# Online Activities 2020

Since March 2020, a number of set theory seminars worldwide have moved online. This provides an unprecedented opportunity for participation.

The talks are listed in reverse chronological order, so please scroll down to find today’s talks!

iesides our list, take a look at the seminar web pages linked below and the following for more seminar listings:

Online seminars and talks in Logic A list of online seminars in logic, collected by Miguel Moreno.

Logic Supergroup (UCONN) An alliance of logicians in quarantine, comprised of logic groups across the world, hosting virtual talks.

Please email new announcements to us at philipp.schlicht@univie.ac.at, boban.velickovic@math.univ-paris-diderot.fr or matteo.viale@unito.it.

Week 6-12 January

CUNY Set Theory Seminar
Time: Friday, 8 January, 2pm New York time (20:00 CET)
Speaker: Thilo Weinert, University of Vienna
Title: Tba
Abstract: Tba
Information: The seminar will take place virtually. Please email Victoria Gitman (vgitman@nylogic.org) for the meeting id.

Week 28 December – 6 January

Hebrew University-Bar Ilan University Set Theory seminar
Time: Wednesday, 30 December, 14:00-16:00 Israel Time (13:00-15:00 CET)
Speaker: Assaf Shani, Harvard
Title:  Actions of tame abelian product groups
Abstract: A Polish group G is tame if for any continuous action of G, the corresponding orbit equivalence relation is Borel. Suppose that G=\prod_n \Gamma_n is a product of countable abelian groups. It follows from results of Solecki and Ding-Gao that if G is tame, then all of its actions are in fact potentially \Pi^0_6. Ding and Gao conjectured that this bound could be improved to \Pi^0_3. We refute this, by finding an action of a tame abelian product group, which is not potentially \Pi^0_5. The proof involves forcing over models where the axiom of choice fails for sequences of finite sets. This is joint work with Shaun Allison.
Information: Contact Menachem Magidor, Asaf Rinot or Omer Ben-Neria ahead of time for the zoom link.

Week 21-27 December

Hebrew University-Bar Ilan University Set Theory seminar
Time: Wednesday, 23 December, 14:00-16:00 Israel Time (13:00-15:00 CET)
Speaker: Roy Shalev, Bar Ilan University
Title: A guessing principle from a Souslin tree, with applications to topology
Abstract: We introduce a new combinatorial principle which we call ♣_AD. This principle asserts the existence of a certain multi-ladder system with guessing and almost-disjointness features, and is shown to be sufficient for carrying out de Caux type constructions of topological spaces.
Our main result states that strong instances of ♣_AD follow from the existence of a Souslin tree.  As an application, we obtain a simple, de Caux type proof of Rudin’s result that if there is a Souslin tree, then there is an S-space which is Dowker.
Information: Contact Menachem Magidor, Asaf Rinot or Omer Ben-Neria ahead of time for the zoom link.

Week 14-20 December

Turin-Udine logic seminar
Time: Friday, 18 December, 16:30pm CET
Speaker: Monroe Eskew, University of Vienna
Title: Weak square from weak presaturation
Abstract: Can we have both a saturated ideal and the tree property on ℵ2? Towards the negative direction, we show that for a regular cardinal κ, if 2<κ≤κ+ and there is a weakly presaturated ideal on κ+ concentrating on cofinality κ, then □∗κ holds. This partially answers a question of Foreman and Magidor about the approachability ideal on ℵ2. A surprising corollary is that if there is a presaturated ideal J on ℵ2 such that P(ℵ2)/J is a semiproper forcing, then CH holds. This is joint work with Sean Cox.
Information: Please see the semiar webpage.

KGRC Research Seminar
Time:
Thursday, 17 December, 15:00 CET
Speaker: Peter Holy, University of Udine
Title: Ramsey-like Operators
Abstract: Starting from measurability upwards, larger large cardinals are usually characterized by the existence of certain elementary embeddings of the universe, or dually, the existence of certain ultrafilters. However, below measurability, we have a somewhat similar picture when we consider certain embeddings with set-sized domain, or ultrafilters for small collections of sets. I will present some new results, and also review some older ones, showing that not only large cardinals below measurability, but also several related concepts can be characterized in such a way, and I will also provide a sample application of these characterizations.
Information: Talk via zoom.

Caltech Logic Seminar
Time: Wednesday, 16 December, 12:00 – 1:00pm Pacific time (22:00 CET)
Speaker: Martino Lupini, Victoria University of Wellington
Title: Classification of extension of C*-algebras and K-homology
Abstract: I will present an introduction from the perspective of Borel complexity theory to the classification problem for extension of C*-algebras, its motivations from operator theory, and its connections with homological algebra.
Information: See the seminar webpage.

Münster research seminar on set theory
Time: Wednesday, 16 December, 15:15-16:45 CET
Speaker: Tba
Title: Tba
Abstract: Tba
Information: Please check the seminar webpage to see if the seminar takes place. Contact rds@wwu.de ahead of time in order to participate.

