# Online Activities June 1 — June 7 2020

June 1

Münster Set Theory Seminar
Time: Tuesday, June 1, 4:15pm CEST
Speaker: No seminar this week
Title: No seminar this week
Abstract: No seminar this week
Information: contact rds@wwu.de ahead of time in order to participate.

Cornell Logic Seminar
Time: Tuesday, June 2, 2:55pm New York time (20:55pm CEST)
Speaker: TBA
Title: TBA
Abstract: TBA
Information: contact solecki@cornell.edu ahead of time to participate.

May 27

Jerusalem Set Theory Seminar
Time: Wednesday, June 3, 11:00am (Israel Time)
Speaker: TBA
Title: TBA
Abstract: TBA
Information: contact omer.bn@mail.huji.ac.il ahead of time in order to participate.

Paris-Lyon Séminaire de Logique
Time:
Wednesday, June 3, 16:00-17:15 CEST
Speaker: Colin Jahel
Title: In 2005, Kechris, Pestov and Todorcevic exhibited a correspondence
between combinatorial properties of structures and dynamical properties
of their automorphism groups. In 2012, Angel, Kechris and Lyons used
this correspondence to show the unique ergodicity of all the actions of
some groups. In this talk, I will give an overview of the aforementioned
results and discuss recent work generalizing results of Angel, Kechris
and Lyons.
Information: Join via the link on the seminar webpage 10 minutes before the talk.

Bristol Logic and Set Theory Seminar
Time:
Wednesday, June 3, 14:00-15:30 (UK time)
Speaker: TBA
Title: TBA
Abstract: TBA

CUNY Set Theory Seminar
Time: Wednesday, June 3, 7pm New York time (1am May 14 CEST)
Speaker: Zachiri McKenzie
Title: Initial self-embeddings of models of set theory: Part I
Abstract: In the 1973 paper ‘Countable models of set theory’, H. Friedman’s investigation of embeddings between countable models of subsystems of ZF yields the following two striking results:

1. Every countable nonstandard model of PA is isomorphic to a proper initial segment of itself.
2. Every countable nonstandard model of a sufficiently strong subsystem of ZF is isomorphic to a proper initial segment that is a union of ranks of the original model.

Note that, in contrast to PA, in the context of set theory there are three alternative notions of ‘initial segment’: transitive subclass, transitive subclass that is closed under subsets and rank-initial segment. Paul Gorbow, in his Ph.D. thesis, systematically studies versions of H. Friedman’s self-embedding that yield isomorphisms between a countable nonstandard model of set theory and a rank-initial segment of itself. In these two talks I will discuss recent joint work with Ali Enayat that investigates models of set theory that are isomorphic to a transitive subclass of itself. We call the maps witnessing these isomorphisms ‘initial self-embeddings’. I will outline a proof of a refinement of H. Friedman’s Theorem that guarantees the existence of initial self-embeddings for certain subsystems of ZF without the powerset axiom. I will then discuss several examples including a nonstandard model of ZFC minus the powerset axiom that admits no initial self-embedding, and models that separate the three different notions of self-embedding for models of set theory. Finally, I will discuss two interesting applications of our version of H. Freidman’s Theorem. The first of these is a refinement of a result due to Quinsey that guarantees the existence of partially elementary proper transitive subclasses of non-standard models of ZF minus the powerset axiom. The second result shows that every countable model of ZF with a nonstandard natural number is isomorphic to a transitive subclass of the hereditarily countable sets of its own constructible universe.​
Information: The seminar will take place virtually. Please email Victoria Gitman (vgitman@nylogic.org) for the meeting id.

Oxford Set Theory Seminar
Time: Wednesday, June 3, 4pm UK time
Speaker: TBA
Title: TBA
Abstract: TBA

Information: The seminar will take place virtually. Please email Sam Adam-Day at me@samadamday.com for the meeting id.

May 28

Kurt Gödel Research Center Seminar (organised by Ben Miller)
Time:
Thursday, June 4, 16:00 CEST
Speaker: Stefan Hoffelner (University of Münster, North Rhine-Westphalia, Germany)
Title: Forcing the Sigma^1_3-separation property

Abstract: The separation property, introduced in the 1920s, is a classical notion in descriptive set theory. It is well-known due to Moschovakis, that Δ12-determinacy implies the Σ13-separation property; yet Δ12-determinacy implies an inner model with a Woodin cardinal. The question whether the Σ13-separation property is consistent relative to just ZFC remained open however since Mathias’ “Surrealist Landscape”-paper. We show that one can force it over L.
Information: Talk via zoom.

May 29

CUNY Set Theory Seminar
Time: Friday, June 5, 2pm New York time (8pm CEST)
Speaker: Michał Godziszewski — Munich Center for Mathematical Philosophy
Title: TBA
Abstract: TBA
Information: The seminar will take place virtually. Please email Victoria Gitman (vgitman@nylogic.org) for the meeting id.

Toronto Set Theory Seminar
Time: Friday, June 5, 1.30pm Toronto time (7.30pm CEST)
Speaker:TBA
Title: TBA
Abstract: TBA
Information: The seminar will take place virtually. ZOOM ID: https://yorku.zoom.us/j/96087161597

# Online activities May 25 — May 31 2020

May 26

Münster Set Theory Seminar
Time: Tuesday, May 26, 4:15pm CEST
Speaker: Liuzhen Wu, Chinese Acad. Sciences, Beijing
Title: BPFA and \Delta_1-definablity of NS_{\omega_1}​.
Abstract: I will discuss a proof of the joint consistency of​ BPFA and \Delta_1-definablity of NS_{\omega_1}. Joint work with Stefan Hoffelner and Ralf Schindler.
Information: contact rds@wwu.de ahead of time in order to participate.

