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 LogiqueTime:** 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 SeminarTime:** 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.

**Information:**Please contact Philip Welch (p.welch@bristol.ac.uk) ahead of time to participate.

**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.**Information:** The seminar will take place virtually. Please email Samuel Adam-Day (me@samadamday.com) for the meeting id.

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