Barcelona Set Theory Seminar
Time: Wednesday, 16 December, 16:00-17:30 CET
Speaker: Victoria Gitman, CUNY
Title: Characterizing large cardinals via abstract logic
Abstract: First-order logic, the commonly accepted formal system underlying mathematics, must draw however minimally on the properties of the set-theoretic universe in which it is defined. Stronger logics such as infinitary logics and second-
order logics require access to much larger chunks of the set-theoretic background. Niceness properties of these logics, such as forms of compactness, are naturally
connected to the existence of large cardinals. Indeed, many large cardinals can be
characterized in terms of compactness properties of strong logics. Strongly compact
and weakly compact cardinals k are precisely the strong and weak compactness
cardinals respectively for the infinitary logic Lkk. Extendible cardinals k are precisely
the strong compactness cardinals for the infinitary second-order logic $\mathbb L2 kk. Vopenka’s Principle holds if and only if every logic has a strong compactness cardinal. In this talk I will review properties of various logics and how their compactness properties characterize various large cardinals. I will discuss joint work with Will Boney, Stamatis Dimopoulos and Menachem Magidor in which we show that the principle Ord is subtle, in the presence of global choice, holds if and only if every logic has a weak compactness cardinal, i.e., it is the analogue of Vopenka’s Principle for weak compactness. We also provide characterizations of the various virtual large cardinals using a new notion of a pseudo-model of a theory. Information: Online. If you wish to attend, please send an email to bagaria@ub.edu asking for the link. Hebrew University-Bar Ilan University Set Theory seminar Time: Wednesday, 16 December, 14:00-16:00 Israel Time (13:00-15:00 CET) Speaker: Roy Shalev, Bar Ilan University Title: S spaces and L spaces, part 1 Abstract: It will be both a survey talk and exposition of new results. Very likely it will be continued the following week. An S-space is a regular topological space that is hereditarily separable but not Lindel\”of. An L-space is a regular topological space that is hereditarily Lindel\”of but not separable. We will survey the history behind the question of their existence and present some constructions. Information: Contact Menachem Magidor, Asaf Rinot or Omer Ben-Neria ahead of time for the zoom link. Week 7-13 December CUNY Set Theory Seminar Time: Friday, December 10, 11am New York time (17:00 CET) Speaker: Dima Sinapova, University of Chicago Title: Iteration, reflection, and singular cardinals Abstract: There is an inherent tension between stationary reflection and the failure of the singular cardinal hypothesis (SCH). The former is a compactness type principle that follows from large cardinals. Compactness is the phenomenon where if a certain property holds for every smaller substructure of an object, then it holds for the entire object. In contrast, failure of SCH is an instance of incompactness. Two classical results of Magidor are: (1) from large cardinals it is consistent to have reflection at ℵω+1, and (2) from large cardinals it is consistent to have the failure of SCH at ℵω. As these principles are at odds with each other, the natural question is whether we can have both. We show the answer is yes. We describe a Prikry style iteration, and use it to force stationary reflection in the presence of not SCH. Then we obtain this situation at ℵω by interleaving collapses. This is joint work with Alejandro Poveda and Assaf Rinot. Information: The seminar will take place virtually. Please email Victoria Gitman (vgitman@nylogic.org) for the meeting id. Turin-Udine logic seminar Time: Friday, 11 December, 16:30pm CET Speaker: Assaf Shani, Harvard University Title: Classification results for countable Archimedean groups Abstract: We study the isomorphism relation for countable ordered Archimedean groups. We locate its complexity with respect to the hierarchy defined by Hjorth, Kechris, and Louveau, showing in particular that its potential complexity is D(Π03), and it cannot be classified using countable sets of reals as invariants. We obtain analogous results for the bi-embeddability relation, and we consider similar problems for circularly ordered groups and ordered divisible Abelian groups. This is joint work with F. Calderoni, D. Marker, and L. Motto Ros. Information: The seminar will take place virtually. Please email Victoria Gitman (vgitman@nylogic.org) for the meeting id. KGRC Research Seminar Time: Thursday, December 10, 15:00 CET Speaker: Michael Hrušák, UNAM, Mexico City Title: Invariant Ideal Axiom Abstract: We shall introduce a consistent set-theoretic axiom which has a profound impact on convergence properties in topological groups. As an application we show that consistently (consequence of IIA) every countable sequential group is either metrizable or kω. Information: Talk via zoom. Caltech Logic Seminar Time: Wednesday, December 9, 12:00 – 1:00pm Pacific time (22:00 CET) Speaker: Jeffrey Bergfalk, University of Vienna Title: The definable content of (co)homological invariants II: definable cohomology and homotopy classification Abstract: In this, the second of a three-part series of talks, we describe a “definable Čech cohomology theory” strictly refining its classical counterpart. As applications, we show that, in strong contrast to its classical counterpart, this definable cohomology theory provides complete homotopy invariants for mapping telescopes of dd-tori and of dd-spheres; we also show that it provides an equivariant homotopy classification of maps from mapping telescopes of dd-tori to spheres, a problem raised in the d=1d=1 case by Borsuk and Eilenberg in 1936. These results build on those of the first talk. They entail, for example, an analysis of the phantom maps from a locally compact Polish space X to a polyhedron P; instrumental in that analysis is the definable lim1lim1functor. They entail more generally an analysis of the homotopy relation on the space of maps from X to P, and we will begin by describing a category particularly germane for this analysis. Time permitting, we will conclude with some discussion and application of a related construction, namely that of the definable homotopy groups of a locally compact Polish space X. This is joint work with Martino Lupini and Aristotelis Panagiotopoulos. Information: See the seminar webpage. Paris-Lyon Séminaire de Logique Time: Wednesday, December 9, 16:00-17:00 CET Speaker: Andrea Vaccaro, IMJ-PRG Title: Set Theory and the Endomorphisms of the Calkin algebra Abstract: Set theory induces a sharp dichotomy in the structure of the set of automorphisms of the Calkin algebra Q(H): under the Open Coloring Axiom (OCA) all the automorphisms of Q(H) are inner (Farah, 2011), whereas the Continuum Hypothesis (CH) implies that there exist uncountably many outer automorphisms of Q(H) (Phillips-Weaver, 2007). After a brief introduction on the line of research that led to these results, I’ll discuss how this dichotomic behavior extends to the semigroup End(Q(H)) of unital endomorphisms of Q(H). In particular, we’ll see that under OCA all unital endomorphisms of Q(H) can be, up to unitary equivalence, lifted to unital endomorphisms of B(H). This fact allows to have an extremely clean picture of End(Q(H)), and has some interesting consequences concerning the class of C*-algebras that embed into Q(H). I will also discuss how the structure of End(Q(H)) completely changes under CH. Information: Zoom ID: 824 8220 9628; sign up for the email list, or contacter silvain.rideau@imj-prg.fr, for the password in advance. Barcelona Set Theory Seminar Time: Wednesday, December 9, 16:00 CET Speaker: Neil Barton, University of Konstanz Title: Intensional classes and intuitionistic topoi Abstract: A popular view in the philosophy of set theory is that of potentialism: the position that the set-theoretic universe unfolds as more sets come into existence. A difficult question for the potentialist is to explain how classes (understood as intensional entities) behave on this framework, and in particular what logic governs them. In this talk we’ll see how category-theoretic resources can be brought to bear on this issue. I’ll first give a brief introduction to topos theory, and then I’ll explain how (drawing on work of Lawvere) we can think of intensional classes for the potentialist as given by a functor category. I’ll suggest some tentative directions for research here, including the possibility that this representation indicates that the logic of intensional classes should be intuitionistic rather than classical, and that the strength of the intuitionistic logic is dependent upon the partial order on the worlds. Information: Online. If you wish to attend, please send an email to bagaria@ub.edu asking for the link. Münster research seminar on set theory Time: Wednesday, December 9, 15:15-16:45 CET Speaker: Tba Title: Tba Abstract: Tba Information: Please check the seminar webpage to see if the seminar takes place. Contact rds@wwu.de ahead of time in order to participate. Computability and application seminar Time: Tuesday, 8 December, 22:00 CET Speaker: Linda Brown Westrick, Pennsylvania State University Title: Luzin’s (N) and randomness reflection Abstract: Tba Information: The seminar will take place virtually via zoom. Week 30 November – 6 December Toronto Set Theory Seminar Time: Friday, December 4, 1.30-3pm Toronto time (19:30-21:00 CET) Speaker: Thomas Gilton, University of Pittsburgh Title: The Abraham-Rubin-Shelah Open Coloring Axiom with a Large Continuum Abstract: Open Coloring Axioms may be viewed as consistent generalizations of Ramsey’s Theorem to ω1 in which topological restrictions are placed on the colorings. The first of these, denoted 𝖮𝖢𝖠ARS, appeared in the 1985 paper by Abraham, Rubin, and Shelah. There the authors showed that 𝖮𝖢𝖠ARS is consistent with 𝖹𝖥𝖢. To ensure that the posets which add the homogeneous sets satisfy the c.c.c., they construct a type of “diagonalization” object (for a continuous coloring χ) called a Preassignment of Colors, which guides the forcing to add the χ-homogeneous sets. However, the only known constructions of effective preassignments require the 𝖢𝖧. Since a forcing iteration of ℵ1-sized posets all of whose proper initial segments satisfy the 𝖢𝖧 results in a model in which 2ℵ0 is at most ℵ2, this leads naturally to the question of whether 𝖮𝖢𝖠ARS is consistent, say, with 2ℵ0=ℵ3. In joint work with Itay Neeman, we answer this question in the affirmative. In light of the 𝖢𝖧 obstacle, we only construct names for preassignments with respect to a small class  of 𝖢𝖧-preserving iterations. However, our preassignments are powerful enough to work even over models in which the 𝖢𝖧 fails. Our final forcing is built by combining the members of  into a new type of forcing, called a Partition Product. A partition product is a type of restricted memory iteration with isomorphism and coherent-overlap conditions on the memories. In particular, each “memory” is isomorphic to a member of . In this talk, we will describe in some detail the definition of a Partition Product. We will then discuss how to construct more general preassignments than those used by Abraham, Rubin, and Shelah, gesturing at the end towards the full construction which we use for our theorem. Information: Email Ivan Ongay Valverde ahead of time for the zoom link. CUNY Set Theory Seminar Time: Friday, December 4, 3pm New York time (21:00 CET) Speaker: Zach Norwood, University of Michigan Title: Tba Abstract: Tba Information: The seminar will take place virtually. Please