span></span></span></span></span></span><span class="gmail-MJX_Assistive_MathML" style="box-sizing:border-box;padding:1px 0px 0px;border:0px;height:1px;width:1px;overflow:hidden;display:block">L<span class="gmail-MJX-TeXAtom-ORD">ω1ω</span></span></span> is obtained by closing finitary first-order logic under countable disjunction and conjunction. There is a kind of normal form for such sentences. For any structure <span id="gmail-MathJax-Element-2-Frame" class="gmail-mjx-chtml gmail-MathJax_CHTML" tabindex="0" style="box-sizing:border-box;display:inline-block;line-height:0;word-spacing:normal;white-space:nowrap;float:none;direction:ltr;max-width:none;max-height:none;min-width:0px;min-height:0px;border:0px;margin:0px;padding:1px 0px"><span id="gmail-MJXc-Node-11" class="gmail-mjx-math" style="box-sizing:border-box;display:inline-block;border-collapse:separate;border-spacing:0px;margin-top:0px"><span id="gmail-MJXc-Node-12" class="gmail-mjx-mrow" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-MJXc-Node-13" class="gmail-mjx-texatom" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-MJXc-Node-14" class="gmail-mjx-mrow" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-MJXc-Node-15" class="gmail-mjx-mi" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-cal-R" style="display:block;white-space:pre;font-family:MJXc-TeX-cal-R,MJXc-TeX-cal-Rw;margin-top:0px;padding-top:0.472em;padding-bottom:0.378em;padding-right:0.021em;box-sizing:content-box">A</span></span></span></span></span></span><span class="gmail-MJX_Assistive_MathML" style="box-sizing:border-box;padding:1px 0px 0px;border:0px;height:1px;width:1px;overflow:hidden;display:block"><span class="gmail-MJX-TeXAtom-ORD"><span class="gmail-MJX-tex-caligraphic">A</span></span></span></span> there is a sentence of <span id="gmail-MathJax-Element-3-Frame" class="gmail-mjx-chtml gmail-MathJax_CHTML" tabindex="0" style="box-sizing:border-box;display:inline-block;line-height:0;word-spacing:normal;white-space:nowrap;float:none;direction:ltr;max-width:none;max-height:none;min-width:0px;min-height:0px;border:0px;margin:0px;padding:1px 0px"><span id="gmail-MJXc-Node-16" class="gmail-mjx-math" style="box-sizing:border-box;display:inline-block;border-collapse:separate;border-spacing:0px;margin-top:0px"><span id="gmail-MJXc-Node-17" class="gmail-mjx-mrow" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-MJXc-Node-18" class="gmail-mjx-msubsup" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-mjx-base" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-MJXc-Node-19" class="gmail-mjx-mi" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-math-I" style="display:block;white-space:pre;font-family:MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw;margin-top:0px;padding-top:0.472em;padding-bottom:0.284em;box-sizing:content-box">L</span></span></span><span class="gmail-mjx-sub" style="display:inline-block;vertical-align:-0.212em;padding-right:0.071em;box-sizing:content-box"><span id="gmail-MJXc-Node-20" class="gmail-mjx-texatom" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-MJXc-Node-21" class="gmail-mjx-mrow" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-MJXc-Node-22" class="gmail-mjx-msubsup" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-mjx-base" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-MJXc-Node-23" class="gmail-mjx-mi" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-math-I" style="display:block;white-space:pre;font-family:MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw;margin-top:0px;padding-top:0.237em;padding-bottom:0.284em;box-sizing:content-box">ω</span></span></span><span class="gmail-mjx-sub" style="display:inline-block;vertical-align:-0.212em;padding-right:0.071em;box-sizing:content-box"><span id="gmail-MJXc-Node-24" class="gmail-mjx-mn" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-main-R" style="display:block;white-space:pre;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.378em;padding-bottom:0.331em;box-sizing:content-box">1</span></span></span></span><span id="gmail-MJXc-Node-25" class="gmail-mjx-mi" style="display:inline-block;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-math-I" style="display:block;white-space:pre;font-family:MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw;margin-top:0px;padding-top:0.237em;padding-bottom:0.284em;box-sizing:content-box">ω</span></span></span></span></span></span></span></span><span class="gmail-MJX_Assistive_MathML" style="box-sizing:border-box;padding:1px 0px 0px;border:0px;height:1px;width:1px;overflow:hidden;display:block">L<span class="gmail-MJX-TeXAtom-ORD">ω1ω</span></span></span>, known as its <em style="box-sizing:border-box">Scott sentence</em>, which describes <span id="gmail-MathJax-Element-4-Frame" class="gmail-mjx-chtml gmail-MathJax_CHTML" tabindex="0" style="box-sizing:border-box;display:inline-block;line-height:0;word-spacing:normal;white-space:nowrap;float:none;direction:ltr;max-width:none;max-height:none;min-width:0px;min-height:0px;border:0px;margin:0px;padding:1px 0px"><span id="gmail-MJXc-Node-26" class="gmail-mjx-math" style="box-sizing:border-box;display:inline-block;border-collapse:separate;border-spacing:0px;margin-top:0px"><span id="gmail-MJXc-Node-27" class="gmail-mjx-mrow" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-MJXc-Node-28" class="gmail-mjx-texatom" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-MJXc-Node-29" class="gmail-mjx-mrow" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-MJXc-Node-30" class="gmail-mjx-mi" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-cal-R" style="display:block;white-space:pre;font-family:MJXc-TeX-cal-R,MJXc-TeX-cal-Rw;margin-top:0px;padding-top:0.472em;padding-bottom:0.378em;padding-right:0.021em;box-sizing:content-box">A</span></span></span></span></span></span><span class="gmail-MJX_Assistive_MathML" style="box-sizing:border-box;padding:1px 0px 0px;border:0px;height:1px;width:1px;overflow:hidden;display:block"><span class="gmail-MJX-TeXAtom-ORD"><span class="gmail-MJX-tex-caligraphic">A</span></span></span></span> up to isomorphism among countable structures. Given a countable scattered linear order <span id="gmail-MathJax-Element-5-Frame" class="gmail-mjx-chtml gmail-MathJax_CHTML" tabindex="0" style="box-sizing:border-box;display:inline-block;line-height:0;word-spacing:normal;white-space:nowrap;float:none;direction:ltr;max-width:none;max-height:none;min-width:0px;min-height:0px;border:0px;margin:0px;padding:1px 0px"><span id="gmail-MJXc-Node-31" class="gmail-mjx-math" style="box-sizing:border-box;display:inline-block;border-collapse:separate;border-spacing:0px;margin-top:0px"><span id="gmail-MJXc-Node-32" class="gmail-mjx-mrow" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-MJXc-Node-33" class="gmail-mjx-mi" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-math-I" style="display:block;white-space:pre;font-family:MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw;margin-top:0px;padding-top:0.472em;padding-bottom:0.284em;box-sizing:content-box">L</span></span></span></span><span class="gmail-MJX_Assistive_MathML" style="box-sizing:border-box;padding:1px 0px 0px;border:0px;height:1px;width:1px;overflow:hidden;display:block">L</span></span> of Hausdorff rank <span id="gmail-MathJax-Element-6-Frame" class="gmail-mjx-chtml gmail-MathJax_CHTML" tabindex="0" style="box-sizing:border-box;display:inline-block;line-height:0;word-spacing:normal;white-space:nowrap;float:none;direction:ltr;max-width:none;max-height:none;min-width:0px;min-height:0px;border:0px;margin:0px;padding:1px 0px"><span id="gmail-MJXc-Node-34" class="gmail-mjx-math" style="box-sizing:border-box;display:inline-block;border-collapse:separate;border-spacing:0px;margin-top:0px"><span id="gmail-MJXc-Node-35" class="gmail-mjx-mrow" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-MJXc-Node-36" class="gmail-mjx-mi" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-math-I" style="display:block;white-space:pre;font-family:MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw;margin-top:0px;padding-top:0.237em;padding-bottom:0.284em;box-sizing:content-box">α</span></span><span id="gmail-MJXc-Node-37" class="gmail-mjx-mo gmail-MJXc-space3" style="display:inline-block;margin-left:0.278em;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-main-R" style="display:block;white-space:pre;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.237em;padding-bottom:0.378em;box-sizing:content-box">&lt;</span></span><span id="gmail-MJXc-Node-38" class="gmail-mjx-msubsup gmail-MJXc-space3" style="display:inline-block;margin-left:0.278em;box-sizing:content-box"><span class="gmail-mjx-base" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-MJXc-Node-39" class="gmail-mjx-mi" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-math-I" style="display:block;white-space:pre;font-family:MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw;margin-top:0px;padding-top:0.237em;padding-bottom:0.284em;box-sizing:content-box">ω</span></span></span><span class="gmail-mjx-sub" style="display:inline-block;vertical-align:-0.212em;padding-right:0.071em;box-sizing:content-box"><span id="gmail-MJXc-Node-40" class="gmail-mjx-mn" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-main-R" style="display:block;white-space:pre;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.378em;padding-bottom:0.331em;box-sizing:content-box">1</span></span></span></span></span></span><span class="gmail-MJX_Assistive_MathML" style="box-sizing:border-box;padding:1px 0px 0px;border:0px;height:1px;width:1px;overflow:hidden;display:block">α&lt;ω1</span></span>, we show that it has a <span id="gmail-MathJax-Element-7-Frame" class="gmail-mjx-chtml gmail-MathJax_CHTML" tabindex="0" style="box-sizing:border-box;display:inline-block;line-height:0;word-spacing:normal;white-space:nowrap;float:none;direction:ltr;max-width:none;max-height:none;min-width:0px;min-height:0px;border:0px;margin:0px;padding:1px 0px"><span id="gmail-MJXc-Node-41" class="gmail-mjx-math" style="box-sizing:border-box;display:inline-block;border-collapse:separate;border-spacing:0px;margin-top:0px"><span id="gmail-MJXc-Node-42" class="gmail-mjx-mrow" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-MJXc-Node-43" class="gmail-mjx-mi" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-math-I" style="display:block;white-space:pre;font-family:MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw;margin-top:0px;padding-top:0.472em;padding-bottom:0.284em;padding-right:0.003em;box-sizing:content-box">d</span></span></span></span><span class="gmail-MJX_Assistive_MathML" style="box-sizing:border-box;padding:1px 0px 0px;border:0px;height:1px;width:1px;overflow:hidden;display:block">d</span></span>-<span id="gmail-MathJax-Element-8-Frame" class="gmail-mjx-chtml gmail-MathJax_CHTML" tabindex="0" style="box-sizing:border-box;display:inline-block;line-height:0;word-spacing:normal;white-space:nowrap;float:none;direction:ltr;max-width:none;max-height:none;min-width:0px;min-height:0px;border:0px;margin:0px;padding:1px 0px"><span id="gmail-MJXc-Node-44" class="gmail-mjx-math" style="box-sizing:border-box;display:inline-block;border-collapse:separate;border-spacing:0px;margin-top:0px"><span id="gmail-MJXc-Node-45" class="gmail-mjx-mrow" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-MJXc-Node-46" class="gmail-mjx-msubsup" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-mjx-base" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-MJXc-Node-47" class="gmail-mjx-mi" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-main-R" style="display:block;white-space:pre;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.378em;padding-bottom:0.331em;box-sizing:content-box">Σ</span></span></span><span class="gmail-mjx-sub" style="display:inline-block;vertical-align:-0.212em;padding-right:0.071em;box-sizing:content-box"><span id="gmail-MJXc-Node-48" class="gmail-mjx-texatom" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-MJXc-Node-49" class="gmail-mjx-mrow" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-MJXc-Node-50" class="gmail-mjx-mn" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-main-R" style="display:block;white-space:pre;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.378em;padding-bottom:0.331em;box-sizing:content-box">2</span></span><span id="gmail-MJXc-Node-51" class="gmail-mjx-mi" style="display:inline-block;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-math-I" style="display:block;white-space:pre;font-family:MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw;margin-top:0px;padding-top:0.237em;padding-bottom:0.284em;box-sizing:content-box">α</span></span><span id="gmail-MJXc-Node-52" class="gmail-mjx-mo" style="display:inline-block;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-main-R" style="display:block;white-space:pre;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.284em;padding-bottom:0.425em;box-sizing:content-box">+</span></span><span id="gmail-MJXc-Node-53" class="gmail-mjx-mn" style="display:inline-block;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-main-R" style="display:block;white-space:pre;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.378em;padding-bottom:0.331em;box-sizing:content-box">1</span></span></span></span></span></span></span></span><span class="gmail-MJX_Assistive_MathML" style="box-sizing:border-box;padding:1px 0px 0px;border:0px;height:1px;width:1px;overflow:hidden;display:block">Σ<span class="gmail-MJX-TeXAtom-ORD">2α+1</span></span></span> Scott sentence. From Ash&#39;s calculation of the back and forth relations for all countable well-orders, we obtain that this upper bound is tight, i.e., for every <span id="gmail-MathJax-Element-9-Frame" class="gmail-mjx-chtml gmail-MathJax_CHTML" tabindex="0" style="box-sizing:border-box;display:inline-block;line-height:0;word-spacing:normal;white-space:nowrap;float:none;direction:ltr;max-width:none;max-height:none;min-width:0px;min-height:0px;border:0px;margin:0px;padding:1px 0px"><span id="gmail-MJXc-Node-54" class="gmail-mjx-math" style="box-sizing:border-box;display:inline-block;border-collapse:separate;border-spacing:0px;margin-top:0px"><span id="gmail-MJXc-Node-55" class="gmail-mjx-mrow" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-MJXc-Node-56" class="gmail-mjx-mi" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-math-I" style="display:block;white-space:pre;font-family:MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw;margin-top:0px;padding-top:0.237em;padding-bottom:0.284em;box-sizing:content-box">α</span></span><span id="gmail-MJXc-Node-57" class="gmail-mjx-mo gmail-MJXc-space3" style="display:inline-block;margin-left:0.278em;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-main-R" style="display:block;white-space:pre;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.237em;padding-bottom:0.378em;box-sizing:content-box">&lt;</span></span><span id="gmail-MJXc-Node-58" class="gmail-mjx-msubsup gmail-MJXc-space3" style="display:inline-block;margin-left:0.278em;box-sizing:content-box"><span class="gmail-mjx-base" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-MJXc-Node-59" class="gmail-mjx-mi" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-math-I" style="display:block;white-space:pre;font-family:MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw;margin-top:0px;padding-top:0.237em;padding-bottom:0.284em;box-sizing:content-box">ω</span></span></span><span class="gmail-mjx-sub" style="display:inline-block;vertical-align:-0.212em;padding-right:0.071em;box-sizing:content-box"><span id="gmail-MJXc-Node-60" class="gmail-mjx-mn" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-main-R" style="display:block;white-space:pre;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.378em;padding-bottom:0.331em;box-sizing:content-box">1</span></span></span></span></span></span><span class="gmail-MJX_Assistive_MathML" style="box-sizing:border-box;padding:1px 0px 0px;border:0px;height:1px;width:1px;overflow:hidden;display:block">α&lt;ω1</span></span> there is a linear order whose optimal Scott sentence has this complexity.</p></div><div style="color:rgb(33,33,33)"><br></div><div style="color:rgb(33,33,33)"><br></div><div style="color:rgb(33,33,33)"><br></div><div style="color:rgb(33,33,33)">Logic Workshop</div><div style="color:rgb(33,33,33)">CUNY Graduate Center, Room 6417</div><div style="color:rgb(33,33,33)">Friday, April 12, 2:00-3:30pm</div><div style="color:rgb(33,33,33)"><p style="font-family:&quot;Open Sans&quot;,&quot;Helvetica Neue&quot;,Helvetica,Arial,sans-serif;box-sizing:border-box;margin-bottom:1em;margin-top:0px;color:rgb(96,108,113)"><strong style="box-sizing:border-box"><a href="https://math.nd.edu/people/faculty/julia-f-knight/" target="_blank" style="font-family:arial,sans-serif;color:rgb(30,107,184);text-decoration-line:none;box-sizing:border-box;background-color:transparent;margin-top:0px">Julia Knight</a></strong>, University of Notre Dame<br style="box-sizing:border-box"><strong style="box-sizing:border-box">Coding structures</strong><br style="box-sizing:border-box"></p><p style="font-family:&quot;Open Sans&quot;,&quot;Helvetica Neue&quot;,Helvetica,Arial,sans-serif;box-sizing:border-box;margin-bottom:1em;color:rgb(96,108,113)">A <em style="box-sizing:border-box;margin-top:0px">Turing computable embedding</em> is a Turing operator that maps one class of structures to another so as to preserve isomorphism. The embedding codes the input structure in the output structure. It is interesting when there is an effective decoding. It is also interesting when the decoding is very difficult. Recently, Harrison-Trainor, Melnikov, R. Miller, and Montalbán have defined very general notions of interpretation, in which the interpreting formulas have no fixed arity. Uniformly defined interpretations give us decoding. I will discuss some known Turing computable embeddings, looking for uniform interpretations that yield effective, or Borel, decoding.</p><ol style="box-sizing:border-box;margin-top:0px;color:rgb(96,108,113);font-family:&quot;Open Sans&quot;,&quot;Helvetica Neue&quot;,Helvetica,Arial,sans-serif"><li style="box-sizing:border-box;margin-top:0px">Marker&#39;s embedding of directed graphs in undirected graphs,</li><li style="box-sizing:border-box">Mal&#39;tsev&#39;s embedding of fields in groups,</li><li style="box-sizing:border-box">Friedman and Stanley&#39;s embedding of graphs in linear orderings.</li></ol><span style="color:rgb(96,108,113);font-family:&quot;Open Sans&quot;,&quot;Helvetica Neue&quot;,Helvetica,Arial,sans-serif">The first two embeddings come with uniform &#39;effective&#39; interpretations, which give uniform effective decoding. For the third, we do not even have uniform interpretation via </span><span id="gmail-m_2794932661880136167gmail-MathJax-Element-1-Frame" style="box-sizing:border-box;display:inline-block;line-height:0;white-space:nowrap;float:none;direction:ltr;max-width:none;max-height:none;min-width:0px;min-height:0px;border:0px;margin:0px;padding:1px 0px;color:rgb(96,108,113);font-family:&quot;Open Sans&quot;,&quot;Helvetica Neue&quot;,Helvetica,Arial,sans-serif"><span id="gmail-m_2794932661880136167gmail-MJXc-Node-1" style="box-sizing:border-box;display:inline-block;border-collapse:separate;border-spacing:0px;margin-top:0px"><span id="gmail-m_2794932661880136167gmail-MJXc-Node-2" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_2794932661880136167gmail-MJXc-Node-3" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_2794932661880136167gmail-MJXc-Node-4" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw;margin-top:0px;padding-top:0.472em;padding-bottom:0.284em;box-sizing:content-box">L</span></span></span><span style="display:inline-block;vertical-align:-0.212em;padding-right:0.071em;box-sizing:content-box"><span id="gmail-m_2794932661880136167gmail-MJXc-Node-5" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_2794932661880136167gmail-MJXc-Node-6" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_2794932661880136167gmail-MJXc-Node-7" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_2794932661880136167gmail-MJXc-Node-8" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw;margin-top:0px;padding-top:0.237em;padding-bottom:0.284em;box-sizing:content-box">ω</span></span></span><span style="display:inline-block;vertical-align:-0.212em;padding-right:0.071em;box-sizing:content-box"><span id="gmail-m_2794932661880136167gmail-MJXc-Node-9" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.378em;padding-bottom:0.331em;box-sizing:content-box">1</span></span></span></span><span id="gmail-m_2794932661880136167gmail-MJXc-Node-10" style="display:inline-block;box-sizing:content-box"><span style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw;margin-top:0px;padding-top:0.237em;padding-bottom:0.284em;box-sizing:content-box">ω</span></span></span></span></span></span></span></span><span style="box-sizing:border-box;padding:1px 0px 0px;border:0px;height:1px;width:1px;overflow:hidden;display:block">Lω1ω</span></span><span style="color:rgb(96,108,113);font-family:&quot;Open Sans&quot;,&quot;Helvetica Neue&quot;,Helvetica,Arial,sans-serif"> formulas.</span><br></div></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><font color="#212121"><br></font></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><font color="#212121"><br></font></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><font color="#212121"><br></font></div><div><font color="#212121"><div><font face="arial, sans-serif">Next Week in Logic at CUNY:</font></div><div><font face="arial, sans-serif"><br></font></div><div><font face="arial, sans-serif">- - - - Monday, Apr 15, 2019 - - - -</font></div><div><font face="arial, sans-serif"><br></font></div><div><font face="arial, sans-serif"><div>Logic and Metaphysics Workshop</div><div>Date: Monday, April 15th, 4.15-6.15  <br></div><div>Place: Room 7314, CUNY Graduate Center <br></div><div>Speaker: Jenn McDonald (CUNY)<br></div><div>Title: Structural Counterfactuals and the Importation Problem<br></div><div><br></div><div>Abstract: Structural causal models lend themselves to an analysis of counterfactuals – a structural semantics of counterfactuals. The basic idea is that a causal model allows for the clear and precise evaluation of any counterfactual encoded by it. Many argue that a structural semantics is superior to a more traditional similarity semantics, in part due to the latter’s independence from any notion of similarity(Galles &amp; Pearl, 1998; Gallow, 2016; Hiddleston, 2005; Hitchcock, 2018; Pearl, 2000; Starr, 2019). I argue, though, that this is too quick. A similarity semantics employs the notion of similarity to answer what Priest (2018) calls the importation problem– the question of what information is to be held fixed in a counterfactual evaluation. I argue that where similarity semantics relies on an unarticulated notion of similarity, a structural semantics relies on an unarticulated notion of aptness. The superiority of structural semantics depends on its ability to deliver on a principled guide to determining which model(s) is apt. In this talk I go some way towards providing this guide.</div><div><br></div><div><br></div><div><br></div></font></div><div><font face="arial, sans-serif"><br></font></div><div><font face="arial, sans-serif">- - - - Tuesday, Apr 16, 2019 - - - -</font></div><div><font face="arial, sans-serif"><br></font></div><div><font face="arial, sans-serif"><br></font></div><div><font face="arial, sans-serif"><br></font></div><div><font face="arial, sans-serif">- - - - Wednesday, Apr 17, 2019 - - - -</font></div><div><font face="arial, sans-serif"><br></font></div><div><font face="arial, sans-serif"><div>The New York City Category Theory Seminar</div><div>Department of Computer Science</div><div>Department of Mathematics </div><div>The Graduate Center of The City University of New York </div><div>Speaker:     Tibor Beke, University of Massachusetts, Lowell.</div><div>Date and Time:     Wednesday April 17, 2019, 7:00 - 8:30 PM., Room 6417.<br></div><div>Title:    Schanuel functors and the Grothendieck (semi)ring of some theories.<br></div><div><br></div><div>Abstract: In a little known article, Schanuel defines a functor from semirings to idempotent semirings and a notion of dimension that is not linearly ordered. He uses it to give an elegant presentation of the Grothendieck semiring of semi-algebraic sets, from which the (much better known) structure of the Grothendieck ring of semi-algebraic sets easily follows. I will review his work and related results on the Grothendieck semiring of algebraically closed fields and similar geometric structures.</div><div><br></div><div><br></div><div><br></div></font></div><div><font face="arial, sans-serif"><br></font></div><div><font face="arial, sans-serif">- - - - Thursday, Apr 18, 2019 - - - -</font></div><div><font face="arial, sans-serif"><br></font></div><div><font face="arial, sans-serif"><br></font></div><div><font face="arial, sans-serif"><br></font></div><div><font face="arial, sans-serif">- - - - Friday, Apr 19, 2019 - - - -</font></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><br></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif">*** CUNY Spring Break April 19-28 ***</div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><br></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><br></div></font></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><font color="#212121">- - - - Other Logic News - - - -</font></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><br></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><font color="#212121"><br></font></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><font color="#212121"><br></font></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><font color="#212121">- - - - Web Site - - - -</font></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><font color="#212121"><br></font></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><font color="#212121">Our new website is located at:  <a href="https://nylogic.github.io/" target="_blank" style="text-decoration-line:none">https://nylogic.github.io/</a></font></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><font color="#212121">Many thanks to Victoria Gitman for her development work!</font></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><font color="#212121"><br></font></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><font color="#212121">--------  ADMINISTRIVIA  --------</font></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><font color="#212121"><br></font></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><font color="#212121">To subscribe/unsubscribe to this list, please email your request to <a href="mailto:jreitz@nylogic.org" target="_blank" style="text-decoration-line:none">jreitz@nylogic.org</a>.</font></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><font color="#212121"><br></font></div><div style="color:rgb(0,0,0);font-family:arial,sans-serif"><font color="#212121">If you have a logic-related event that you would like included in future mailings, please email <a href="mailto:jreitz@nylogic.org" target="_blank" style="text-decoration-line:none">jreitz@nylogic.org</a>. </font></div></div></div></div></div></div></div></div></div>
56
4/4/2019 20:26:01NUS Logic SeminarGao Ziyuan: Finitely distinguishable erasing pattern languages with bounded
variable frequency
Invitation to the Logic Seminar at the National University of Singapore

Date: Wednesday, 10 April 2019, 17:00 hrs

Room: S17#04-06, Department of Mathematics, NUS

Speaker: Gao Ziyuan

Title: Finitely distinguishable erasing pattern languages with bounded
variable frequency

URL: http://www.comp.nus.edu.sg/~fstephan/logicseminar.html

Abstract: A pattern is a nonempty string made up of symbols from two
disjoint sets, an alphabet $\Sigma$ of constant symbols and a
countably infinite set $X$ of variables. The erasing pattern
language $L(\pi)$ generated by a pattern $\pi$ is the set of all words
over $\Sigma$ obtained by replacing all the variables of $\pi$ with
arbitrary words over $\Sigma$, with the proviso that identical
variables be replaced with the same string. In particular,
patterns allow for repetitions of variable substrings, a feature known
as backreferencing in programming languages. A distinguishing
set for any pattern $\pi$ w.r.t. a class $\Pi$ of patterns containing
$\pi$ is a set $D$ of words over $\Sigma$ that distinguishes $\pi$
from every other pattern $\tau$ in $\Pi$ with $L(\tau) \neq L(\pi)$
(i.e., $D \cap L(\pi) \neq D \cap L(\tau)$ whenever $L(\tau) \neq
L(\pi)$).