Cornell Logic Seminar
Time: Tuesday, May 26, 2:55pm New York time (20:55pm CEST)
Speaker: TBA
Title: TBA
Abstract: TBA
Information: contact solecki@cornell.edu ahead of time to participate.

May 27

Jerusalem Set Theory Seminar
Time: Wednesday, May 27, 11:00am (Israel Time)
Speaker: TBA
Title: TBA
Abstract: TBA
Information: contact omer.bn@mail.huji.ac.il ahead of time in order to participate.

Paris-Lyon Séminaire de Logique
Time:
Wednesday, May 27, 16:00-17:15 CEST
Speaker: Eliott Kaplan – University of Illinois at Urbana-Champaign
Title: Model completeness for the differential field of transseries with exponentiation
Abstract: I will discuss the expansion of the differential field of logarithmic-exponential transseries by its natural exponential function. This expansion is model complete and locally o-minimal. I give an axiomatization of the theory of this expansion that is effective relative to the theory of the real exponential field. These results build on Aschenbrenner, van den Dries, and van der Hoeven’s model completeness result for the differential field of transseries. My method can be adapted to show that the differential field of transseries with its restricted sine and cosine and its unrestricted exponential is also model complete and locally o-minimal.
Information: Join via the link on the seminar webpage 10 minutes before the talk.

Bristol Logic and Set Theory Seminar
Time:
Wednesday, May 27, 14:00-15:30 (UK time)
Speaker: Philip Welch, University of Bristol
Title: Higher type recursion for Infinite time Turing Machines IV
Abstract: This is part of a series informal working seminars on an extension of Kleene’s early 1960’s on recursion in higher types. (This formed a central theme on the borders of set theory and recursion theory in the 60’s and early 70’s, although now unfortunately not much discussed. Amongst the main names here were Gandy, Aczel, Moschovakis, Harrington, Normann.) We aim to present a coherent version of type-2 recursion for the infinite time Turing machine model. We aim to be somewhat (but not entirely) self-contained. Basic descriptive set theory, and recursion theory, together with admissibility theory will be assumed.

CUNY Set Theory Seminar
Time: Wednesday, May 27, 7pm New York time (1am May 14 CEST)
Speaker: Bartosz Wcisło, University of Warsaw
Title: Tarski boundary II
Abstract: Truth theories investigate the notion of truth with axiomatic methods. To a fixed base theory (typically Peano Arithmetic PA) we add a unary predicate T(x) with the intended interpretation ‘x is a (code of a) true sentence.’ Then we analyse how adding various possible sets of axioms for that predicate affects its behaviour. One of the aspects we are trying to understand is which truth-theoretic principles make the added truth predicate ‘strong’ in that the resulting theory is not conservative over the base theory. Ali Enayat proposed to call this ‘demarcating line’ between conservative and non-conservative truth theories ‘the Tarski boundary.’ Research on Tarski boundary revealed that natural truth theoretic principles extending compositional axioms tend to be either conservative over PA or exactly equivalent to the principle of global reflection over PA. It says that sentences provable in PA are true in the sense of the predicate T. This in turn is equivalent to Δ0-induction for the compositional truth predicate which turns out to be a surprisingly robust theory.
In our talk, we will try to sketch proofs representative of research on Tarski boundary. We will present the proof by Enayat and Visser showing that the compositional truth predicate is conservative over PA. We will also try to discuss how this proof forms a robust basis for further conservativeness results.
On the non-conservative side of Tarski boundary, the picture seems less organised, since more arguments are based on ad hoc constructions. However, we will try to show some themes which occur rather repeatedly in these proofs: iterated truth predicates and the interplay between properties of good truth-theoretic behaviour and induction. To this end, we will present the argument that disjunctive correctness together with the internal induction principle for a compositional truth predicate yields the same consequences as Δ0-induction for the compositional truth predicate (as proved by Ali Enayat) and that it shares arithmetical consequences with global reflection. The presented results are currently known to be suboptimal.
This talk is intended as a continuation of ‘Tarski boundary’ presentation. However, we will try to avoid excessive assumptions on familiarity with the previous part.
Information: The seminar will take place virtually. Please email Victoria Gitman (vgitman@nylogic.org) for the meeting id.

Oxford Set Theory Seminar
Time: Wednesday, May 27, 4pm UK time
Speaker: Ali Enayat, Gothenberg
Title: Leibnizian and anti-Leibnizian motifs in set theory
Abstract: Leibniz’s principle of identity of indiscernibles at first sight appears completely unrelated to set theory, but Mycielski (1995) formulated a set-theoretic axiom nowadays referred to as LM (for Leibniz-Mycielski) which captures the spirit of Leibniz’s dictum in the following sense: LM holds in a model M of ZF iff M is elementarily equivalent to a model M* in which there is no pair of indiscernibles. LM was further investigated in a 2004 paper of mine, which includes a proof that LM is equivalent to the global form of the Kinna-Wagner selection principle in set theory. On the other hand, one can formulate a strong negation of Leibniz’s principle by first adding a unary predicate I(x) to the usual language of set theory, and then augmenting ZF with a scheme that ensures that I(x) describes a proper class of indiscernibles, thus giving rise to an extension ZFI of ZF that I showed (2005) to be intimately related to Mahlo cardinals of finite order. In this talk I will give an expository account of the above and related results that attest to a lively interaction between set theory and Leibniz’s principle of identity of indiscernibles.

May 28

Kurt Gödel Research Center Seminar (organised by Ben Miller)
Time:
Thursday, May 28, 16:00 CEST
Speaker: Diego Mejía, Shizuoka University, Japan
Title: Preserving splitting families
Abstract: We present a method to force splitting families that can be preserved by a large class of finite support iterations of ccc posets. As an application, we show how to force several cardinal characteristics of the continuum to be pairwise different.
Information: Talk via zoom.

