Set Theory Talks

Hello everyone,

This week our weekly Nankai Logic Colloquium is going to be in the afternoon.

Our speaker this week will be Victor Hugo Yanez from Nanjing Normal University. This talk is going to take place this Friday, Dec 01, from 4pm to 5pm(UTC+8, Beijing time). 

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Best wishes,

Ming Xiao

Scott Mutchnik, University of Illinois at Chicago
(N)SOP2n+1+1 Theories

Among the classical properties of unstable theories defined by Shelah, our understanding of the strict order hierarchy, NSOPn, has remained relatively limited past n=4 at the greatest. Methods originating from stability theory have given insight into the structure of stronger unstable classes, including simple and NSOP1 theories. In particular, syntactic information about formulas in a first-order theory often corresponds to semantic information about independence in a theory's models, which generalizes phenomena such as linear independence in vector spaces and algebraic independence in algebraically closed fields. We discuss how the fine structure of this independence reveals exponential behavior within the strict order hierarchy, particularly at the levels SOP2n+1+1 for positive integers n. Our results suggest a potential theory of independence for NSOPn theories, for arbitrarily large values of n.

Hello everyone,

Welcome back to Nankai Logic Colloquium! This week our weekly Nankai Logic Colloquium is going to be in the morning.

Our speaker this week will be Marcin Sabok from McGill University. This talk is going to take place this Friday, Nov 17, from 9am to 10am(UTC+8, Beijing time). 

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Title ：The 34th Nankai Logic Colloquium --Marcin Sabok

Time ：9:00am, Nov. 17, 2023(Beijing Time)

Zoom Number ：872 7448 5609

Passcode ：448066

Link ：https://zoom.us/j/87274485609?pwd=z90Pn2KFasUa3KbbvQ1d7xSl3eP6rc.1

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Best wishes,

Ming Xiao

Alex Citkin (Metropolitan Telecommunications).
Title: On logics of acceptance and rejection

Abstract: In his book Formalization of Logic, Carnap suggested the following process of refutation: for any set of formulas Γ and any formula α, if Γ ⊢ α and α is rejected, reject Γ. Thus, in contrast to the Łukasiewicz’s approach to refutation, the predicate of rejection is defined on sets of formulas rather than just formulas. In addition to a predicate of rejection, we introduce a predicate of acceptance which is also defined on sets of formulas, and this leads us to constructing two-layered logical systems, the ground layer of which is a conventional deductive system (providing us with means for derivation), and the top layer having predicates of acceptance and rejection. In the case when the set of accepted formulas coincides with the set of theorems of the underlying logic and the set of rejected formulas coincides with the sets of non-theorems, we obtain a conventional deductive system. The predicate of acceptance can be non-adjunctive, and this allows us to use such systems as an alternative approach to defining Jaśkowski style discursive logics.

Speaker:     Larry Moss, Indiana University, Bloomington .

Date and Time:     Wednesday November 8, 2023, 7:00 - 8:30 PM. ZOOM TALK

Title:     On Kripke, Vietoris, and Hausdorff Polynomial Functors.

Abstract: The Vietoris space of compact subsets of a given Hausdorff space yields an endofunctor V on the category of Hausdorff spaces. Vietoris polynomial endofunctors on that category are built from V, the identity and constant functors by forming products, coproducts and compositions. These functors are known to have terminal coalgebras and we deduce that they also have initial algebras. We present an analogous class of endofunctors on the category of extended metric spaces, using in lieu of V the Hausdorff functor H. We prove that the ensuing Hausdorff polynomial functors have terminal coalgebras and initial algebras. Whereas the canonical constructions of terminal coalgebras for Vietoris polynomial functors takes omega steps, one needs \omega + \omega steps in general for Hausdorff ones. We also give a new proof that the closed set functor on metric spaces has no fixed points.

Asymptotics of the Spencer-Shelah Random Graph Sequence

In combinatorics, the Spencer-Shelah random graph sequence is a variation on the independent-edge random graph model. We fix an irrational number a∈(0,1), and we probabilistically generate the n-th Spencer-Shelah graph (with parameter a) by taking n vertices, and for every pair of distinct vertices, deciding whether they are connected with a biased coin flip, with success probability n−a. On the other hand, in model theory, an R-mac is a class of finite structures, where the cardinalities of definable subsets are particularly well-behaved. In this talk, we will introduce the notion of 'probabalistic R-mac' and present an incomplete proof that the Spencer Shelah random graph sequence is an example of one.