Two basic types of counting problems will be discussed: first, given
any pattern $\pi$ belonging to a class $\Pi$ of patterns, does $\pi$
have a finite distinguishing set w.r.t. $\Pi$; second, what is the
minimum size of a distinguishing set for $\pi$ w.r.t. $\Pi$?
The latter quantity gives a measure of the information complexity of
$\pi$ w.r.t. $\Pi$, and it is known as the (classical) teaching
dimension of $\pi$ w.r.t. $\Pi$ in computational learning theory.

We also consider the problem of determining, for any given strict
partial order $\prec$ on $\Pi$ (up to equivalence of patterns) and any
pattern $\pi$ belonging to $\Pi$, the minimum size of a distinguishing
set for $\pi$ w.r.t. the subclass of all $\tau$ in $\Pi$ such that
$\tau \not\prec \pi$; this quantity is known as the preference-based
teaching dimension of $\pi$ w.r.t. $(\Pi,\prec)$. We study
how the classical and preference-based teaching dimensions of patterns
w.r.t. various "naturally" defined classes of patterns and strict
partial orders depend on the alphabet size and the maximum number of
variable repetitions in any single pattern belonging to the class.

Gao Ziyuanhttp://www.comp.nus.edu.sg/~fstephan/logicseminar.html
57
Wed, 03 Apr 2019 14:20:37Wrocław University of TechnologySzymon Żeberski: Mycielski theorem and Miller treesTuesday, April 9, 2019, 17:15
Wrocław University of Technology, 215 D-1

Speaker: Szymon Żeberski (Wrocław University of Technology,)

Title: Mycielski theorem and Miller trees

Abstract:

The classical Mycielski theorem says that for comeager $A\subseteq [0,1]^2$ one can find a perfect set $P$ such that $P\times P\subseteq A\cup\Delta$. (The same is true if we start with $A$ of measure 1.)

We will discuss how far this can be generalized if we replace perfect set by superperfect set, i.e a body of a Miller tree.

It turns out that there is a comeager $A\subseteq (\omega^\omega)^2$ such that $A\cup \Delta$ does not contain any set of the form $M\times M$, where $M$ is superperfect.

However, for comeager $A\subseteq [0,1]^2$ one can find a perfect set $P$ and a superperfect set $M\supseteq P$ such that $P\times M\subseteq A\cup\Delta$.

We will also discuss measure case, where results are slightly different.
Szymon Zeberski9714
58
Tue, 02 Apr 2019 8:57:22Kurt Godel Research CenterDiana Carolina Montoya - The equality $\mathfrak{p} = \mathfrak{t}$ and the generalized characteristicsTalk held by Diana Carolina Montoya (KGRC) at the KGRC seminar on 2019-04-04.

<strong>Abstract</strong>: Malliaris and Shelah solved in the positive the longstanding problem of whether the two cardinal invariants <span style="color: #000000;font-family: sans-serif"><span style="font-size: 16px">$\mathfrak{p}$ </span></span>(the pseudointersection number) and <span style="color: #000000;font-family: sans-serif"><span style="font-size: 16px">$\mathfrak{t}$</span></span> (the tower number) are equal. In this talk, I will review some essential points in their proof in order to motivate the study of the analogous question for the generalized characteristics <span style="color: #000000;font-family: sans-serif"><span style="font-size: 16px">$\mathfrak{p}(\kappa)$</span></span> and <span style="color: #000000;font-family: sans-serif"><span style="font-size: 16px">$\mathfrak{t}(\kappa)$</span></span>. I will present some results of Garti regarding this generalization and finally some recent progress (joint work with Fischer, Schilhan, and Soukup) towards answering this question.
Diana Carolina Montoya9706
59
4/1/2019 21:35:35NUS Logic SeminarWu Guohua: wtt-=degrees of dre setsInvitation to the Logic Seminar at the National University of Singapore

Date: Wednesday, 3 April 2019, 17:00 hrs

Room: S17#04-06, Department of Mathematics, NUS

Speaker: Wu Guohua

Title: wtt-=degrees of dre sets

URL: http://www.comp.nus.edu.sg/~fstephan/logicseminar.html

Abstract: In this talk, I will present some recent work on wtt-degrees
of dre set. In particular, I will give a rough idea of showing the density
of this structure. I will also give a few projects in this direction.

Wu Guohuahttp://www.comp.nus.edu.sg/~fstephan/logicseminar.html
60
Mon, 01 Apr 2019 11:49:18Bristol Logic SeminarDorottya Sziraki: Open graphs and hypergraphs on definable subsets of generalized Baire spacesTuesday, April 2, 2019, 15.00
Howard House 4th Floor Seminar Room, University of Bristol

Speaker: Dorottya Sziraki (Central European University, Budapest)

Title: Open graphs and hypergraphs on definable subsets of generalized Baire spaces

Abstract:

The open graph dichotomy for a given subset $X$ of the Baire space $\omega^\omega$ is a generalization of the perfect set property for $X$ which can be viewed as the perfect set version of the Open Coloring axiom restricted to $X$. In joint work with P. Schlicht, we extend a theorem of Q. Feng's about the open graph dichotomy for definable subsets of the Baire space to the generalized Baire space $\kappa^\kappa$, where $\kappa$ is any uncountable cardinal with $\kappa^{&lt;\kappa}=\kappa$. More concretely, we show that the $\kappa$-analogue of the open graph dichotomy for all subsets of $\kappa^\kappa$ which are definable from a $\kappa$-sequence of ordinals is consistent relative to the existence of an inaccessible cardinal above $\kappa$. In the talk, I will sketch a proof of this result.

If time allows, I will also report on the progress of possible generalizations of Q. Feng's and our above mentioned theorems for certain definable infinite dimensional hypergraphs. These concern (special cases of) an infinite dimensional version of the open graph dichotomy which was recently introduced by R. Carroy, B.D. Miller and D.T. Soukup.
Dorottya Sziráki9702
61
4/1/2019 0:21:36Carnegie Mellon Logic Seminarcmu math logic seminar, coming attractions April 2 --- No seminar