May 29

CUNY Set Theory Seminar
Time: Friday, May 29, 2pm New York time (8pm CEST)
Speaker: Kameryn Williams University of Hawaii at Mānoa
Title: The geology of inner mantles
Abstract: An inner model is a ground if V is a set forcing extension of it. The intersection of the grounds is the mantle, an inner model of ZFC which enjoys many nice properties. Fuchs, Hamkins, and Reitz showed that the mantle is highly malleable. Namely, they showed that every model of set theory is the mantle of a bigger, better universe of sets. This then raises the possibility of iterating the definition of the mantle—the mantle, the mantle of the mantle, and so on, taking intersections at limit stages—to obtain even deeper inner models. Let’s call the inner models in this sequence the inner mantles.
In this talk I will present some results about the sequence of inner mantles, answering some questions of Fuchs, Hamkins, and Reitz. Specifically, I will present the following results, analogues of classic results about the sequence of iterated HODs.
1. (Joint with Reitz) Consider a model of set theory and consider an ordinal eta in that model. Then this model has a class forcing extension whose eta-th inner mantle is the model we started out with, where the sequence of inner mantles does not stabilize before eta.
2. It is consistent that the omega-th inner mantle is an inner model of ZF + ¬AC.
3. It is consistent that the omega-th inner mantle is not a definable class, and indeed fails to satisfy Collection.
Information: The seminar will take place virtually. Please email Victoria Gitman (vgitman@nylogic.org) for the meeting id.

Toronto Set Theory Seminar
Time: Friday, May 29, 1.30pm Toronto time (7.30pm CEST)
Speaker:TBA
Title: TBA
Abstract: TBA
Information: The seminar will take place virtually. ZOOM ID: https://yorku.zoom.us/j/96087161597

Udine online activities
Time: Friday, May 29, 16:30-18:30 CEST
Speaker: Peter Holy, University of Udine
Title: Generalized topologies on 2^kappa, Silver forcing, and the diamond principle
Abstract:I will talk about the connections between topologies on 2^kappa induced by ideals on kappa and topologies on 2^kappa induced by certain tree forcing notions, highlighting the connection of the topology induced by the nonstationary ideal with kappa-Silver forcing. Assuming that Jensen’s diamond principle holds at kappa, we then generalize results on kappa-Silver forcing of Friedman, Khomskii and Kulikov that were originally shown for inaccessible kappa: In particular, I will present a proof that also in our situation, kappa-Silver forcing satisfies a strong form of Axiom A. By a result of Friedman, Khomskii and Kulikov, this implies that meager sets are nowhere dense in the nonstationary topology. If time allows, I will also sketch a proof of the consistency of the statement that every Delta^1_1 set (in the standard bounded topology on 2^kappa) has the Baire property in the nonstationary topology, again assuming the diamond principle to hold at kappa (rather than its inaccessibility). This is joint work with Marlene Koelbing, Philipp Schlicht and Wolfgang Wohofsky.
Information: Via Microsoft Teams, to participate contact vincenzo.dimonte@uniud.it

# Online activities May 18 — May 22 2020

May 19

Münster Set Theory Seminar
Time: Tuesday, May 12, 4:15pm CEST
Speaker: Stefan Hoffelner, University of Muenster
Title: Forcing the $\bf{\Sigma^1_3}$-separation property.
Abstract: The separation property, introduced in the 1920s, is a classical notion in descriptive set theory. It is well-known due to Moschovakis, that $\bf{\Delta^1_2}$-determinacy implies the $\bf{\Sigma^1_3}$-separation property; yet $\bf{\Delta^1_2}$-determinacy implies an inner model with a Woodin cardinal. The question whether the $\bf{\Sigma^1_3}$-separation property is consistent relative to just ZFC remained open however since Mathias’ „Surrealist Landscape“-paper from 1968. We show that one can force it over L.
Information: contact rds@wwu.de ahead of time in order to participate.

Cornell Logic Seminar
Time: Tuesday, May 19, 2:55pm New York time (20:55pm CEST)
Speaker: TBA
Title: TBA
Abstract: TBA
Information: contact solecki@cornell.edu ahead of time to participate.

May 20

Jerusalem Set Theory Seminar
Time: Wednesday, May 20, 11:00am (Israel Time)
Speaker: TBA
Title: TBA
Abstract: TBA
Information: contact omer.bn@mail.huji.ac.il ahead of time in order to participate.

Paris-Lyon Séminaire de Logique
Time:
Wednesday, May 13, 16:00-17:15 CEST
Speaker: Michael Hrusak, Universidad Nacional Autónoma de México
Title: Strong measure zero in Polish groups (joint with W. Wohofsky, J. Zapletal and/or O. Zindulka)
Abstract: We study the extent to which the Galvin-Mycielski-Solovay
characterization of strong measure zero subsets of the real line
extends to arbitrary Polish group. In particular, we show that
an abelian Polish group satisfies the GMS characterization if and only
if it is locally compact. We shall also consider the non-abelian case,
and discuss the use and existence of invariant submeasures on Polish groups.
Information: Join via the link on the seminar webpage 10 minutes before the talk.

Bristol Logic and Set Theory Seminar
Time:
Wednesday, May 20, 14:00-15:30 (UK time)
Speaker: Philip Welch (University of Bristol)
Title: Higher type recursion for infinite time Turing machines IV
Abstract: This is part of a series informal working seminars on an extension of Kleene’s early 1960’s on recursion in higher types. (This formed a central theme on the borders of set theory and recursion theory in the 60’s and early 70’s, although now unfortunately not much discussed. Amongst the main names here were Gandy, Aczel, Moschovakis, Harrington, Normann.) We aim to present a coherent version of type-2 recursion for the infinite time Turing machine model. We aim to be somewhat (but not entirely) self-contained. Basic descriptive set theory, and recursion theory, together with admissibility theory will be assumed.