Alex Citkin (Metropolitan Telecommunications).
Title: On logics of acceptance and rejection

Abstract: In his book Formalization of Logic, Carnap suggested the following process of refutation: for any set of formulas Γ and any formula α, if Γ ⊢ α and α is rejected, reject Γ. Thus, in contrast to the Łukasiewicz’s approach to refutation, the predicate of rejection is defined on sets of formulas rather than just formulas. In addition to a predicate of rejection, we introduce a predicate of acceptance which is also defined on sets of formulas, and this leads us to constructing two-layered logical systems, the ground layer of which is a conventional deductive system (providing us with means for derivation), and the top layer having predicates of acceptance and rejection. In the case when the set of accepted formulas coincides with the set of theorems of the underlying logic and the set of rejected formulas coincides with the sets of non-theorems, we obtain a conventional deductive system. The predicate of acceptance can be non-adjunctive, and this allows us to use such systems as an alternative approach to defining Jaśkowski style discursive logics.

Speaker:     Larry Moss, Indiana University, Bloomington .

Date and Time:     Wednesday November 8, 2023, 7:00 - 8:30 PM. ZOOM TALK

Title:     On Kripke, Vietoris, and Hausdorff Polynomial Functors.

Abstract: The Vietoris space of compact subsets of a given Hausdorff space yields an endofunctor V on the category of Hausdorff spaces. Vietoris polynomial endofunctors on that category are built from V, the identity and constant functors by forming products, coproducts and compositions. These functors are known to have terminal coalgebras and we deduce that they also have initial algebras. We present an analogous class of endofunctors on the category of extended metric spaces, using in lieu of V the Hausdorff functor H. We prove that the ensuing Hausdorff polynomial functors have terminal coalgebras and initial algebras. Whereas the canonical constructions of terminal coalgebras for Vietoris polynomial functors takes omega steps, one needs \omega + \omega steps in general for Hausdorff ones. We also give a new proof that the closed set functor on metric spaces has no fixed points.

Alessandro Berarducci and Marcello Mamino, University of Pisa
Provability logic: models within models in Peano Arithmetic

In 1994 Jech gave a model theoretic proof of Gödel's second incompleteness theorem for Zermelo-Fraenkel set theory in the following form: ZF does not prove that ZF has a model. Kotlarski showed that Jech's proof can be adapted to Peano Arithmetic with the role of models being taken by complete consistent extensions. In this note we take another step in the direction of replacing proof-theoretic by model-theoretic arguments. We show, without passing through the arithmetized completeness theorem, that the existence of a model of PA of complexity Σ02 is independent of PA, where a model is identified with the set of formulas with parameters which hold in the model. Our approach is based on a new interpretation of the provability logic of Peano Arithmetic with the modal operator interpreted as truth in every Σ02-model.

Speaker:     Emilio Minichiello, CUNY Graduate Center.

Date and Time:     Wednesday October 25, 2023, 7:00 - 8:30 PM. IN PERSON TALK (GC 6417)

Title:     A Mathematical Model of Package Management Systems.

Abstract: In this talk, I will review some recent joint work with Gershom Bazerman and Raymond Puzio. The motivation is simple: provide a mathematical model of package management systems, such as the Hackage package respository for Haskell, or Homebrew for Mac users. We introduce Dependency Structures with Choice (DSC) which are sets equipped with a collection of possible dependency sets for every element and satisfying some simple conditions motivated from real life use cases. We define a notion of morphism of DSCs, and prove that the resulting category of DSCs is equivalent to the category of antimatroids, which are mathematical structures found in combinatorics and computer science. We analyze this category, proving that it is finitely complete, has coproducts and an initial object, but does not have all coequalizers. Further, we construct a functor from a category of DSCs equipped with a certain subclass of morphisms to the opposite of the category of finite distributive lattices, making use of a simple finite characterization of the Bruns-Lakser completion.