April 9 --- Vahagn Aslanyan

April 16 --- Garrett Ervin
62
3/31/2019 22:43:14This Week in Logic at CUNYThis Week in Logic at CUNY<div dir="ltr"><div dir="ltr"><div dir="ltr"><div dir="ltr"><div dir="ltr"><div style="color:rgb(33,33,33)">This Week in Logic at CUNY:</div><div><div><div><font color="#212121"><br></font></div><div><font color="#212121">- - - - Monday, Apr 1, 2019 - - - -</font></div><div><font color="#212121"><br></font></div><div><font color="#212121"><div>Logic and Metaphysics Workshop</div><div>Date: Monday, April 1st, 4.15-6.15  <br></div><div>Place: Room 7314, CUNY Graduate Center <br></div><div>Speaker: Elena Ficara (Paderborn)<br></div><div>Title: What does it mean that Contradiction is the Norm of Truth?<br></div><div><br></div><div>Abstract: In my talk I argue for the thesis CT: contradiction is the norm of truth, and ask about its relevance for contemporary philosophical logic. I first present three positions in the history of philosophy that have advocated some versions of CT, namely Plato’s idea of the “dialectical gymnastics” in the Parmenides (Plato, Parmenides 136 B-E), Aristotle’s notion of dialectics in the Topics (Aristotle, Topics I, 2-3) and Metaphysics (Aristotle, Met III 1, 995 a 24-29), and Hegel’s contradictio est regula veri (Hegel Werke 2, 533), then derive from them some answers to the questions:</div><div><br></div><div>What is meant by “contradiction” in CT?</div><div>What is meant by “truth” in CT?<br></div><div>What is meant by “norm” in CT?<br></div><div><br></div><div>I will show that to examine the meaning of CT in historical perspective is useful to understand the seeds of genuine glut theories.</div><div><br></div><div><br></div></font></div><div><font color="#212121"><br></font></div><div><font color="#212121">- - - - Tuesday, Apr 2, 2019 - - - -</font></div><div><font color="#212121"><br></font></div><div><font color="#212121"><br></font></div><div><font color="#212121"><br></font></div><div><font color="#212121">- - - - Wednesday, Apr 3, 2019 - - - -</font></div><div><font color="#212121"><div><br></div><div>MOPA (Models of Peano Arithmetic)</div><div>CUNY Graduate Center, Room 4213.03 (Math Thesis Room)</div><div>Wednesday, April 3, 6:30-8:00pm</div><div><p style="box-sizing:border-box;margin-bottom:1em;margin-top:0px;color:rgb(96,108,113);font-family:&quot;Open Sans&quot;,&quot;Helvetica Neue&quot;,Helvetica,Arial,sans-serif">Michał Tomasz Godziszewski, University of Warsaw<br style="box-sizing:border-box"><strong style="box-sizing:border-box"><span id="gmail-MathJax-Element-1-Frame" class="gmail-mjx-chtml gmail-MathJax_CHTML" tabindex="0" style="box-sizing:border-box;display:inline-block;line-height:0;font-weight:normal;word-spacing:normal;white-space:nowrap;float:none;direction:ltr;max-width:none;max-height:none;min-width:0px;min-height:0px;border:0px;margin:0px;padding:1px 0px"><span id="gmail-MJXc-Node-1" class="gmail-mjx-math" style="box-sizing:border-box;display:inline-block;border-collapse:separate;border-spacing:0px;margin-top:0px"><span id="gmail-MJXc-Node-2" class="gmail-mjx-mrow" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-MJXc-Node-3" class="gmail-mjx-msubsup" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-mjx-base" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-MJXc-Node-4" class="gmail-mjx-mi" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-main-R" style="display:block;white-space:pre;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.378em;padding-bottom:0.331em;box-sizing:content-box">Π</span></span></span><span class="gmail-mjx-stack" style="display:inline-block;vertical-align:-0.315em;box-sizing:content-box"><span class="gmail-mjx-sup" style="display:block;margin-top:0px;padding-bottom:0.255em;padding-left:0px;padding-right:0.071em;box-sizing:content-box"><span id="gmail-MJXc-Node-6" class="gmail-mjx-mn" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-main-R" style="display:block;white-space:pre;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.378em;padding-bottom:0.378em;box-sizing:content-box">0</span></span></span><span class="gmail-mjx-sub" style="display:block;padding-right:0.071em;box-sizing:content-box"><span id="gmail-MJXc-Node-5" class="gmail-mjx-mn" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-main-R" style="display:block;white-space:pre;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.378em;padding-bottom:0.331em;box-sizing:content-box">1</span></span></span></span></span></span></span><span class="gmail-MJX_Assistive_MathML" style="box-sizing:border-box;padding:1px 0px 0px;border:0px;height:1px;width:1px;overflow:hidden;display:block">Π10</span></span>-computable quotient presentations of nonstandard models of arithmetic</strong><br style="box-sizing:border-box"></p><p style="box-sizing:border-box;margin-bottom:1em;color:rgb(96,108,113);font-family:&quot;Open Sans&quot;,&quot;Helvetica Neue&quot;,Helvetica,Arial,sans-serif">A computable quotient presentation of a mathematical structure <span id="gmail-MathJax-Element-2-Frame" class="gmail-mjx-chtml gmail-MathJax_CHTML" tabindex="0" style="box-sizing:border-box;display:inline-block;line-height:0;word-spacing:normal;white-space:nowrap;float:none;direction:ltr;max-width:none;max-height:none;min-width:0px;min-height:0px;border:0px;margin:0px;padding:1px 0px"><span id="gmail-MJXc-Node-7" class="gmail-mjx-math" style="box-sizing:border-box;display:inline-block;border-collapse:separate;border-spacing:0px;margin-top:0px"><span id="gmail-MJXc-Node-8" class="gmail-mjx-mrow" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-MJXc-Node-9" class="gmail-mjx-texatom" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-MJXc-Node-10" class="gmail-mjx-mrow" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-MJXc-Node-11" class="gmail-mjx-mi" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-cal-R" style="display:block;white-space:pre;font-family:MJXc-TeX-cal-R,MJXc-TeX-cal-Rw;margin-top:0px;padding-top:0.472em;padding-bottom:0.378em;padding-right:0.021em;box-sizing:content-box">A</span></span></span></span></span></span><span class="gmail-MJX_Assistive_MathML" style="box-sizing:border-box;padding:1px 0px 0px;border:0px;height:1px;width:1px;overflow:hidden;display:block"><span class="gmail-MJX-TeXAtom-ORD"><span class="gmail-MJX-tex-caligraphic">A</span></span></span></span> consists of a computable structure on the natural numbers <span id="gmail-MathJax-Element-3-Frame" class="gmail-mjx-chtml gmail-MathJax_CHTML" tabindex="0" style="box-sizing:border-box;display:inline-block;line-height:0;word-spacing:normal;white-space:nowrap;float:none;direction:ltr;max-width:none;max-height:none;min-width:0px;min-height:0px;border:0px;margin:0px;padding:1px 0px"><span id="gmail-MJXc-Node-12" class="gmail-mjx-math" style="box-sizing:border-box;display:inline-block;border-collapse:separate;border-spacing:0px;margin-top:0px"><span id="gmail-MJXc-Node-13" class="gmail-mjx-mrow" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-MJXc-Node-14" class="gmail-mjx-mo" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-main-R" style="display:block;white-space:pre;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.472em;padding-bottom:0.613em;box-sizing:content-box">⟨</span></span><span id="gmail-MJXc-Node-15" class="gmail-mjx-texatom" style="display:inline-block;box-sizing:content-box"><span id="gmail-MJXc-Node-16" class="gmail-mjx-mrow" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-MJXc-Node-17" class="gmail-mjx-mi" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-ams-R" style="display:block;white-space:pre;font-family:MJXc-TeX-ams-R,MJXc-TeX-ams-Rw;margin-top:0px;padding-top:0.425em;padding-bottom:0.331em;box-sizing:content-box">N</span></span></span></span><span id="gmail-MJXc-Node-18" class="gmail-mjx-mo" style="display:inline-block;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-main-R" style="display:block;white-space:pre;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;padding-bottom:0.519em;box-sizing:content-box">,</span></span><span id="gmail-MJXc-Node-19" class="gmail-mjx-mo gmail-MJXc-space1" style="display:inline-block;margin-left:0.167em;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-main-R" style="display:block;white-space:pre;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.19em;padding-bottom:0.331em;box-sizing:content-box">⋆</span></span><span id="gmail-MJXc-Node-20" class="gmail-mjx-mo" style="display:inline-block;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-main-R" style="display:block;white-space:pre;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;padding-bottom:0.519em;box-sizing:content-box">,</span></span><span id="gmail-MJXc-Node-21" class="gmail-mjx-mo gmail-MJXc-space1" style="display:inline-block;margin-left:0.167em;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-main-R" style="display:block;white-space:pre;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.19em;padding-bottom:0.331em;box-sizing:content-box">∗</span></span><span id="gmail-MJXc-Node-22" class="gmail-mjx-mo" style="display:inline-block;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-main-R" style="display:block;white-space:pre;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;padding-bottom:0.519em;box-sizing:content-box">,</span></span><span id="gmail-MJXc-Node-23" class="gmail-mjx-mo gmail-MJXc-space1" style="display:inline-block;margin-left:0.167em;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-main-R" style="display:block;white-space:pre;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;padding-bottom:0.331em;box-sizing:content-box">…</span></span><span id="gmail-MJXc-Node-24" class="gmail-mjx-mo" style="display:inline-block;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-main-R" style="display:block;white-space:pre;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.472em;padding-bottom:0.613em;box-sizing:content-box">⟩</span></span></span></span><span class="gmail-MJX_Assistive_MathML" style="box-sizing:border-box;padding:1px 0px 0px;border:0px;height:1px;width:1px;overflow:hidden;display:block">⟨<span class="gmail-MJX-TeXAtom-ORD">N</span>,⋆,∗,…⟩</span></span>, meaning that the operations and relations of the structure are computable, and an equivalence relation <span id="gmail-MathJax-Element-4-Frame" class="gmail-mjx-chtml gmail-MathJax_CHTML" tabindex="0" style="box-sizing:border-box;display:inline-block;line-height:0;word-spacing:normal;white-space:nowrap;float:none;direction:ltr;max-width:none;max-height:none;min-width:0px;min-height:0px;border:0px;margin:0px;padding:1px 0px"><span id="gmail-MJXc-Node-25" class="gmail-mjx-math" style="box-sizing:border-box;display:inline-block;border-collapse:separate;border-spacing:0px;margin-top:0px"><span id="gmail-MJXc-Node-26" class="gmail-mjx-mrow" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-MJXc-Node-27" class="gmail-mjx-mi" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-math-I" style="display:block;white-space:pre;font-family:MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw;margin-top:0px;padding-top:0.472em;padding-bottom:0.284em;padding-right:0.026em;box-sizing:content-box">E</span></span></span></span><span class="gmail-MJX_Assistive_MathML" style="box-sizing:border-box;padding:1px 0px 0px;border:0px;height:1px;width:1px;overflow:hidden;display:block">E</span></span> on <span id="gmail-MathJax-Element-5-Frame" class="gmail-mjx-chtml gmail-MathJax_CHTML" tabindex="0" style="box-sizing:border-box;display:inline-block;line-height:0;word-spacing:normal;white-space:nowrap;float:none;direction:ltr;max-width:none;max-height:none;min-width:0px;min-height:0px;border:0px;margin:0px;padding:1px 0px"><span id="gmail-MJXc-Node-28" class="gmail-mjx-math" style="box-sizing:border-box;display:inline-block;border-collapse:separate;border-spacing:0px;margin-top:0px"><span id="gmail-MJXc-Node-29" class="gmail-mjx-mrow" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-MJXc-Node-30" class="gmail-mjx-texatom" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-MJXc-Node-31" class="gmail-mjx-mrow" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-MJXc-Node-32" class="gmail-mjx-mi" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-ams-R" style="display:block;white-space:pre;font-family:MJXc-TeX-ams-R,MJXc-TeX-ams-Rw;margin-top:0px;padding-top:0.425em;padding-bottom:0.331em;box-sizing:content-box">N</span></span></span></span></span></span><span class="gmail-MJX_Assistive_MathML" style="box-sizing:border-box;padding:1px 0px 0px;border:0px;height:1px;width:1px;overflow:hidden;display:block"><span class="gmail-MJX-TeXAtom-ORD">N</span></span></span>, not necessarily computable but which is a congruence with respect to this structure, such that the quotient <span id="gmail-MathJax-Element-6-Frame" class="gmail-mjx-chtml gmail-MathJax_CHTML" tabindex="0" style="box-sizing:border-box;display:inline-block;line-height:0;word-spacing:normal;white-space:nowrap;float:none;direction:ltr;max-width:none;max-height:none;min-width:0px;min-height:0px;border:0px;margin:0px;padding:1px 0px"><span id="gmail-MJXc-Node-33" class="gmail-mjx-math" style="box-sizing:border-box;display:inline-block;border-collapse:separate;border-spacing:0px;margin-top:0px"><span id="gmail-MJXc-Node-34" class="gmail-mjx-mrow" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-MJXc-Node-35" class="gmail-mjx-mo" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-main-R" style="display:block;white-space:pre;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.472em;padding-bottom:0.613em;box-sizing:content-box">⟨</span></span><span id="gmail-MJXc-Node-36" class="gmail-mjx-texatom" style="display:inline-block;box-sizing:content-box"><span id="gmail-MJXc-Node-37" class="gmail-mjx-mrow" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-MJXc-Node-38" class="gmail-mjx-mi" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-ams-R" style="display:block;white-space:pre;font-family:MJXc-TeX-ams-R,MJXc-TeX-ams-Rw;margin-top:0px;padding-top:0.425em;padding-bottom:0.331em;box-sizing:content-box">N</span></span></span></span><span id="gmail-MJXc-Node-39" class="gmail-mjx-mo" style="display:inline-block;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-main-R" style="display:block;white-space:pre;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;padding-bottom:0.519em;box-sizing:content-box">,</span></span><span id="gmail-MJXc-Node-40" class="gmail-mjx-mo gmail-MJXc-space1" style="display:inline-block;margin-left:0.167em;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-main-R" style="display:block;white-space:pre;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.19em;padding-bottom:0.331em;box-sizing:content-box">⋆</span></span><span id="gmail-MJXc-Node-41" class="gmail-mjx-mo" style="display:inline-block;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-main-R" style="display:block;white-space:pre;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;padding-bottom:0.519em;box-sizing:content-box">,</span></span><span id="gmail-MJXc-Node-42" class="gmail-mjx-mo gmail-MJXc-space1" style="display:inline-block;margin-left:0.167em;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-main-R" style="display:block;white-space:pre;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.19em;padding-bottom:0.331em;box-sizing:content-box">∗</span></span><span id="gmail-MJXc-Node-43" class="gmail-mjx-mo" style="display:inline-block;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-main-R" style="display:block;white-space:pre;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;padding-bottom:0.519em;box-sizing:content-box">,</span></span><span id="gmail-MJXc-Node-44" class="gmail-mjx-mo gmail-MJXc-space1" style="display:inline-block;margin-left:0.167em;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-main-R" style="display:block;white-space:pre;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;padding-bottom:0.331em;box-sizing:content-box">…</span></span><span id="gmail-MJXc-Node-45" class="gmail-mjx-mo" style="display:inline-block;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-main-R" style="display:block;white-space:pre;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.472em;padding-bottom:0.613em;box-sizing:content-box">⟩</span></span></span></span><span class="gmail-MJX_Assistive_MathML" style="box-sizing:border-box;padding:1px 0px 0px;border:0px;height:1px;width:1px;overflow:hidden;display:block">⟨<span class="gmail-MJX-TeXAtom-ORD">N</span>,⋆,∗,…⟩</span></span> is isomorphic to the given structure <span id="gmail-MathJax-Element-7-Frame" class="gmail-mjx-chtml gmail-MathJax_CHTML" tabindex="0" style="box-sizing:border-box;display:inline-block;line-height:0;word-spacing:normal;white-space:nowrap;float:none;direction:ltr;max-width:none;max-height:none;min-width:0px;min-height:0px;border:0px;margin:0px;padding:1px 0px"><span id="gmail-MJXc-Node-46" class="gmail-mjx-math" style="box-sizing:border-box;display:inline-block;border-collapse:separate;border-spacing:0px;margin-top:0px"><span id="gmail-MJXc-Node-47" class="gmail-mjx-mrow" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-MJXc-Node-48" class="gmail-mjx-texatom" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-MJXc-Node-49" class="gmail-mjx-mrow" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-MJXc-Node-50" class="gmail-mjx-mi" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-cal-R" style="display:block;white-space:pre;font-family:MJXc-TeX-cal-R,MJXc-TeX-cal-Rw;margin-top:0px;padding-top:0.472em;padding-bottom:0.378em;padding-right:0.021em;box-sizing:content-box">A</span></span></span></span></span></span><span class="gmail-MJX_Assistive_MathML" style="box-sizing:border-box;padding:1px 0px 0px;border:0px;height:1px;width:1px;overflow:hidden;display:block"><span class="gmail-MJX-TeXAtom-ORD"><span class="gmail-MJX-tex-caligraphic">A</span></span></span></span>. Thus, one may consider computable quotient presentations of graphs, groups, orders, rings and so on.</p><p style="box-sizing:border-box;margin-bottom:1em;color:rgb(96,108,113);font-family:&quot;Open Sans&quot;,&quot;Helvetica Neue&quot;,Helvetica,Arial,sans-serif">A natural question asked by B. Khoussainov in 2016, is if the Tennenbaum Thoerem extends to the context of computable presentations of nonstandard models of arithmetic. In a joint work with J.D. Hamkins we have proved that no nonstandard model of arithmetic admits a computable quotient presentation by a computably enumerable equivalence relation on the natural numbers.</p><p style="box-sizing:border-box;margin-bottom:1em;color:rgb(96,108,113);font-family:&quot;Open Sans&quot;,&quot;Helvetica Neue&quot;,Helvetica,Arial,sans-serif">However, as it happens, there exists a nonstandard model of arithmetic admitting a computable quotient presentation by a co-c.e. equivalence relation. Actually, there are infinitely many of those. The idea of the proof consists in simulating the Henkin construction via finite injury priority argument. What is quite surprising, the construction works (i.e. injury lemma holds) by Hilbert&#39;s Basis Theorem. During the talk I&#39;ll present ideas of the proof of the latter result, which is joint work with T. Slaman and L. Harington.</p></div><div><br></div><div><br></div><div><br></div><div><br></div></font></div><div><font color="#212121"><div>The New York City Category Theory Seminar</div><div>Department of Computer Science</div><div>Department of Mathematics </div><div>Date and Time:     Wednesday April 3, 2019, 7:45 - 9:00 PM., Room 6417. (NOTICE SPECIAL TIME.)<br></div><div><div>Speaker:     Eoin Moore, The Graduate Center, CUNY.</div><div>Title:    The Arithmetical Completeness and Soundness of the Logic of Proofs.<br></div><div><br></div><div>Abstract: BHK semantics were introduced to provide a semantics of intuitionistic logic (Int) whereby the truth of a proposition was demonstrated by exhibiting a proof of it. This semi-rigorous approach had some serious difficulties with its exact formalization. The major hurdle of formalization was completed with Artemov&#39;s Logic of Proofs (LP), which provided the desired provability semantics. Int was embedded into LP, which was then embedded into Peano Arithmetic (PA). In this talk I will discuss the second inclusion. I will show that there is a complete and sound interpretation of LP into PA, in which LP proof terms are mapped to PA provability formulas.</div></div><div><br></div><div><br></div><div><br></div><div><br></div></font></div><div><font color="#212121"><br></font></div><div><font color="#212121">- - - - Thursday, Apr 4, 2019 - - - -</font></div><div><font color="#212121"><br></font></div><div><font color="#212121"><br></font></div><div><font color="#212121"><br></font></div><div><font color="#212121">- - - - Friday, Apr 5, 2019 - - - -</font></div><div><font color="#212121"><br></font></div><div><div><font color="#212121">Set Theory Seminar</font></div><div><font color="#212121">CUNY Graduate Center, Room 6417</font></div><div><font color="#212121">Friday, </font><span style="color:rgb(33,33,33)">April 5</span><font color="#212121">, 10:00-11:45am</font></div><div><span style="color:rgb(96,108,113);font-family:&quot;Open Sans&quot;,&quot;Helvetica Neue&quot;,Helvetica,Arial,sans-serif">Michał Tomasz Godziszewski, University of Warsaw</span><br style="box-sizing:border-box;color:rgb(96,108,113);font-family:&quot;Open Sans&quot;,&quot;Helvetica Neue&quot;,Helvetica,Arial,sans-serif"><p style="box-sizing:border-box;margin-bottom:1em;margin-top:0px;color:rgb(96,108,113);font-family:&quot;Open Sans&quot;,&quot;Helvetica Neue&quot;,Helvetica,Arial,sans-serif"><strong style="box-sizing:border-box">Set-Theoretic Independence and Machine Learning</strong><br style="box-sizing:border-box"></p><p style="box-sizing:border-box;margin-bottom:1em;color:rgb(96,108,113);font-family:&quot;Open Sans&quot;,&quot;Helvetica Neue&quot;,Helvetica,Arial,sans-serif">In a recent exciting paper <em style="box-sizing:border-box;margin-top:0px">Learnability can be undecidable</em> by S. Ben-David et. al. published in <em style="box-sizing:border-box">Nature Machine Intelligence</em> the authors argue that certain abstract learnability questions are undecidable by ZFC axioms. The general learning problem considered there is to find a way of choosing a finite set that maximizes a particular expected value (within a certain range of error) with an obstacle that the probability distribution is unknown, or more formally:<br style="box-sizing:border-box"><em style="box-sizing:border-box">given a family of functions <span id="gmail-MathJax-Element-1-Frame" class="gmail-mjx-chtml gmail-MathJax_CHTML" tabindex="0" style="box-sizing:border-box;display:inline-block;line-height:0;font-style:normal;word-spacing:normal;white-space:nowrap;float:none;direction:ltr;max-width:none;max-height:none;min-width:0px;min-height:0px;border:0px;margin:0px;padding:1px 0px"><span id="gmail-MJXc-Node-1" class="gmail-mjx-math" style="box-sizing:border-box;display:inline-block;border-collapse:separate;border-spacing:0px;margin-top:0px"><span id="gmail-MJXc-Node-2" class="gmail-mjx-mrow" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-MJXc-Node-3" class="gmail-mjx-texatom" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-MJXc-Node-4" class="gmail-mjx-mrow" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-MJXc-Node-5" class="gmail-mjx-mi" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-cal-R" style="display:block;white-space:pre;font-family:MJXc-TeX-cal-R,MJXc-TeX-cal-Rw;margin-top:0px;padding-top:0.425em;padding-bottom:0.331em;padding-right:0.11em;box-sizing:content-box">F</span></span></span></span></span></span><span class="gmail-MJX_Assistive_MathML" style="box-sizing:border-box;padding:1px 0px 0px;border:0px;height:1px;width:1px;overflow:hidden;display:block"><span class="gmail-MJX-TeXAtom-ORD"><span class="gmail-MJX-tex-caligraphic">F</span></span></span></span> from some fixed domain <span id="gmail-MathJax-Element-2-Frame" class="gmail-mjx-chtml gmail-MathJax_CHTML" tabindex="0" style="box-sizing:border-box;display:inline-block;line-height:0;font-style:normal;word-spacing:normal;white-space:nowrap;float:none;direction:ltr;max-width:none;max-height:none;min-width:0px;min-height:0px;border:0px;margin:0px;padding:1px 0px"><span id="gmail-MJXc-Node-6" class="gmail-mjx-math" style="box-sizing:border-box;display:inline-block;border-collapse:separate;border-spacing:0px;margin-top:0px"><span id="gmail-MJXc-Node-7" class="gmail-mjx-mrow" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-MJXc-Node-8" class="gmail-mjx-mi" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-math-I" style="display:block;white-space:pre;font-family:MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw;margin-top:0px;padding-top:0.472em;padding-bottom:0.284em;padding-right:0.024em;box-sizing:content-box">X</span></span></span></span><span class="gmail-MJX_Assistive_MathML" style="box-sizing:border-box;padding:1px 0px 0px;border:0px;height:1px;width:1px;overflow:hidden;display:block">X</span></span> to the real numbers and an unknown probability distribution <span id="gmail-MathJax-Element-3-Frame" class="gmail-mjx-chtml gmail-MathJax_CHTML" tabindex="0" style="box-sizing:border-box;display:inline-block;line-height:0;font-style:normal;word-spacing:normal;white-space:nowrap;float:none;direction:ltr;max-width:none;max-height:none;min-width:0px;min-height:0px;border:0px;margin:0px;padding:1px 0px"><span id="gmail-MJXc-Node-9" class="gmail-mjx-math" style="box-sizing:border-box;display:inline-block;border-collapse:separate;border-spacing:0px;margin-top:0px"><span id="gmail-MJXc-Node-10" class="gmail-mjx-mrow" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-MJXc-Node-11" class="gmail-mjx-mi" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-math-I" style="display:block;white-space:pre;font-family:MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw;margin-top:0px;padding-top:0.237em;padding-bottom:0.472em;box-sizing:content-box">μ</span></span></span></span><span class="gmail-MJX_Assistive_MathML" style="box-sizing:border-box;padding:1px 0px 0px;border:0px;height:1px;width:1px;overflow:hidden;display:block">μ</span></span> over <span id="gmail-MathJax-Element-4-Frame" class="gmail-mjx-chtml gmail-MathJax_CHTML" tabindex="0" style="box-sizing:border-box;display:inline-block;line-height:0;font-style:normal;word-spacing:normal;white-space:nowrap;float:none;direction:ltr;max-width:none;max-height:none;min-width:0px;min-height:0px;border:0px;margin:0px;padding:1px 0px"><span id="gmail-MJXc-Node-12" class="gmail-mjx-math" style="box-sizing:border-box;display:inline-block;border-collapse:separate;border-spacing:0px;margin-top:0px"><span id="gmail-MJXc-Node-13" class="gmail-mjx-mrow" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-MJXc-Node-14" class="gmail-mjx-mi" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-math-I" style="display:block;white-space:pre;font-family:MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw;margin-top:0px;padding-top:0.472em;padding-bottom:0.284em;padding-right:0.024em;box-sizing:content-box">X</span></span></span></span><span class="gmail-MJX_Assistive_MathML" style="box-sizing:border-box;padding:1px 0px 0px;border:0px;height:1px;width:1px;overflow:hidden;display:block">X</span></span>, find, based on a finite sample generated by <span id="gmail-MathJax-Element-5-Frame" class="gmail-mjx-chtml gmail-MathJax_CHTML" tabindex="0" style="box-sizing:border-box;display:inline-block;line-height:0;font-style:normal;word-spacing:normal;white-space:nowrap;float:none;direction:ltr;max-width:none;max-height:none;min-width:0px;min-height:0px;border:0px;margin:0px;padding:1px 0px"><span id="gmail-MJXc-Node-15" class="gmail-mjx-math" style="box-sizing:border-box;display:inline-block;border-collapse:separate;border-spacing:0px;margin-top:0px"><span id="gmail-MJXc-Node-16" class="gmail-mjx-mrow" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-MJXc-Node-17" class="gmail-mjx-mi" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-math-I" style="display:block;white-space:pre;font-family:MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw;margin-top:0px;padding-top:0.237em;padding-bottom:0.472em;box-sizing:content-box">μ</span></span></span></span><span class="gmail-MJX_Assistive_MathML" style="box-sizing:border-box;padding:1px 0px 0px;border:0px;height:1px;width:1px;overflow:hidden;display:block">μ</span></span>, a function in <span id="gmail-MathJax-Element-6-Frame" class="gmail-mjx-chtml gmail-MathJax_CHTML" tabindex="0" style="box-sizing:border-box;display:inline-block;line-height:0;font-style:normal;word-spacing:normal;white-space:nowrap;float:none;direction:ltr;max-width:none;max-height:none;min-width:0px;min-height:0px;border:0px;margin:0px;padding:1px 0px"><span id="gmail-MJXc-Node-18" class="gmail-mjx-math" style="box-sizing:border-box;display:inline-block;border-collapse:separate;border-spacing:0px;margin-top:0px"><span id="gmail-MJXc-Node-19" class="gmail-mjx-mrow" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-MJXc-Node-20" class="gmail-mjx-texatom" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-MJXc-Node-21" class="gmail-mjx-mrow" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-MJXc-Node-22" class="gmail-mjx-mi" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-cal-R" style="display:block;white-space:pre;font-family:MJXc-TeX-cal-R,MJXc-TeX-cal-Rw;margin-top:0px;padding-top:0.425em;padding-bottom:0.331em;padding-right:0.11em;box-sizing:content-box">F</span></span></span></span></span></span><span class="gmail-MJX_Assistive_MathML" style="box-sizing:border-box;padding:1px 0px 0px;border:0px;height:1px;width:1px;overflow:hidden;display:block"><span class="gmail-MJX-TeXAtom-ORD"><span class="gmail-MJX-tex-caligraphic">F</span></span></span></span>whose expectation with respect to <span id="gmail-MathJax-Element-7-Frame" class="gmail-mjx-chtml gmail-MathJax_CHTML" tabindex="0" style="box-sizing:border-box;display:inline-block;line-height:0;font-style:normal;word-spacing:normal;white-space:nowrap;float:none;direction:ltr;max-width:none;max-height:none;min-width:0px;min-height:0px;border:0px;margin:0px;padding:1px 0px"><span id="gmail-MJXc-Node-23" class="gmail-mjx-math" style="box-sizing:border-box;display:inline-block;border-collapse:separate;border-spacing:0px;margin-top:0px"><span id="gmail-MJXc-Node-24" class="gmail-mjx-mrow" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-MJXc-Node-25" class="gmail-mjx-mi" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-math-I" style="display:block;white-space:pre;font-family:MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw;margin-top:0px;padding-top:0.237em;padding-bottom:0.472em;box-sizing:content-box">μ</span></span></span></span><span class="gmail-MJX_Assistive_MathML" style="box-sizing:border-box;padding:1px 0px 0px;border:0px;height:1px;width:1px;overflow:hidden;display:block">μ</span></span> is (close to) maximal.