CUNY Set Theory Seminar
Time: Wednesday, May 20, 7pm New York time (1am May 14 CEST)
Speaker: TBA
Title: TBA
Abstract: TBA
Information: The seminar will take place virtually. Please email Victoria Gitman (vgitman@nylogic.org) for the meeting id.

Oxford Set Theory Seminar
Time: Wednesday, May 20, 4pm UK time
Speaker: Joel David Hamkins, OXFORD
Title: Bi-interpretation of weak set theories
Abstract: Set theory exhibits a truly robust mutual interpretability phenomenon: in any model of one set theory we can define models of diverse other set theories and vice versa. In any model of ZFC, we can define models of ZFC + GCH and also of ZFC + ¬CH and so on in hundreds of cases. And yet, it turns out, in no instance do these mutual interpretations rise to the level of bi-interpretation. Ali Enayat proved that distinct theories extending ZF are never bi-interpretable, and models of ZF are bi-interpretable only when they are isomorphic. So there is no nontrivial bi-interpretation phenomenon in set theory at the level of ZF or above.  Nevertheless, for natural weaker set theories, we prove, including ZFC- without power set and Zermelo set theory Z, there are nontrivial instances of bi-interpretation. Specifically, there are well-founded models of ZFC- that are bi-interpretable, but not isomorphic—even (H(\omega_1),\in) and (H(\omega_2),\in) can be bi-interpretable—and there are distinct bi-interpretable theories extending ZFC-. Similarly, using a construction of Mathias, we prove that every model of ZF is bi-interpretable with a model of Zermelo set theory in which the replacement axiom fails. This is joint work with Alfredo Roque Freire.

May 21

Kurt Gödel Research Center Seminar (organised by Ben Miller)
Time:
Thursday, May 21, 16:00 CEST
Speaker: TBA
Title: TBA
Abstract: TBA
Information: Talk via zoom.

May 22

CUNY Set Theory Seminar
Time: Friday, May 22, 2pm New York time (8pm CEST)
Speaker: Ali Enayat, University of Gothenburg
Title: Recursively saturated models of set theory and their close relatives: Part II
Abstract: A model M of set theory is said to be ‘condensable’ if there is an ‘ordinal’ α of M such that the rank initial segment of M determined by α is both isomorphic to M, and an elementary submodel of M for infinitary formulae appearing in the well-founded part of M. Clearly if M is condensable, then M is ill-founded. The work of Barwise and Schlipf in the mid 1970s showed that countable recursively saturated models of ZF are condensable.
In this two-part talk, we present a number of new results related to condensable models, including the following two theorems.
Theorem 1. Assuming that there is a well-founded model of ZFC plus ‘there is an inaccessible cardinal’, there is a condensable model M of ZFC which has the property that every definable element of M is in the well-founded part of M (in particular, M is ω-standard, and therefore not recursively saturated).
Theorem 2. The following are equivalent for an ill-founded model M of ZF of any cardinality:
(a) M is expandable to Gödel-Bernays class theory plus Δ11-Comprehension.
(b) There is a cofinal subset of ‘ordinals’ α of M such that the rank initial segment of M determined by α is an elementary submodel of M for infinitary formulae appearing in the well-founded part of M.
Moreover, if M is a countable ill-founded model of ZFC, then conditions (a) and (b) above are equivalent to:
(c) M is expandable to Gödel-Bernays class theory plus Δ11-Comprehension + Σ12-Choice.
Information: The seminar will take place virtually. Please email Victoria Gitman (vgitman@nylogic.org) for the meeting id.

Toronto Set Theory Seminar
Time: Friday, May 22, 1.30pm Toronto time (7.30pm CEST)
Speaker: Vinicius de Oliveira Rodrigues, University of São Paulo and University of São Paulo, Institute of Mathematics and Statistics
Title: Pseudocompact hyperspaces of Isbell-Mrówka spaces
Abstract: J. Ginsburg has asked what is the relation between the pseudocompactness of the ω-th power of a topological space Xand the pseudocompactness of its Vietoris Hyperspace, exp(X). M. Hrusak, I. Martínez-Ruiz and F. Hernandez-Hernandez studied this question restricted to Isbell-Mrówka spaces, that is, spaces of the form Ψ(A) where A is an almost disjoint family. Regarding these spaces, if exp(X) is pseudocompact, then Xω is also pseudocompact, and Xω is pseudocompact iff A is a MAD family. They showed that if the cardinal characteristic 𝔭 is 𝔠, then for every MAD family A, exp(Ψ(A)) is pseudocompact, and if the cardinal characteristic 𝔥 is less than 𝔠, there exists a MAD family A such that exp(Ψ(A)) is not pseudocompact. They asked if there exists a MAD family A (in ZFC) such that exp(Ψ(A)) is pseudocompact.
In this talk, we present some new results on the (consistent) existence of MAD families whose hyperspaces of their Isbell-Mrówka spaces are (or are not) pseudocompact by constructing new examples. Moreover, we give some combinatorial equivalences for every Isbell-Mrówka space from a MAD family having pseudocompact hyperspace. This is a joint work with, O. Guzman, M. Hrusak, S. Todorcevic and A. Tomita.
Information: The seminar will take place virtually. ZOOM ID: https://yorku.zoom.us/j/96087161597

# Online activities May 11 — May 17 2020

May 12

Münster Set Theory Seminar
Time: Tuesday, May 12, 4:15pm CEST
Speaker: Farmer Schlutzenberg, University of Muenster
Title: $j:V_\delta\to V_\delta$ in $L(V_\delta)$
Abstract: Assuming $\mathrm{ZF}+V=L(V_\delta)$ where $\delta$ is a
limit ordinal of uncountable cofinality, we show there is no
non-trivial $\Sigma_1$-elementary $j:V_\delta\to V_\delta$. Reference:
Section 8 of “Reinhardt cardinals and non-definability”, arXiv
2002.01215.
Information: contact rds@wwu.de ahead of time in order to participate.