email Victoria Gitman (vgitman@nylogic.org) for the meeting id. Set Theory in the UK @ Home Time: Friday, December 4, 9:30am-2pm UK time (10:30-15:00 CET) Schedule: 9:30-9:55 Yair Hayut: Generics via ultrapowers 10.00-10.50 Arno Pauly: Luzin’s (N) and randomness reflection 11.00-11.50 Peter Holy: Ramsey-like operators lunch break 13.30-13.55 Jiachen Yuan: Indestructibility of supercompactness and large cardinals Information: Please see the meeting’s webpage at http://www1.maths.leeds.ac.uk/~pmtadb/STUK6/STUK6.html KGRC Research Seminar Time: Thursday, December 3, 15:00 CET Speaker: Mirna Džamonja, CNRS & Panthéon Sorbonne, Paris and Czech Academy of Sciences, Prague Title: On logics that make a bridge from the Discrete to the Continuous Abstract: The talk starts with a surveys of some recent connections between logic and discrete mathematics. Then we discuss logics which model the passage between an infinite sequence of finite models to an uncountable limiting object, such as is the case in the context of graphons. Of particular interest is the connection between the countable and the uncountable object that one obtains as the union versus the combinatorial limit of the same sequence. We compare such logics and discuss some consequences of such comparisons, as well as some hopes for further results in this research project. Information: Talk via zoom. Caltech Logic Seminar Time: Wednesday, December 2, 12:00 – 1:00pm Pacific time (22:00 CET) Speaker: Aristotelis Panagiotopoulos, University of Münster Title: The definable content of homological invariants I: Ext(−,−)Ext⁡(−,−) and lim1(−) Abstract: This is the first talk in a three-part series in which we illustrate how classical invariants of homological algebra and algebraic topology can be enriched with additional descriptive set-theoretic information. In the first talk we will focus on the “definable enrichment” of the first derived functors of Hom(−,−)Hom⁡(−,−) and lim(−)lim⁡(−). We will show that the resulting “definable Ext(B,F)Ext⁡(B,F)” for pairs of countable abelian groups B,FB,F; and the “definable lim1(A)lim1⁡(A)” for towers AA of Polish abelian groups substantially refine their purely algebraic counterparts. In the process, we will develop an Ulam stability framework for quotients of Polish groups GG by Polishable subgroups HH and we will provide several rigidity results in the case where the ambient Polish group GG is abelian and non-archimedean. A special case of our rigidity results answers a question of Kanovei and Reeken regarding quotients of the pp-adic groups. This is joint work with Jeffrey Bergfalk and Martino Lupini. Information: See the seminar webpage. Bristol Logic and Set Theory Seminar/Oxford Set Theory Seminar Time: Wednesday, December 2, 16:00-17:30 UK time (17:00-18:30 CET) Speaker: Kameryn Williams, University of Hawai’i at Manoa 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, both positive and negative, about the sequence of inner mantles, answering some questions of Fuchs, Hamkins, and Reitz, results which are analogues of classic results about the sequence of iterated HODs. On the positive side: (Joint with Reitz) Every model of set theory is the eta-th inner mantle of a class forcing extension for any ordinal eta in the model. On the negative side: The sequence of inner mantles may fail to carry through at limit stages. Specifically, it is consistent that the omega-th inner mantle not be a definable class and it is consistent that it be a definable inner model of ¬AC. Information: For the Zoom access code, contact Samuel Adam-Day me@samadamday.com. Link: https://zoom.us/j/96803195711 (open 30 minutes before) Barcelona Set Theory Seminar Time: Wednesday, December 2, 16:00 CET Speaker: Toby Meadows, UC Irvine Title: What set theory could not be Abstract: I am going to argue that set theory – playing its foundational role – is not particularly concerned with sets. Despite appearances, I don’t think this is a frivolous claim. Among other things, even if the conclusion is doubted, it reveals an important split in attitudes researchers have toward set theory. Information: Online. If you wish to attend, please send an email to bagaria@ub.edu asking for the link. Münster research seminar on set theory Time: Wednesday, December 2, 15:15-16:45 CET Speaker: Tba Title: Tba Abstract: Tba Information: Please check the seminar webpage to see if the seminar takes place. Contact rds@wwu.de ahead of time in order to participate. Hebrew University-Bar Ilan University Set Theory seminar Time: Wednesday, December 2, 14:00-16:00 Israel Time (13:00-15:00 CET) Speaker: Menachem Magidor, Hebrew University of Jerusalem Title: Woodin’s extender algebra and its applications (part 2) Abstract: I’ll continue to talk about Woodin’s extender algebra and absoluteness Information: Contact Menachem Magidor, Asaf Rinot or Omer Ben-Neria ahead of time for the zoom link. Logic Seminar, Carnegie Mellon University Time: Tuesday, December 1, 3:30 – 4:30pm Eastern Standard Time (21:30 – 22:30 CET) Speaker: Nam Trang, University of North Texas Title: Ideals and determinacy Abstract: We present some ideas involved in the proof of the equiconsistency of AD_\reals + Theta is regular and the existence of a strong, pseudo-homogeneous ideal on P_{\omega_1}(\reals). Some variations of this hypothesis are also shown to be equiconsistent with AD_\reals + Theta is regular. This work is related to and partially answers a long-standing conjecture of Woodin regarding the equiconsistency of AD_\reals + Theta is regular and CH + the nonstationary ideal on \omega_1 is \omega_1-dense. We put this result in a broader context of the general program of understanding connections between canonical models of large cardinals, models of determinacy, and strong forcing axioms (e.g. PFA, MM). Information: Zoom link https://cmu.zoom.us/j/621951121, meeting ID: 621 951 121 Week 23-29 November Toronto Set Theory Seminar Time: Friday, November 27, 1.30-3pm Toronto time (19:30-21:00 CET) Speaker: Sandra Müller, University of Vienna Title: The Large Cardinal Strength of Determinacy Axioms Abstract: The study of inner models was initiated by Gödel’s analysis of the constructible universe L. Later, it became necessary to study canonical inner models with large cardinals, e.g. measurable cardinals, strong cardinals or Woodin cardinals, which were introduced by Jensen, Mitchell, Steel, and others. Around the same time, the study of infinite two-player games was driven forward by Martin’s proof of analytic determinacy from a measurable cardinal, Borel determinacy from ZFC, and Martin and Steel’s proof of levels of projective determinacy from Woodin cardinals with a measurable cardinal on top. First Woodin and later Neeman improved the result in the projective hierarchy by showing that in fact the existence of a countable iterable model, a mouse, with Woodin cardinals and a top measure suffices to prove determinacy in the projective hierarchy. This opened up the possibility for an optimal result stating the equivalence between local determinacy hypotheses and the existence of mice in the projective hierarchy, just like the equivalence of analytic determinacy and the existence of x♯for every real x which was shown by Martin and Harrington in the 70’s. The existence of mice with Woodin cardinals and a top measure from levels of projective determinacy was shown by Woodin in the 90’s. Together with his earlier and Neeman’s results this estabilishes a tight connection between descriptive set theory in the projective hierarchy and inner model theory. In this talk, we will outline some of the main results connecting determinacy hypotheses with the existence of mice with large cardinals and discuss a number of more recent results in this area. Information: Email Ivan Ongay Valverde ahead of time for the zoom link. Münster research seminar on set theory Time: Thursday, November 26, 16:15-17:45 CET Speaker: Grigor Sargsyan, Gdansk Title: Determinacy, forcing axioms and inner models (part 4) Abstract: We will exposit some recent results of the speaker and others that connect determinacy axioms, forcing axioms and inner models. A culmination of this work is a recent proof that the most liberally backgrounded construction of a model build from an extender sequence cannot be shown to converge in ZFC alone. In this construction, which is a type of Kc construction, one uses extenders that are certified by a Mostowski collapse. This result challenges common perceptions of the role of the model Kc in the inner model program. We will mention a specific consistency result showing that the failure of □ω3 and □(ω3) with 2ω=2ω1=ω2 and 2ω2=ω3 is weaker than a Woodin cardinal that is a limit of Woodin cardinals. Many people have been involved in this project. The work is heavily based on the efforts of Steel, Jensen, Woodin, Schindler, Mitchell, Schimmerling, Trang, Larson, Neeman, Zeman, Schlutzenberg, the speaker and many others. Information: contact rds@wwu.de ahead of time in order to participate. KGRC Research Seminar Time: Thursday, November 26, 15:00 CET Speaker: Damian Sobota, KGRC Title: Convergence of Borel measures and filters on omega Abstract: The celebrated Josefson–Nissenzweig theorem asserts, under certain interpretations, that for every infinite compact space K there exists a sequence of normalized signed Borel measures on K which converges to 0 with respect to every continuous real-valued function (i.e. the corresponding integrals converge to 0). We showed that in the case of products of two infinite compact spaces K and L one can construct such a sequence of measures with an additional property that every measure has finite support—let us call such a sequence “an fsJN-sequence” (i.e. a finitely supported Josefson–Nissenzweig sequence). We then studied the case when the spaces K and L are only pseudocompact and we proved in ZFC that if the product of K and L is pseudocompact, then it also admits an fsJN-sequence. On the other hand, we showed that under the Continuum Hypothesis, or Martin’s axiom, or even some weaker set-theoretic assumptions concerning weak P-points, there exists a pseudocompact space X such that its square is not pseudocompact and it does not admit any fsJN-sequences. During my talk I will discuss these as well as other results concerning the topic and obtained during a joint work with various combinations of J. Kakol, W. Marciszewski and L. Zdomskyy. Information: Talk via zoom. Barcelona Set Theory Seminar Time: Wednesday, November 25, 16:00 CET Speaker: Andrew Brooke-Taylor, University of Leeds Title: Categorifying Borel reducibility Abstract: Borel reducibility is a framework that has been very successful in showing that classification programmes in different areas of mathematics are not possible to complete. However, a feature of many such classification programmes that is not accounted for in the standard Borel reducibility framework is functoriality – a good classification function is expected to respect all maps between objects, not just the isomorphisms. I will present an extension of the Borel reducibility framework that takes functoriality into account, and give some initial results showing that this is a meaningful refinement of the standard framework. Information: Online. If you wish to attend, please send an email to bagaria@ub.edu asking for the link. Hebrew University-Bar Ilan University Set Theory seminar Time: Wednesday, November 25, 14:00-16:00 Israel Time (13:00-15:00 CET) Speaker: Menachem Magidor, Hebrew University of Jerusalem Title: Woodin’s extender algebra and its applications Abstract: This talk will survey known results and will be the first of several talks which will not necessarily