</em></p><p style="box-sizing:border-box;margin-bottom:1em;color:rgb(96,108,113);font-family:&quot;Open Sans&quot;,&quot;Helvetica Neue&quot;,Helvetica,Arial,sans-serif">The authors then provide a translation from this statistical framework to infinite comibnatorics: namely, they prove that existence of certain learning functions corresponding to the problem above (the so-called <em style="box-sizing:border-box;margin-top:0px">estimating the maximum</em> learners, or EMX-learners) translates into the existence of the so-called monotone compression schemes, which in turn is equivalent to a statement in cardinal arithmetic that is indeed independent of ZFC. Specifically, let <span id="gmail-MathJax-Element-8-Frame" class="gmail-mjx-chtml gmail-MathJax_CHTML" tabindex="0" style="box-sizing:border-box;display:inline-block;line-height:0;word-spacing:normal;white-space:nowrap;float:none;direction:ltr;max-width:none;max-height:none;min-width:0px;min-height:0px;border:0px;margin:0px;padding:1px 0px"><span id="gmail-MJXc-Node-26" class="gmail-mjx-math" style="box-sizing:border-box;display:inline-block;border-collapse:separate;border-spacing:0px;margin-top:0px"><span id="gmail-MJXc-Node-27" class="gmail-mjx-mrow" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-MJXc-Node-28" class="gmail-mjx-mi" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-math-I" style="display:block;white-space:pre;font-family:MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw;margin-top:0px;padding-top:0.472em;padding-bottom:0.284em;padding-right:0.024em;box-sizing:content-box">X</span></span></span></span><span class="gmail-MJX_Assistive_MathML" style="box-sizing:border-box;padding:1px 0px 0px;border:0px;height:1px;width:1px;overflow:hidden;display:block">X</span></span> be an infinite set, <span id="gmail-MathJax-Element-9-Frame" class="gmail-mjx-chtml gmail-MathJax_CHTML" tabindex="0" style="box-sizing:border-box;display:inline-block;line-height:0;word-spacing:normal;white-space:nowrap;float:none;direction:ltr;max-width:none;max-height:none;min-width:0px;min-height:0px;border:0px;margin:0px;padding:1px 0px"><span id="gmail-MJXc-Node-29" class="gmail-mjx-math" style="box-sizing:border-box;display:inline-block;border-collapse:separate;border-spacing:0px;margin-top:0px"><span id="gmail-MJXc-Node-30" class="gmail-mjx-mrow" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-MJXc-Node-31" class="gmail-mjx-mi" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-math-I" style="display:block;white-space:pre;font-family:MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw;margin-top:0px;padding-top:0.472em;padding-bottom:0.284em;padding-right:0.106em;box-sizing:content-box">F</span></span><span id="gmail-MJXc-Node-32" class="gmail-mjx-mi" style="display:inline-block;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-math-I" style="display:block;white-space:pre;font-family:MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw;margin-top:0px;padding-top:0.425em;padding-bottom:0.284em;box-sizing:content-box">i</span></span><span id="gmail-MJXc-Node-33" class="gmail-mjx-mi" style="display:inline-block;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-math-I" style="display:block;white-space:pre;font-family:MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw;margin-top:0px;padding-top:0.237em;padding-bottom:0.284em;box-sizing:content-box">n</span></span><span id="gmail-MJXc-Node-34" class="gmail-mjx-mo" style="display:inline-block;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-main-R" style="display:block;white-space:pre;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.472em;padding-bottom:0.613em;box-sizing:content-box">(</span></span><span id="gmail-MJXc-Node-35" class="gmail-mjx-mi" style="display:inline-block;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-math-I" style="display:block;white-space:pre;font-family:MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw;margin-top:0px;padding-top:0.472em;padding-bottom:0.284em;padding-right:0.024em;box-sizing:content-box">X</span></span><span id="gmail-MJXc-Node-36" class="gmail-mjx-mo" style="display:inline-block;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-main-R" style="display:block;white-space:pre;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.472em;padding-bottom:0.613em;box-sizing:content-box">)</span></span></span></span><span class="gmail-MJX_Assistive_MathML" style="box-sizing:border-box;padding:1px 0px 0px;border:0px;height:1px;width:1px;overflow:hidden;display:block">Fin(X)</span></span> be the family of its finite subsets, and let <span id="gmail-MathJax-Element-10-Frame" class="gmail-mjx-chtml gmail-MathJax_CHTML" tabindex="0" style="box-sizing:border-box;display:inline-block;line-height:0;word-spacing:normal;white-space:nowrap;float:none;direction:ltr;max-width:none;max-height:none;min-width:0px;min-height:0px;border:0px;margin:0px;padding:1px 0px"><span id="gmail-MJXc-Node-37" class="gmail-mjx-math" style="box-sizing:border-box;display:inline-block;border-collapse:separate;border-spacing:0px;margin-top:0px"><span id="gmail-MJXc-Node-38" class="gmail-mjx-mrow" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-MJXc-Node-39" class="gmail-mjx-mi" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-math-I" style="display:block;white-space:pre;font-family:MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw;margin-top:0px;padding-top:0.237em;padding-bottom:0.284em;box-sizing:content-box">m</span></span><span id="gmail-MJXc-Node-40" class="gmail-mjx-mo gmail-MJXc-space3" style="display:inline-block;margin-left:0.278em;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-main-R" style="display:block;white-space:pre;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.237em;padding-bottom:0.378em;box-sizing:content-box">&gt;</span></span><span id="gmail-MJXc-Node-41" class="gmail-mjx-mi gmail-MJXc-space3" style="display:inline-block;margin-left:0.278em;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-math-I" style="display:block;white-space:pre;font-family:MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw;margin-top:0px;padding-top:0.472em;padding-bottom:0.284em;box-sizing:content-box">k</span></span></span></span><span class="gmail-MJX_Assistive_MathML" style="box-sizing:border-box;padding:1px 0px 0px;border:0px;height:1px;width:1px;overflow:hidden;display:block">m&gt;k</span></span> be natural nubers. A <em style="box-sizing:border-box">monotone compressions scheme</em> for <span id="gmail-MathJax-Element-11-Frame" class="gmail-mjx-chtml gmail-MathJax_CHTML" tabindex="0" style="box-sizing:border-box;display:inline-block;line-height:0;word-spacing:normal;white-space:nowrap;float:none;direction:ltr;max-width:none;max-height:none;min-width:0px;min-height:0px;border:0px;margin:0px;padding:1px 0px"><span id="gmail-MJXc-Node-42" class="gmail-mjx-math" style="box-sizing:border-box;display:inline-block;border-collapse:separate;border-spacing:0px;margin-top:0px"><span id="gmail-MJXc-Node-43" class="gmail-mjx-mrow" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-MJXc-Node-44" class="gmail-mjx-mo" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-main-R" style="display:block;white-space:pre;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.472em;padding-bottom:0.613em;box-sizing:content-box">(</span></span><span id="gmail-MJXc-Node-45" class="gmail-mjx-mi" style="display:inline-block;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-math-I" style="display:block;white-space:pre;font-family:MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw;margin-top:0px;padding-top:0.472em;padding-bottom:0.284em;padding-right:0.024em;box-sizing:content-box">X</span></span><span id="gmail-MJXc-Node-46" class="gmail-mjx-mo" style="display:inline-block;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-main-R" style="display:block;white-space:pre;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;padding-bottom:0.519em;box-sizing:content-box">,</span></span><span id="gmail-MJXc-Node-47" class="gmail-mjx-mi gmail-MJXc-space1" style="display:inline-block;margin-left:0.167em;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-math-I" style="display:block;white-space:pre;font-family:MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw;margin-top:0px;padding-top:0.237em;padding-bottom:0.284em;box-sizing:content-box">m</span></span><span id="gmail-MJXc-Node-48" class="gmail-mjx-mo" style="display:inline-block;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-main-R" style="display:block;white-space:pre;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;padding-bottom:0.519em;box-sizing:content-box">,</span></span><span id="gmail-MJXc-Node-49" class="gmail-mjx-mi gmail-MJXc-space1" style="display:inline-block;margin-left:0.167em;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-math-I" style="display:block;white-space:pre;font-family:MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw;margin-top:0px;padding-top:0.472em;padding-bottom:0.284em;box-sizing:content-box">k</span></span><span id="gmail-MJXc-Node-50" class="gmail-mjx-mo" style="display:inline-block;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-main-R" style="display:block;white-space:pre;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.472em;padding-bottom:0.613em;box-sizing:content-box">)</span></span></span></span><span class="gmail-MJX_Assistive_MathML" style="box-sizing:border-box;padding:1px 0px 0px;border:0px;height:1px;width:1px;overflow:hidden;display:block">(X,m,k)</span></span> is a function <span id="gmail-MathJax-Element-12-Frame" class="gmail-mjx-chtml gmail-MathJax_CHTML" tabindex="0" style="box-sizing:border-box;display:inline-block;line-height:0;word-spacing:normal;white-space:nowrap;float:none;direction:ltr;max-width:none;max-height:none;min-width:0px;min-height:0px;border:0px;margin:0px;padding:1px 0px"><span id="gmail-MJXc-Node-51" class="gmail-mjx-math" style="box-sizing:border-box;display:inline-block;border-collapse:separate;border-spacing:0px;margin-top:0px"><span id="gmail-MJXc-Node-52" class="gmail-mjx-mrow" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-MJXc-Node-53" class="gmail-mjx-mi" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-math-I" style="display:block;white-space:pre;font-family:MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw;margin-top:0px;padding-top:0.472em;padding-bottom:0.472em;padding-right:0.06em;box-sizing:content-box">f</span></span><span id="gmail-MJXc-Node-54" class="gmail-mjx-mo gmail-MJXc-space3" style="display:inline-block;margin-left:0.278em;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-main-R" style="display:block;white-space:pre;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.143em;padding-bottom:0.331em;box-sizing:content-box">:</span></span><span id="gmail-MJXc-Node-55" class="gmail-mjx-mo gmail-MJXc-space3" style="display:inline-block;margin-left:0.278em;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-main-R" style="display:block;white-space:pre;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.472em;padding-bottom:0.613em;box-sizing:content-box">[</span></span><span id="gmail-MJXc-Node-56" class="gmail-mjx-mi" style="display:inline-block;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-math-I" style="display:block;white-space:pre;font-family:MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw;margin-top:0px;padding-top:0.472em;padding-bottom:0.284em;padding-right:0.024em;box-sizing:content-box">X</span></span><span id="gmail-MJXc-Node-57" class="gmail-mjx-msubsup" style="display:inline-block;box-sizing:content-box"><span class="gmail-mjx-base" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-MJXc-Node-58" class="gmail-mjx-mo" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-main-R" style="display:block;white-space:pre;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.472em;padding-bottom:0.613em;box-sizing:content-box">]</span></span></span><span class="gmail-mjx-sup" style="display:inline-block;vertical-align:0.513em;padding-left:0px;padding-right:0.071em;box-sizing:content-box"><span id="gmail-MJXc-Node-59" class="gmail-mjx-mi" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-math-I" style="display:block;white-space:pre;font-family:MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw;margin-top:0px;padding-top:0.472em;padding-bottom:0.284em;box-sizing:content-box">k</span></span></span></span><span id="gmail-MJXc-Node-60" class="gmail-mjx-mo gmail-MJXc-space3" style="display:inline-block;margin-left:0.278em;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-main-R" style="display:block;white-space:pre;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.237em;padding-bottom:0.331em;box-sizing:content-box">→</span></span><span id="gmail-MJXc-Node-61" class="gmail-mjx-mi gmail-MJXc-space3" style="display:inline-block;margin-left:0.278em;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-math-I" style="display:block;white-space:pre;font-family:MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw;margin-top:0px;padding-top:0.472em;padding-bottom:0.284em;padding-right:0.106em;box-sizing:content-box">F</span></span><span id="gmail-MJXc-Node-62" class="gmail-mjx-mi" style="display:inline-block;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-math-I" style="display:block;white-space:pre;font-family:MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw;margin-top:0px;padding-top:0.425em;padding-bottom:0.284em;box-sizing:content-box">i</span></span><span id="gmail-MJXc-Node-63" class="gmail-mjx-mi" style="display:inline-block;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-math-I" style="display:block;white-space:pre;font-family:MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw;margin-top:0px;padding-top:0.237em;padding-bottom:0.284em;box-sizing:content-box">n</span></span><span id="gmail-MJXc-Node-64" class="gmail-mjx-mo" style="display:inline-block;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-main-R" style="display:block;white-space:pre;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.472em;padding-bottom:0.613em;box-sizing:content-box">(</span></span><span id="gmail-MJXc-Node-65" class="gmail-mjx-mi" style="display:inline-block;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-math-I" style="display:block;white-space:pre;font-family:MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw;margin-top:0px;padding-top:0.472em;padding-bottom:0.284em;padding-right:0.024em;box-sizing:content-box">X</span></span><span id="gmail-MJXc-Node-66" class="gmail-mjx-mo" style="display:inline-block;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-main-R" style="display:block;white-space:pre;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.472em;padding-bottom:0.613em;box-sizing:content-box">)</span></span></span></span><span class="gmail-MJX_Assistive_MathML" style="box-sizing:border-box;padding:1px 0px 0px;border:0px;height:1px;width:1px;overflow:hidden;display:block">f:[X]k→Fin(X)</span></span> such that<span class="gmail-mjx-chtml gmail-MJXc-display" style="box-sizing:border-box;display:block;line-height:0;text-align:center;word-spacing:normal;white-space:nowrap;float:none;direction:ltr;max-width:none;max-height:none;min-width:0px;min-height:0px;border:0px;margin:1em 0px;padding:0px"><span id="gmail-MathJax-Element-13-Frame" class="gmail-mjx-chtml gmail-MathJax_CHTML" tabindex="0" style="box-sizing:border-box;display:inline-block;line-height:0;word-spacing:normal;float:none;direction:ltr;max-width:none;max-height:none;min-width:0px;min-height:0px;border:0px;margin:0px;padding:1px 0px"><span id="gmail-MJXc-Node-67" class="gmail-mjx-math" style="box-sizing:border-box;display:inline-block;border-collapse:separate;border-spacing:0px;margin-top:0px"><span id="gmail-MJXc-Node-68" class="gmail-mjx-mrow" style="display:inline-block;text-align:left;margin-top:0px;box-sizing:content-box"><span id="gmail-MJXc-Node-69" class="gmail-mjx-mi" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-main-R" style="display:block;white-space:pre;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.425em;padding-bottom:0.378em;box-sizing:content-box">∀</span></span><span id="gmail-MJXc-Node-70" class="gmail-mjx-mi" style="display:inline-block;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-math-I" style="display:block;white-space:pre;font-family:MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw;margin-top:0px;padding-top:0.472em;padding-bottom:0.284em;box-sizing:content-box">A</span></span><span id="gmail-MJXc-Node-71" class="gmail-mjx-mo gmail-MJXc-space3" style="display:inline-block;margin-left:0.278em;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-main-R" style="display:block;white-space:pre;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.237em;padding-bottom:0.378em;box-sizing:content-box">∈</span></span><span id="gmail-MJXc-Node-72" class="gmail-mjx-mo gmail-MJXc-space3" style="display:inline-block;margin-left:0.278em;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-main-R" style="display:block;white-space:pre;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.472em;padding-bottom:0.613em;box-sizing:content-box">[</span></span><span id="gmail-MJXc-Node-73" class="gmail-mjx-mi" style="display:inline-block;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-math-I" style="display:block;white-space:pre;font-family:MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw;margin-top:0px;padding-top:0.472em;padding-bottom:0.284em;padding-right:0.024em;box-sizing:content-box">X</span></span><span id="gmail-MJXc-Node-74" class="gmail-mjx-msubsup" style="display:inline-block;box-sizing:content-box"><span class="gmail-mjx-base" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-MJXc-Node-75" class="gmail-mjx-mo" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-main-R" style="display:block;white-space:pre;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.472em;padding-bottom:0.613em;box-sizing:content-box">]</span></span></span><span class="gmail-mjx-sup" style="display:inline-block;vertical-align:0.584em;padding-left:0px;padding-right:0.071em;box-sizing:content-box"><span id="gmail-MJXc-Node-76" class="gmail-mjx-mi" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-math-I" style="display:block;white-space:pre;font-family:MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw;margin-top:0px;padding-top:0.237em;padding-bottom:0.284em;box-sizing:content-box">m</span></span></span></span><span id="gmail-MJXc-Node-77" class="gmail-mjx-mi" style="display:inline-block;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-main-R" style="display:block;white-space:pre;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.425em;padding-bottom:0.331em;box-sizing:content-box">∃</span></span><span id="gmail-MJXc-Node-78" class="gmail-mjx-mi" style="display:inline-block;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-math-I" style="display:block;white-space:pre;font-family:MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw;margin-top:0px;padding-top:0.472em;padding-bottom:0.284em;box-sizing:content-box">B</span></span><span id="gmail-MJXc-Node-79" class="gmail-mjx-mo gmail-MJXc-space3" style="display:inline-block;margin-left:0.278em;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-main-R" style="display:block;white-space:pre;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.237em;padding-bottom:0.378em;box-sizing:content-box">∈</span></span><span id="gmail-MJXc-Node-80" class="gmail-mjx-mo gmail-MJXc-space3" style="display:inline-block;margin-left:0.278em;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-main-R" style="display:block;white-space:pre;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.472em;padding-bottom:0.613em;box-sizing:content-box">[</span></span><span id="gmail-MJXc-Node-81" class="gmail-mjx-mi" style="display:inline-block;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-math-I" style="display:block;white-space:pre;font-family:MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw;margin-top:0px;padding-top:0.472em;padding-bottom:0.284em;padding-right:0.024em;box-sizing:content-box">X</span></span><span id="gmail-MJXc-Node-82" class="gmail-mjx-msubsup" style="display:inline-block;box-sizing:content-box"><span class="gmail-mjx-base" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-MJXc-Node-83" class="gmail-mjx-mo" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-main-R" style="display:block;white-space:pre;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.472em;padding-bottom:0.613em;box-sizing:content-box">]</span></span></span><span class="gmail-mjx-sup" style="display:inline-block;vertical-align:0.584em;padding-left:0px;padding-right:0.071em;box-sizing:content-box"><span id="gmail-MJXc-Node-84" class="gmail-mjx-mi" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-math-I" style="display:block;white-space:pre;font-family:MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw;margin-top:0px;padding-top:0.472em;padding-bottom:0.284em;box-sizing:content-box">k</span></span></span></span><span id="gmail-MJXc-Node-85" class="gmail-mjx-mspace" style="display:inline-block;width:0.222em;height:0px;box-sizing:content-box"></span><span id="gmail-MJXc-Node-86" class="gmail-mjx-mo" style="display:inline-block;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-main-R" style="display:block;white-space:pre;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.472em;padding-bottom:0.613em;box-sizing:content-box">(</span></span><span id="gmail-MJXc-Node-87" class="gmail-mjx-mi" style="display:inline-block;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-math-I" style="display:block;white-space:pre;font-family:MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw;margin-top:0px;padding-top:0.472em;padding-bottom:0.284em;box-sizing:content-box">B</span></span><span id="gmail-MJXc-Node-88" class="gmail-mjx-mo gmail-MJXc-space3" style="display:inline-block;margin-left:0.278em;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-main-R" style="display:block;white-space:pre;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.331em;padding-bottom:0.472em;box-sizing:content-box">⊆</span></span><span id="gmail-MJXc-Node-89" class="gmail-mjx-mi gmail-MJXc-space3" style="display:inline-block;margin-left:0.278em;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-math-I" style="display:block;white-space:pre;font-family:MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw;margin-top:0px;padding-top:0.472em;padding-bottom:0.284em;box-sizing:content-box">A</span></span><span id="gmail-MJXc-Node-90" class="gmail-mjx-mo gmail-MJXc-space3" style="display:inline-block;margin-left:0.278em;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-main-R" style="display:block;white-space:pre;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.331em;padding-bottom:0.472em;box-sizing:content-box">⊆</span></span><span id="gmail-MJXc-Node-91" class="gmail-mjx-mi gmail-MJXc-space3" style="display:inline-block;margin-left:0.278em;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-math-I" style="display:block;white-space:pre;font-family:MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw;margin-top:0px;padding-top:0.472em;padding-bottom:0.472em;padding-right:0.06em;box-sizing:content-box">f</span></span><span id="gmail-MJXc-Node-92" class="gmail-mjx-mo" style="display:inline-block;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-main-R" style="display:block;white-space:pre;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.472em;padding-bottom:0.613em;box-sizing:content-box">(</span></span><span id="gmail-MJXc-Node-93" class="gmail-mjx-mi" style="display:inline-block;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-math-I" style="display:block;white-space:pre;font-family:MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw;margin-top:0px;padding-top:0.472em;padding-bottom:0.284em;box-sizing:content-box">B</span></span><span id="gmail-MJXc-Node-94" class="gmail-mjx-mo" style="display:inline-block;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-main-R" style="display:block;white-space:pre;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.472em;padding-bottom:0.613em;box-sizing:content-box">)</span></span><span id="gmail-MJXc-Node-95" class="gmail-mjx-mo" style="display:inline-block;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-main-R" style="display:block;white-space:pre;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.472em;padding-bottom:0.613em;box-sizing:content-box">)</span></span><span id="gmail-MJXc-Node-96" class="gmail-mjx-mo" style="display:inline-block;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-main-R" style="display:block;white-space:pre;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;padding-bottom:0.331em;box-sizing:content-box">.</span></span></span></span><span class="gmail-MJX_Assistive_MathML gmail-MJX_Assistive_MathML_Block" style="box-sizing:border-box;width:351.766px;padding:1px 0px 0px;border:0px;height:1px;overflow:hidden;display:block">∀A∈[X]m∃B∈[X]k<span width="mediummathspace"></span>(B⊆A⊆f(B)).