Cornell Logic Seminar
Time: Tuesday, May 12, 2:55pm New York time (20:55pm CEST)
Speaker: Konstantin Slutsky, University of Paris 7
Title: Smooth Orbit equivalence relation of free Borel R^d-actions
Abstract: Smooth Orbit Equivalence (SOE) is an orbit equivalence relation between free ℝd-flows that acts by diffeomorphisms between orbits. This idea originated in ergodic theory of ℝ-flows under the name of time-change equivalence, where it is closely connected with the concept of Kakutani equivalence of induced transformations. When viewed from the ergodic theoretical viewpoint, SOE has a rich structure in dimension one, but, as discovered by Rudolph, all ergodic measure-preserving ℝd-flows, d > 1, are SOE. Miller and Rosendal initiated the study of this concept from the point of view of descriptive set theory, where phase spaces of flows aren’t endowed with any measures. This significantly enlarges the class of potential orbit equivalences, and they proved that all nontrivial free Borel ℝ-flows are SOE. They posed a question of whether the same remains to be true in dimension d>1. In this talk, we answer their question in the affirmative, and show that all nontrivial Borel ℝd-flows are SOE.
Information: contact solecki@cornell.edu ahead of time to participate.

May 13

Jerusalem Set Theory Seminar
Time: Wednesday, May 13, 11:00am (Israel Time)
Speaker: Alejandro Poveda (Universitat de Barcelona)
Title: Sigma-Prikry forcings and their iterations (Part II)
Abstract:In the previous talk, we introduced the notion of \Sigma-Prikry forcing and showed that many classical Prikry-type forcing which center on countable cofinalities fall into this framework.
The aim of this talk is to present our iteration scheme for \Sigma-Prikry forcings.
In case time permits, we will also show how to use this general iteration theorem to derive as a corollary the following strengthening of Sharon’s theorem: starting with \omega-many supercompact cardinals one can force a generic extension where Refl(<\omega,\kappa^+) holds and the SCH_\kappa fails, for \kappa being a strong limit cardinal with cofinality \omega
Information: contact omer.bn@mail.huji.ac.il ahead of time in order to participate.

Paris-Lyon Séminaire de Logique
Time:
Wednesday, May 13, 16:00-17:15 CEST
Speaker: Caroline Terry (University of Chicago)
Title: Speeds of hereditary properties and mutual algebricity
Abstract: A hereditary graph property is a class of finite graphs closed under isomorphism and induced subgraphs. Given a hereditary graph property H, the speed of H is the function which sends an integer n to the number of distinct elements in H with underlying set {1,…,n}. Not just any function can occur as the speed of hereditary graph property. Specifically, there are discrete “jumps” in the possible speeds. Study of these jumps began with work of Scheinerman and Zito in the 90’s, and culminated in a series of papers from the 2000’s by Balogh, Bollob\'{a}s, and Weinreich, in which essentially all possible speeds of a hereditary graph property were characterized. In contrast to this, many aspects of this problem in the hypergraph setting remained unknown. In this talk we present new hypergraph analogues of many of the jumps from the graph setting, specifically those involving the polynomial, exponential, and factorial speeds. The jumps in the factorial range turned out to have surprising connections to the model theoretic notion of mutual algebricity, which we also discuss.
Information: Join via the link on the seminar webpage 10 minutes before the talk.

Bristol Logic and Set Theory Seminar
Time:
Wednesday, May 13, 14:00-15:30 (UK time)
Speaker: Philip Welch (University of Bristol)
Title: Higher type recursion for infinite time Turing machines III
Abstract:This is part of a series informal working seminars on an extension of Kleene’s early 1960’s on recursion in higher types. (This formed a central theme on the borders of set theory and recursion theory in the 60’s and early 70’s, although now unfortunately not much discussed. Amongst the main names here were Gandy, Aczel, Moschovakis, Harrington, Normann.) We aim to present a coherent version of type-2 recursion for the infinite time Turing machine model. We aim to be somewhat (but not entirely) self-contained. Basic descriptive set theory, and recursion theory, together with admissibility theory will be assumed.

CUNY Set Theory Seminar
Time: Wednesday, May 13, 7pm New York time (1am May 14 CEST)
Speaker: Laurence Kirby, CUNY
Title: Bounded finite set theory
Abstract: There is a well-known close logical connection between PA and finite set theory. Is there a set theory that corresponds in an analogous way to bounded arithmetic IΔ0? I propose a candidate for such a theory, called IΔ0S, and consider the questions: what set-theoretic axioms can it prove? And given a model M of IΔ0 is there a model of IΔ0S whose ordinals are isomorphic to M? The answer is yes if M is a model of Exp; to obtain the answer we use a new way of coding sets by numbers.
Information: The seminar will take place virtually. Please email Victoria Gitman (vgitman@nylogic.org) for the meeting id.