follow in the consecutive weeks. Information: Contact Menachem Magidor, Asaf Rinot or Omer Ben-Neria ahead of time for the zoom link. Helsinki Logic Seminar Time: Wednesday, November 25, 12:15 Helsinki Time (11:15 CET) Speaker: Carolin Antos-Kuby, University of Konstanz Title: Two aspects of explanatoriness Abstract: The phenomenon of explanation in mathematics is an interesting one: If there are different proofs for one theorem, all of them show that the theorem holds but often only some also show why the theorem holds, i.e. additionally also explain the theorem. Unlike in the natural sciences this phenomenon is not easily reducible to the phenomenon of causation. It is even unclear if there is only one form of explanatoriness or if it is a pluralistic notion. Here we give an example from recent descriptive set theory where we study two approaches to proving set-theoretic dichotomy theorems. We will see that both approaches provide explanations for the theorem, albeit in very different ways. We will use this to highlight two ways in which explanatoriness can be spelled out and distinguish between agent-dependent and agent-independent notions of explanatoriness. Information: See the seminar webpage. Logic Seminar, Carnegie Mellon University Time: Tuesday, November 24, 3:30 – 4:30pm Eastern Standard Time (21:30 – 22:30 CET) Speaker: Shaun Allison Title: An anticlassification result for TSI Polish groups Abstract: We give a dynamical obstruction to classification by TSI Polish groups, and apply it to an equivalence relation of Clemens and Coskey. This yields the first known example of an equivalence relation that is classifiable by a CLI Polish group but not by TSI Polish groups. This work is joint with Aristotelis Panagiotopoulos. Information: Zoom link https://cmu.zoom.us/j/621951121, meeting ID: 621 951 121 Week 16-22 November CUNY Set Theory Seminar Time: Friday, November 20, 3pm New York time (21:00 CET) Speaker: Philipp Schlicht, University of Vienna Title: The recognisable universe in the presence of measurable cardinals Abstract: A set x of ordinals is called recognisable if it is defined, as a singleton, by a formula phi(y) with ordinal parameters that is evaluated in L[y]. The evaluation is always forcing absolute, in contrast to even Sigma_1-formulas with ordinal parameters evaluated in V. Furthermore, this notion is closely related to similar concepts in infinite computation and Hamkins’ and Leahy’s implicitly definable sets. It is conjectured that the recognisable universe generated by all recognisable sets is forcing absolute, given sufficient large cardinals. Our goal is thus to determine the recognisable universe in the presence of large cardinals. The new main result, joint with Philip Welch, is a computation of the recognisable universe within the least inner model with infinitely many measurable cardinals. Information: The seminar will take place virtually. Please email Victoria Gitman (vgitman@nylogic.org) for the meeting id. Toronto Set Theory Seminar Time: Friday, November 20, 1.30pm Toronto time (19:30 CET) Speaker: Andrea Medini, University of Vienna Title: Topological applications of Wadge theory Abstract: Wadge theory provides an exhaustive analysis of the topological complexity of the subsets of a zero-dimensional Polish space. Fons van Engelen pioneered its applications to topology by obtaining a classification of the zero-dimensional homogeneous Borel spaces (recall that a space$X$is homogeneous if for all$x,y\in X$there exists a homeomorphism$h:X\longrightarrow X$such that$h(x)=y$). As a corollary, he showed that all such spaces (apart from trivial exceptions) are in fact strongly homogeneous (recall that a space$X$is strongly homogeneous if all non-empty clopen subspaces of$X$are homeomorphic to each other). In a joint work with the other members of the “Wadge Brigade” (namely, Raphaël Carroy and Sandra Müller), we showed that this last result extends beyond the Borel realm if one assumes AD. We intend to sketch the proof of this theorem, with a view towards a complete classification of the zero-dimensional homogeneous spaces under AD. Information: Email Ivan Ongay Valverde ahead of time for the zoom link. Southern Illinois University Logic Seminar Time: Thursday, November 19, 1pm CET (20:00 CET) Speaker: Anush Tserunyan, Mc Gill University Title: Tba Abstract: Tba Information: See the seminar webpage. KGRC Research Seminar Time: Thursday, November 19, 15:00 CET Speaker: Gabriel Fernandes, Bar-Ilan University Title: Local club condensation in extender models Abstract: Local club condensation is a condensation principle defined by Friedman and Holy. It is a theorem due to Friedman and Holy that local club condensation holds in most of the extender models that are weakly iterable. We prove that in any weakly iterable extender model with λ-indexing, given a cardinal κκ, the sequence ⟨Lα[E]∣α<κ++⟩ witnesses local club condensation on the interval (κ+,κ++) iff κ is not a subcompact cardinal in L[E]. We also prove that if κ is subcompact, then there is no sequence ⟨Mα∣α<κ++⟩∈L[E] with Mκ=(Hκ)L[E] and Mκ++=(Hκ++)L[E] which witnesses local club condensation in (κ+,κ++). Using the equivalence between subcompact cardinals and ¬◻κ, due to Schimmerling and Zeman, it follows that ◻κ holds iff the sequence ⟨Lα[E]∣α<κ++⟩ witnesses local club condensation on the interval (κ+,κ++). Information: Talk via zoom. Caltech Logic Seminar Time: Wednesday, November 18, 12:00 – 1:00pm Pacific time (22:00 CET) Speaker: Asger Törnquist, University of Copenhagen Title: A new proof of Thoma’s theorem on type I groups Abstract: In the theory of unitary group representations, the following theorem of Elmar Thoma from the early 1960s is fundamental: A countable discrete group is “type I” if and only if it has an abelian finite index subgroup. By way of a celebrated theorem of Glimm from the same period, a group being “type I” is equivalent to saying that the irreducible unitary representations of the group admits a smooth classification in the familiar sense of Borel reducibility, and in fact they are all finite-dimensional in this case. Glimm’s theorem, and later work by Hjorth, Farah and Thomas, implies that if a group is not type I, then it is quite hard to classify the irreducible unitary representations. In this talk I will give an overview of the descriptive set-theoretic perspective on the classification of irreducible representations, and I will discuss a new proof of Thoma’s theorem due to F.E. Tonti and the speaker. Information: See the seminar webpage. Bristol Logic and Set Theory Seminar/Oxford Set Theory Seminar Time: Wednesday, November 18, 16:00-17:30 UK time (17:00-18:30 CET) Speaker: Gabriel Goldberg, Harvard University Title: Even ordinals and the Kunen inconsistency Abstract: The Burali-Forti paradox suggests that the transfinite cardinals “go on forever,” surpassing any conceivable bound one might try to place on them. The traditional Zermelo-Frankel axioms for set theory fall into a hierarchy of axiomatic systems formulated by reasserting this intuition in increasingly elaborate ways: the large cardinal hierarchy. Or so the story goes. A serious problem for this already naive account of large cardinal set theory is the Kunen inconsistency theorem, which seems to impose an upper bound on the extent of the large cardinal hierarchy itself. If one drops the Axiom of Choice, Kunen’s proof breaks down and a new hierarchy of choiceless large cardinal axioms emerges. These axioms, if consistent, represent a challenge for those “maximalist” foundational stances that take for granted both large cardinal axioms and the Axiom of Choice. This talk concerns some recent advances in our understanding of the weakest of the choiceless large cardinal axioms and the prospect, as yet unrealized, of establishing their consistency and reconciling them with the Axiom of Choice. Information: For the Zoom access code, contact Samuel Adam-Day me@samadamday.com. Link: https://zoom.us/j/96803195711 (open 30 minutes before) Barcelona Set Theory Seminar Time: Wednesday, November 18, 16:00 CET Speaker: Monroe Eskew, University of Vienna Title: Uncommon systems of embeddings Abstract: We will survey some results from two papers about systems of elementary embeddings that “narrowly avoid” Kunen’s inconsistency. The first involves systems where the embedding is amenable to the target model. The uncommon features are that the system can be densely ordered (even isomorphic to the reals) and branch off into non-amalgamable models. The second paper focuses on collections of distinct models that are pairwise mutually embeddable. Some restrictions on the structure of the system are imposed by requiring the models to satisfy V=HOD. This is joint work with Sy Friedman, Yair Hayut, and Farmer Schlutzenberg. Information: Online. If you wish to attend, please send an email to bagaria@ub.edu asking for the link. Paris-Lyon Séminaire de Logique Time: Wednesday, November 18, 16:00-17:00 CET Speaker: Denis Osin Title: A topological zero-one law and elementary equivalence of finitely generated groups Abstract: The space of finitely generated marked groups, denoted by G, is a locally compact Polish space whose elements are groups with fixed finite generating sets; the topology on G is induced by the local convergence of the corresponding Caley graphs. We will discuss equivalent characterizations of closed subspaces S of G satisfying the following zero-one law: for any sentence sigma in the infinitary logic L_{\omega_1, \omega}, the set of all models of sigma in S is either meager or comeager. In particular, this zero-one law holds for certain natural spaces associated to hyperbolic groups and their generalizations. We will also discuss some open problems. Information: Zoom ID: 824 8220 9628; sign up for the email list, or contacter silvain.rideau@imj-prg.fr, for the password in advance. Münster research seminar on set theory Time: Wednesday, November 18, 15:15-16:45 CET Speaker: Grigor Sargsyan, Gdansk Title: Determinacy, forcing axioms and inner models (part 3) Abstract: We will exposit some recent results of the speaker and others that connect determinacy axioms, forcing axioms and inner models. A culmination of this work is a recent proof that the most liberally backgrounded construction of a model build from an extender sequence cannot be shown to converge in ZFC alone. In this construction, which is a type of Kc construction, one uses extenders that are certified by a Mostowski collapse. This result challenges common perceptions of the role of the model Kc in the inner model program. We will mention a specific consistency result showing that the failure of □ω3 and □(ω3) with 2ω=2ω1=ω2 and 2ω2=ω3 is weaker than a Woodin cardinal that is a limit of Woodin cardinals. Many people have been involved in this project. The work is heavily based on the efforts of Steel, Jensen, Woodin, Schindler, Mitchell, Schimmerling, Trang, Larson, Neeman, Zeman, Schlutzenberg, the speaker and many others. Information: contact rds@wwu.de ahead of time in order to participate. Hebrew University-Bar Ilan University Set Theory seminar Time: Wednesday, November 18, 14:00-16:00 Israel Time (13:00-15:00 CET) Speaker: Vera Fischer, University of Vienna Title: Independent families in the countable and the uncountable Abstract: Independent families on$\omega$are families of infinite sets of integers with the property that for any two finite subfamilies$A$and$B$the set$\bigcap A\backslash \bigcup B$is infinite. Of particular interest are the sets of the possible cardinalities of maximal independent families, which we refer to as the spectrum of independence. Even though we do have the tools