</span></span></span></p><p style="box-sizing:border-box;margin-bottom:1em;color:rgb(96,108,113);font-family:&quot;Open Sans&quot;,&quot;Helvetica Neue&quot;,Helvetica,Arial,sans-serif">The main result of the paper then is that there exists a monotone compressions scheme for <span id="gmail-MathJax-Element-14-Frame" class="gmail-mjx-chtml gmail-MathJax_CHTML" tabindex="0" style="box-sizing:border-box;display:inline-block;line-height:0;word-spacing:normal;white-space:nowrap;float:none;direction:ltr;max-width:none;max-height:none;min-width:0px;min-height:0px;border:0px;margin:0px;padding:1px 0px"><span id="gmail-MJXc-Node-97" class="gmail-mjx-math" style="box-sizing:border-box;display:inline-block;border-collapse:separate;border-spacing:0px;margin-top:0px"><span id="gmail-MJXc-Node-98" class="gmail-mjx-mrow" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-MJXc-Node-99" class="gmail-mjx-mo" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-main-R" style="display:block;white-space:pre;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.472em;padding-bottom:0.613em;box-sizing:content-box">(</span></span><span id="gmail-MJXc-Node-100" class="gmail-mjx-mo" style="display:inline-block;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-main-R" style="display:block;white-space:pre;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.472em;padding-bottom:0.613em;box-sizing:content-box">[</span></span><span id="gmail-MJXc-Node-101" class="gmail-mjx-mn" style="display:inline-block;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-main-R" style="display:block;white-space:pre;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.378em;padding-bottom:0.378em;box-sizing:content-box">0</span></span><span id="gmail-MJXc-Node-102" class="gmail-mjx-mo" style="display:inline-block;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-main-R" style="display:block;white-space:pre;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;padding-bottom:0.519em;box-sizing:content-box">,</span></span><span id="gmail-MJXc-Node-103" class="gmail-mjx-mn gmail-MJXc-space1" style="display:inline-block;margin-left:0.167em;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-main-R" style="display:block;white-space:pre;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.378em;padding-bottom:0.331em;box-sizing:content-box">1</span></span><span id="gmail-MJXc-Node-104" class="gmail-mjx-mo" style="display:inline-block;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-main-R" style="display:block;white-space:pre;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.472em;padding-bottom:0.613em;box-sizing:content-box">]</span></span><span id="gmail-MJXc-Node-105" class="gmail-mjx-mo" style="display:inline-block;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-main-R" style="display:block;white-space:pre;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;padding-bottom:0.519em;box-sizing:content-box">,</span></span><span id="gmail-MJXc-Node-106" class="gmail-mjx-mi gmail-MJXc-space1" style="display:inline-block;margin-left:0.167em;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-math-I" style="display:block;white-space:pre;font-family:MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw;margin-top:0px;padding-top:0.237em;padding-bottom:0.284em;box-sizing:content-box">m</span></span><span id="gmail-MJXc-Node-107" class="gmail-mjx-mo gmail-MJXc-space2" style="display:inline-block;margin-left:0.222em;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-main-R" style="display:block;white-space:pre;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.284em;padding-bottom:0.425em;box-sizing:content-box">+</span></span><span id="gmail-MJXc-Node-108" class="gmail-mjx-mn gmail-MJXc-space2" style="display:inline-block;margin-left:0.222em;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-main-R" style="display:block;white-space:pre;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.378em;padding-bottom:0.331em;box-sizing:content-box">1</span></span><span id="gmail-MJXc-Node-109" class="gmail-mjx-mo" style="display:inline-block;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-main-R" style="display:block;white-space:pre;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;padding-bottom:0.519em;box-sizing:content-box">,</span></span><span id="gmail-MJXc-Node-110" class="gmail-mjx-mi gmail-MJXc-space1" style="display:inline-block;margin-left:0.167em;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-math-I" style="display:block;white-space:pre;font-family:MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw;margin-top:0px;padding-top:0.237em;padding-bottom:0.284em;box-sizing:content-box">m</span></span><span id="gmail-MJXc-Node-111" class="gmail-mjx-mo" style="display:inline-block;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-main-R" style="display:block;white-space:pre;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.472em;padding-bottom:0.613em;box-sizing:content-box">)</span></span></span></span><span class="gmail-MJX_Assistive_MathML" style="box-sizing:border-box;padding:1px 0px 0px;border:0px;height:1px;width:1px;overflow:hidden;display:block">([0,1],m+1,m)</span></span> for some <span id="gmail-MathJax-Element-15-Frame" class="gmail-mjx-chtml gmail-MathJax_CHTML" tabindex="0" style="box-sizing:border-box;display:inline-block;line-height:0;word-spacing:normal;white-space:nowrap;float:none;direction:ltr;max-width:none;max-height:none;min-width:0px;min-height:0px;border:0px;margin:0px;padding:1px 0px"><span id="gmail-MJXc-Node-112" class="gmail-mjx-math" style="box-sizing:border-box;display:inline-block;border-collapse:separate;border-spacing:0px;margin-top:0px"><span id="gmail-MJXc-Node-113" class="gmail-mjx-mrow" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-MJXc-Node-114" class="gmail-mjx-mi" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-math-I" style="display:block;white-space:pre;font-family:MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw;margin-top:0px;padding-top:0.237em;padding-bottom:0.284em;box-sizing:content-box">m</span></span></span></span><span class="gmail-MJX_Assistive_MathML" style="box-sizing:border-box;padding:1px 0px 0px;border:0px;height:1px;width:1px;overflow:hidden;display:block">m</span></span> if and only if <span id="gmail-MathJax-Element-16-Frame" class="gmail-mjx-chtml gmail-MathJax_CHTML" tabindex="0" style="box-sizing:border-box;display:inline-block;line-height:0;word-spacing:normal;white-space:nowrap;float:none;direction:ltr;max-width:none;max-height:none;min-width:0px;min-height:0px;border:0px;margin:0px;padding:1px 0px"><span id="gmail-MJXc-Node-115" class="gmail-mjx-math" style="box-sizing:border-box;display:inline-block;border-collapse:separate;border-spacing:0px;margin-top:0px"><span id="gmail-MJXc-Node-116" class="gmail-mjx-mrow" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-MJXc-Node-117" class="gmail-mjx-msubsup" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-mjx-base" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-MJXc-Node-118" class="gmail-mjx-mn" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-main-R" style="display:block;white-space:pre;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.378em;padding-bottom:0.331em;box-sizing:content-box">2</span></span></span><span class="gmail-mjx-sup" style="display:inline-block;vertical-align:0.591em;padding-left:0px;padding-right:0.071em;box-sizing:content-box"><span id="gmail-MJXc-Node-119" class="gmail-mjx-texatom" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-MJXc-Node-120" class="gmail-mjx-mrow" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-MJXc-Node-121" class="gmail-mjx-msubsup" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-mjx-base" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-MJXc-Node-122" class="gmail-mjx-mi" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-main-R" style="display:block;white-space:pre;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.425em;padding-bottom:0.331em;box-sizing:content-box">ℵ</span></span></span><span class="gmail-mjx-sub" style="display:inline-block;vertical-align:-0.212em;padding-right:0.071em;box-sizing:content-box"><span id="gmail-MJXc-Node-123" class="gmail-mjx-mn" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-main-R" style="display:block;white-space:pre;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.378em;padding-bottom:0.378em;box-sizing:content-box">0</span></span></span></span></span></span></span></span><span id="gmail-MJXc-Node-124" class="gmail-mjx-mo gmail-MJXc-space3" style="display:inline-block;margin-left:0.278em;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-main-R" style="display:block;white-space:pre;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.237em;padding-bottom:0.378em;box-sizing:content-box">&lt;</span></span><span id="gmail-MJXc-Node-125" class="gmail-mjx-msubsup gmail-MJXc-space3" style="display:inline-block;margin-left:0.278em;box-sizing:content-box"><span class="gmail-mjx-base" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-MJXc-Node-126" class="gmail-mjx-mi" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-main-R" style="display:block;white-space:pre;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.425em;padding-bottom:0.331em;box-sizing:content-box">ℵ</span></span></span><span class="gmail-mjx-sub" style="display:inline-block;vertical-align:-0.212em;padding-right:0.071em;box-sizing:content-box"><span id="gmail-MJXc-Node-127" class="gmail-mjx-mi" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-math-I" style="display:block;white-space:pre;font-family:MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw;margin-top:0px;padding-top:0.237em;padding-bottom:0.284em;box-sizing:content-box">ω</span></span></span></span></span></span><span class="gmail-MJX_Assistive_MathML" style="box-sizing:border-box;padding:1px 0px 0px;border:0px;height:1px;width:1px;overflow:hidden;display:block">2<span class="gmail-MJX-TeXAtom-ORD">ℵ0</span>&lt;ℵω</span></span>.</p><p style="box-sizing:border-box;margin-bottom:1em;color:rgb(96,108,113);font-family:&quot;Open Sans&quot;,&quot;Helvetica Neue&quot;,Helvetica,Arial,sans-serif">K.P. Hart immediately observed that the main combinatorial content of the results in the paper is related to Kuratowski&#39;s theorem on decompositions of finite powers of sets and that the monotone compression functions on the unit interval cannot, in a certain sense, be constructive or descriptively nice - namely, they cannot be Borel measurable. During the talk I will introduce the subject of the paper in question, and present the set-theoretic aspects of the main results.</p></div><div><br></div><div><br></div><div><br></div><div><br></div><div><font color="#212121"><br></font></div><div><font color="#212121">Model Theory Seminar</font></div><div><font color="#212121">CUNY Graduate Center, Room 6417</font></div><div><font color="#212121">Friday, </font><span style="color:rgb(33,33,33)">April 5</span><font color="#212121">, 12:30-2:00pm</font></div><div><span style="box-sizing:border-box;color:rgb(96,108,113);font-family:&quot;Open Sans&quot;,&quot;Helvetica Neue&quot;,Helvetica,Arial,sans-serif"><a href="https://faculty.math.illinois.edu/~pynncoa2/" target="_blank" style="color:rgb(30,107,184);box-sizing:border-box;background-color:transparent;text-decoration-line:none;margin-top:0px">Nigel Pynn-Coates</a></span><span style="color:rgb(96,108,113);font-family:&quot;Open Sans&quot;,&quot;Helvetica Neue&quot;,Helvetica,Arial,sans-serif">, University of Illinois at Urbana-Champaign</span><br style="box-sizing:border-box;color:rgb(96,108,113);font-family:&quot;Open Sans&quot;,&quot;Helvetica Neue&quot;,Helvetica,Arial,sans-serif"><p style="box-sizing:border-box;margin-bottom:1em;margin-top:0px;color:rgb(96,108,113);font-family:&quot;Open Sans&quot;,&quot;Helvetica Neue&quot;,Helvetica,Arial,sans-serif"><strong style="box-sizing:border-box">Model companions of theories of valued differential fields</strong><br style="box-sizing:border-box"></p><p style="box-sizing:border-box;margin-bottom:1em;color:rgb(96,108,113);font-family:&quot;Open Sans&quot;,&quot;Helvetica Neue&quot;,Helvetica,Arial,sans-serif">I will survey what is known about model companions of theories of (ordered) valued differential fields and discuss my ongoing work towards isolating a model companion for a certain theory of ordered valued differential fields, including positive results at the level of the value group.</p></div><div><font color="#212121"><br></font></div><div><font color="#212121"><br></font></div><div><font color="#212121"><br></font></div><div><font color="#212121">Logic Workshop</font></div><div><font color="#212121">CUNY Graduate Center, Room 6417</font></div><div><font color="#212121">Friday, </font><span style="color:rgb(33,33,33)">April 5</span><font color="#212121">, 2:00-3:30pm</font></div><div style="color:rgb(33,33,33)"><p style="box-sizing:border-box;margin-bottom:1em;margin-top:0px;color:rgb(96,108,113);font-family:&quot;Open Sans&quot;,&quot;Helvetica Neue&quot;,Helvetica,Arial,sans-serif"><span style="box-sizing:border-box"><a href="https://www3.nd.edu/~cholak/" target="_blank" style="color:rgb(30,107,184);box-sizing:border-box;background-color:transparent;text-decoration-line:none;margin-top:0px">Peter Cholak</a></span>, University of Notre Dame<br style="box-sizing:border-box"><span style="box-sizing:border-box">Computability-Theoretic Aspects of Ramsey&#39;s Theorem</span><br style="box-sizing:border-box"></p><p style="box-sizing:border-box;margin-bottom:1em;color:rgb(96,108,113);font-family:&quot;Open Sans&quot;,&quot;Helvetica Neue&quot;,Helvetica,Arial,sans-serif">Ramsey&#39;s Theorem for pairs and 2 colors says that for every 2-coloring, <span id="gmail-m_-1880571767338173311gmail-MathJax-Element-1-Frame" class="gmail-m_-1880571767338173311gmail-mjx-chtml gmail-m_-1880571767338173311gmail-MathJax_CHTML" style="box-sizing:border-box;display:inline-block;line-height:0;word-spacing:normal;white-space:nowrap;float:none;direction:ltr;max-width:none;max-height:none;min-width:0px;min-height:0px;border:0px;margin:0px;padding:1px 0px"><span id="gmail-m_-1880571767338173311gmail-MJXc-Node-1" class="gmail-m_-1880571767338173311gmail-mjx-math" style="box-sizing:border-box;display:inline-block;border-collapse:separate;border-spacing:0px;margin-top:0px"><span id="gmail-m_-1880571767338173311gmail-MJXc-Node-2" class="gmail-m_-1880571767338173311gmail-mjx-mrow" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_-1880571767338173311gmail-MJXc-Node-3" class="gmail-m_-1880571767338173311gmail-mjx-mi" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-m_-1880571767338173311gmail-mjx-char gmail-m_-1880571767338173311gmail-MJXc-TeX-math-I" style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw;margin-top:0px;padding-top:0.472em;padding-bottom:0.284em;padding-right:0.045em;box-sizing:content-box">C</span></span></span></span><span class="gmail-m_-1880571767338173311gmail-MJX_Assistive_MathML" style="box-sizing:border-box;padding:1px 0px 0px;border:0px;height:1px;width:1px;overflow:hidden;display:block">C</span></span>, of pairs of natural numbers, there is an infinite set <span id="gmail-m_-1880571767338173311gmail-MathJax-Element-2-Frame" class="gmail-m_-1880571767338173311gmail-mjx-chtml gmail-m_-1880571767338173311gmail-MathJax_CHTML" style="box-sizing:border-box;display:inline-block;line-height:0;word-spacing:normal;white-space:nowrap;float:none;direction:ltr;max-width:none;max-height:none;min-width:0px;min-height:0px;border:0px;margin:0px;padding:1px 0px"><span id="gmail-m_-1880571767338173311gmail-MJXc-Node-4" class="gmail-m_-1880571767338173311gmail-mjx-math" style="box-sizing:border-box;display:inline-block;border-collapse:separate;border-spacing:0px;margin-top:0px"><span id="gmail-m_-1880571767338173311gmail-MJXc-Node-5" class="gmail-m_-1880571767338173311gmail-mjx-mrow" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_-1880571767338173311gmail-MJXc-Node-6" class="gmail-m_-1880571767338173311gmail-mjx-mi" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-m_-1880571767338173311gmail-mjx-char gmail-m_-1880571767338173311gmail-MJXc-TeX-math-I" style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw;margin-top:0px;padding-top:0.472em;padding-bottom:0.284em;padding-right:0.057em;box-sizing:content-box">H</span></span></span></span><span class="gmail-m_-1880571767338173311gmail-MJX_Assistive_MathML" style="box-sizing:border-box;padding:1px 0px 0px;border:0px;height:1px;width:1px;overflow:hidden;display:block">H</span></span>, such that all pairs from <span id="gmail-m_-1880571767338173311gmail-MathJax-Element-3-Frame" class="gmail-m_-1880571767338173311gmail-mjx-chtml gmail-m_-1880571767338173311gmail-MathJax_CHTML" style="box-sizing:border-box;display:inline-block;line-height:0;word-spacing:normal;white-space:nowrap;float:none;direction:ltr;max-width:none;max-height:none;min-width:0px;min-height:0px;border:0px;margin:0px;padding:1px 0px"><span id="gmail-m_-1880571767338173311gmail-MJXc-Node-7" class="gmail-m_-1880571767338173311gmail-mjx-math" style="box-sizing:border-box;display:inline-block;border-collapse:separate;border-spacing:0px;margin-top:0px"><span id="gmail-m_-1880571767338173311gmail-MJXc-Node-8" class="gmail-m_-1880571767338173311gmail-mjx-mrow" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_-1880571767338173311gmail-MJXc-Node-9" class="gmail-m_-1880571767338173311gmail-mjx-mi" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-m_-1880571767338173311gmail-mjx-char gmail-m_-1880571767338173311gmail-MJXc-TeX-math-I" style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw;margin-top:0px;padding-top:0.472em;padding-bottom:0.284em;padding-right:0.057em;box-sizing:content-box">H</span></span></span></span><span class="gmail-m_-1880571767338173311gmail-MJX_Assistive_MathML" style="box-sizing:border-box;padding:1px 0px 0px;border:0px;height:1px;width:1px;overflow:hidden;display:block">H</span></span> have the same constant color. <span id="gmail-m_-1880571767338173311gmail-MathJax-Element-4-Frame" class="gmail-m_-1880571767338173311gmail-mjx-chtml gmail-m_-1880571767338173311gmail-MathJax_CHTML" style="box-sizing:border-box;display:inline-block;line-height:0;word-spacing:normal;white-space:nowrap;float:none;direction:ltr;max-width:none;max-height:none;min-width:0px;min-height:0px;border:0px;margin:0px;padding:1px 0px"><span id="gmail-m_-1880571767338173311gmail-MJXc-Node-10" class="gmail-m_-1880571767338173311gmail-mjx-math" style="box-sizing:border-box;display:inline-block;border-collapse:separate;border-spacing:0px;margin-top:0px"><span id="gmail-m_-1880571767338173311gmail-MJXc-Node-11" class="gmail-m_-1880571767338173311gmail-mjx-mrow" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_-1880571767338173311gmail-MJXc-Node-12" class="gmail-m_-1880571767338173311gmail-mjx-mi" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-m_-1880571767338173311gmail-mjx-char gmail-m_-1880571767338173311gmail-MJXc-TeX-math-I" style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw;margin-top:0px;padding-top:0.472em;padding-bottom:0.284em;padding-right:0.057em;box-sizing:content-box">H</span></span></span></span><span class="gmail-m_-1880571767338173311gmail-MJX_Assistive_MathML" style="box-sizing:border-box;padding:1px 0px 0px;border:0px;height:1px;width:1px;overflow:hidden;display:block">H</span></span> is called a <em style="box-sizing:border-box">homogeneous</em> set for <span id="gmail-m_-1880571767338173311gmail-MathJax-Element-5-Frame" class="gmail-m_-1880571767338173311gmail-mjx-chtml gmail-m_-1880571767338173311gmail-MathJax_CHTML" style="box-sizing:border-box;display:inline-block;line-height:0;word-spacing:normal;white-space:nowrap;float:none;direction:ltr;max-width:none;max-height:none;min-width:0px;min-height:0px;border:0px;margin:0px;padding:1px 0px"><span id="gmail-m_-1880571767338173311gmail-MJXc-Node-13" class="gmail-m_-1880571767338173311gmail-mjx-math" style="box-sizing:border-box;display:inline-block;border-collapse:separate;border-spacing:0px;margin-top:0px"><span id="gmail-m_-1880571767338173311gmail-MJXc-Node-14" class="gmail-m_-1880571767338173311gmail-mjx-mrow" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_-1880571767338173311gmail-MJXc-Node-15" class="gmail-m_-1880571767338173311gmail-mjx-mi" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-m_-1880571767338173311gmail-mjx-char gmail-m_-1880571767338173311gmail-MJXc-TeX-math-I" style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw;margin-top:0px;padding-top:0.472em;padding-bottom:0.284em;padding-right:0.045em;box-sizing:content-box">C</span></span></span></span><span class="gmail-m_-1880571767338173311gmail-MJX_Assistive_MathML" style="box-sizing:border-box;padding:1px 0px 0px;border:0px;height:1px;width:1px;overflow:hidden;display:block">C</span></span>. We will also consider Ramsey&#39;s Theorem in other settings. For example, we can be given countably many <span id="gmail-m_-1880571767338173311gmail-MathJax-Element-6-Frame" class="gmail-m_-1880571767338173311gmail-mjx-chtml gmail-m_-1880571767338173311gmail-MathJax_CHTML" style="box-sizing:border-box;display:inline-block;line-height:0;word-spacing:normal;white-space:nowrap;float:none;direction:ltr;max-width:none;max-height:none;min-width:0px;min-height:0px;border:0px;margin:0px;padding:1px 0px"><span id="gmail-m_-1880571767338173311gmail-MJXc-Node-16" class="gmail-m_-1880571767338173311gmail-mjx-math" style="box-sizing:border-box;display:inline-block;border-collapse:separate;border-spacing:0px;margin-top:0px"><span id="gmail-m_-1880571767338173311gmail-MJXc-Node-17" class="gmail-m_-1880571767338173311gmail-mjx-mrow" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_-1880571767338173311gmail-MJXc-Node-18" class="gmail-m_-1880571767338173311gmail-mjx-mn" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-m_-1880571767338173311gmail-mjx-char gmail-m_-1880571767338173311gmail-MJXc-TeX-main-R" style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.378em;padding-bottom:0.331em;box-sizing:content-box">1</span></span></span></span><span class="gmail-m_-1880571767338173311gmail-MJX_Assistive_MathML" style="box-sizing:border-box;padding:1px 0px 0px;border:0px;height:1px;width:1px;overflow:hidden;display:block">1</span></span>-colorings of natural numbers, <span id="gmail-m_-1880571767338173311gmail-MathJax-Element-7-Frame" class="gmail-m_-1880571767338173311gmail-mjx-chtml gmail-m_-1880571767338173311gmail-MathJax_CHTML" style="box-sizing:border-box;display:inline-block;line-height:0;word-spacing:normal;white-space:nowrap;float:none;direction:ltr;max-width:none;max-height:none;min-width:0px;min-height:0px;border:0px;margin:0px;padding:1px 0px"><span id="gmail-m_-1880571767338173311gmail-MJXc-Node-19" class="gmail-m_-1880571767338173311gmail-mjx-math" style="box-sizing:border-box;display:inline-block;border-collapse:separate;border-spacing:0px;margin-top:0px"><span id="gmail-m_-1880571767338173311gmail-MJXc-Node-20" class="gmail-m_-1880571767338173311gmail-mjx-mrow" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_-1880571767338173311gmail-MJXc-Node-21" class="gmail-m_-1880571767338173311gmail-mjx-mo" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-m_-1880571767338173311gmail-mjx-char gmail-m_-1880571767338173311gmail-MJXc-TeX-main-R" style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.472em;padding-bottom:0.613em;box-sizing:content-box">{</span></span><span id="gmail-m_-1880571767338173311gmail-MJXc-Node-22" class="gmail-m_-1880571767338173311gmail-mjx-msubsup" style="display:inline-block;box-sizing:content-box"><span class="gmail-m_-1880571767338173311gmail-mjx-base" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_-1880571767338173311gmail-MJXc-Node-23" class="gmail-m_-1880571767338173311gmail-mjx-mi" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-m_-1880571767338173311gmail-mjx-char gmail-m_-1880571767338173311gmail-MJXc-TeX-math-I" style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw;margin-top:0px;padding-top:0.237em;padding-bottom:0.284em;box-sizing:content-box">c</span></span></span><span class="gmail-m_-1880571767338173311gmail-mjx-sub" style="display:inline-block;vertical-align:-0.212em;padding-right:0.071em;box-sizing:content-box"><span id="gmail-m_-1880571767338173311gmail-MJXc-Node-24" class="gmail-m_-1880571767338173311gmail-mjx-mi" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-m_-1880571767338173311gmail-mjx-char gmail-m_-1880571767338173311gmail-MJXc-TeX-math-I" style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw;margin-top:0px;padding-top:0.425em;padding-bottom:0.284em;box-sizing:content-box">i</span></span></span></span><span id="gmail-m_-1880571767338173311gmail-MJXc-Node-25" class="gmail-m_-1880571767338173311gmail-mjx-mo" style="display:inline-block;box-sizing:content-box"><span class="gmail-m_-1880571767338173311gmail-mjx-char gmail-m_-1880571767338173311gmail-MJXc-TeX-main-R" style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.472em;padding-bottom:0.613em;box-sizing:content-box">}</span></span></span></span><span class="gmail-m_-1880571767338173311gmail-MJX_Assistive_MathML" style="box-sizing:border-box;padding:1px 0px 0px;border:0px;height:1px;width:1px;overflow:hidden;display:block">{ci}</span></span>, and ask for a set <span id="gmail-m_-1880571767338173311gmail-MathJax-Element-8-Frame" class="gmail-m_-1880571767338173311gmail-mjx-chtml gmail-m_-1880571767338173311gmail-MathJax_CHTML" style="box-sizing:border-box;display:inline-block;line-height:0;word-spacing:normal;white-space:nowrap;float:none;direction:ltr;max-width:none;max-height:none;min-width:0px;min-height:0px;border:0px;margin:0px;padding:1px 0px"><span id="gmail-m_-1880571767338173311gmail-MJXc-Node-26" class="gmail-m_-1880571767338173311gmail-mjx-math" style="box-sizing:border-box;display:inline-block;border-collapse:separate;border-spacing:0px;margin-top:0px"><span id="gmail-m_-1880571767338173311gmail-MJXc-Node-27" class="gmail-m_-1880571767338173311gmail-mjx-mrow" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_-1880571767338173311gmail-MJXc-Node-28" class="gmail-m_-1880571767338173311gmail-mjx-mi" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-m_-1880571767338173311gmail-mjx-char gmail-m_-1880571767338173311gmail-MJXc-TeX-math-I" style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw;margin-top:0px;padding-top:0.472em;padding-bottom:0.284em;padding-right:0.045em;box-sizing:content-box">C</span></span></span></span><span class="gmail-m_-1880571767338173311gmail-MJX_Assistive_MathML" style="box-sizing:border-box;padding:1px 0px 0px;border:0px;height:1px;width:1px;overflow:hidden;display:block">C</span></span>such that, for each <span id="gmail-m_-1880571767338173311gmail-MathJax-Element-9-Frame" class="gmail-m_-1880571767338173311gmail-mjx-chtml gmail-m_-1880571767338173311gmail-MathJax_CHTML" style="box-sizing:border-box;display:inline-block;line-height:0;word-spacing:normal;white-space:nowrap;float:none;direction:ltr;max-width:none;max-height:none;min-width:0px;min-height:0px;border:0px;margin:0px;padding:1px 0px"><span id="gmail-m_-1880571767338173311gmail-MJXc-Node-29" class="gmail-m_-1880571767338173311gmail-mjx-math" style="box-sizing:border-box;display:inline-block;border-collapse:separate;border-spacing:0px;margin-top:0px"><span id="gmail-m_-1880571767338173311gmail-MJXc-Node-30" class="gmail-m_-1880571767338173311gmail-mjx-mrow" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_-1880571767338173311gmail-MJXc-Node-31" class="gmail-m_-1880571767338173311gmail-mjx-mi" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-m_-1880571767338173311gmail-mjx-char gmail-m_-1880571767338173311gmail-MJXc-TeX-math-I" style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw;margin-top:0px;padding-top:0.425em;padding-bottom:0.284em;box-sizing:content-box">i</span></span></span></span><span class="gmail-m_-1880571767338173311gmail-MJX_Assistive_MathML" style="box-sizing:border-box;padding:1px 0px 0px;border:0px;height:1px;width:1px;overflow:hidden;display:block">i</span></span>, all but finitely many elements of <span id="gmail-m_-1880571767338173311gmail-MathJax-Element-10-Frame" class="gmail-m_-1880571767338173311gmail-mjx-chtml gmail-m_-1880571767338173311gmail-MathJax_CHTML" style="box-sizing:border-box;display:inline-block;line-height:0;word-spacing:normal;white-space:nowrap;float:none;direction:ltr;max-width:none;max-height:none;min-width:0px;min-height:0px;border:0px;margin:0px;padding:1px 0px"><span id="gmail-m_-1880571767338173311gmail-MJXc-Node-32" class="gmail-m_-1880571767338173311gmail-mjx-math" style="box-sizing:border-box;display:inline-block;border-collapse:separate;border-spacing:0px;margin-top:0px"><span id="gmail-m_-1880571767338173311gmail-MJXc-Node-33" class="gmail-m_-1880571767338173311gmail-mjx-mrow" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_-1880571767338173311gmail-MJXc-Node-34" class="gmail-m_-1880571767338173311gmail-mjx-mi" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-m_-1880571767338173311gmail-mjx-char gmail-m_-1880571767338173311gmail-MJXc-TeX-math-I" style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw;margin-top:0px;padding-top:0.472em;padding-bottom:0.284em;padding-right:0.045em;box-sizing:content-box">C</span></span></span></span><span class="gmail-m_-1880571767338173311gmail-MJX_Assistive_MathML" style="box-sizing:border-box;padding:1px 0px 0px;border:0px;height:1px;width:1px;overflow:hidden;display:block">C</span></span> have the same color with respect to <span id="gmail-m_-1880571767338173311gmail-MathJax-Element-11-Frame" class="gmail-m_-1880571767338173311gmail-mjx-chtml