Set Theory Talks

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.

Brian Wynne, CUNY
Recent developments in the model theory of Abelian lattice-ordered groups

An Abelian lattice-ordered group (ℓ-group) is an Abelian group with a partial ordering, invariant under translations, that is a lattice ordering. A prototypical example of an ℓ-group is C(X), the continuous real-valued functions on the topological space X with pointwise operations and ordering. Let A be the class of ℓ-groups, viewed as structures for the first-order language L={+,−,0,∧,∨}. After giving more background on ℓ-groups, I will survey what is known about the ℓ-groups existentially closed (e.c.) in A, including some new examples I constructed using Fraïssé limits. Then I will discuss some recently published work of Scowcroft concerning the ℓ-groups e.c. in W+, the class of nonzero Archimedean ℓ-groups with distinguished strong order unit (viewed as structures for L1=L∪{1}). Building on Scowcroft's results, I will present new axioms for the ℓ-groups e.c. in W+ and show how they allow one to characterize those spaces X for which (C(X),1X) is e.c. in W+.

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.

Hello everyone,

Sorry for the interrupting again. I would like to apologize(again) for a mistake in the previous announcement. There was a serious mistake in the time mentioned. The correct time of the Nankai Logic Colloquium this week is in the afternoon, 4pm to 5pm (instead of morning mentioned in the last email), Friday Beijing time. I am very very sorry for the confuse it may cause.

The following is a corrected version of the announcement for this week:

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

Our speaker this week will be Ilijas Farah from York University. This talk is going to take place this Friday,  May 5th,  from 4pm to 5pm(UTC+8, Beijing time). 

Best wishes,

Ming Xiao

Title:  Corona rigidity

Abstract. Reduced powers associated with the Frech\'et filter are well-known to be countably saturated (that is, $\aleph_1$-saturated). Because of this the Continuum Hypothesis implies that the reduced power of every countable structure has $2^{2^{\aleph_0}}$ automorphisms, and that for such reduced powers elementary equivalence is a sufficient condition for isomorphism. On the other hand, forcing axioms imply that some reduced powers (e.g., those of finite Boolean algebras) have only trivial automorphisms while some other reduced powers are saturated and they $2^{2^{\aleph_0}}$ automorphisms, provably in ZFC (e.g., those of the 2-element cyclic group).

This begs two questions: Which structures have saturated reduced powers, provably in ZFC? For which structures forcing axioms imply the `corona rigidity', that their reduced powers have only trivial automorphisms?
I will give a complete answer to the first question and a partial (rather surprising) answer to the second.

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Speaker:     Gemma De las Cuevas, University of Innsbruck.

Date and Time:     Wednesday May 3, 2023, 7:00 - 8:30 PM. ZOOM TALK.

Title:     A framework for universality across disciplines.

Abstract: What is the scope of universality across disciplines? And what is its relation to undecidability? To address these questions, we build a categorical framework for universality. Its instances include Turing machines, spin models, and others. We introduce a hierarchy of universality and argue that it distinguishes universal Turing machines as a non-trivial form of universality. We also outline the relation to undecidability by drawing a connection to Lawvere’s Fixed Point Theorem. Joint work with Sebastian Stengele, Tobias Reinhart and Tomas Gonda.

Classification via effective lists

'Classifying' a natural collection of structures is a common goal in mathematics. Providing a classification can mean different things, e.g., identifying a set of invariants that settle the isomorphism problem or creating a list of all structures of a given kind without repetition of isomorphism type. Here we discuss recent work on classifications of the latter kind from the perspective of computable structure theory. We’ll consider natural classes of computable structures such as vector spaces, equivalence relations, algebraic fields, and trees to better understand the nuances of classification via effective lists and its relationship to other forms of classification in this setting.

Brian Wynne, CUNY
Recent developments in the model theory of Abelian lattice-ordered groups

An Abelian lattice-ordered group (ℓ-group) is an Abelian group with a partial ordering, invariant under translations, that is a lattice ordering. A prototypical example of an ℓ-group is C(X), the continuous real-valued functions on the topological space X with pointwise operations and ordering. Let A be the class of ℓ-groups, viewed as structures for the first-order language L={+,−,0,∧,∨}. After giving more background on ℓ-groups, I will survey what is known about the ℓ-groups existentially closed (e.c.) in A, including some new examples I constructed using Fraïssé limits. Then I will discuss some recently published work of Scowcroft concerning the ℓ-groups e.c. in W+, the class of nonzero Archimedean ℓ-groups with distinguished strong order unit (viewed as structures for L1=L∪{1}). Building on Scowcroft's results, I will present new axioms for the ℓ-groups e.c. in W+ and show how they allow one to characterize those spaces X for which (C(X),1X) is e.c. in W+.

Speaker:     Dusko Pavlovic, University of Hawai‘i at Mānoa.

Date and Time:     Wednesday April 26, 2023, 7:00 - 8:30 PM. ZOOM TALK.

Title:     Program-closed categories.