Elliot Glazer, Harvard University
Coin flipping on models of arithmetic to define the standard cut

We will discuss the following claim: 'The standard cut of a model M of PA (or even Q) is uniformly definable with respect to a randomly chosen predicate.' Restricting our consideration to countable models, this claim is true in the usual sense, i.e. there is a formula φ such that for any countable model of arithmetic M, the set SφM:={P⊂M:ω={x∈M:(M,P)⊨φ(x)}} is Lebesgue measure 1. However, if M is countably saturated, then there is no φ such that SφM is measured by the completed product measure on 2M. We will identify various combinatorial ideals on 2M that can be used to formalize the original claim with no restriction on the cardinality of M, and discuss the relationship between closure properties of these ideals and principles of choice.

Speaker:     Michael Shulman, University of San Diego.

Date and Time:     Wednesday October 18, 2023, 7:00 - 8:30 PM.  ZOOM TALK

Title:     The derivator of setoids.

Abstract: The question of "what is a homotopy theory" or "what is a higher category" is already interesting in classical mathematics, but in constructive mathematics (such as the internal logic of a topos) it becomes even more subtle. In particular, existing constructive attempts to formulate a homotopy theory of spaces (infinity-groupoids) have the curious property that their "0-truncated objects" are more general than ordinary sets, being instead some kind of "free exact completion" of the category of sets (a.k.a. "setoids"). It is at present unclear whether this is a necessary feature of a constructive homotopy theory or whether it can be avoided somehow. One way to find some evidence about this question is to use the "derivators" of Heller, Franke, and Grothendieck, as they give us access to higher homotopical structure without depending on a preconcieved notion of what such a thing should be. It turns out that constructively, the free exact completion of the category of sets naturally forms a derivator that has a universal property analogous to the classical category of sets and to the classical homotopy theory of spaces: it is the "free cocompletion of a point" in a certain universe. This suggests that either setoids are an unavoidable aspect of constructive homotopy theory, or more radical modifications to the notion of homotopy theory are needed.

The number of ergodic models of an infinitary sentence

Given an Lω1ω-sentence φ in a countable language, we call an ergodic S∞-invariant probability measure on the Borel space of countable models of φ (having fixed underlying set) an ergodic model of φ. I will discuss the number of ergodic models of such a sentence φ, including the case when φ is a Scott sentence. This is joint work with N. Ackerman, C. Freer, A. Kruckman and A. Kwiatkowska.

Alessandro Berarducci and Marcello Mamino, University of Pisa
Provability logic: models within models in Peano Arithmetic

In 1994 Jech gave a model theoretic proof of Gödel's second incompleteness theorem for Zermelo-Fraenkel set theory in the following form: ZF does not prove that ZF has a model. Kotlarski showed that Jech's proof can be adapted to Peano Arithmetic with the role of models being taken by complete consistent extensions. In this note we take another step in the direction of replacing proof-theoretic by model-theoretic arguments. We show, without passing through the arithmetized completeness theorem, that the existence of a model of PA of complexity Σ02 is independent of PA, where a model is identified with the set of formulas with parameters which hold in the model. Our approach is based on a new interpretation of the provability logic of Peano Arithmetic with the modal operator interpreted as truth in every Σ02-model.

Speaker:     Emilio Minichiello, CUNY Graduate Center.

Date and Time:     Wednesday October 25, 2023, 7:00 - 8:30 PM. IN PERSON TALK (GC 6417)

Title:     A Mathematical Model of Package Management Systems.

Abstract: In this talk, I will review some recent joint work with Gershom Bazerman and Raymond Puzio. The motivation is simple: provide a mathematical model of package management systems, such as the Hackage package respository for Haskell, or Homebrew for Mac users. We introduce Dependency Structures with Choice (DSC) which are sets equipped with a collection of possible dependency sets for every element and satisfying some simple conditions motivated from real life use cases. We define a notion of morphism of DSCs, and prove that the resulting category of DSCs is equivalent to the category of antimatroids, which are mathematical structures found in combinatorics and computer science. We analyze this category, proving that it is finitely complete, has coproducts and an initial object, but does not have all coequalizers. Further, we construct a functor from a category of DSCs equipped with a certain subclass of morphisms to the opposite of the category of finite distributive lattices, making use of a simple finite characterization of the Bruns-Lakser completion.