gmail-m_-1880571767338173311gmail-MathJax_CHTML" style="box-sizing:border-box;display:inline-block;line-height:0;word-spacing:normal;white-space:nowrap;float:none;direction:ltr;max-width:none;max-height:none;min-width:0px;min-height:0px;border:0px;margin:0px;padding:1px 0px"><span id="gmail-m_-1880571767338173311gmail-MJXc-Node-35" class="gmail-m_-1880571767338173311gmail-mjx-math" style="box-sizing:border-box;display:inline-block;border-collapse:separate;border-spacing:0px;margin-top:0px"><span id="gmail-m_-1880571767338173311gmail-MJXc-Node-36" class="gmail-m_-1880571767338173311gmail-mjx-mrow" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_-1880571767338173311gmail-MJXc-Node-37" class="gmail-m_-1880571767338173311gmail-mjx-msubsup" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-m_-1880571767338173311gmail-mjx-base" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_-1880571767338173311gmail-MJXc-Node-38" class="gmail-m_-1880571767338173311gmail-mjx-mi" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-m_-1880571767338173311gmail-mjx-char gmail-m_-1880571767338173311gmail-MJXc-TeX-math-I" style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw;margin-top:0px;padding-top:0.237em;padding-bottom:0.284em;box-sizing:content-box">c</span></span></span><span class="gmail-m_-1880571767338173311gmail-mjx-sub" style="display:inline-block;vertical-align:-0.212em;padding-right:0.071em;box-sizing:content-box"><span id="gmail-m_-1880571767338173311gmail-MJXc-Node-39" class="gmail-m_-1880571767338173311gmail-mjx-mi" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-m_-1880571767338173311gmail-mjx-char gmail-m_-1880571767338173311gmail-MJXc-TeX-math-I" style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw;margin-top:0px;padding-top:0.425em;padding-bottom:0.284em;box-sizing:content-box">i</span></span></span></span></span></span><span class="gmail-m_-1880571767338173311gmail-MJX_Assistive_MathML" style="box-sizing:border-box;padding:1px 0px 0px;border:0px;height:1px;width:1px;overflow:hidden;display:block">ci</span></span>. Call <span id="gmail-m_-1880571767338173311gmail-MathJax-Element-12-Frame" class="gmail-m_-1880571767338173311gmail-mjx-chtml gmail-m_-1880571767338173311gmail-MathJax_CHTML" style="box-sizing:border-box;display:inline-block;line-height:0;word-spacing:normal;white-space:nowrap;float:none;direction:ltr;max-width:none;max-height:none;min-width:0px;min-height:0px;border:0px;margin:0px;padding:1px 0px"><span id="gmail-m_-1880571767338173311gmail-MJXc-Node-40" class="gmail-m_-1880571767338173311gmail-mjx-math" style="box-sizing:border-box;display:inline-block;border-collapse:separate;border-spacing:0px;margin-top:0px"><span id="gmail-m_-1880571767338173311gmail-MJXc-Node-41" class="gmail-m_-1880571767338173311gmail-mjx-mrow" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_-1880571767338173311gmail-MJXc-Node-42" class="gmail-m_-1880571767338173311gmail-mjx-mi" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-m_-1880571767338173311gmail-mjx-char gmail-m_-1880571767338173311gmail-MJXc-TeX-math-I" style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw;margin-top:0px;padding-top:0.472em;padding-bottom:0.284em;padding-right:0.045em;box-sizing:content-box">C</span></span></span></span><span class="gmail-m_-1880571767338173311gmail-MJX_Assistive_MathML" style="box-sizing:border-box;padding:1px 0px 0px;border:0px;height:1px;width:1px;overflow:hidden;display:block">C</span></span> a <em style="box-sizing:border-box">cohesive</em> set for <span id="gmail-m_-1880571767338173311gmail-MathJax-Element-13-Frame" class="gmail-m_-1880571767338173311gmail-mjx-chtml gmail-m_-1880571767338173311gmail-MathJax_CHTML" style="box-sizing:border-box;display:inline-block;line-height:0;word-spacing:normal;white-space:nowrap;float:none;direction:ltr;max-width:none;max-height:none;min-width:0px;min-height:0px;border:0px;margin:0px;padding:1px 0px"><span id="gmail-m_-1880571767338173311gmail-MJXc-Node-43" class="gmail-m_-1880571767338173311gmail-mjx-math" style="box-sizing:border-box;display:inline-block;border-collapse:separate;border-spacing:0px;margin-top:0px"><span id="gmail-m_-1880571767338173311gmail-MJXc-Node-44" class="gmail-m_-1880571767338173311gmail-mjx-mrow" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_-1880571767338173311gmail-MJXc-Node-45" class="gmail-m_-1880571767338173311gmail-mjx-mo" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-m_-1880571767338173311gmail-mjx-char gmail-m_-1880571767338173311gmail-MJXc-TeX-main-R" style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.472em;padding-bottom:0.613em;box-sizing:content-box">{</span></span><span id="gmail-m_-1880571767338173311gmail-MJXc-Node-46" class="gmail-m_-1880571767338173311gmail-mjx-msubsup" style="display:inline-block;box-sizing:content-box"><span class="gmail-m_-1880571767338173311gmail-mjx-base" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_-1880571767338173311gmail-MJXc-Node-47" class="gmail-m_-1880571767338173311gmail-mjx-mi" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-m_-1880571767338173311gmail-mjx-char gmail-m_-1880571767338173311gmail-MJXc-TeX-math-I" style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw;margin-top:0px;padding-top:0.237em;padding-bottom:0.284em;box-sizing:content-box">c</span></span></span><span class="gmail-m_-1880571767338173311gmail-mjx-sub" style="display:inline-block;vertical-align:-0.212em;padding-right:0.071em;box-sizing:content-box"><span id="gmail-m_-1880571767338173311gmail-MJXc-Node-48" class="gmail-m_-1880571767338173311gmail-mjx-mi" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-m_-1880571767338173311gmail-mjx-char gmail-m_-1880571767338173311gmail-MJXc-TeX-math-I" style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw;margin-top:0px;padding-top:0.425em;padding-bottom:0.284em;box-sizing:content-box">i</span></span></span></span><span id="gmail-m_-1880571767338173311gmail-MJXc-Node-49" class="gmail-m_-1880571767338173311gmail-mjx-mo" style="display:inline-block;box-sizing:content-box"><span class="gmail-m_-1880571767338173311gmail-mjx-char gmail-m_-1880571767338173311gmail-MJXc-TeX-main-R" style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.472em;padding-bottom:0.613em;box-sizing:content-box">}</span></span></span></span><span class="gmail-m_-1880571767338173311gmail-MJX_Assistive_MathML" style="box-sizing:border-box;padding:1px 0px 0px;border:0px;height:1px;width:1px;overflow:hidden;display:block">{ci}</span></span>. Among the questions we will explore here is what can be computed from homogeneous sets. For example, given countably many <span id="gmail-m_-1880571767338173311gmail-MathJax-Element-14-Frame" class="gmail-m_-1880571767338173311gmail-mjx-chtml gmail-m_-1880571767338173311gmail-MathJax_CHTML" style="box-sizing:border-box;display:inline-block;line-height:0;word-spacing:normal;white-space:nowrap;float:none;direction:ltr;max-width:none;max-height:none;min-width:0px;min-height:0px;border:0px;margin:0px;padding:1px 0px"><span id="gmail-m_-1880571767338173311gmail-MJXc-Node-50" class="gmail-m_-1880571767338173311gmail-mjx-math" style="box-sizing:border-box;display:inline-block;border-collapse:separate;border-spacing:0px;margin-top:0px"><span id="gmail-m_-1880571767338173311gmail-MJXc-Node-51" class="gmail-m_-1880571767338173311gmail-mjx-mrow" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_-1880571767338173311gmail-MJXc-Node-52" class="gmail-m_-1880571767338173311gmail-mjx-mn" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-m_-1880571767338173311gmail-mjx-char gmail-m_-1880571767338173311gmail-MJXc-TeX-main-R" style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.378em;padding-bottom:0.331em;box-sizing:content-box">1</span></span></span></span><span class="gmail-m_-1880571767338173311gmail-MJX_Assistive_MathML" style="box-sizing:border-box;padding:1px 0px 0px;border:0px;height:1px;width:1px;overflow:hidden;display:block">1</span></span>-colorings <span id="gmail-m_-1880571767338173311gmail-MathJax-Element-15-Frame" class="gmail-m_-1880571767338173311gmail-mjx-chtml gmail-m_-1880571767338173311gmail-MathJax_CHTML" style="box-sizing:border-box;display:inline-block;line-height:0;word-spacing:normal;white-space:nowrap;float:none;direction:ltr;max-width:none;max-height:none;min-width:0px;min-height:0px;border:0px;margin:0px;padding:1px 0px"><span id="gmail-m_-1880571767338173311gmail-MJXc-Node-53" class="gmail-m_-1880571767338173311gmail-mjx-math" style="box-sizing:border-box;display:inline-block;border-collapse:separate;border-spacing:0px;margin-top:0px"><span id="gmail-m_-1880571767338173311gmail-MJXc-Node-54" class="gmail-m_-1880571767338173311gmail-mjx-mrow" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_-1880571767338173311gmail-MJXc-Node-55" class="gmail-m_-1880571767338173311gmail-mjx-mo" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-m_-1880571767338173311gmail-mjx-char gmail-m_-1880571767338173311gmail-MJXc-TeX-main-R" style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.472em;padding-bottom:0.613em;box-sizing:content-box">{</span></span><span id="gmail-m_-1880571767338173311gmail-MJXc-Node-56" class="gmail-m_-1880571767338173311gmail-mjx-msubsup" style="display:inline-block;box-sizing:content-box"><span class="gmail-m_-1880571767338173311gmail-mjx-base" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_-1880571767338173311gmail-MJXc-Node-57" class="gmail-m_-1880571767338173311gmail-mjx-mi" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-m_-1880571767338173311gmail-mjx-char gmail-m_-1880571767338173311gmail-MJXc-TeX-math-I" style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw;margin-top:0px;padding-top:0.237em;padding-bottom:0.284em;box-sizing:content-box">c</span></span></span><span class="gmail-m_-1880571767338173311gmail-mjx-sub" style="display:inline-block;vertical-align:-0.212em;padding-right:0.071em;box-sizing:content-box"><span id="gmail-m_-1880571767338173311gmail-MJXc-Node-58" class="gmail-m_-1880571767338173311gmail-mjx-mi" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-m_-1880571767338173311gmail-mjx-char gmail-m_-1880571767338173311gmail-MJXc-TeX-math-I" style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw;margin-top:0px;padding-top:0.425em;padding-bottom:0.284em;box-sizing:content-box">i</span></span></span></span><span id="gmail-m_-1880571767338173311gmail-MJXc-Node-59" class="gmail-m_-1880571767338173311gmail-mjx-mo" style="display:inline-block;box-sizing:content-box"><span class="gmail-m_-1880571767338173311gmail-mjx-char gmail-m_-1880571767338173311gmail-MJXc-TeX-main-R" style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.472em;padding-bottom:0.613em;box-sizing:content-box">}</span></span></span></span><span class="gmail-m_-1880571767338173311gmail-MJX_Assistive_MathML" style="box-sizing:border-box;padding:1px 0px 0px;border:0px;height:1px;width:1px;overflow:hidden;display:block">{ci}</span></span>, is there a 2-coloring <span id="gmail-m_-1880571767338173311gmail-MathJax-Element-16-Frame" class="gmail-m_-1880571767338173311gmail-mjx-chtml gmail-m_-1880571767338173311gmail-MathJax_CHTML" style="box-sizing:border-box;display:inline-block;line-height:0;word-spacing:normal;white-space:nowrap;float:none;direction:ltr;max-width:none;max-height:none;min-width:0px;min-height:0px;border:0px;margin:0px;padding:1px 0px"><span id="gmail-m_-1880571767338173311gmail-MJXc-Node-60" class="gmail-m_-1880571767338173311gmail-mjx-math" style="box-sizing:border-box;display:inline-block;border-collapse:separate;border-spacing:0px;margin-top:0px"><span id="gmail-m_-1880571767338173311gmail-MJXc-Node-61" class="gmail-m_-1880571767338173311gmail-mjx-mrow" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_-1880571767338173311gmail-MJXc-Node-62" class="gmail-m_-1880571767338173311gmail-mjx-mi" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-m_-1880571767338173311gmail-mjx-char gmail-m_-1880571767338173311gmail-MJXc-TeX-math-I" style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw;margin-top:0px;padding-top:0.472em;padding-bottom:0.284em;padding-right:0.045em;box-sizing:content-box">C</span></span></span></span><span class="gmail-m_-1880571767338173311gmail-MJX_Assistive_MathML" style="box-sizing:border-box;padding:1px 0px 0px;border:0px;height:1px;width:1px;overflow:hidden;display:block">C</span></span>, such that every homogeneous set for <span id="gmail-m_-1880571767338173311gmail-MathJax-Element-17-Frame" class="gmail-m_-1880571767338173311gmail-mjx-chtml gmail-m_-1880571767338173311gmail-MathJax_CHTML" style="box-sizing:border-box;display:inline-block;line-height:0;word-spacing:normal;white-space:nowrap;float:none;direction:ltr;max-width:none;max-height:none;min-width:0px;min-height:0px;border:0px;margin:0px;padding:1px 0px"><span id="gmail-m_-1880571767338173311gmail-MJXc-Node-63" class="gmail-m_-1880571767338173311gmail-mjx-math" style="box-sizing:border-box;display:inline-block;border-collapse:separate;border-spacing:0px;margin-top:0px"><span id="gmail-m_-1880571767338173311gmail-MJXc-Node-64" class="gmail-m_-1880571767338173311gmail-mjx-mrow" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_-1880571767338173311gmail-MJXc-Node-65" class="gmail-m_-1880571767338173311gmail-mjx-mi" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-m_-1880571767338173311gmail-mjx-char gmail-m_-1880571767338173311gmail-MJXc-TeX-math-I" style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw;margin-top:0px;padding-top:0.472em;padding-bottom:0.284em;padding-right:0.045em;box-sizing:content-box">C</span></span></span></span><span class="gmail-m_-1880571767338173311gmail-MJX_Assistive_MathML" style="box-sizing:border-box;padding:1px 0px 0px;border:0px;height:1px;width:1px;overflow:hidden;display:block">C</span></span> computes a <em style="box-sizing:border-box">cohesive</em> set for <span id="gmail-m_-1880571767338173311gmail-MathJax-Element-18-Frame" class="gmail-m_-1880571767338173311gmail-mjx-chtml gmail-m_-1880571767338173311gmail-MathJax_CHTML" style="box-sizing:border-box;display:inline-block;line-height:0;word-spacing:normal;white-space:nowrap;float:none;direction:ltr;max-width:none;max-height:none;min-width:0px;min-height:0px;border:0px;margin:0px;padding:1px 0px"><span id="gmail-m_-1880571767338173311gmail-MJXc-Node-66" class="gmail-m_-1880571767338173311gmail-mjx-math" style="box-sizing:border-box;display:inline-block;border-collapse:separate;border-spacing:0px;margin-top:0px"><span id="gmail-m_-1880571767338173311gmail-MJXc-Node-67" class="gmail-m_-1880571767338173311gmail-mjx-mrow" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_-1880571767338173311gmail-MJXc-Node-68" class="gmail-m_-1880571767338173311gmail-mjx-mo" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-m_-1880571767338173311gmail-mjx-char gmail-m_-1880571767338173311gmail-MJXc-TeX-main-R" style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.472em;padding-bottom:0.613em;box-sizing:content-box">{</span></span><span id="gmail-m_-1880571767338173311gmail-MJXc-Node-69" class="gmail-m_-1880571767338173311gmail-mjx-msubsup" style="display:inline-block;box-sizing:content-box"><span class="gmail-m_-1880571767338173311gmail-mjx-base" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-m_-1880571767338173311gmail-MJXc-Node-70" class="gmail-m_-1880571767338173311gmail-mjx-mi" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-m_-1880571767338173311gmail-mjx-char gmail-m_-1880571767338173311gmail-MJXc-TeX-math-I" style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw;margin-top:0px;padding-top:0.237em;padding-bottom:0.284em;box-sizing:content-box">c</span></span></span><span class="gmail-m_-1880571767338173311gmail-mjx-sub" style="display:inline-block;vertical-align:-0.212em;padding-right:0.071em;box-sizing:content-box"><span id="gmail-m_-1880571767338173311gmail-MJXc-Node-71" class="gmail-m_-1880571767338173311gmail-mjx-mi" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-m_-1880571767338173311gmail-mjx-char gmail-m_-1880571767338173311gmail-MJXc-TeX-math-I" style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw;margin-top:0px;padding-top:0.425em;padding-bottom:0.284em;box-sizing:content-box">i</span></span></span></span><span id="gmail-m_-1880571767338173311gmail-MJXc-Node-72" class="gmail-m_-1880571767338173311gmail-mjx-mo" style="display:inline-block;box-sizing:content-box"><span class="gmail-m_-1880571767338173311gmail-mjx-char gmail-m_-1880571767338173311gmail-MJXc-TeX-main-R" style="display:block;white-space:pre-wrap;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.472em;padding-bottom:0.613em;box-sizing:content-box">}</span></span></span></span><span class="gmail-m_-1880571767338173311gmail-MJX_Assistive_MathML" style="box-sizing:border-box;padding:1px 0px 0px;border:0px;height:1px;width:1px;overflow:hidden;display:block">{ci}</span></span>?</p></div></div></div><div style="color:rgb(33,33,33)"><br></div><div style="color:rgb(33,33,33)"><br></div><div style="color:rgb(33,33,33)"><br></div><div><div><font color="#212121">Next Week in Logic at CUNY:</font></div><div><font color="#212121"><br></font></div><div><font color="#212121">- - - - Monday, Apr 8, 2019 - - - -</font></div><div><font color="#212121"><br></font></div><div><font color="#212121"><br></font></div><div><font color="#212121"><br></font></div><div><font color="#212121">- - - - Tuesday, Apr 9, 2019 - - - -</font></div><div><font color="#212121"><br></font></div><div><font color="#212121"><br></font></div><div><font color="#212121"><br></font></div><div><font color="#212121">- - - - Wednesday, Apr 10, 2019 - - - -</font></div><div><font color="#212121"><br></font></div><div><font color="#212121"><br></font></div><div><font color="#212121"><br></font></div><div><font color="#212121">- - - - Thursday, Apr 11, 2019 - - - -</font></div><div><font color="#212121"><br></font></div><div><font color="#212121"><br></font></div><div><font color="#212121"><br></font></div><div><font color="#212121">- - - - Friday, Apr 12, 2019 - - - -</font></div></div><div><font color="#212121"><br></font></div><div><div style="color:rgb(33,33,33)">Set Theory Seminar</div><div style="color:rgb(33,33,33)">CUNY Graduate Center, Room 6417</div><div style=""><font color="#000000">Friday, April 12, 10:00-11:45am</font></div><div><font color="#000000"><span style="font-family:&quot;Open Sans&quot;,&quot;Helvetica Neue&quot;,Helvetica,Arial,sans-serif">Jonas Reitz, CUNY</span><br style="box-sizing:border-box;font-family:&quot;Open Sans&quot;,&quot;Helvetica Neue&quot;,Helvetica,Arial,sans-serif"></font><div><font face="Open Sans, Helvetica Neue, Helvetica, Arial, sans-serif" color="#000000"><b>Generalized Cohen Iterations</b></font></div><div><font face="Open Sans, Helvetica Neue, Helvetica, Arial, sans-serif" color="#000000"><br></font></div><div><font face="Open Sans, Helvetica Neue, Helvetica, Arial, sans-serif" color="#000000">Adding Cohen subsets to each of a class of cardinals in turn is a common construction in set theory, and underlies many fundamental results.  The construction comes in two basic flavors, products (as in Easton’s Theorem on the powers of regular cardinals) and iterations (forcing the GCH).  These flavors are  apparently quite similar, forcing at stage kappa to add subsets via the Cohen partial order Add(kappa,lambda).  They differ only in the universe over which Add(kappa,lambda) is defined - in the case of products the ground model poset is used at each stage, whereas in typical iterations the poset is taken from the partial extension up to kappa.  In this talk I will consider an alternative, in which we allow Add(kappa,lambda) to be defined over an arbitrary inner model (lying between the ground model and the extension up to kappa) at each stage.   These generalized Cohen iterations are ZFC-preserving, although neither the proof for products nor for traditional iterations transfers directly.   They allow constructions such as class iterations of class products of Cohen forcing, with applications including new work with Kameryn Williams on iterating the Mantle.</font></div></div><div style=""><font color="#000000"><br></font></div><div style=""><font color="#000000"><br></font></div><div style=""><font color="#000000"><br></font></div><div style=""><font color="#000000">Model Theory Seminar</font></div><div style="color:rgb(33,33,33)">CUNY Graduate Center, Room 6417</div><div style="color:rgb(33,33,33)">Friday, April 12, 12:30-2:00pm</div><div style="color:rgb(33,33,33)"><strong style="box-sizing:border-box;color:rgb(96,108,113);font-family:&quot;Open Sans&quot;,&quot;Helvetica Neue&quot;,Helvetica,Arial,sans-serif"><a href="http://sites.nd.edu/rachaelalvir/" target="_blank" style="box-sizing:border-box;background-color:transparent;color:rgb(30,107,184);text-decoration-line:none;margin-top:0px">Rachael Alvir</a></strong><span style="color:rgb(96,108,113);font-family:&quot;Open Sans&quot;,&quot;Helvetica Neue&quot;,Helvetica,Arial,sans-serif">, University of Notre Dame</span><br style="box-sizing:border-box;color:rgb(96,108,113);font-family:&quot;Open Sans&quot;,&quot;Helvetica Neue&quot;,Helvetica,Arial,sans-serif"><strong style="box-sizing:border-box;color:rgb(96,108,113);font-family:&quot;Open Sans&quot;,&quot;Helvetica Neue&quot;,Helvetica,Arial,sans-serif">TBA</strong><br></div><div style="color:rgb(33,33,33)"><br></div><div style="color:rgb(33,33,33)"><br></div><div style="color:rgb(33,33,33)"><br></div><div style="color:rgb(33,33,33)">Logic Workshop</div><div style="color:rgb(33,33,33)">CUNY Graduate Center, Room 6417</div><div style="color:rgb(33,33,33)">Friday, April 12, 2:00-3:30pm</div><div style="color:rgb(33,33,33)"><p style="box-sizing:border-box;margin-bottom:1em;margin-top:0px;color:rgb(96,108,113);font-family:&quot;Open Sans&quot;,&quot;Helvetica Neue&quot;,Helvetica,Arial,sans-serif"><strong style="box-sizing:border-box"><a href="https://math.nd.edu/people/faculty/julia-f-knight/" target="_blank" style="box-sizing:border-box;background-color:transparent;color:rgb(30,107,184);text-decoration-line:none;margin-top:0px">Julia Knight</a></strong>, University of Notre Dame<br style="box-sizing:border-box"><strong style="box-sizing:border-box">Coding structures</strong><br style="box-sizing:border-box"></p><p style="box-sizing:border-box;margin-bottom:1em;color:rgb(96,108,113);font-family:&quot;Open Sans&quot;,&quot;Helvetica Neue&quot;,Helvetica,Arial,sans-serif">A <em style="box-sizing:border-box;margin-top:0px">Turing computable embedding</em> is a Turing operator that maps one class of structures to another so as to preserve isomorphism. The embedding codes the input structure in the output structure. It is interesting when there is an effective decoding. It is also interesting when the decoding is very difficult. Recently, Harrison-Trainor, Melnikov, R. Miller, and Montalbán have defined very general notions of interpretation, in which the interpreting formulas have no fixed arity. Uniformly defined interpretations give us decoding. I will discuss some known Turing computable embeddings, looking for uniform interpretations that yield effective, or Borel, decoding.</p><ol style="box-sizing:border-box;margin-top:0px;color:rgb(96,108,113);font-family:&quot;Open Sans&quot;,&quot;Helvetica Neue&quot;,Helvetica,Arial,sans-serif"><li style="box-sizing:border-box;margin-top:0px">Marker&#39;s embedding of directed graphs in undirected graphs,</li><li style="box-sizing:border-box">Mal&#39;tsev&#39;s embedding of fields in groups,</li><li style="box-sizing:border-box">Friedman and Stanley&#39;s embedding of graphs in linear orderings.</li></ol><span style="color:rgb(96,108,113);font-family:&quot;Open Sans&quot;,&quot;Helvetica Neue&quot;,Helvetica,Arial,sans-serif">The first two embeddings come with uniform &#39;effective&#39; interpretations, which give uniform effective decoding. For the third, we do not even have uniform interpretation via </span><span id="gmail-MathJax-Element-1-Frame" class="gmail-mjx-chtml gmail-MathJax_CHTML" tabindex="0" style="box-sizing:border-box;display:inline-block;line-height:0;white-space:nowrap;float:none;direction:ltr;max-width:none;max-height:none;min-width:0px;min-height:0px;border:0px;margin:0px;padding:1px 0px;color:rgb(96,108,113);font-family:&quot;Open Sans&quot;,&quot;Helvetica Neue&quot;,Helvetica,Arial,sans-serif"><span id="gmail-MJXc-Node-1" class="gmail-mjx-math" style="box-sizing:border-box;display:inline-block;border-collapse:separate;border-spacing:0px;margin-top:0px"><span id="gmail-MJXc-Node-2" class="gmail-mjx-mrow" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-MJXc-Node-3" class="gmail-mjx-msubsup" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-mjx-base" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-MJXc-Node-4" class="gmail-mjx-mi" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-math-I" style="display:block;white-space:pre;font-family:MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw;margin-top:0px;padding-top:0.472em;padding-bottom:0.284em;box-sizing:content-box">L</span></span></span><span class="gmail-mjx-sub" style="display:inline-block;vertical-align:-0.212em;padding-right:0.071em;box-sizing:content-box"><span id="gmail-MJXc-Node-5" class="gmail-mjx-texatom" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-MJXc-Node-6" class="gmail-mjx-mrow" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-MJXc-Node-7" class="gmail-mjx-msubsup" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-mjx-base" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span id="gmail-MJXc-Node-8" class="gmail-mjx-mi" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-math-I" style="display:block;white-space:pre;font-family:MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw;margin-top:0px;padding-top:0.237em;padding-bottom:0.284em;box-sizing:content-box">ω</span></span></span><span class="gmail-mjx-sub" style="display:inline-block;vertical-align:-0.212em;padding-right:0.071em;box-sizing:content-box"><span id="gmail-MJXc-Node-9" class="gmail-mjx-mn" style="display:inline-block;margin-top:0px;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-main-R" style="display:block;white-space:pre;font-family:MJXc-TeX-main-R,MJXc-TeX-main-Rw;margin-top:0px;padding-top:0.378em;padding-bottom:0.331em;box-sizing:content-box">1</span></span></span></span><span id="gmail-MJXc-Node-10" class="gmail-mjx-mi" style="display:inline-block;box-sizing:content-box"><span class="gmail-mjx-char gmail-MJXc-TeX-math-I" style="display:block;white-space:pre;font-family:MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw;margin-top:0px;padding-top:0.237em;padding-bottom:0.284em;box-sizing:content-box">ω</span></span></span></span></span></span></span></span><span class="gmail-MJX_Assistive_MathML" style="box-sizing:border-box;padding:1px 0px 0px;border:0px;height:1px;width:1px;overflow:hidden;display:block">L<span class="gmail-MJX-TeXAtom-ORD">ω1ω</span></span></span><span style="color:rgb(96,108,113);font-family:&quot;Open Sans&quot;,&quot;Helvetica Neue&quot;,Helvetica,Arial,sans-serif"> formulas.</span><br></div></div><div><font color="#212121"><br></font></div><div><font color="#212121"><br></font></div><div><font color="#212121"><br></font></div><div><font color="#212121"><br></font></div><div><font color="#212121">- - - - Other Logic News - - - -</font></div><div><br></div><div><font color="#212121"><br></font></div><div><font color="#212121"><br></font></div><div><font color="#212121">- - - - Web Site - - - -</font></div><div><font color="#212121"><br></font></div><div><font color="#212121">Our new website is located at:  <a href="https://nylogic.github.io/" target="_blank">https://nylogic.github.io/</a></font></div><div><font color="#212121">Many thanks to Victoria Gitman for her development work!</font></div><div><font color="#212121"><br></font></div><div><font color="#212121">--------  ADMINISTRIVIA  --------</font></div><div><font color="#212121"><br></font></div><div><font color="#212121">To subscribe/unsubscribe to this list, please email your request to <a href="mailto:jreitz@nylogic.org" target="_blank">jreitz@nylogic.org</a>.</font></div><div><font color="#212121"><br></font></div><div><font color="#212121">If you have a logic-related event that you would like included in future mailings, please email <a href="mailto:jreitz@nylogic.org" target="_blank">jreitz@nylogic.org</a>. </font></div></div></div></div></div></div></div>
63
Fri, 29 Mar 2019 18:08:50Barcelona Set Theory SeminarJoan Bagaria: Finite trees and stationary setsDate: Thursday 4 April 2019
Time: 16:00
Place: Room S-1*
<p style="text-align: center;">BARCELONA RESEARCH GROUP IN SET THEORY
BARCELONA SET THEORY SEMINAR</p>
Title: Finite trees and stationary sets