Abstract: > Let CC be a symmetric monoidal category with a comonoid on every object. Let CC* be the cartesian subcategory with the same objects and just the comonoid homomorphisms. A *programming language* is a well-ordered object P with a *program closure*: a family of X-natural surjections
CC(XA,B) <<--run_X-- CC*(X,P)
one for every pair A,B. In this talk, I will sketch a proof that program closure is a property: Any two programming languages are isomorphic along run-preserving morphisms. The result counters Kleene's interpretation of the Church-Turing Thesis, which has been formalized categorically as the suggestion that computability is a structure, like a group presentation, and not a property, like completeness. We prove that it is like completeness. The draft of a book on categorical computability is available from the web site dusko.org.

Speaker:     Gemma De las Cuevas, University of Innsbruck.

Date and Time:     Wednesday May 3, 2023, 7:00 - 8:30 PM. ZOOM TALK.

Title:     A framework for universality across disciplines.

Abstract: What is the scope of universality across disciplines? And what is its relation to undecidability? To address these questions, we build a categorical framework for universality. Its instances include Turing machines, spin models, and others. We introduce a hierarchy of universality and argue that it distinguishes universal Turing machines as a non-trivial form of universality. We also outline the relation to undecidability by drawing a connection to Lawvere’s Fixed Point Theorem. Joint work with Sebastian Stengele, Tobias Reinhart and Tomas Gonda.

Classification via effective lists

'Classifying' a natural collection of structures is a common goal in mathematics. Providing a classification can mean different things, e.g., identifying a set of invariants that settle the isomorphism problem or creating a list of all structures of a given kind without repetition of isomorphism type. Here we discuss recent work on classifications of the latter kind from the perspective of computable structure theory. We’ll consider natural classes of computable structures such as vector spaces, equivalence relations, algebraic fields, and trees to better understand the nuances of classification via effective lists and its relationship to other forms of classification in this setting.

_____________________________________________________________________

Best wishes,

Ming Xiao

Katarzyna W. Kowalik, University of Warsaw
The chain-antichain principle and proof size

The chain-antichain principle CAC is a well-known consequence of Ramsey's theorem for pairs and two colours RT22. It says that for every partial order on N there exists an infinite chain or antichain with respect to this order. Both of these principles are Π03-conservative over the weak base theory RCA∗0. Such conservation results usually prompt to ask about lengths of proofs. Kołodziejczyk, Wong and Yokoyama proved that RT22 has a non-elementary speedup over RCA∗0 for proofs of Σ1 sentences. We show that the behaviour of CAC is the opposite: it can be polynomially simulated by RCA∗0 with respect to Π03 sentences. Our argument uses a technique of forcing interpretation developed by Avigad. In the first step we syntactically simulate a construction of a generic computable ultrapower of a model of RCA∗0. Then we find a generic cut satisfying CAC inside the ultrapower.

Speaker:     Walter Tholen, York University.

Date and Time:     Wednesday April 19, 2023, 7:00 - 8:30 PM. ZOOM TALK.

Title:     What does “smallness” mean in categories of topological spaces?

Abstract: Quillen’s notion of small object and the Gabriel-Ulmer notion of finitely presentable or generated object are fundamental in homotopy theory and categorical algebra. Do these notions always lead to rather uninteresting classes of objects in categories of topological spaces, such as the class of finite discrete spaces, or just the empty space , as the examples and remarks in the existing literature may suggest?

In this talk we will demonstrate that the establishment of full characterizations of these notions (and some natural variations thereof) in many familiar categories of spaces, such as those of T_i-spaces (i= 0, 1, 2), can be quite challenging and may lead to unexpected surprises. In fact, we will show that there are significant differences in this regard even amongst the categories defined by the standard separation conditions, with the T1-separation condition standing out. The findings about these specific categories lead us to insights also when considering rather arbitrary full reflective subcategories of Top.

(Based on joint work with J. Adamek, M. Husek, and J. Rosicky.)

The proper forcing axiom for ℵ1-sized posets and the continuum

We discuss Shelah's memory iteration technique and use it to show that the PFA for posets of size ℵ1 is consistent with large continuum. This is joint work with David Aspero.

Speaker:     Dusko Pavlovic, University of Hawai‘i at Mānoa.

Date and Time:     Wednesday April 26, 2023, 7:00 - 8:30 PM. ZOOM TALK.

Title:     Program-closed categories.

Abstract: > Let CC be a symmetric monoidal category with a comonoid on every object. Let CC* be the cartesian subcategory with the same objects and just the comonoid homomorphisms. A *programming language* is a well-ordered object P with a *program closure*: a family of X-natural surjections
CC(XA,B) <<--run_X-- CC*(X,P)
one for every pair A,B. In this talk, I will sketch a proof that program closure is a property: Any two programming languages are isomorphic along run-preserving morphisms. The result counters Kleene's interpretation of the Church-Turing Thesis, which has been formalized categorically as the suggestion that computability is a structure, like a group presentation, and not a property, like completeness. We prove that it is like completeness. The draft of a book on categorical computability is available from the web site dusko.org.

Speaker:     Walter Tholen, York University.

Date and Time:     Wednesday April 19, 2023, 7:00 - 8:30 PM. ZOOM TALK.