Vincent Guingona, Towson University
Indivisibility of Classes of Graphs

This talk will discuss my work with Miriam Parnes and four undergraduates which took place last summer at an REU at Towson University. We say that a class of structures in some fixed language is indivisible if, for all structures A in the class and number of colors k, there is a structure B in the class such that, no matter how we color B with k colors, there is a monochromatic copy of A in B. Parnes and I became interested in this property when studying the classification of theories via positive combinatorial configurations. In this talk, following the work with our students, I will examine indivisibility on classes of graphs. In particular, we will look at hereditarily sparse graphs, cographs, perfect graphs, threshold graphs, and a few other classes. This work is joint with Felix Nusbaum, Zain Padamsee, Miriam Parnes, Christian Pippin, and Ava Zinman.

Elliot Glazer, Harvard University
Coin flipping on models of arithmetic to define the standard cut

We will discuss the following claim: 'The standard cut of a model M of PA (or even Q) is uniformly definable with respect to a randomly chosen predicate.' Restricting our consideration to countable models, this claim is true in the usual sense, i.e. there is a formula φ such that for any countable model of arithmetic M, the set SφM:={P⊂M:ω={x∈M:(M,P)⊨φ(x)}} is Lebesgue measure 1. However, if M is countably saturated, then there is no φ such that SφM is measured by the completed product measure on 2M. We will identify various combinatorial ideals on 2M that can be used to formalize the original claim with no restriction on the cardinality of M, and discuss the relationship between closure properties of these ideals and principles of choice.

Speaker:     Michael Shulman, University of San Diego.

Date and Time:     Wednesday October 18, 2023, 7:00 - 8:30 PM.

Title:     The derivator of setoids.

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Title ：The 32nd Nankai Logic Colloquium --Shaun Allison

Time ：9:00am, Jun. 9, 2023(Beijing Time)

Zoom Number ：893 1745 8343

Passcode ： 283146

Link ：https://us02web.zoom.us/j/89317458343?pwd=L01Hc28yc0J2OGk3c3VPS3gvVjVndz09

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Title: An anti-classification theorem for the topological conjugacy
problem for Cantor minimal systems

Abstract:
The isomorphism problem in dynamics  dates back to a question of von
Neumann from 1932: Is it possible to classify (in some reasonable
sense) the ergodic measure-preserving diffeomorphisms of a compact
manifold up to isomorphism? We would like to study a similar problem:
let C be the Cantor set and let Min(C) stand for the space of all
minimal homeomorphisms of the Cantor set. Recall that a Cantor set
homeomorphism T is in Min(C) if every orbit of T is dense in C.  We
say that S and T in Min(C) are topologically conjugate if there exists
a Cantor set homeomorphism h such that Sh=hT. We prove an
anti-classification result showing that even for very liberal
interpretations of what a "reasonable'' classification scheme might
be, a classification of minimal Cantor set homeomorphism up to
topological conjugacy is impossible. We see is as a consequence of the
following: we prove that the topological conjugacy relation of Cantor
minimal systems TopConj treated as a subset of Min(C) x Min(C) is
complete analytic. In particular, TopConj is a non-Borel subset of
Min(C) x Min(C). Roughly speaking, it means that it is impossible to
tell if two minimal Cantor set homeomorphisms are topologically
conjugate  using only a countable amount of information and
computation.

Our result is proved by applying a Foreman-Rudolph-Weiss-type
construction used for an anti-classification theorem for ergodic
automorphisms of the Lebesgue space. We find a continuous map F from
the space of all subtrees over non-negative integers N with
arbitrarily long branches into the class of minimal homeomorphisms of
the Cantor set. Furthermore, F is a reduction, which means that a tree
T is ill-founded if and only if F(T) is topologically conjugate to its
inverse. Since the set of ill-founded trees with arbitrarily long
branches is a well-known example of a complete analytic set, we see
that it is essentially impossible to classify which minimal Cantor set
homeomorphisms are topologically conjugate to their inverses.

This is joint work with Konrad Deka, Felipe García-Ramos, Kosma
Kapsrzak, Philipp Kunde (all from the Jagiellonian University in
Kraków).