Speaker: Joan Bagaria

Abstract: We shall present a proof of Andreas Blass’ embedding
theorem for finite rooted trees, which assigns pairwise-disjoint
stationary-reflecting sets of ordinals to the nodes of such a tree
in a coherent way. The proof uses Jensen’s square principle.
After identifying the main obstacles for the theorem’s
generalization to hyperstationary sets, we will show how one of
the obstacles may be removed by the use of the new notion of
hypercofinality.
Joan Bagaria9699
64
Thu, 28 Mar 2019 15:40:43Israeli Logic TalksAsger Törnquist: All about mad families<strong>BIU Infinite Combinatorics Seminar</strong>

Mon, 01/04/2019 - 13:00

Speaker: Asger Törnquist

Title: All about mad families

Abstract. I will give an overview of the developments in the past 5 years regarding mad families.

We'll study families of subsets of the natural numbers, and say that such a family is almost disjoint if any two distinct elements intersect finitely. The Axiom of Choice implies the existence of infinite almost disjoint family which is maximal under inclusion.

Mathias proved in the late 1960s that it is consistent with ZF+DC that there are no mad families. He needed a Mahlo cardinal to do this. In 2014 I showed that the classical Solovay-Lévy model has no infinite mad families, and shortly thereafter, in 2016, Horowitz and Shelah showed that you don't even need an inaccessible to get a model of ZF+DC+no infinite mad families.

A wealth of related questions have also been settled, most recently, I have shown with David Schrittesser that "All sets are Ramsey"+"Ramsey uniformization" implies "no infinite mad families ".

I'll also discuss open problems. The talk will not assume any prior knowledge of mad families.
Asger Tornquist9695
65
Thu, 28 Mar 2019 13:28:38Prague Set Theory SeminarEgbert Thümmel: Topologies on Boolean algebrasDear all,

The seminar meets on Wednesday April 3rd at 11:00 in the Institute of
Mathematics CAS, Zitna 25, seminar room, 3rd floor, front building.

Program: Egbert Thümmel -- Topologies on Boolean algebras

We define the sequential topology on complete Boolean algebra which was
successfully applied especially for ccc and Maharam algebras. B. Balcar
asked whether in this topology zero has always a neighbourhood base of
downward closed sets.
We give a precise answer to this question (equivalence in forcing terms)
and connections to other problems.

Best,
David
Egbert Thümmel9692
66
Thu, 28 Mar 2019 9:48:24Kurt Godel Research CenterMiha Habič: Capturing by normal ultrapowersTalk held by Miha Habič (Czech Technical University in Prague, Czech Republic and Charles University, Prague, Czech Republic) at the KGRC seminar on 2019-03-28.

<strong>Abstract</strong>: If $\kappa$ is measurable and GCH holds, then any ultrapower by a normal measure on $\kappa$ will be missing some subset of $\kappa^+$. On the other hand, Cummings showed that, starting from a $(\kappa+2)$-strong $\kappa$, one can force to a model (without collapsing cardinals) where $\kappa$ carries a normal measure whose ultrapower captures the entire powerset of $\kappa^+$. Moreover, the large cardinal hypothesis is optimal. I will present an improvement of Cummings' result and show that this capturing property can consistently hold at the least measurable cardinal.

This is joint work with Radek Honzík.

&nbsp;
Miha Habic9689
67
Tue, 26 Mar 2019 14:52:19ConferenceLogic Colloquium 2019 (reminder)<a href="https://settheory.mathtalks.org/wp-content/uploads/2019/03/carousel.png"><img class="alignnone wp-image-9684 size-large" src="https://settheory.mathtalks.org/wp-content/uploads/2019/03/carousel-1024x307.png" alt="" width="500" height="150"></a>

This is a reminder that the deadline for early bird registrations for the <strong>Logic Colloquium 2019</strong>, <strong>August 11-16, Prague, Czech Republic </strong>is approaching.
<h3>Deadlines</h3>
<ul>
<li>Abstract submission&nbsp; &nbsp; &nbsp; &nbsp; &nbsp;<strong>April 30, 2019</strong></li>
<li>Application for support&nbsp; &nbsp;&nbsp;<strong>April 15, 2019</strong></li>
<li>Early bird registration&nbsp; &nbsp; &nbsp; &nbsp;<strong>May 15, 2019</strong></li>
</ul>
The conference now has a website with more info, registration and abstract submission
<h3 style="text-align: center"><a href="https://www.lc2019.cz/" target="_blank" rel="nofollow noreferrer noopener"><strong>www.lc2019.cz</strong></a></h3>
A poster for the conference can be downloaded <a href="https://settheory.mathtalks.org/wp-content/uploads/2019/03/poster-charles-bridge-variant-dark-blurred-A3-screen.pdf">here</a>.
<h3>Invited &amp; Tutorial Speakers</h3>
<strong>Tutorials</strong>
<ul>
<li>Michael Rathjen (Leeds)</li>
<li>Dilip Raghavan (Singapore)</li>
</ul>
<strong>Plenary lectures</strong>
<ul>
<li>Hannes Leitgeb (Munich)</li>
<li>Ulrich Kohlenbach (Darmstadt)</li>
<li>Samson Abramsky (Oxford)</li>
<li>Zoé Chatzidakis (Paris)</li>
<li>Osvaldo Guzman (Toronto)</li>
<li>Matthew Harrison-Trainor (Wellington)</li>
<li>Jan Krajíček (Prague)</li>
<li>Vincenzo de Risi (Paris)</li>
<li>Gil Sagi (Haifa)</li>
<li>Thomas Scanlon (Berkeley)</li>
<li>Rineke Verbrugge (Groningen)</li>
<li>Martin Ziegler (KAIST, Korea)</li>
</ul>
<h3>Special Sessions</h3>
<ul>
<li><strong>Set Theory</strong>, organized by David Chodounský and Osvaldo Guzmán</li>
<li><strong>Model Theory</strong>, organized by Thomas Scanlon and Maryanthe Malliaris</li>
<li><strong>Reflection Principles and Modal Logic</strong>, organized by Joost Joosten and David Fernández-Duque</li>
<li><strong>Proof Theory and Proof Complexity</strong>, organized by Ulrich Kohlenbach and Samuel Buss</li>
<li><strong>Computability</strong>, organized by Damir Dzhafarov and Alexander Shen</li>
<li><strong>Foundations of Geometry</strong>, organized by John T. Baldwin and Victor Pambuccian</li>
</ul>
<h3>SESSION&nbsp; ON SET THEORY</h3>
<ul>
<li><strong>Yair Hayut</strong> (Vienna)</li>
<li><strong>Heike Mildenberger</strong> (Freiburg)</li>
<li><strong>Daniel Soukup</strong> (Vienna)</li>
<li><strong>Andy Zucker</strong> (Paris)</li>
</ul>
<h3>Conference Venue</h3>
Czech Technical University
Thákurova 9
166 34&nbsp;Prague 6

Czech Republic
Andy Zucker,Daniel Soukup,Dilip Raghavan,Heike Mildenberger,Yair Hayut
https://drive.google.com/drive/folders/1LuYK2iWKoJxjNBG0-d-BIJRtjIIiIZ4_?usp=sharing
9682
68
Tue, 26 Mar 2019 6:49:08Toronto Set Theory SeminarSlawomir Solecki: Transfinite sequences of topologies and descriptive complexityPlace: Fields Institute (Room 210)
Date: March 39, 2019 (13:30-15:00)
Speaker:&nbsp; Slawomir Solecki
Title: Transfinite sequences of topologies and descriptive complexity
Abstract: We introduce a notion of filtration from one topology $\sigma$ to another $\tau$ assuming that $\tau$ contains $\sigma$. Such filtrations are certain transfinite sequence of topologies interpolating between $\sigma$ and $\tau$. We consider the question of whether a filtration succeeds in reaching $\tau$, and, if it does, at what stage it happens. Answers to these questions involve descriptive set theoretic conditions on the relationship between $\sigma$ and $\tau$. This theme arose in investigations concerning the Scott analysis of certain definable equivalence relations, but the talk will be independent of these considerations.
Slawomir Solecki9680
69
Sat, 23 Mar 2019 8:50:59Israeli Logic TalksRalf Schindler: Paradoxical" sets with no well-ordering of the reals"<span style="text-decoration: underline;">HUJI Set Theory Seminar</span>

On Wednesday, March 27, (14:00-15:30 in Ross 63) Ralf Schindler (Münster) will talk about "Paradoxical" sets with no well-ordering of the reals.

Abstract: By a Hamel basis we mean a basis for the reals, R, construed as a vecor space over the field of rationals. In 1905, G. Hamel constructed such a basis from a well-ordering of R. In 1975, D. Pincus and K. Prikry asked "whether a Hamel basis exists in any model in which R cannot be well ordered." About two years ago, we answered this positively in a joint paper with M. Beriashvili, L. Wu, and L. Yu. In more recent joint work, additionally with J. Brendle and F. Castiblanco we constructed a model of ZF with a Luzin set, a Sierpiński set, a Burstin basis, and a Mazurkiewicz set, but with no well-ordering of R. In joint work with V. Kanovei, we constructed such a model in which even all those "paradoxical" sets are projective.
Ralf Schindler9674
70
Fri, 22 Mar 2019 10:05:30Barcelona Set Theory SeminarMoritz Müller: Provability and consistency of circuit lower boundsDate: Thursday 28 March 2019
Time: 16:00
Place: Room S-1*
Departament de Matemàtiques i Informàtica
Universitat de Barcelona
Gran Via de les Corts Catalanes 585, 08007 Barcelona
<p style="text-align: center;">BARCELONA RESEARCH GROUP IN SET THEORY
BARCELONA SET THEORY SEMINAR</p>
&nbsp;

Speaker: Moritz Müller (UPC)

Title: Provability and consistency of circuit lower bounds

Abstract. In 1995 Razborov asked for the right fragment of
bounded arithmetic capturing existing techniques to prove
circuit lower bounds for explicit Boolean functions. The talk
reports some new developments.

&nbsp;
Moritz Müller9676
71
Fri, 22 Mar 2019 7:51:54Wrocław University of TechnologyDamian Sobota: Josefson-Nissenzweig theorem for $C(K)$-spacesTuesday, March 26, 2019, 17:15
Wrocław University of Technology, 215 D-1

Speaker: Damian Sobota (Universityof Viena)

Title: Josefson-Nissenzweig theorem for $C(K)$-spaces

Abstract:

The Josefson-Nissenzweig theorem is a powerful tool in Banach space theory. Its special version for Banach spaces of continuous functions reads as follows: for a given infinite compact space K there exists a sequence $(\mu_n)$ of normalized signed Radon measures on K such that the integrals $\mu_n(f)$ converge to $0$ for any function $f$ in $C(K)$. During my talk I will investigate when the sequence $(\mu_n)$ can be chosen in such a way that every $\mu_n$ is just a finite linear combination of Dirac point measures (in other words, $\mu_n$ has finite support). This will appear to have connections with the Grothendieck property of Banach spaces and complementability of the space $c_0$. In particular, I'll present a very elementary proof that $c_0$ is always complemented in a space $C(K\times K)$.
Damian Sobota9671
72
Wed, 20 Mar 2019 14:08:17Prague Set Theory SeminarSaeed Ghasemi: AF-algebras with Cantor-set propertyDear all,

The seminar meets on Wednesday March 27th at 11:00 in the Institute of
Mathematics CAS, Zitna 25, seminar room, 3rd floor, front building.

Program: Saeed Ghasemi -- AF-algebras with Cantor-set property

A separable AF-algebra is a C*-algebra which is (isomorphic to) the
inductive limit of a direct sequence of finite dimensional C*-algebras.
We introduce a class of separable AF-algebras, called AF-algebras with
Cantor-set property, which are, in some contexts, suitable
noncommutative analogues of the Cantor set. One of the main features of
AF-algebras with Cantor-set property is that they are all Fraisse
limits of some category of finite dimensional C*-algebras and left
invertible embeddings. As a consequence of this, many properties of the
Cantor set that can be proved using the Fraisse theory, such as the
homogeneity and universality, also can also be proved for AF-algebras
with Cantor-set property. In fact, the category of all finite
dimensional C*-algebras and left invertible embeddings is a Fraisse
category and its Fraisse limit is a separable AF-algebra with Cantor-set
property which has the universality property that maps surjectively onto
any separable AF-algebra.*
This is a joint work with Wieslaw Kubis.

*- All of these results can be restated and proved in the language of
partially ordered abelian groups without mentioning any C*-algebras.

Best,
David
Saeed Ghasemi9669
73
Wed, 20 Mar 2019 4:26:41NUS Logic SeminarAshutosh Kumar: Order dimension of Turing degreesInvitation to the Logic Seminar at the National University of Singapore

Date: Wednesday, 27 March 2019, 17:00 hrs

Room: S17#04-06, Department of Mathematics, NUS

Speaker: Ashutosh Kumar

Title: Order dimension of Turing degrees

URL: http://www.comp.nus.edu.sg/~fstephan/logicseminar.html

The order dimension of a partially ordered set (P,&lt;)
is the smallest size of a family F of linear orders,
each extending &lt;, such that the intersection of F is
the given ordering &lt;.

Higuchi, Lempp, Raghavan and Stephan asked if the order dimension
of Turing degrees could be decided in ZFC. We show that the answer is no.

This is joint work with Dilip Raghavan.
Ashutosh Kumar9667
74
Tue, 19 Mar 2019 14:44:56Israeli Logic TalksSpencer Unger: Stationary reflection and the singular cardinals hypothesis<span style="text-decoration: underline;">HUJI Logic Seminar</span>

Tomorrow, Spencer Unger will speak in our logic seminar about Stationary reflection and the singular cardinals hypothesis. As usual we meet at 11am in Ross 63. Looking forward to seeing you there,

Title: Stationary reflection and the singular cardinals hypothesis.

Abstact. We examine reflection of stationary sets at successors of singular cardinals and its connection with cardinal arithmetic. For instance it has been open whether the failure of the singular cardinal hypothesis at a singular cardinal mu of uncountable cofinality implies the existence of a nonreflecting stationary subset of mu^+. In recent joint work with Omer Ben-Neria and Yair Hayut we have shown that the answer is no modulo the consistency of some large cardinals. In this talk, we survey some instances of methods used in the proof. In particular, we show how to construct Prikry sequences over iterated ultrapowers and exploit them for combinatorial proofs.
Spencer Unger9648
75
Tue, 19 Mar 2019 14:43:43Israeli Logic TalksRalf Schindler: Variants of the extender algebra and their applications<span style="text-decoration: underline;">BIU Infinite Combinatorics Seminar</span>

Mon, 25/03/2019 - 13:00

Speaker: Ralf Schindler (Münster)

Title: Variants of the extender algebra and their applications

Abstract. In the 1970'ies, Bukowský identified a beautiful and handy criterion for when V is a forcing extension of a given inner model, which proved very useful recently in set theoretical geology. In the 1990'ies, Woodin isolated his extender algebra which makes use of a large cardinal, a Woodin cardinal. It turns out that Bukowský's theorem and Woodin's extender algebra may be presented in a uniform fashion - one proof and one forcing gives both results. We will present the proof and then discuss its application in inner model theoretic geology. This is joint work with Grigor Sargsyan and Farmer Schlutzenberg.

&nbsp;
Ralf Schindler9646
76
Tue, 19 Mar 2019 6:46:31Toronto Set Theory SeminarAntonio Aviles: Twisted sums of spaces of continuous functionsPlace: Fields Institute (Room 210)
Date: March 22, 2019 (13:30-15:00)
Speaker: Antonio Aviles
Title: Twisted sums of spaces of continuous functions
Abstract: Given two Banach spaces $Z$ and $X$, can we find a Banach space $Y$ that contains $X$ as an uncomplemented subspace and $Y/X = Z$? We will mention two instances of this problem connected to set theoretic questions. When $X = c_0$ and $Z=C(K)$ is a space of continuous functions on a nonmetric compactum, the answer may be negative under $MA_{\omega_1}$ but it is always positive under CH (joint work with W. Marciszewski and G. Plebanek). When $X = \ell_\infty/c_0$ and $Z=c_0(\mathfrak{c})$, the answer is positive provided splitting chains exist in $\mathcal{P}(\omega)/fin$ (joint work with P. Borodulin-Nadzieja, F. Cabello, D. Chodounsk\'{y} and O. Guzm\'{a}n)
Antonio Avilés9637
77
Mon, 18 Mar 2019 10:12:15Kurt Godel Research CenterYair Hayut: Strong compactness and the filter extension propertyTalk held by Yair Hayut (KGRC) at the KGRC seminar on 2019-03-21.

<strong>Abstract</strong>

The notion of strongly compact cardinal is one of the earliest large cardinal axioms, yet it is still poorly understood.

I will review some classical and semi-classical connections between partial strong compactness, the strong tree property and the filter extension property, getting a level-by-level equivalence and an elementary embedding characterization.

This analysis is especially interesting for the property "every&nbsp;<span id="MathJax-Element-1-Frame" class="MathJax" style="font-size: 16px; font-family: sans-serif; font-style: normal; font-weight: normal; line-height: normal; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px;" role="presentation" data-mathml="<math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;><mi>κ</mi></math>"><span id="MathJax-Span-1" class="math"><span id="MathJax-Span-2" class="mrow"><span id="MathJax-Span-3" class="mi">κ</span></span></span><span class="MJX_Assistive_MathML" role="presentation">κ</span></span>-complete filter on <span style="font-family: sans-serif;"><span style="font-size: 16px;">$\kappa$ </span></span>can be extended to a <span style="font-family: sans-serif;"><span style="font-size: 16px;">$\kappa$ </span></span>-complete ultrafilter" (where <span style="font-family: sans-serif;"><span style="font-size: 16px;">$\kappa$ </span></span>&nbsp;is uncountable). This property was isolated by Mitchell and was named "<span style="font-family: sans-serif;"><span style="font-size: 16px;">$\kappa$ </span></span>-compactness" by Gitik. In his recent paper, Gitik showed that some definable versions of it have a relatively low consistency strength, yet others provide an inner models with a Woodin cardinal. Applying the equivalence above to this case, I will improve the previously known lower bound for <span style="font-family: sans-serif;"><span style="font-size: 16px;">$\kappa$ </span></span>-compactness.

Then, I'll move to a more speculative area, and conjecture that <span style="font-family: sans-serif;"><span style="font-size: 16px;">$\kappa$ </span></span>-compactness is equiconsistent with a certain large cardinal axiom in the realm of subcompact cardinals. I will give a few arguments in favour of this conjecture.
Yair Hayut9630
78
Mon, 18 Mar 2019 10:05:19Kurt Godel Research CenterEuropean Set Theory Conference 2019 - registration reminderLet us remind you that registration is open (still with the Early Fee) for the&nbsp; European Set Theory Conference. We welcome contributed talks and encourage you to take advantage of the support from ASL.

Details:<a href="https://sites.google.com/view/estc2019/estc-2019/registration"> European Set Theory Conference</a>

Please note that the registration for the Advanced Class 2019 (Young Set Theory Workshop) is separate.
9628
79
Mon, 18 Mar 2019 10:01:00ConferenceAdvanced Class 2019 (Young Set Theory Workshop) - registration reminderLet us remind you that registration is open (still with the Early Fee) for the Advanced Class 2019 (Young Set Theory Workshop). We welcome poster submissions and encourage you to take advantage of the support from ASL.

Details: <a href="https://sites.google.com/view/estc2019/advanced-class-yst/registration">Advanced Class 2019 (Young Set Theory Workshop)&nbsp;</a>

Please note that the registration for the European Set Theory Conference is separate.
9626
80
Fri, 15 Mar 2019 18:52:42Carnegie Mellon Logic SeminarJing Zhang: Poset dimension and singular cardinals<div class="moz-text-flowed" lang="x-unicode">
Mathematical logic seminar - Mar 19 2019
Time:&nbsp;&nbsp;&nbsp;&nbsp; 3:30pm - 4:30 pm

Room:&nbsp;&nbsp;&nbsp;&nbsp; Wean Hall 8220

Speaker:&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; Jing Zhang
CMU

Title:&nbsp;&nbsp;&nbsp;&nbsp; Poset dimension and singular cardinals

Abstract:

The dimension of a poset (P, ≤P) is defined as the least cardinal λ such that there exists a λ-sized collection of linear extensions of P realizing P, that is to say a ≤P b if and only a ≤ b in any linear extension in the collection. We will focus on the poset Pα(κ), that is the poset of subsets of κ of size less than α partially ordered by inclusion, and determine completely the dimension of such posets under GCH. Then we will mention a few consistency results when GCH fails. In particular, we point out the connection between the dimension of the poset Pα (2κ) and the density of 2κ under the &lt;α-box product topology, and show it is consistent that they are different.</div>
Jing Zhang9623
81
Fri, 15 Mar 2019 8:16:08Barcelona Set Theory SeminarSamuel Gomes da Silva: Reductions between certain incidenceproblems and the Continuum HypothesisBCNSETS
BARCELONA RESEARCH GROUP IN SET THEORY
BARCELONA SET THEORY SEMINAR
Reductions between certain incidence
problems and the Continuum Hypothesis
Samuel Gomes da Silva
UFBA, Brazil
Abstract: We consider two families of incidence problems, C 1 and C 2 ,
which are related to real numbers and countable subsets of the real
line. Instances of problems in C 1 are as follows: given a real number x,
pick randomly a countable set A of reals hoping that x is in A, whereas
instances of problems in C 2 are as follows: given a countable set A of
reals, pick randomly a real number x hoping that x is not in A. One could
arguably defend that, at least intuitively, problems in C 2 are easier to
solve than problems in C 1 . Indeed, we show that, after some suitable
formalization, one can prove (in ZFC) that, on the one hand, problems in
C 2 are at least as simple to solve as problems in C 1 . On the other hand,
the statement ``Problems in C 1 have the exact same complexity as
problems in C 2 '' is equivalent to the Continuum Hypothesis.
Date: Thursday 21 March 2019
Time: 16:00
Place: Room S-1*
Departament de Matemàtiques i Informàtica
Universitat de Barcelona
Gran Via de les Corts Catalanes 585, 08007 Barcelona
* Enter the University building through the door 20 meters to the right of the main
door and, as you enter the courtyard, turn left, go to the end of the corridor, and
then downstairs.
Samuel Gomes da Silva9622
82
Thu, 14 Mar 2019 12:09:51Prague Set Theory SeminarJonathan Verner: Towers in filters, cardinal invariants, and Luzin type families, part IIDear all,

The seminar meets on Wednesday March 20th at 11:00 in the Institute of
Mathematics CAS, Zitna 25, seminar room, 3rd floor, front building.

Program: Jonathan Verner will continue his talk from the last seminar:

Towers in filters, cardinal invariants, and Luzin type families

Jonathan will present results from his recent paper (with J. Brendle, B.
Farkas);
We investigate which filters on ω can contain towers, that is, a modulo
finite descending sequence without any pseudointersection. We prove the
following results:
(1) Many classical examples of nice tall filters contain no towers.
(2) It is consistent that tall analytic P-filters contain towers of
arbitrary regular height.
(3) It is consistent that all towers generate non-meager filters.
(4) The statement “Every ultrafilter contains towers.” is independent of
ZFC.

Best,
David
Jonathan Verner9618
83
Tue, 12 Mar 2019 6:40:08Toronto Set Theory SeminarFrancisco Guevara Parra: Analytic spaces and their Tukey typesPlace: Fields Institute (Room 210)
Date: March 15, 2018 (13:30-15:00)
Speaker:&nbsp;Francisco Guevara Parra
Title:&nbsp;Analytic spaces and their Tukey types
Abstract:&nbsp;In this Thesis we study topologies on countable sets from the
perspective of Tukey reductions of their neighbourhood
filters. In particular we will study $k$-analytic group topologies
on $\omega$. This will allow us to obtain a metrization theorem
for analytic sequential group topologies on $\omega$, as well
as a classification of such groups in terms of the Tukey type of
the filter of neighbourhoods of the identity. Recall that
a countable topological space is analytic if the topology is analytic
as a subset of the Cantor space.
Francisco Guevara Parra9615
84
Mon, 11 Mar 2019 18:43:27Kurt Godel Research CenterChristopher Lambie-Hanson: Chromatic numbers of finite subgraphsTalk held by Christopher Lambie-Hanson (Virginia Commonwealth University, Richmond, USA) at the KGRC seminar on 2019-03-14.