May 14

Kurt Gödel Research Center Seminar (organised by Ben Miller)
Time:
Thursday, May 14, 16:00 CEST
Speaker: Andrew Brooke Taylor, University of Leeds (UK)
Title: Product of CW complexes
Abstract: CW spaces are often presented as the “spaces of choice” in algebraic topology courses, being relatively nice spaces built up by successively gluing on Euclidean balls of increasing dimension. However, the product of CW complexes need not be a CW complex, as shown by Dowker soon after CW complexes were introduced. Work in the 1980s characterised when the product is a CW complex under the assumption of CH, or just b=ℵ1. In this talk I will give and prove a complete characterisation of when the product of CW complexes is a CW complex, valid under ZFC. The characterisation however involves b; the proof is point-set-topological (I won’t assume any knowledge of algebraic topology) and uses Hechler conditions.
Information: Talk via zoom.

May 15

CUNY Set Theory Seminar
Time: Friday, May 15, 2pm New York time (8pm CEST)
Speaker: Ali Enayat, University of Gothenburg
Title: Recursively saturated models of set theory and their close relatives: Part I
Abstract: A model M of set theory is said to be ‘condensable’ if there is an ‘ordinal’ α of M such that the rank initial segment of M determined by α is both isomorphic to M, and an elementary submodel of M for infinitary formulae appearing in the well-founded part of M. Clearly if M is condensable, then M is ill-founded. The work of Barwise and Schlipf in the mid 1970s showed that countable recursively saturated models of ZF are condensable.
In this two-part talk, we present a number of new results related to condensable models, including the following two theorems.
Theorem 1. Assuming that there is a well-founded model of ZFC plus ‘there is an inaccessible cardinal’, there is a condensable model M of ZFC which has the property that every definable element of M is in the well-founded part of M (in particular, M is ω-standard, and therefore not recursively saturated).
Theorem 2. The following are equivalent for an ill-founded model M of ZF of any cardinality:
(a) M is expandable to Gödel-Bernays class theory plus Δ11-Comprehension.
(b) There is a cofinal subset of ‘ordinals’ α of M such that the rank initial segment of M determined by α is an elementary submodel of M for infinitary formulae appearing in the well-founded part of M.
Moreover, if M is a countable ill-founded model of ZFC, then conditions (a) and (b) above are equivalent to:
(c) M is expandable to Gödel-Bernays class theory plus Δ11-Comprehension + Σ12-Choice.
Information: The seminar will take place virtually. Please email Victoria Gitman (vgitman@nylogic.org) for the meeting id.

# Online activities May 4 — May 10 2020

May 5

Münster Set Theory Seminar
Time: Tuesday, May 5, 4:15pm CEST
Speaker: Andreas Lietz, University of Muenster
Title: How to force (*) from less than a supercompact
Abstract: Asperò-Schindler have shown that Woodin’s axiom (*) is a consequence of MM^{++} and the latter is known to be forceable from a supercompact cardinal. (*) however has consistency strength of \omega-many Woodin cardinals, so it should be possible to force it from a much weaker assumption. We present a construction that does so from strictly less than a \kappa^{+++}-supercompact cardinal \kappa (+GCH). The strategy will be to iterate the forcing from the proof of MM^{++}\Rightarrow(\ast). Two main difficulties arise: Whenever we want to use that forcing we will have to make sure that it is semiproper and that NS_{\omega_1} is saturated. We hope that the large cardinal assumption can be lowered to around the region of an inaccessible limit of Woodin cardinals. This is joint work with Ralf Schindler.

Cornell Logic Seminar
Time: Tuesday, May 5, 2:55pm New York time (20:55pm CEST)
Speaker: Andrew Zucker, University of Lyon
Title: Topological dynamics beyond Polish groups
Abstract: When G is a Polish group, one way of knowing that G has “nice” dynamics is to show that M(G), the universal minimal flow of G, is metrizable. However, works of Bartosova, Gheysens, and Krupinski–Pillay investigate groups beyond the Polish realm, such as Sym(κ), Homeo(ω1), and automorphism groups of uncountable, ω-homogeneous structures. For example, Bartosova shows that the universal minimal flow of Sym(κ) is the space of linear orders on κ–not a metrizable space, but still “nice.” In this talk, we seek to put these results into a general framework which encompasses all topological groups.
This is joint work with Gianluca Basso

May 6

Paris-Lyon Séminaire de Logique
Time:
Wednesday, May 6, 16:00-17:15 CEST
Speaker: Ilijas Farah – York University (Toronto)
Title: Between ultrapowers and reduced products.
Abstract: Ultrapowers and reduced powers are two popular tools for studying countable (and separable metric) structures. Once an ultrafilter on N is fixed, these constructions are functors into the category of countably saturated structures of the language of the original structure. The question of the exact relation between these two functors has been raised only recently by Schafhauser and Tikuisis, in the context of Elliott’s classification programme. Is there an ultrafilter on N such that the quotient map from the reduced product associated with the Fréchet filter onto the ultrapower has the right inverse? The answer to this question involves both model theory and set theory. Although these results were motivated by the study of C*-algebras, all of the results and proofs will be given in the context of classical (discrete) model theory.
Information: Join via the link on the seminar webpage 10 minutes before the talk.

Oxford Set Theory Seminar
Time:
Wednesday, May 6, 16:00-17:30 UK time (17:00-18:30 CEST)
Speaker: Victoria Gitman, City University of New York
Title: Elementary embeddings and smaller large cardinals
Abstract: A common theme in the definitions of larger large cardinals is the existence of elementary embeddings from the universe into an inner model. In contrast, smaller large cardinals, such as weakly compact and Ramsey cardinals, are usually characterized by their combinatorial properties such as existence of large homogeneous sets for colorings. It turns out that many familiar smaller large cardinals have elegant elementary embedding characterizations. The embeddings here are correspondingly ‘small’; they are between transitive set models of set theory, usually the size of the large cardinal in question. The study of these elementary embeddings has led us to isolate certain important properties via which we have defined robust hierarchies of large cardinals below a measurable cardinal. In this talk, I will introduce these types of elementary embeddings and discuss the large cardinal hierarchies that have come out of the analysis of their properties. The more recent results in this area are a joint work with Philipp Schlicht.