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Title ：The 31st Nankai Logic Colloquium --Dominik Kwietniak

Time ：16:00pm, Jun. 2, 2023(Beijing Time)

Zoom Number ： 876 3579 6414

Passcode ： 318535

Link ：https://us02web.zoom.us/j/87635796414?pwd=M1hZSEFvL0FzMUZQcHVCQ0w2QlhtUT09

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title: Definability of the nonstationary ideal on $\omega_1$

abstract: The nonstationary ideals are natural nontrivial ideals defined on all uncountable regular cardinals. In this talk, various aspects of definability of nonstationary ideals on uncountable cardinals are explored. The main focus is  the definability of nonstationary ideal on $\omega_1$ ($NS_{\omega_1}$ for short) in some canonical models of set theory. In particular, under MM or (*) axiom, $NS_{\omega_1}$ is not $\Pi_1$ definable. On the other hand, it is consistent that in some model of $PFA$, $NS_{\omega_1}$ is $\Pi_1$ definable. This is based on the accumulated work of Aspero, Hoffelner, Larson, Schindler, Sun, Wu.

Title ： The 30th Nankai Logic Colloquium --Liuzhen Wu

Time ：16:00pm, May. 26, 2023 (Beijing Time) 

Zoom Number ： 851 5601 8255

Passcode ： 136440

Link ：https://us02web.zoom.us/j/85156018255?pwd=UjFUb3cwT0poY0JYakRub2kyNGdSdz09

Title: A surjection from square onto power

Abstract: In 1892, Cantor proved that, for all sets $A$, there are no bijections between $A$ and the power set of $A$. Cantor's construction is so explicit that it can be carried out in ZF (the Zermelo--Fraenkel set theory without the axiom of choice). In 1906, by virtue of Zermelo's well-ordering theorem, Hessenberg proved the idempotency theorem, which states that there is a bijection between $A$ and the square of $A$ for all infinite sets $A$. (Another proof of the idempotency theorem was given by Zorn in 1944 using Zorn's lemma.) In 1924, Tarski proved that the idempotency theorem is in fact equivalent to the axiom of choice. On the other hand, in 1954, Specker proved in ZF a surprising generalization of Cantor's theorem, which states that, for all infinite sets $A$, there are no injections from the power set of $A$ into the square of $A$. It is then natural to ask whether it is provable in ZF that, for all infinite sets $A$, there are no surjections from the square of $A$ onto the power set of $A$. This question is known as the dual Specker problem and was proposed by Truss in 1973. In this talk, we give a negative answer to this question; that is, the existence of an infinite set $A$ such that the square of $A$ maps onto the power set of $A$ is consistent with ZF. This is joint work with Yinhe Peng and Liuzhen Wu.

Title ：The 29th Nankai Logic Colloquium --Guozhen Shen

Time ：16:00pm, May. 19, 2023 (Beijing Time) 

Zoom Number ：856 2849 0880

Passcode ： 073635

Link ：https://us02web.zoom.us/j/85628490880?pwd=dTBrV0NLc0l1bmFTY1RHR0d0TUNDZz09

Maciej Sendłak (Warsaw).
Title: Explanatory realism and counterfactuals

Abstract: In my talk, I want to propose a novel approach to the question of counterfactuals. This is grounded in two assumptions, imported from the philosophy of science. The first one has it that to explain a phenomenon is to show how it depends on something else. The second states that the correct explanation ought to be contrastive. This means that a good explanation justifies the occurrence of a phenomenon and – at the same time – excludes occurrence of some other states of affairs. I am going to argue that – together with the assumption that conditionals express a dependence relation between A and C – the above gives ground for analysis of counterfactuals. According to this proposal: “A>C” is true at the world of evaluation iff there is a relation of dependence that hold between referents of A and C, and the same relation of dependence holds in the world of evaluation.

Miha Habič, Bard College at Simon's Rock
Some old and new results on nonamalgamable forcing extensions

Fixing some countable transitive model M of set theory, we can consider its generic multiverse, the family of all models obtainable from M by taking any sequence of forcing extensions and ground models. There is an attractive similarity between the generic multiverse and the Turing degrees, but the multiverse has the drawback (or feature?) that it contains nonamalgamable models, that is, models with no common upper bound, as was observed by several people, going back to at least Mostowski. In joint work with Hamkins, Klausner, Verner, and Williams in 2019, we studied the order-theoretic properties of the generic multiverse and, among other results, gave a characterization of which partial orders embed nicely into the multiverse. I will present our results in the simplest case of Cohen forcing, as well as existing generalizations to wide forcing, and some new results on non-Cohen ccc forcings.