to control the spectrum of independence at$\omega$(at least to a large extent), there are many relevant questions regarding higher counterparts of independence in generalised Baire spaces, which remain widely open. Information: Contact Menachem Magidor, Asaf Rinot or Omer Ben-Neria ahead of time for the zoom link. Helsinki Logic Seminar Time: Wednesday, November 18, 12:15 Helsinki Time (11:15 CET) 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 -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 a byproduct of the groundbreaking result of Schindler and Asperò showing that Woodin’s axiom (*) can be forced by a stationary set preserving forcing. Information: See the seminar webpage. Week 9-15 November Toronto Set Theory Seminar Time: Friday, November 13, 11am Toronto time (17:00 CET) Speaker: Ralf Schindler, University of Münster Title: Martin’s Maximum^++ implies the P_max axiom (*) Abstract: Forcing axioms spell out the dictum that if a statement can be forced, then it is already true. The P_max axiom (*) goes beyond that by claiming that if a statement is consistent, then it is already true. Here, the statement in question needs to come from a resticted class of statements, and “consistent” needs to mean “consistent in a strong sense.” It turns out that (*) is actually equivalent to a forcing axiom, and the proof is by showing that the (strong) consistency of certain theories gives rise to a corresponding notion of forcing producing a model of that theory. This is joint work with D. Asperó building upon earlier work of R. Jensen and (ultimately) Keisler’s “consistency properties.” Information: Email Ivan Ongay Valverde ahead of time for the zoom link. CUNY Set Theory Seminar Time: Friday, November 13, 3pm New York time (21:00 CET) Speaker: Diana Montoya, University of Vienna Title: Independence and uncountable cardinals Abstract: The classical concept of independence, first introduced by Fichtenholz and Kantorovic has been of interest within the study of combinatorics of the subsets of the real line. In particular the study of the cardinal characteristic i defined as the minimum size of a maximal independent family of subsets of ω. In the first part of the talk, we will review the basic theory, as well as the most important results regarding the independence number. We will also point out our construction of a poset P forcing a maximal independent family of minimal size which turns out to be indestructible after forcing with a countable support iteration of Sacks forcing. In the second part, we will talk about the generalization (or possible generalizations) of the concept of independence in the generalized Baire spaces, i.e. within the space κκ when κ is a regular uncountable cardinal and the new challenges this generalization entails. Moreover, for a specific version of generalized independence, we can have an analogous result to the one mentioned in the paragraph above. This is joint work with Vera Fischer. Information: The seminar will take place virtually. Please email Victoria Gitman (vgitman@nylogic.org) for the meeting id. Ghent-Leeds Virtual Logic Seminar Time: Thursday, November 12, 15:00 CET Speaker: Paul Levy, University of Birmingham Title: Broad infinity and generation principles Abstract: In set theory, Mahlo’s principle (also called “Ord is Mahlo”) states that the class of regular cardinals is stationary. Although it has many intuitive consequences, such as the existence of Grothendieck universes, Mahlo’s principle itself is not so intuitive. To resolve this situation, we give a new axiom scheme called Broad Infinity, resembling the axiom of Infinity. It is equivalent over ZFC to Mahlo’s principle, but arguably is more intuitive. We see that both Infinity and Broad Infinity give us principles for generating a set, a family or an ordinal. We also track the use of Choice and Excluded Middle in proving these results. Information: Please contact Paul Shafer in advance to participate. KGRC Research Seminar Time: Thursday, November 12, 15:00 CET Speaker: Hossein Lamei Ramandi, University of Toronto, Ontario, Canada Title: Can You Take Komjath’s Inaccessible Away? Abstract: In this talk we aim to compare Kurepa trees and Aronszajn trees. Moreover, we talk about the affect of large cardinal assumptions on this comparison. Using the the method of walks on ordinals, we will show it is consistent with ZFC that there is a Kurepa tree and every Kurepa tree contains a Souslin subtree, if there is an inaccessible cardinal. This is stronger than Komjath’s theorem which asserts the same consistency from two inaccessible cardinals. We will briefly sketch the ideas to prove that our large cardinal assumption is optimal. If time permits, we talk about the comparison of Kurepa trees and Aronszajn trees in the presence of no large cardinal. Information: Talk via zoom. Caltech Logic Seminar Time: Wednesday, November 11, 12:00 – 1:00pm Pacific time (22:00 CET) Speaker: Miroslav Zelený, Charles University Title: On the Luzin-Novikov theorem Abstract: We show that for every ordinal α∈[1,ω1)α∈[1,ω1), there is a closed set F⊂2ω×ωωF⊂2ω×ωω such that for every x∈2ωx∈2ω, the section {y∈ωω:(x,y)∈F}{y∈ωω:(x,y)∈F} is a two-point set and FF cannot be covered by countably many graphs B(n)⊂2ω×ωωB(n)⊂2ω×ωω of functions of the variable x∈2ωx∈2ω such that each B(n)B(n) is in the additive Borel class Σ0αΣα0. This rules out the possibility to have a quantitative version of the Luzin-Novikov theorem. The construction is a modification of the method of Harrington who invented it to show that there exists a countable Π01Π10 set in ωωωω containing a non-arithmetic singleton. By another application of the same method, we get closed sets excluding a quantitative version of the Saint Raymond theorem on Borel sets with σσ-compact sections. (Joint work with P. Holický) Information: See the seminar webpage. MOPA (Models of Peano Arithmetic), CUNY Time: Wednesday, November 11, 12pm New York time (18:00 CET) Speaker: Joel David Hamkins, Oxford University Title: Continuous models of arithmetic Abstract: Ali Enayat had asked whether there is a model of Peano arithmetic (PA) that can be represented as ⟨Q,⊕,⊗⟩, where ⊕ and ⊗ are continuous functions on the rationals Q. We prove, affirmatively, that indeed every countable model of PA has such a continuous presentation on the rationals. More generally, we investigate the topological spaces that arise as such topological models of arithmetic. The reals R, the reals in any finite dimension Rn, the long line and the Cantor space do not, and neither does any Suslin line; many other spaces do; the status of the Baire space is open. This is joint work with Ali Enayat, myself and Bartosz Wcisło. Information: The seminar will take place virtually. Please email Victoria Gitman (vgitman@nylogic.org) for the meeting id. Barcelona Set Theory Seminar Time: Wednesday, November 11, 16:00 CET Speaker: Gabriel Goldberg, UC Berkeley Title: On the uniqueness of elementary embeddings Abstract: This talk will explore the connection between Woodin’s HOD Conjecture and certain uniqueness properties of elementary embeddings of models of set theory, which will lead to some consequences of the failure of the HOD Conjecture reminiscent of the consequences of choiceless large cardinal axioms. Information: Online. If you wish to attend, please send an email to bagaria@ub.edu asking for the link. Münster research seminar on set theory Time: Wednesday, November 11, 15:15-1645 CET Speaker: Grigor Sargsyan, Gdansk Title: Determinacy, forcing axioms and inner models Abstract: We will exposit some recent results of the speaker and others that connect determinacy axioms, forcing axioms and inner models. A culmination of this work is a recent proof that the most liberally backgrounded construction of a model build from an extender sequence cannot be shown to converge in ZFC alone. In this construction, which is a type of Kc construction, one uses extenders that are certified by a Mostowski collapse. This result challenges common perceptions of the role of the model Kc in the inner model program. We will mention a specific consistency result showing that the failure of □ω3 and □(ω3) with 2ω=2ω1=ω2 and 2ω2=ω3 is weaker than a Woodin cardinal that is a limit of Woodin cardinals. Many people have been involved in this project. The work is heavily based on the efforts of Steel, Jensen, Woodin, Schindler, Mitchell, Schimmerling, Trang, Larson, Neeman, Zeman, Schlutzenberg, the speaker and many others. Information: contact rds@wwu.de ahead of time in order to participate. Hebrew University-Bar Ilan University Set Theory seminar Time: Wednesday, November 11, 14:00-16:00 Israel Time (13:00-15:00 CET) Speaker: Yair Hayut, Hebrew University Title: Higher Chang Conjecture Abstract: In this talk, I will focus on a joint work with Eskew. The main result is the consistency of (\kappa^+, \kappa) –>> (\lambda^+, \lambda). Information: Contact Menachem Magidor, Asaf Rinot or Omer Ben-Neria ahead of time for the zoom link. Helsinki Logic Seminar Time: Wednesday, November 11, 12:15 Helsinki Time (11:15 CET) Speaker: Jeffrey Schatz Title: Axiom Selection after Large Cardinals: Maximize and the Question of CH Abstract: There are two noted mathematical programs providing axioms extending the theory of Zermelo-Fraenkel set theory with large cardinals: the inner model program and the forcing axiom program. While these programs historically developed to serve different mathematical goals and ends, proponents of each have attempted to justify their preferred axiom candidate on the basis of its supposed maximization potential. Since ‘maximize’ proves central to the justification of ZFC+LCs itself, and shows up centrally in the current debate over how to best extend this theory, any attempt to resolve this debate will need to investigate the relationship between maximization notions and the candidates for a strong theory of sets. This talk will survey this project, discussing the history of ‘maximization’ considerations in set theory, introducing the main candidates for extending ZFC+LCs, and conclude by presenting recent results toward a resolution of these questions. Information: See the seminar webpage. Week 2-8 November CUNY Set Theory Seminar Time: Friday, November 6, 3pm New York time (21:00 CET) Speaker: Ernest Schimmerling, Carnegie Mellon University Title: Covering at limit cardinals of K Abstract: Theorem (Mitchell and Schimmerling, submitted for publication) Assume there is no transitive class model of ZFC with a Woodin cardinal. Let ν be a singular ordinal such that ν>ω_2 and cf(ν)<|ν|. Suppose ν is a regular cardinal in K. Then ν is a measurable cardinal in K. Moreover, if cf(ν)>ω, then oK(ν)≥cf(ν). I will say something intuitive and wildly incomplete but not misleading about the meaning of the theorem, how it is proved, and the history of results behind it. Information: The seminar will take place virtually. Please email Victoria Gitman (vgitman@nylogic.org) for the meeting id. Toronto Set Theory Seminar Time: Friday, November 6, 1.30pm Toronto time (19:30 CET) Speaker: Jonathan Schilhan, University of Vienna Title: Definable maximal families of reals in forcing extensions Abstract: Many types of combinatorial, algebraic or measure-theoretic families of reals, such as mad families, Hamel bases or Vitali sets, can be framed as maximal independent sets in analytic hypergraphs on Polish spaces. Their existence is guaranteed by the Axiom of Choice, but low-projective witnesses ($\mathbf{Delta}^1_2$) were only known to exist in general in models of the form$L[a]$for a real$a$. Our main result is that, after a countable support iteration of Sacks forcing or for example