Abstract: By the De Bruijn-Erdős Compactness Theorem, if a graph <span style="font-family: sans-serif;"><span style="font-size: 16px;">$G$&nbsp;</span></span>has infinite chromatic number, then it has finite subgraphs of arbitrarily large finite chromatic number. We can therefore define an increasing function $f_G:\omega\to \omega$&nbsp;by letting $f_G(n)$ be the least number of vertices in a subgraph of&nbsp; $G$ with chromatic number $n$. We will show in ZFC that, for every function <span style="font-family: sans-serif;"><span style="font-size: 16px;">$f:\omega\to \omega$</span></span>&nbsp;there is a graph <span style="font-family: sans-serif;"><span style="font-size: 16px;">$G$</span></span>&nbsp;with chromatic number <span style="font-family: sans-serif;"><span style="font-size: 16px;">$\aleph_1$&nbsp;</span></span>such that $f_G$&nbsp;grows faster than <span style="font-family: sans-serif;"><span style="font-size: 16px;">$f$</span></span>. This answers a question of Erdős, Hajnal, and Szemeredi. Time permitting, we will discuss connections between our proof and various diamond and club-guessing principles.
Chris Lambie-Hansonhttps://www.youtube.com/watch?v=HrVGQOR6tTM9612
85
Sun, 10 Mar 2019 14:40:51Israeli Logic TalksMiguel Moreno: The Main Gap in the generalized Borel-reducibility hierarchy<span style="text-decoration: underline;">BIU Infinite Combinatorics Seminar</span>

Mon, 11/03/2019 - 13:00

Speaker: Miguel Moreno (BIU)

Title: The Main Gap in the generalized Borel-reducibility hierarchy

Abstract. During this talk we will discuss where in the generalized Borel-reducibility hierarchy are the isomorphism relation of first order complete theories. These theories are divided in two kind:classifiable and non-classifiable. To study the classifiable theories case is needed the use of Ehrenfeucht-Fraïssé games. On the other hand the study of the non-classifiable theories is done by using colored trees. The goal of the talk is to see the classifiable theories case and start the non-classifiable theories case by proving that it is possible to map every element of the generalized Baire, f, into a colored tree, J(f), such that; for every f and g elements of the generalized Baire space, J(f) and J(g) are isomorphic as colored trees if and only if f and g coincide on a club.
Miguel Moreno9643
86
Fri, 08 Mar 2019 5:18:45ConferenceBoise Extravaganza in Set Theory (BEST) 2019 Conference, June 19-21We are pleased to announce that the <strong>2019 Boise Extravaganza in Set Theory</strong> will take place in <strong>Ashland, Oregon</strong>, on the campus of <strong>Southern Oregon University</strong>, during <strong>June 19-21</strong>.

BEST is an international conference featuring talks on a broad range of recent advances in set theory research. It particularly aims to support the careers of young researchers in set theory. The conference is organized by the Set Theory group at Boise State University and is structured as a symposium of the 100th annual meeting of the American Association for the Advancement of Science, Pacific Division (AAAS-PD).

Travel grants will be available for students and postdocs participating in the conference.

Please visit the <a href="https://math.boisestate.edu/best/">BEST web site</a>, which will be updated with more details as they become available.

<strong>Organizers</strong> Liljana Babinkostova (Boise State University), John Clemens (Boise State University), Samuel Coskey (Boise State University), Marion Scheepers (Boise State University)

<strong>Scientific Committee</strong> Natasha Dobrinen (University of Denver), Simon Thomas (Rutgers University)
Dana Bartosova,Steve Jackson,Reese Johnston,Assaf Shani,Piotr SzewczakWed Jun 19 2019https://math.boisestate.edu/best/
https://drive.google.com/drive/folders/1eFMs8TS65lmpZowjELmhN0OjlNvw72Fq?usp=sharing
9607
87
Thu, 07 Mar 2019 18:02:12ConferenceSet Theory: Bridging Mathematics and Philosophy, Konstanz, July 29-31, 2019<h1 style="text-align: center;"><a href="https://fpnc2019.forcing-project.com/">Set Theory: Bridging Mathematics and Philosophy</a></h1>
<div>July 29-31, 2019, Zukunftskolleg, University of Konstanz, Germany&nbsp;

2nd instalment of the Forcing Project Networking Conferences seriesWebsite:&nbsp;<a class="autolinked" href="https://fpnc2019.forcing-project.com/" target="_blank" rel="noopener noreferrer" data-behavior="truncate">https://fpnc2019.forcing-project.com</a>

Organization: Carolin Antos, Neil Barton, Deborah Kant, Daniel Kuby (University of Konstanz)

</div>
<h1>Invited Speakers</h1>
<ul>
<li>Joan Bagaria (University of Barcelona)</li>
<li>Mirna Džamonja (University of East Anglia)</li>
<li>Leon Horsten (University of Bristol)</li>
<li>Juliette Kennedy (University of Helsinki)</li>
<li>Godehard Link (MCMP, Munich)</li>
<li>Marianna Antonutti Marfori (MCMP, Munich)</li>
<li>Toby Meadows (University of California, Irvine)</li>
</ul>
<div></div>
<h1>Call for Papers</h1>
<div>

The project “<a href="https://forcing-project.com/" target="_blank" rel="noopener noreferrer">Forcing: Conceptual Change in the Foundations of Mathematics</a>” (2018-2023) aims to analyse the development of modern set theory since the introduction of the forcing technique both from a historical and philosophical point of view. It brings together methods and research questions from different research areas in the history and philosophy of mathematics to investigate if and how the extensive use of the forcing method brought about a conceptual change in set theory; and in which ways this may influence the philosophy of set theory and the foundations of mathematics.

</div>
<div>

The research group organises a series of Networking Conferences with the goal of reaching out to researchers from these different areas. The second instalment will be devoted to the topic of&nbsp;<strong>recent set theory as a bridge between mathematics and philosophy</strong>&nbsp;and focuses on the interaction between mathematical and philosophical arguments and views in set theory. Set theory has long been both a mathematical discipline and a program with foundational motivations. It seems that this dual character makes it a natural crossway between mathematics and philosophy, possibly more so than other mathematical disciplines.

</div>
<h1>Topics</h1>
<div>

We welcome contributions which

a) add to current discussions in the philosophy of set theory (set-theoretic pluralism, height and width potentialism/actualism, the universe/multiverse debate, the forcing technique, justification of new axioms, contrasts with other foundational frameworks) by relating philosophical and mathematical arguments to one another; by working out the philosophical import of set-theoretic results; or by giving set-theoretic explications of philosophical concepts;

b) question or uphold the relevance of philosophical arguments in set theory. For example, according to Penelope Maddy's naturalism, first philosophical arguments play no justificatory role in set theory. Should (mathematical) naturalism be understood in Maddy's style? Are there other forms of naturalism that are more tolerant of traditional philosophical questions?

</div>
<div>c) analyse the mathematical and philosophical content of the concept "set-theoretic practice" as used in recent set-theoretic programs. For example, do the different foundational programmes offered by the likes of Friedman, Hamkins and Woodin constitute different set-theoretic practices?d) investigate how the inclusion of alternative set theories (constructive set theory, class theories, set theories based on non-classical logic, categorial theories of sets) impact the philosophy of set theory.

</div>
<h1>Submissions</h1>
<div>

Abstracts of 300-500 words should be submitted in PDF (with LaTeX source) or Word format no later than March 31, 2019, via email to &lt;<a href="mailto:submissions@forcing-project.com?subject=Submission%20FPNC2019" target="_blank" rel="noopener noreferrer">submissions@forcing-project.com</a>&gt;. Notifications of acceptance will be issued by April 15, 2019.

</div>
<h1>Financial support</h1>
<div>

As we would like to enable early career researchers (including PhD students) to apply, we are in the process of organizing funding for travel and accommodation for the contributed speakers. Please contact the organizers for further information.

</div>
<h1>Conference registration</h1>
<div>

The conference is free (no conference fee) and everyone is welcome to attend. For logistical reasons, please register by sending an email to &lt;<a href="mailto:registration@forcing-project.com?subject=Registration%20FPNC2019" target="_blank" rel="noopener noreferrer">registration@forcing-project.com</a>&gt; before July 1, 2019.

</div>
<h1>Dates</h1>
<ul>
<li>March 31, 2019: Deadline for submissions to CfP</li>
<li>April 15, 2019: Notification of acceptance</li>
<li>July 1, 2019: Conference registration deadline</li>
<li>July 29-31, 2019: Conference</li>
</ul>
<h1>Contact</h1>
<div>For inquiries please send an email to the organizers &lt;<a href="mailto:fpnc2018@forcing-project.com?subject=Information%20regarding%20FPNC2019" target="_blank" rel="noopener noreferrer">fpnc2019@forcing-project.com</a>&gt;.</div>
Godehard Link,Joan Bagaria,Juliette Kennedy,Leon Horsten,Marianna Antonutti Marfori,Mirna Džamonja,Toby MeadowsMon Jul 29 2019https://fpnc2019.forcing-project.com/9603
88
Thu, 07 Mar 2019 13:19:16Prague Set Theory SeminarJonathan Verner: Towers in filters, cardinal invariants, and Luzin type familiesDear all,

The seminar meets on Wednesday March 13th at 11:00 in the Institute of
Mathematics CAS, Zitna 25, seminar room, 3rd floor, front building.

Program: Jonathan Verner -- Towers in filters, cardinal invariants, and
Luzin type families

Jonathan will present results from his recent paper (with J. Brendle, B.
Farkas);
We investigate which filters on ω can contain towers, that is, a modulo
finite descending sequence without any pseudointersection. We prove the
following results:
(1) Many classical examples of nice tall filters contain no towers.
(2) It is consistent that tall analytic P-filters contain towers of
arbitrary regular height.
(3) It is consistent that all towers generate non-meager filters.
(4) The statement “Every ultrafilter contains towers.” is independent of
ZFC.

Best,
David
Jonathan Verner9600
89
Thu, 07 Mar 2019 7:42:18NUS Logic SeminarWong Tin Lok: End extensions and subsystems of second-order arithmeticInvitation to the Logic Seminar at the National University of Singapore

Date: Wednesday, 13 March 2019, 17:00 hrs

Room: S17#04-06, Department of Mathematics, NUS

Speaker: Wong Tin Lok

Title: End extensions and subsystems of second-order arithmetic

URL: http://www.comp.nus.edu.sg/~fstephan/logicseminar.html

Abstract:
Investigations in reverse mathematics reveal that most naturally
occurring theorems in mathematics are equivalent to one of five
arithmetic axioms nowadays known as the BIG FIVE. These provide
strong empirical evidence for the importance of the Big Five.
In the talk, I will attempt to explain their importance mathematically
in terms of the characteristics of their models.

The work to be presented is joint with Stephen G. Simpson (Vanderbilt).
Wong Tin Lok9599
90
Wed, 06 Mar 2019 13:37:30Kurt Godel Research CenterSerhii Bardyla - Complete topological semigroupsTalk held by Serhii Bardyla (KGRC) at the KGRC seminar on 2019-03-07.

<strong>Abstract</strong>: The first part of the talk will be devoted to the investigation of completeness in the class of topological semigroups.

Then we shall discuss a topologization of semigroups of partial isomorphisms between principal ideals of a tree.
Serhii Bardylahttps://youtu.be/j5tX4Au8DtYhttps://drive.google.com/file/d/11gll1l0yBxMrGPQcvVQjyOQ-v-da3yno/view?usp=drivesdk9594
91
Tue, 05 Mar 2019 17:35:36NUS Logic SeminarDilip Raghavan: A small ultrafilter numberInvitation to the Logic Seminar at the National University of Singapore

Date: Wednesday, 6 March 2019, 17:00 hrs

Room: S17#04-06, Department of Mathematics, NUS

Speaker: Dilip Raghavan

Title: A small ultrafilter number

URL: http://www.comp.nus.edu.sg/~fstephan/logicseminar.html

Abstract: It is proved to be consistent relative to a measurable
cardinal that there is a uniform ultrafilter on the real numbers which
is generated by fewer than the maximum possible number of sets. It is
also shown to be consistent relative to a supercompact cardinal that
there is a uniform ultrafilter on aleph_{omega+1} which is generated
by fewer than 2^{aleph_{omega+1}} sets.

This is joint work with Saharon Shelah.
Dilip Raghavan9588
92
Tue, 05 Mar 2019 7:20:01Toronto Set Theory SeminarYann Pequignot: Finite versus infinite: an insufficient shiftPlace: Fields Institute (Room 210)
Date: , 2018 (13:30-15:00)
Speaker:&nbsp;Yann Pequignot
Title:&nbsp; &nbsp;&nbsp;Finite versus infinite: an insufficient shift
Abstract:&nbsp;The Borel chromatic number – introduced by Kechris, Solecki, and Todorcevic (1999) – generalizes the chromatic number to Borel graphs. While the G_0 dichotomy states that there exists a minimal graph with uncountable Borel chromatic number, it turns out that characterizing when a graph has infinite Borel chromatic number is far more intricate. Even in the case of graphs generated by a single function, the situation is quite complicated. The Shift Graph on the space of infinite subsets of natural numbers is generated by the function that removes the minimum element. It is acyclic but has infinite Borel chromatic number. In 1999, Kechris, Solecki, and Todorcevic asked whether the Shift Graph is minimal among the graphs generated by a single Borel function that have infinite Borel chromatic number. I will sketch a proof that the answer is negative using descriptive complexity considerations and a representation theorem for Sigma^1_2 sets due to Marcone (1994). This result has recently been considerably strengthened by Todorcevic and Vidnyanszky who proved that the set of closed subsets of the Shift Graph that have infinite Borel Chromatic number is Pi^1_2 complete, therefore ruling out most interesting basis results for this class of Borel graphs.
Yann Pequignot9586
93
Fri, 01 Mar 2019 20:39:32Carnegie Mellon Logic SeminarHector Alonzo Barriga-Acosta: Some combinatorics on the normality of the countable box product of the convergent sequenceMathematical logic seminar - Mar 5 2019
Time: 3:30pm - 4:30 pm

Room: Wean Hall 8220

Speaker: Hector Alonzo Barriga Acosta
Universidad Nacional Autónoma de México

Title: Some combinatorics on the normality of the countable box product
of the convergent sequence

Abstract:

The normality of □ (ω + 1)^ω is a question raised in the 40's (it is
known
that consistently this space is normal). Through the years many different
tecniques have been developed, but non of them have solved the question in
ZFC. We'll take a look to a combinatorial point of view, given by Judy
Roitman, of this problem.
Hector Alonzo Barriga-Acosta9583
94
Fri, 01 Mar 2019 9:41:10Prague Set Theory SeminarMichal Doucha: Definable pseudometrics and Borel reductions between themDear all,

The seminar meets on Wednesday March 6th at 11:00 in the Institute of
Mathematics CAS, Zitna 25, seminar room, 3rd floor, front building.

Program: Michal Doucha -- Definable pseudometrics and Borel reductions
between them

I will introduce a ``continuous generalizatio'' of the theory of
definable equivalence relations and Borel reductions between them.
Equivalence relations will be replaced by pseudometrics and reductions
between them will be replaced by certain uniformly continuous maps. I
will explain our motivation and prove some basic results. I will present
some open problems whose solutions may require completely new ideas from
invariant descriptive set theory. It will be based on a joint paper with
Marek Cúth and Ondřej Kurka.
Michal Doucha9580
95
Thu, 28 Feb 2019 20:52:23Barcelona Set Theory SeminarAlejandro Poveda: Prikry-type forcing and the failure of the Singular Cardinal HypothesisBCNSETS
BARCELONA RESEARCH GROUP IN SET THEORY
BARCELONA SET THEORY SEMINAR


Prikry-type forcing and the failure of the
Singular Cardinal Hypothesis

Alejandro Poveda
Universitat de Barcelona

Abstract: Prikry-type forcing plays a central role in
combinatorics due to its close links with central set-theoretic
principles such as the Singular Cardinal Hypothesis (SCH) or the
Tree Property. The original Prikry forcing was devised to change
the cofinality of a measurable cardinal to ω, but there are now
many other more sophisticated constructions that yield more
powerful applications. Among them we can find Magidor and
Radin forcing for changing cofinalities to uncountable cardinals.
In this session we will describe the Prikry forcing with collapses
and present a proof of Magidor’s theorem on the consistency,
relative to appropriate large cardinal hypotheses, of the failure
of the SCH at the first singular cardinal.

Date: Thursday 7 March 2019
Time: 16:00
Place: Room S-1*
Departament de Matemàtiques i Informàtica
Universitat de Barcelona
Gran Via de les Corts Catalanes 585, 08007 Barcelona

* Enter the University building through the door 20 meters to the right of the
main door and, as you enter the courtyard, turn left, go to the end of the
corridor, and then downstairs.
Alejandro Poveda9578
96
Sat, 23 Feb 2019 15:38:36Israeli Logic TalksSandra Müller: Projective determinacy for games of length $\omega^2$ and longer<span style="text-decoration: underline;">BIU Infinite Combinatorics Seminar</span>
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<div class="views-field views-field--block views-field-field-speaker-value-1 view-upcoming-lectures-default__field-field-speaker-value-1 view-upcoming-lectures__field"><label class="views-field-label views-field-label-- views-label-field-speaker-value-1 view-upcoming-lectures-default__field-field-speaker-value-1">Speaker: </label> Sandra Müller (KGRC)</div>
<div class="views-field views-field--block views-field-field-when-value-1 view-upcoming-lectures-default__field-field-when-value-1 view-upcoming-lectures__field"><label class="views-field-label views-field-label-- views-label-field-when-value-1 view-upcoming-lectures-default__field-field-when-value-1"> Date : </label> <span class="date-display-single">25/02/2019 - <span class="date-display-start">13:00</span><span class="date-display-separator"> - </span><span class="date-display-end">15:00</span></span></div>
<div class="views-field views-field--block views-field-title-1 view-upcoming-lectures-default__field-title-1 view-upcoming-lectures__field"><label class="views-field-label views-field-label-- views-label-title-1 view-upcoming-lectures-default__field-title-1"> Title : </label> <a href="https://math.biu.ac.il/en/node/2743"><strong>Projective determinacy for games of length $\omega^2$ and longer</strong></a></div>
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<label class="views-field-label views-field-label-- views-label-field-abstract-value view-upcoming-lectures-default__field-field-abstract-value"> Abstract: </label>
We will study infinite two player games and the large cardinal strength corresponding to their determinacy. For games of length $\omega$ this is well understood and there is a tight connection between the determinacy of projective games and the existence of canonical inner models with Woodin cardinals. For games of arbitrary countable length, Itay Neeman proved the determinacy of analytic games of length $\omega \cdot \theta$ for countable $\theta\&gt; \omega$ from a sharp for $\theta$ Woodin cardinals.

We aim for a converse at successor ordinals. In joint work with Juan P. Aguilera we showed that determinacy of $\boldsymbol\Pi^1\_{n+1}$ games of length $\omega^2$ implies the existence of a premouse with $\omega+n$ Woodin cardinals. This generalizes to a premouse with $\omega+\omega$ Woodin cardinals from the determinacy of games of length $\omega^2$ with $\Game^{\mathbb{R}}\boldsymbol\Pi^1\_1$ payoff.

If time allows, we will also sketch how these methods can be adapted to, in combination with results of Nam Trang, obtain $\omega^\alpha+n$ Woodin cardinals for countable ordinals $\alpha$ and natural numbers $n$ from the determinacy of sufficiently long projective games.

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Sandra Müllerhttps://muellersandra.github.io/upcomingtalk/talk/invsemtalk/2019/01/25/TalkBarIlan.html9565
97
Sat, 23 Feb 2019 2:51:01Carnegie Mellon Logic SeminarRaphaël Carroy: Strongly surjective linear ordersMathematical logic seminar - Feb 26 2019
Time: 3:30pm - 4:30 pm

Room: Wean Hall 8220

Speaker: Raphaël Carroy (KGRC, Vienna)

Title: Strongly surjective linear orders

Abstract:

When a linear order has an increasing surjection onto each of its
suborders we say that it is strongly surjective. We prove that countable
strongly surjective orders are the union of an analytic and a coanalytic
set, and that moreover they are complete for this class of sets. If time
allows it, I'll also discuss the existence of uncountable strongly
surjective orders. This is a joint work with Riccardo Camerlo and Alberto
Marcone.
Raphaël Carroy9522
98
Fri, 22 Feb 2019 15:51:37Prague Set Theory SeminarJan Grebík: Vizing's Theorem for GraphingsDear all,

The seminar meets on Wednesday February 27th at 11:00 in the Institute
of Mathematics CAS, Zitna 25, seminar room, 3rd floor, front building.

Program: Jan Grebík -- Vizing's Theorem for Graphings

I will show that the measurable edge chromatic index of a graphing G
(bounded degree Borel graph with invariant measure) with maximum degree
D is D+1. This is a joint work with Oleg Pikhurko.

Best,
David
Jan Grebík9520
99
Fri, 22 Feb 2019 7:19:37Wrocław University of TechnologyBarnabas Farkas: Degrees of destructionTuesday, February 26, 2019, 17:15
Wrocław University of Technology, 215 D-1

Speaker: Barnabas Farkas (TU Wien)

Title: Degrees of destruction

Abstract:

I'm going to present a survey on our results (joint with L. Zdomskyy) about the following strong notion of destroying Borel ideals: We say that the forcing notion $\mathbb{P}$ $+$-destroys the Borel ideal $\mathcal{I}$ if $\mathbb{P}$ adds an $\mathcal{I}$-positive $\dot{X}$ which has finite intersection with every $A\in \mathcal{I}\cap V$. I will
talk about the following:

(1) Examples when usual destruction (that is, when $\dot{X}$ required to be infinite only) implies $+$-destruction, and when it does not.

(2) Characterization of those Borel ideals which can be $+$-destroyed, in particular, we will see that if $\mathcal{I}$ can be $+$-destroyed then the associated Mathias-Prikry forcing $+$-destroys it.

(3) Characterization of those analytic P-ideals which are $+$-destroyed by the associated Laver-Prikry forcing.
Barnabas Farkas9517
100
Wed, 20 Feb 2019 13:34:30Barcelona Set Theory SeminarAlejandro Poveda: Prikry-type forcing: properties and applicationsBCNSETS
BARCELONA RESEARCH GROUP IN SET THEORY
BARCELONA SET THEORY SEMINAR


Prikry-type forcing: properties and applications

Alejandro Poveda
Universitat de Barcelona

Abstract: Prikry-type forcing plays a central role in
combinatorics due to its close links with central set-theoretic
principles such as the Singular Cardinal Hypothesis (SCH) or the
Tree Property. The original Prikry forcing was devised to change
the cofinality of a measurable cardinal to ω, but there are now
many other more sophisticated constructions that yield more
powerful applications. Among them we can find Magidor and
Radin forcing for changing cofinalities to uncountable cardinals,
or the Diagonal supercompact Prikry forcing with collapses, due
also to Magidor, which can be used to force the failure of SCH
at the first singular cardinal. In this session we will prove some
of the key properties of Prikry forcing and will see how it is
used to prove the consistency of the negation of SCH.

Date: Thursday 28 February 2019
Time: 16:00
Place: Room S-1*
Departament de Matemàtiques i Informàtica
Universitat de Barcelona
Gran Via de les Corts Catalanes 585, 08007 Barcelona

* Enter the University building through the door 20 meters to the right of the
main door and, as you enter the courtyard, turn left, go to the end of the
corridor, and then downstairs.
Alejandro Poveda9512
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