May 7

Kurt Gödel Research Center Seminar (organised by Ben Miller)
Time:
Thursday, May 7, 16:00 CEST
Speaker: Stefan Hoffelner, University of Muenster
Title: TBA
Abstract: TBA
Information: Talk via zoom.

May 8

Toronto Set Theory Seminar
Time: Friday, May 8, 1:30-3:00pm EDT (19:30-21:00 CEST)
Speaker: Dima Sinapova, University of Chicago
Title: Iteration, reflection, and Prikry forcing
Abstract: There is an inherent tension between stationary reflection and the failure of SCH. The former is a compactness type principle that follows from large cardinals. The latter is an instance of incompactness, and usually obtained using Prikry forcing. We describe a Prikry style iteration, and use it to force stationary reflection in the presence of not SCH. Then we discuss the situation at smaller cardinals. This is joint work with Alejandro Poveda and Assaf Rinot.
Information: The talk will take place via zoom: https://yorku.zoom.us/j/96087161597.

CUNY Set Theory Seminar
Time: Friday, May 8, 2pm New York time (8pm CEST)
Speaker: Sandra Mueller, KGRC Vienna
Title: How to obtain lower bounds in set theory
Abstract: Computing the large cardinal strength of a given statement is one of the key research directions in set theory. Fruitful tools to tackle such questions are given by inner model theory. The study of inner models was initiated by Gödel’s analysis of the constructible universe L. Later, it was extended to canonical inner models with large cardinals, e.g. measurable cardinals, strong cardinals or Woodin cardinals, which were introduced by Jensen, Mitchell, Steel, and others.
We will outline two recent applications where inner model theory is used to obtain lower bounds in large cardinal strength for statements that do not involve inner models. The first result, in part joint with J. Aguilera, is an analysis of the strength of determinacy for certain infinite two player games of fixed countable length, and the second result, joint with Y. Hayut, involves combinatorics of infinite trees and the perfect subtree property for weakly compact cardinals κ. Finally, we will comment on obstacles, questions, and conjectures for lifting these results higher up in the large cardinal hierarchy.
Information: The seminar will take place virtually. Please email Victoria Gitman (vgitman@nylogic.org) for the meeting id.

# Online Activities 27 April – 3 May 2020

This is a list of online talks in set theory next week. For a complete list, see online activities. In case we missed an upcoming talk, please email matteo.viale@unito.it, boban.velickovic@math.univ-paris-diderot.fr or philipp.schlicht@bristol.ac.uk.

April 28

Münster Set Theory Seminar
Time: Tuesday, April 28, 4:15pm CEST
Speaker: Matteo Viale, University of Torino
Title: Tameness for set theory
Abstract: We show that (assuming large cardinals) set theory is a tractable (and we dare to say tame) first order theory when formalized in a first order signature with natural predicate symbols for the basic definable concepts of second and third order arithmetic, and appealing to the model-theoretic notions of model completeness and model companionship.
Specifically we develop a general framework linking generic absoluteness results to model companionship and show that (with the required care in details) a Pi_2-property formalized in an appropriate language for second or third order number theory is forcible from some T extending ZFC + large cardinals if and only if it is consistent with the universal fragment of T if and only if it is realized in the model companion of T.
Part (but not all) of our results are conditional to the proof of Schindler and Asperò that Woodin’s axiom (*) can be forced by a stationary set preserving forcing.

Cornell Logic Seminar
Time: Tuesday, April 28, 2:55pm New York time (20:55pm CEST)
Speaker: Anton Bernsteyn, Carnegie Mellon University
Title: Descriptive combinatorics and distributed algorithms
Abstract: Descriptive combinatorics is the study of combinatorial problems (such as graph coloring) under additional topological or measure-theoretic regularity restrictions. It turns out that there is a close relationship between descriptive combinatorics and distributed computing, i.e., the area of computer science concerned with problems that can be solved efficiently by a decentralized network of processors. In this talk I will outline this relationship and present a number of applications.

April 29

Paris-Lyon Séminaire de Logique
Time:
Wednesday, April 29, 16:00-17:15 CEST
Speaker: Christian Rosendal – University of Illinois at Chicago
Title: Continuity of universally measurable homomorphisms
Abstract: We show that a universally measurable homomorphism between Polish groups is automatically continuous. Using our general analysis of continuity of group homomorphisms, this result is used to calibrate the strength of the existence of a discontinuous homomorphism between Polish groups. In particular, it is shown that, modulo ZF+DC, the existence of a discontinuous homomorphism between Polish groups implies that the Hamming graph on {0, 1}N has finite chromatic number. This solves a classical problem originating in JPR Christensen’s work on Haar null sets.
Information: Join via the link on the seminar webpage 10 minutes before the talk.

April 30

Kurt Gödel Research Center Seminar (organised by Ben Miller)
Time:
Thursday, April 23, 16:00 CEST
Speaker: Sandra Müller, KGRC
Title: How to obtain lower bounds in set theory
Abstract: Computing the large cardinal strength of a given statement is one of the key research directions in set theory. Fruitful tools to tackle such questions are given by inner model theory. The study of inner models was initiated by Gödel’s analysis of the constructible universe LL. Later, it was extended to canonical inner models with large cardinals, e.g. measurable cardinals, strong cardinals or Woodin cardinals, which were introduced by Jensen, Mitchell, Steel, and others.
We will outline two recent applications where inner model theory is used to obtain lower bounds in large cardinal strength for statements that do not involve inner models. The first result, in part joint with J. Aguilera, is an analysis of the strength of determinacy for certain infinite two player games of fixed countable length, and the second result, joint with Y. Hayut, involves combinatorics of infinite trees and the perfect subtree property for weakly compact cardinals κκ.
Information: Talk via zoom.