splitting forcing (a less known forcing adding splitting reals) over L, every analytic hypergraph on a Polish space has a$\mathbf{\Delta}^1_2$maximal independent set. As a corollary, this solves an open problem of Brendle, Fischer and Khomskii by providing a model with a$\Pi^1_1$mif (maximal independent family) while the independence number$\mathfrak{i}$is bigger than$\aleph_1$. Information: Email Ivan Ongay Valverde ahead of time for the zoom link. Ghent-Leeds Virtual Logic Seminar Time: Thursday, November 5, 15:00 CET Speaker: Sandra Müller, University of Vienna Title: Determinacy and inner models Abstract: The study of inner models was initiated by Gödel’s analysis of the constructible universe L. Later, it became necessary to study canonical inner models with large cardinals, e.g. measurable cardinals, strong cardinals or Woodin cardinals, which were introduced by Jensen, Mitchell, Steel, and others. Around the same time, the study of infinite two-player games was driven forward by Martin’s proof of analytic determinacy from a measurable cardinal, Borel determinacy from ZFC, and Martin and Steel’s proof of levels of projective determinacy from Woodin cardinals with a measurable cardinal on top. First Woodin and later Neeman improved the result in the projective hierarchy by showing that in fact the existence of a countable iterable model, a mouse, with Woodin cardinals and a top measure suffices to prove determinacy in the projective hierarchy. This opened up the possibility for an optimal result stating the equivalence between local determinacy hypotheses and the existence of mice in the projective hierarchy, just like the equivalence of analytic determinacy and the existence of X^# for every real X which was shown by Martin and Harrington in the 70’s. The existence of mice with Woodin cardinals and a top measure from levels of projective determinacy was shown by Woodin in the 90’s. Together with his earlier and Neeman’s results this establishes a tight connection between descriptive set theory in the projective hierarchy and inner model theory. In this talk, we will outline some of the main results connecting determinacy hypotheses with the existence of mice with large cardinals and discuss a number of more recent results in this area, some of which are joint work with Juan Aguilera. Information: Please contact Paul Shafer in advance to participate. KGRC Research Seminar Time: Thursday, November 5, 15:00 CET Speaker: Omer Ben-Neria, Hebrew University Title: On Continuous Tree-Like Scales and related properties of Internally Approachable structures Abstract: In his PhD thesis, Luis Pereira isolated and developed several principles of singular cardinals that emerge from Shelah’s PCF theory; principles which involve properties of scales, such as the inexistence of continuous Tree-Like scales, and properties of internally approachable structures such as the Approachable Free Subset Property. In the talk, we will discuss these principles and their relations, and present new results from a joint work with Dominik Adolf concerning their consistency and consistency strength. Information: Talk via zoom. CUNY Logic Seminar (MOPA) Time: Wednesday, November 4, 3pm New York time (21:00 CET) Speaker: Victoria Gitman, CUNY Title: A model of second-order arithmetic satisfying AC but not DC: Part II Abstract: One of the strongest second-order arithmetic systems is full second-order arithmetic Z2 which asserts that every second-order formula (with any number of set quantifiers) defines a set. We can augment Z2 with choice principles such as the choice scheme and the dependent choice scheme. The Σ1n-choice scheme asserts for every Σ1n-formula φ(n,X) that if for every n, there is a set Xwitnessing φ(n,X), then there is a single set Z whose n-th slice Zn is a witness for φ(n,X). The Σ1n-dependent choice scheme asserts that every Σ1n-relation φ(X,Y) without terminal nodes has an infinite branch: there is a set Z such that φ(Zn,Zn+1) holds for all n. The system Z2 proves the Σ12-choice scheme and the Σ12-dependent choice scheme. The independence of Π12-choice scheme from Z2 follows by taking a model of Z2 whose sets are the reals of the Feferman-Levy model of ZF in which every ℵLn is countable and ℵLω is the first uncountable cardinal. We construct a model of ZF+ACω whose reals give a model of Z2 together with the full choice scheme in which Π12-dependent choice fails. This result was first proved by Kanovei in 1979 and published in Russian. It was rediscovered by Sy Friedman and myself with a slightly simplified proof. Information: The seminar will take place virtually. Please email Victoria Gitman (vgitman@nylogic.org) for the meeting id. Caltech Logic Seminar Time: Wednesday, November 4, 12:00 – 1:00pm Pacific time (22:00 CEST) Speaker: Anush Tserunyan, McGill University Title: A backward ergodic theorem and its forward implications Abstract: In the classical pointwise ergodic theorem for a probability measure preserving (pmp) transformation TT, one takes averages of a given integrable function over the intervals {x,T(x),T2(x),…,Tn(x)}{x,T(x),T2(x),…,Tn(x)} in front of the point xx. We prove a “backward” ergodic theorem for a countable-to-one pmp TT, where the averages are taken over subtrees of the graph of TT that are rooted at xx and lie behind xx (in the direction of T−1T−1). Surprisingly, this theorem yields forward ergodic theorems for countable groups, in particular, for pmp actions of finitely generated groups, where the averages are taken along set-theoretic (but backward) trees on the generating set. This strengthens Bufetov’s theorem from 2000, which was the leading result in this vein. This is joint work with Jenna Zomback. Information: Zoom, please see the seminar webpage for login information. Bristol Logic and Set Theory Seminar/Oxford Set Theory Seminar Time: Wednesday, November 4, 16:00-17:30 UK time (17:00-18:30 CET) Speaker: Mirna Džamonja, CNRS & Panthéon Sorbonne, Paris and Czech Academy of Sciences, Prague Title: On Wide Aronszajn Trees Abstract: Aronszajn trees are a staple of set theory, but there are applications where the requirement of all levels being countable is of no importance. This is the case in set-theoretic model theory, where trees of height and size ω1 but with no uncountable branches play an important role by being clocks of Ehrenfeucht–Fraïssé games that measure similarity of model of size ℵ1. We call such trees wide Aronszajn. In this context one can also compare trees T and T’ by saying that T weakly embeds into T’ if there is a function f that map T into T’ while preserving the strict order <_T. This order translates into the comparison of winning strategies for the isomorphism player, where any winning strategy for T’ translates into a winning strategy for T’. Hence it is natural to ask if there is a largest such tree, or as we would say, a universal tree for the class of wood Aronszajn trees with weak embeddings. It was known that there is no such a tree under CH, but in 1994 Mekler and Väänanen conjectured that there would be under MA(ω1). In our upcoming JSL paper with Saharon Shelah we prove that this is not the case: under MA(ω1) there is no universal wide Aronszajn tree. The talk will discuss that paper. The paper is available on the arxiv and on line at JSL in the preproof version doi: 10.1017/jsl.2020.42. Information: For the Zoom access code, contact Samuel Adam-Day me@samadamday.com. Link: https://zoom.us/j/96803195711 (open 30 minutes before) Paris-Lyon Séminaire de Logique Time: Wednesday, November 4, 16:00-17:00 CET Speaker: Jeffrey Bergfalk, University of Vienna Title: Set theory and strong homology: an overview Abstract: Motivated by several recent advances, we will provide a research history of the main set-theoretic problems arising in the study of strong homology. We will presume no knowledge, in our audience, of the latter. The aforementioned advances close out a second major phase of research in this area, leaving just a few conspicuous last “first questions,” and our aim is to provide some context for engaging them. This research centers on multidimensional combinatorial phenomena generalizing the classical theme of \emph{nontrivial coherent families indexed by ωω}; its progress has involved an intriguing mix of classical (forcing axioms, iterations of large cardinal length) and novel (higher-dimensional Δ-systems, simplicial combinatorics) set-theoretic techniques. Information: Zoom ID: 824 8220 9628; sign up for the email list, or contacter silvain.rideau@imj-prg.fr, for the password in advance. Hebrew University-Bar Ilan University Set Theory seminar Time: Wednesday, November 4, 14:00-16:00 Israel Time (13:00-15:00 CET) Speaker: Yair Hayut, Hebrew University Title: Higher Chang Conjecture Abstract: In this talk I will present some results regarding the consistency strength of Higher variants of Chang’s Conjecture. I will start with the classical result by Silver of Chang’s Conjecture from$\omega_1$-Erdos cardinal. Then, I will give an upper bound for the consistency strength of$(\aleph_{\omega+1}, \aleph_{\omega}) –>>(\aleph_1, \aleph_0)$and$(\aleph_4, \aleph_3) –>> (\aleph_2, \aleph_1)$(joint with Eskew)from supercompactness assumptions. If time permits, I will describe the strategy for obtaining a global result:(\kappa^+,\kappa) –>> (\mu^+, \mu)for all regular$\kappa$, and$\mu < \kappa$, and talk about the barriers that we face when trying to extend this result. Information: Contact Menachem Magidor, Asaf Rinot or Omer Ben-Neria ahead of time for the zoom link. Helsinki Logic Seminar Time: Wednesday, November 4, 12:15 Helsinki Time (11:15 CET) Speaker: Ralf Schindler, University of Münster Title: Martin’s Maximum^++ implies the P_max axiom (*). Abstract: Forcing axioms spell out the dictum that if a statement can be forced, then it is already true. The P_max axiom (*) goes beyond that by claiming that if a statement is consistent, then it is already true. Here, the statement in question needs to come from a resticted class of statements, and “consistent” needs to mean “consistent in a strong sense.” It turns out that (*) is actually equivalent to a forcing axiom, and the proof is by showing that the (strong) consistency of certain theories gives rise to a corresponding notion of forcing producing a model of that theory. This is joint work with D. Asperó building upon earlier work of R. Jensen and (ultimately) Keisler’s “consistency properties.” Information: Zoom, see the seminar webpage for login information. University of Wisconsin Logic Seminar Time: Tuesday, November 3, 3:00pm CST (22:00 CET) Speaker: Vera Fischer, University of Vienna Title: Independent families in the countable and the uncountable Abstract: Independent families on ω are families of infinite sets of integers with the property that for any two disjoint finite subfamilies A and B, the set ⋂ A \ ⋃ B is infinite. Of particular interest are the sets of the possible cardinalities of maximal independent families, which we refer to as the spectrum of independence. Even though we do have the tools to control the spectrum of independence at ω (at least to a large extent), there are many relevant questions regarding higher counterparts of independence in generalized Baire spaces, which remain wide open. Information: Zoom Meeting ID: 970 9130 0913, Passcode: 926119 Week 26 October – 1 November Toronto Set Theory Seminar Time: Friday, October 30, 1.30pm Toronto time (18:30pm CET) Speaker: Spencer Unger Title: Reflection properties at successors of singulars Abstract: We survey some recent advances in techniques for getting reflection properties at successors of singulars with particular attention to the tree property