Title:     What does “smallness” mean in categories of topological spaces?

Abstract: Quillen’s notion of small object and the Gabriel-Ulmer notion of finitely presentable or generated object are fundamental in homotopy theory and categorical algebra. Do these notions always lead to rather uninteresting classes of objects in categories of topological spaces, such as the class of finite discrete spaces, or just the empty space , as the examples and remarks in the existing literature may suggest?

In this talk we will demonstrate that the establishment of full characterizations of these notions (and some natural variations thereof) in many familiar categories of spaces, such as those of T_i-spaces (i= 0, 1, 2), can be quite challenging and may lead to unexpected surprises. In fact, we will show that there are significant differences in this regard even amongst the categories defined by the standard separation conditions, with the T1-separation condition standing out. The findings about these specific categories lead us to insights also when considering rather arbitrary full reflective subcategories of Top.

(Based on joint work with J. Adamek, M. Husek, and J. Rosicky.)

_____________________________________________________________________

Best wishes,

Ming Xiao

Speaker: Shawn Simpson (Pitt)
Title: Logic and inference in the sender-receiver model

Abstract: The sender-receiver model was developed by David Lewis to tackle the question of the conventionality of meaning. But many people who cared about the conventionality of meaning did so because they thought it was intimately connected to the conventionality of logic. Since Lewis’s work, only a few attempts have been made to say anything about the nature of logic and inference from the perspective of the sender-receiver model. This talk will look at the what’s been said in that regard, by Skyrms and others, and suggest a few general lessons.

Parameter-free comprehension in second-order arithmetic

Second-order arithmetic has two types of objects: numbers and sets of numbers, which we think of as the reals. The second-order arithmetic framework has been used successfully to investigate what kinds of real numbers need to exist to prove various significant results in analysis. One of the strongest second-order arithmetic axiomatizations is the theory Z2 consisting of the axioms PA (for numbers), the set induction axiom, and comprehension for all second-order formulas with set parameters. How significant is the inclusion of set parameters in the comprehension scheme? Let Z−p2 be like Z2, but where set parameters are not allowed in the comprehension scheme. Harvey Friedman showed that Z2 and Z−p2 are equiconsistent because parameter-free comprehension suffices to build a model's version of the constructible universe L inside the model and the 'constructible' reals satisfy Z2. Kanovei recently showed that models of Z−p2 can be very badly behaved, for example, their sets may not even be closed under complement. Kanovei also showed that there can be nicely behaved models of Z−p2 in which Σ12-comprehension (with set parameters) holds. He constructed his model in a forcing extension by a tree iteration of Sacks forcing. In Kanovei's model, Σ14-comprehension (with set parameters) fails and he asked whether this can be improved to Σ13-comprehension. In this talk, I will show how to construct a model of Σ12-comprehension and Z−p2 in which Σ13-comprehension fails. The model will be constructed in a forcing extension by a tree iteration of Jensen's forcing. Jensen's forcing is a sub-poset of Sacks forcing constructed by Jensen to show that it is consistent to have a non-constructible Π12-definable singleton real (every Σ12-definable set of reals is constructible by Shoenfield's Absoluteness).

Jonathan Osinski, University of Hamburg
Model-Theoretic Characterizations of Weak Vopěnka's Principle

It has been known since the 1980s that Vopěnka's Principle (VP) is equivalent to certain statements about logics, e.g. to the schema 'Every logic has a compactness cardinal.' On the other hand, it was only recently shown by Trevor Wilson that a related statement statement called Weak Vopěnka's Principle (WVP) is strictly weaker than VP. In fact, Joan Bagaria and Wilson showed that WVP is equivalent to the existence of Πn-strong cardinals for all natural numbers n. We generalize logical characterizations of strong cardinals to achieve a characterization of Πn-strong cardinals and therefore of WVP in terms of properties of strong logics. This is partly joint work with Will Boney and partly with Trevor Wilson.

Filippo Calderoni, Rutgers University
Rotation equivalence and rigidity

The theory of countable Borel equivalence relations analyzes the actions of countable groups on Polish spaces. The main question studied is how much information is encoded by the corresponding orbit space. The amount of encoded information reflects the extent to which the action is rigid.

In this talk we will discuss rigidity results for the action of the group of rational rotations. In particular we will analyze the rotation equivalence on spheres in higher dimension. This is connected to superrigidity results of Margulis and to Zimmer’s program about the actions of discrete subgroups of Lie groups on manifolds.

Gregory Taylor (CUNY)
Title: First-order logics over fixed domain

Abstract: What we call first-order logic over fixed domain was initiated, in a certain guise, by Peirce around 1885 and championed, albeit in idiosyncratic form, by Zermelo in papers from the 1930s.  We characterize such logics model- and proof-theoretically and argue that they constitute exploration of a clearly circumscribed conception of domain-dependent generality.  Whereas a logic, or family of such, can be of interest for any of a variety of reasons, we suggest that one of those reasons might be that said logic fosters some clarification regarding just what qualifies as a logical concept, a logical operation, or a logical law.