May 1

Toronto Set Theory Seminar
Time: Friday, May 1, 1:30-3:00pm EDT (19:30-21:00 CEST)
Speaker: Paul Szeptycki
Title: Strong convergence properties and an example from a square-sequence
Abstract: We present an example of a space constructed from square(kappa), answering some questions of Arhangel’skii. Coauthors Bill Chen and Cesar Corral-Rojas.
Information: The talk will take place via zoom: https://yorku.zoom.us/j/925557716.

CUNY Set Theory Seminar
Time: Friday, May 1, 2pm New York time (8pm CEST)
Speaker: Joan Bagaria, Universitat de Barcelona
Title: From Strong to Woodin cardinals: A level-by-level analysis of the Weak Vopenka Principle
Abstract: In May 2019 Trevor Wilson proved that the Weak Vopenka Principle (WVP), which asserts that the opposite of the category of Ordinals cannot be fully embedded into the category of Graphs, is equivalent to the class of ordinals being Woodin. In particular this implies that WVP is not equivalent to Vopenka’s Principle, thus solving an important long-standing open question in category theory. I will report on a joint ensuing work with Trevor Wilson in which we analyse the strength of WVP for definable classes of full subcategories of Graphs, obtaining exact level-by-level characterisations in terms of a natural hierarchy of strong cardinals.
Information: The seminar will take place virtually. Please email Victoria Gitman (vgitman@nylogic.org) for the meeting id.

# Online Activities 20-26 April 2020

This is a list of online talks in set theory next week.

For a complete list see online activities. In case we missed an upcoming talk, please email matteo.viale@unito.it, boban.velickovic@math.univ-paris-diderot.fr or philipp.schlicht@bristol.ac.uk.

April 21

Münster Set Theory Seminar
Time: Tuesday, April 21, 4:15pm CEST
Speaker: Farmer Schlutzenberg, University of Münster
Title: Non-definability of embeddings $j:V_\lambda\to V_\lambda$
Abstract: Assume $\ZF$. We show that there is no limit ordinal $\lambda$ and $\Sigma_1$-elementary $j:V_\lambda\to V_\lambda$ which is definable from parameters over $V_\lambda$.

April 22

New York Logic Seminar (MOPA)
Time: Wednesday, April 22, 7pm New York time (Thursday, April 23, 1am CEST)
Speaker: Corey Switzer, CUNY
Title: Hanf Numbers of Arithmetics
Abstract: Recall that given a complete theory T and a type p(x) the Hanf number for p(x) is the least cardinal κ so that any model of T of size κ realizes p(x) (if such a κ exists and ∞ otherwise). The Hanf number for T, denoted H(T), is the supremum of the successors of the Hanf numbers for all possible types p(x) whose Hanf numbers are <∞. We have seen so far in the seminar that for any complete, consistent T in a countable language H(T)≤ℶω1 (a result due to Morley). In this talk I will present the following theorems: (1) The Hanf number for true arithmetic is ℶω (Abrahamson-Harrington-Knight) but (2) the Hanf number for False Arithmetic is ℶω1 (Abrahamson-Harrington)
Information: The seminar will take place virtually. Please email Victoria Gitman for the meeting id.

Bar-Ilan University and Hebrew University Set Theory Seminar
Time: Wednesday, April 22, 11am IST (10am CEST)
Speaker: Jing Zhang
Title: Transformations of the transfinite plane
Abstract: We discuss the existence of certain transformation functions turning pairs of ordinals into triples (or pairs) of ordinals, that allows reductions of complicated Ramsey theoretic problems into simpler ones. We will focus on the existence of various kinds of strong colorings. The basic technique is Todorcevic’s walks on ordinals. Joint work with Assaf Rinot.
Information: The zoom meeting ID is 243-676-331 and no password.

April 23

Kurt Gödel Research Center Seminar (organised by Ben Miller)
Time:
Thursday, April 23, 16:00 CEST
Speaker: Noé de Rancourt, KGRC
Title: Weakly Ramsey ultrafilters
Abstract: Weakly Ramsey ultrafilters are ultrafilters on ωω satisfying a weak local version of Ramsey’s theorem; they naturally generalize Ramsey ultrafilters. It is well known that an ultrafilter on ωω is Ramsey if and only if it is minimal in the Rudin-Keisler ordering; in joint work with Jonathan Verner, we proved that similarly, weakly Ramsey ultrafilters are low in this ordering: there are no infinite chains below them. This generalizes a result of Laflamme’s. In this talk, I will outline a proof of this result, and the construction of a counterexample to the converse of this fact, namely a non-weakly Ramsey ultrafilter having exactly one Rudin-Keisler predecessor. This construction is partly based on finite combinatorics.
Information: Talk via zoom.

April 24

CUNY Set Theory Seminar
Time: Friday, April 24, 2pm New York time (8pm CEST)
Speaker: Arthur Apter, CUNY
Title: Indestructibility and the First Two Strongly Compact Cardinals
Abstract: Starting from a model of ZFC with two supercompact cardinals, I will discuss how to force and construct a model in which the first two strongly compact cardinals κ1 and κ2 are also the first two measurable cardinals. In this model, κ1’s strong compactness is indestructible under arbitrary κ1-directed closed forcing, and κ2’s strong compactness is indestructible under Add(κ2,λ) for any ordinal λ. This answers a generalized version of a question of Sargsyan.
Information: The seminar will take place virtually. Please email Victoria Gitman (vgitman@nylogic.org) for the meeting id.