and stationary reflection Information: Email Ivan Ongay Valverde ahead of time for the zoom link. CUNY Set Theory Seminar Time: Friday, October 30, 1pm New York time (18:00 CET) Speaker: Benedikt Löwe, University of Hamburg Title: Analysis in higher analogues of the reals Abstract: The real numbers are up to isomorphism the only completely ordered field with a countable dense subset. We consider non-Archimedean ordered fields whose smallest dense subset has cardinality kappa and investigate whether anything resembling ordinary analysis works on these fields. In particular, we look at generalisations of the intermediate value theorem and the Bolzano-Weierstrass theorem, and realise that there is some mathematical tension between these theorems: the intermediate value theorem requires some saturation whereas Bolzano-Weierstrass fails if the field is saturated. We consider weakenings of Bolzano-Weierstrass compatible with saturation and realise that these are equivalent to the weak compactness of kappa. This is joint work with Merlin Carl, Lorenzo Galeotti, and Aymane Hanafi. Information: The seminar will take place virtually. Please email Victoria Gitman (vgitman@nylogic.org) for the meeting id. KGRC Research Seminar Time: Thursday, October 29, 15:00 CET Speaker: Philipp Lücke, University of Barcelona, Spain Title: Structural reflection and shrewd cardinals Abstract: In my talk, I want to present work dealing with the interplay between extensions of the Downward Löwenheim–Skolem Theorem to strong logics, large cardinal axioms and set-theoretic reflection principles, focussing on the characterization of large cardinal notions through model- and set-theoretic reflection properties. The work of Bagaria and his collaborators shows that various important objects in the middle and upper reaches of the large cardinal hierarchy can be characterized through principles of structural reflection. I will discuss recent results dealing with possible characterizations of notions from the lower part of this hierarchy through the principle SR−SR−, introduced by Bagaria and Väänänen. These results show that the principle SR−SR− is closely connected to the notion of shrewd cardinals, introduced by Rathjen in a proof-theoretic context, and embedding characterizations of these cardinals that resembles Magidor’s classical characterization of supercompactness. Information: Talk via zoom. Caltech Logic Seminar Time: Wednesday, October 28, 12:00 – 1:00pm Pacific time (21:00 CET) Speaker: Tyler Arant, UCLA Title: From recursively presented metric spaces to recursive Polish spaces Abstract: Beyond the Baire space, recursively presented metric spaces are structures which serve as a setting for effective descriptive set theory. Motivated by the classical distinction between a complete separable metric space and its corresponding Polish space topological structure, we will explore the notions and issues involved in moving from a recursively presented metric space to its effective Polish space structure. We will survey different approaches to these issues, in particular work by Moschovakis on recursive frames and work by Louveau on effective topology, and prove some original results which clarify some foundational problems in the area. Information: Online talk https://caltech.zoom.us/j/89937794322?pwd=d2kzZEkrSWo0QW93RWdJMnVucE83Zz09 CUNY Logic Seminar (MOPA) Time: Wednesday, October 28, 3pm New York time (20:00 CET) Speaker: Victoria Gitman, CUNY Title: A model of second-order arithmetic satisfying AC but not DC Abstract: One of the strongest second-order arithmetic systems is full second-order arithmetic Z2 which asserts that every second-order formula (with any number of set quantifiers) defines a set. We can augment Z2 with choice principles such as the choice scheme and the dependent choice scheme. The Σ1n-choice scheme asserts for every Σ1n-formula φ(n,X) that if for every n, there is a set Xwitnessing φ(n,X), then there is a single set Z whose n-th slice Zn is a witness for φ(n,X). The Σ1n-dependent choice scheme asserts that every Σ1n-relation φ(X,Y) without terminal nodes has an infinite branch: there is a set Z such that φ(Zn,Zn+1) holds for all n. The system Z2 proves the Σ12-choice scheme and the Σ12-dependent choice scheme. The independence of Π12-choice scheme from Z2 follows by taking a model of Z2 whose sets are the reals of the Feferman-Levy model of ZF in which every ℵLn is countable and ℵLω is the first uncountable cardinal. We construct a model of ZF+ACω whose reals give a model of Z2 together with the full choice scheme in which Π12-dependent choice fails. This result was first proved by Kanovei in 1979 and published in Russian. It was rediscovered by Sy Friedman and myself with a slightly simplified proof. Information: The seminar will take place virtually. Please email Victoria Gitman (vgitman@nylogic.org) for the meeting id. Paris-Lyon Séminaire de Logique Time: Wednesday, October 28, 16:00-17:00 CET Speaker: Marcin Sabok, Mc Gill University, Montreal Title: Hyperfiniteness at Gromov boundaries Abstract: I will discuss recent results establishing hyperfiniteness of equivalence relations induced by actions on Gromov boundaries of various hyperbolic spaces. This includes boundary actions of hyperbolic groups (joint work with T. Marquis) and actions of the mapping class group on boundaries of the arc graph and the curve graph (joint work with P. Przytycki) Information: Zoom ID: 824 8220 9628; sign up for the email list, or contacter silvain.rideau@imj-prg.fr, for the password in advance. Barcelona Set Theory Seminar Time: Wednesday, October 28, 16:00 CET Speaker: Joel David Hamkins, University of Oxford Title: A new proof of the Barwise extension theorem, and the universal finite sequence Abstract: The Barwise extension theorem, asserting that every countable model of ZF set theory admits an end-extension to a model of ZFC+V=L, is both a technical culmination of the pioneering methods of Barwise in admissible set theory and infinitary logic and also one of those rare mathematical theorems that is saturated with philosophical significance. In this talk, I shall describe a new proof of the theorem that omits any need for infinitary logic and relies instead only on classical methods of descriptive set theory. This proof leads directly to the universal finite sequence, a Sigma_1-definable finite sequence, which can be extended arbitrarily as desired in suitable end-extensions of the universe. The result has strong consequences for the nature of set-theoretic potentialism. This work is joint with Kameryn J. Williams. Information: Online. If you wish to attend, please send an email to bagaria@ub.edu asking for the link. Bar-Ilan-Jerusalem Set Theory Seminar Time: Wednesday, October 28, 14:00-16:00 Israel Time (13:00-15:00 CET) Speaker: Omer Ben Neria, Hebrew University Title: On Continuous Tree-Like Scales and related properties of Internally Approachable structures Abstract: In his PhD thesis, Luis Pereira isolated and developed several principles of singular cardinals that emerge from Shelah’s PCF theory; principles which involve properties of scales, such as the inexistence of continuous Tree-Like scales, and properties of internally approachable structures such as the Approachable Free Subset Property. In the talk, we will discuss these principles and their relations, and present new results from a joint work with Dominik Adolf concerning their consistency and consistency strength. Information: Contact Menachem Magidor, Asaf Rinot or Omer Ben-Neria ahead of time for the zoom link. Helsinki Logic Seminar Time: Wednesday, October 28, 12:15 Helsinki Time (11:15 CET) Speaker: Yurii Khomskii, Amsterdam University College and Universität Hamburg Title: Bounded Symbiosis and Upwards Reflection Abstract: In [1], Bagaria and Väänänen developed a framework for studying the large cardinal strength of Löwenheim-Skolem theorems of strong logics using the notion of Symbiosis (originally introduced by Väänänen in [2]). Symbiosis provides a way of relating model theoretic properties of strong logics to definability in set theory. We continue the systematic investigation of Symbiosis and apply it to upwards Löwenheim-Skolem theorems and upwards reflection principles. To achieve this, the notion of Symbiosis is adapted to what we call “Bounded Symbiosis”. As an application, we provide some upper and lower bounds for the large cardinal strength of upwards Löwenheim-Skolem principles of second order logic. This is joint work with Lorenzo Galeotti and Jouko Väänänen. [1] Joan Bagaria and Jouko Väänänen, “On the Symbiosis Between Model-Theoretic and Set-Theoretic Properties of Large Cardinals”, Journal of Symbolic Logic 81 (2) P. 584-604 [2] Jouko Väänänen, “Abstract logic and set theory. I. Definability.” In Logic Colloquium ’78 (Mons, 1978), volume 97 of Stud. Logic Foundations Math., pages 391–421. North-Holland, Amsterdam-New York, 1979. Information: Zoom Meeting ID: 476 210 6037 Passcode: HLGrp Week 19-25 October CUNY Set Theory Seminar Time: Friday, October 23, 15:00 New York time (21:00 CEST) Speaker: Gabriel Goldberg, University of Berkeley Title: Ultrapowers and the approximation property Abstract: Countably complete ultrafilters are the combinatorial manifestation of strong large cardinal axioms, but many of their basic properties are undecidable no matter the large cardinal axioms one is willing to adopt. The Ultrapower Axiom (UA) is a set theoretic principle that permits the development of a much clearer picture of countably complete ultrafilters and, consequently, the large cardinals from which they derive. It is not known whether UA is (relatively) consistent with very large cardinals, but assuming there is a canonical inner model with a supercompact cardinal, the answer should be yes: this inner model should satisfy UA and yet inherit all large cardinals present in the universe of sets. The predicted resemblance between the large cardinal structure of this model and that of the universe itself is so extreme as to suggest that certain consequences of UA must in fact be provable outright from large cardinal axioms. While the inner model theory of supercompact cardinals remains a major open problem, this talk will describe a technique that already permits a number of consequences of UA to be replicated from large cardinals alone. Still, the technique rests on the existence of inner models that absorb large cardinals, but instead of building canonical inner models, one takes ultrapowers. Information: The seminar will take place virtually. Please email Victoria Gitman (vgitman@nylogic.org) for the meeting id. Kurt Gödel Research Center Seminar (organised by Ben Miller) Time: Thursday, October 22, 15:00 CEST Speaker: Philipp Schlicht, KGRC Title: Tree forcings, sharps and absoluteness Abstract: In joint results with Fabiana Castiblanco from 2018, we showed that several classical tree forcings preserve sharps for reals and levels of projective determinacy, and studied their impact on definable equivalence relations (in particular, the question whether they add equivalence classes to thin projective equivalence relations). I will discuss these results and natural open problems on tree forcings and absoluteness that arise from them. Information: Talk via zoom. Bar-Ilan-Jerusalem Set Theory Seminar Time: Thursday, October 22, 10:00am Israel Time (10:00 CEST) Speaker: Gabriel Fernandes, Bar-Ilan University Title: Local club condensation in extender models Abstract: Local club condensation is an abstraction of the condensation properties of the constructible hierarchy. We will prove that for extender models that are countably iterable, given a cardinal kappa, the J_alpha^{E} hierarchy witnesses local club condensation in the interval From the above and the equivalence between subcompact cardinals and square, due to Schimmerling and Zeman, it follows that in such extender models \square_kappa holds iff the J_alpha^{E} hierarchy witnesses that local club condensation holds in the interval (kappa^+,kappa^++). Information: Contact Menachem Magidor, Asaf Rinot or Omer Ben-Neria ahead of time for the zoom link. Caltech Logic Seminar Time: Wednesday, October 21, 12:00 – 1:00pm Pacific time (21:00 CEST) Speaker: Ronnie Chen, UIUC Title: Borel and analytic sets in locales Abstract: A locale is, informally, a topological space without an underlying set of points, with only an abstract lattice of “open sets”. Various results in the literature suggest that locale theory behaves in many ways like a generalization of descriptive set theory with countability restrictions removed. This talk will introduce locale theory from a descriptive set-theoretic point of view, and survey some known and new results which are common to both contexts. In particular, we will introduce the “∞∞-Borel hierarchy” of a locale, and sketch the existence of “σσ-analytic, non-∞∞-Borel sets”. Information: Online talk https://caltech.zoom.us/j/84553965424?pwd=NHRyclZlZ1cydjZBNWkvTlF5QVFmdz09 Paris-Lyon Séminaire de Logique Time: Wednesday, October 21, 17:00-18:00 CEST Speaker: Dima Sinapova, University of Illinois at Chicago Title: Iteration, reflection, and singular cardinals Abstract: There is an inherent tension between stationary reflection and the failure of the singular cardinal hypothesis (SCH). The former is a compactness type principle that follows from large cardinals. Compactness is the phenomenon where if a certain property holds for every smaller substructure of an object, then it holds for the entire object. In contrast, failure of SCH is an instance of incompactness. Two classical results of Magidor are: (1) from large cardinals it is consistent to have reflection at ℵω+1, and (2) from large cardinals it is consistent to have the failure of SCH at ℵω. As these principles are at odds with each other, the natural question is whether we can have both. We show the answer is yes. We describe a Prikry style iteration, and use it to force stationary reflection in the presence of not SCH. Then we obtain this situation at ℵω by interleaving collapses. This is joint work with Alejandro Poveda and Assaf Rinot. Information: Zoom ID: 824 8220 9628; sign up for the email list, or contacter silvain.rideau@imj-prg.fr, for the password in advance. Bristol Logic and Set Theory Seminar/Oxford Set Theory Seminar Time: Wednesday, October 21, 16:00-17:30 UK time (17:00-18:30 CEST) Speaker: Andreas Blass, University of Michigan Title: tba Abstract: tba Information: For the Zoom access code, contact Samuel Adam-Day me@samadamday.com. Link: https://zoom.us/j/96803195711 (open 30 minutes before) Barcelona Set Theory Seminar Time: Wednesday, October 21, 16:00 CEST Speaker: Mirna Dzamonja, Université Panthéon-Sorbonne, Paris and Czech Academy of Science Title: On wide Aronszajn trees Abstract: Aronszajn trees are a staple of set theory, but there are applications where the requirement of all levels being countable is of no importance. This is the case in set-theoretic model theory, where trees of height and size ω1 but with no uncountable branches play an important role by being clocks of Ehrenfeucht-Fraïssé games that measure similarity of models of size א1. We call such trees wide Aronszajn. In this context one can also compare trees T and T’ by saying that T weakly embeds into T’ if there is a function f that maps T into T’ while preserving the strict order <T. This order translates into the comparison of winning strategies for the isomorphism player, where any winning strategy for T’ translates into a winning strategy for T’. Hence it is natural to ask if there is a largest such tree, or as we would say, a universal tree for the class of wide Aronszajn trees with weak embeddings. It was known that there is no such a tree under CH, but in 1994 Mekler and Väänanen conjectured that there would be under MA(ω1). In our upcoming JSL paper with Saharon Shelah we prove that this is not the case: under MA(ω1) there is no universal wide Aronszajn tree. The talk will discuss that paper, which is available on the arxiv and online at JSL in the preproof version DOI: 10.1017/jsl.2020.42. Information: Online. If you wish to attend, please send an email to bagaria@ub.edu asking for the link. Logic Seminar, Carnegie Mellon University Time: Tuesday, October 20, 3:30 – 4:30pm Eastern Daylight Time (21:00 CEST) Speaker: Garrett Ervin, Carnegie Mellon University Title: Maximally splitting pruned trees in locally finite graphs Abstract: Let G be an infinite but locally finite connected graph, and let x be a vertex in G. We prove that G contains a pruned tree T rooted at x that splits as early and as often as possible. While this tree isn’t completely canonical, its levels are. The main point of the talk will be that the phrases “as early” and “as often” here are actually meaningful. Toward seeing this, we’ll show that in many graphs there is actually a maximal set of minimal boundary containing a given vertex x. We call this set the diamond of x, and denote it D(x). Diamonds can be used to prove many of the classical duality results in graph theory, including Hall’s matching theorem, König’s lemma, and Menger’s theorem. We’ll use diamonds to establish the existence of the tree T. Information: Zoom link https://cmu.zoom.us/j/621951121, meeting ID: 621 951 121 Week 12-18 October CUNY Set Theory Seminar Time: Friday, October 16, 15:00 New York time (21:00 CEST) Speaker: Richard Matthews, University of Leeds Title: Taking Reinhardt’s Power Away Abstract: Many large cardinals can be defined through elementary embeddings from the set-theoretic universe to some inner model, with the guiding principle being that the closer the inner model is to the universe the stronger the resulting theory. Under ZFC, the Kunen Inconsistency places a hard limit on how close this can be. One is then naturally led to the question of what theory is necessary to derive this inconsistency with the primary focus having historically been embeddings in ZF without Choice. In this talk we take a different approach to weakening the required theory, which is to study elementary embeddings from the universe into itself in ZFC without Power Set. We shall see that I1, one of the largest large cardinal axioms not known to be inconsistent with ZFC, gives an upper bound to the naive version of this question. However, under reasonable assumptions, we can reobtain this inconsistency in our weaker theory. Information: The seminar will take place virtually. Please email Victoria Gitman (vgitman@nylogic.org) for the meeting id. Kurt Gödel Research Center Seminar Time: Thursday, October 15, 15:00 CEST Speaker: Ziemowit Kostana, University of Warsaw Title: Fraïssé theory, and forcing absoluteness of rigidity for linear orders Abstract: During the talk I would like to introduce the theory of Cohen-like first-order structures. These are countable or uncountable structures which are “generic” much in the same sense as the Cohen reals. They can be added to the universe of set theory using finite or, say, countable conditions and exhibit different properties. I will focus on the construction of a rigid linear order, whose rigidity is absolute for ccc extensions. Information: Talk via zoom. Caltech Logic Seminar Time: Wednesday, October 14, 12:00 – 1:00pm Pacific time (21:00 CEST) Speaker: Jindřich Zapletal, University of Florida Title: Geometric Set Theory Abstract: The talk will be an outline of the book we published with Paul Larson recently. In particular, I will show how amalgamation problems in algebra naturally appear in consistency results for the choiceless set theory ZF+DCZF+DC, and how they can be stratified from a set-theoretic point of view. Information: Online talk https://caltech.zoom.us/j/88977992358?pwd=UjJjdFhZZEVuMjFmZ0VLZFd5dGhqQT09 Illinois Logic Group Logic Seminar Time: Wednesday, October 14, 1pm US Central Daylight Time (20:00 CEST) Speaker: Jenna Zomback, UIUC Math Title: A backward ergodic theorem and its forward implications Abstract: A pointwise ergodic theorem for the action of a transformation T on a probability space equates the global property of ergodicity of the transformation to its pointwise combinatorics. Our main result is a backward (in the direction of T−1) ergodic theorem for countable-to-one probability measure preserving (pmp) transformations T. We discuss various examples of such transformations, including the shift map on Markov chains, which yields a new (forward) pointwise ergodic theorem for pmp actions of finitely generated countable groups, as well as one for the (non-pmp) actions of free groups on their boundary. This is joint work with Anush Tserunyan. Information: The seminar will take place remotely. Paris-Lyon Séminaire de Logique Time: Wednesday, October 14, 16:00-17:15 CEST Speaker: Tomás Ibarlucía, University of Paris Title: Automorphism groups acting on Hilbert spaces without almost invariant vectors Abstract: We will discuss, first, how to construct automorphisms of countable/separable saturated models (and, more interestingly, pairs of automorphisms) that act “very freely” on the structure, in a sense given by stability theory. Then we will see how to use this to show that automorphism groups of aleph_0-categorical metric structures have Kazhdan’s Property (T), which roughly means that their unitary actions on Hilbert spaces do not have almost invariant vectors in non-trivial ways. Information: Zoom ID: 824 8220 9628; sign up for the email list, or contacter silvain.rideau@imj-prg.fr, for the password in advance. Helsinki Logic Seminar Time: Wednesday, October 14, 12:15 Helsinki Time (11:15 CEST) Speaker: Miguel Moreno, University of Vienna Title: Filter Reflection and Generalised Descriptive Set Theory Abstract: Filter reflection is an abstract version of stationary reflection motivated from many results in generalised descriptive set theory. In this talk we will define filter reflection and different avatars of it. We will focus on its consequences in generalised descriptive set theory. We will also discuss how to force filter reflection and how to force the failure of filter reflection. This is a joint work with Gabriel Fernandes and Assaf Rinot. Information: Zoom meeting ID: 684 3587 2772, Passcode: 110592 Logic Seminar, Carnegie Mellon University Time: Tuesday, October 13, 3:30 – 4:30pm Eastern Daylight Time (21:00 CEST) Speaker: Farmer Schlutzenberg, University of Muenster Title: Definability of elementary embeddings beyond the axiom of choice Abstract: Large cardinal axioms play a central role in set theory and our understanding of relative consistency strength. Such axioms are typically exhibited by elementary (truth-preserving) embeddings of the form$j:V\to M$, where$V$is the universe of all sets and$M$a sub-universe. Demanding greater resemblance between$V$and$M$tends to lead to stronger large cardinal notions. Kunen famously showed that the strongest possible resemblance —$V=M$— is inconsistent with the assumption that$V$models ZFC (including Choice). Suzuki showed later that just assuming$V$models ZF, there can be no elementary$j:V\to V$which is definable from parameters. Now one can ask whether (or to what extent) Suzuki’s result still holds if we weaken ZF. In this talk we will discuss various results — and counterexamples — along these lines. Some of the results I’ll mention were first due to Gabe Goldberg. (References: arXiv 2006.10574, 2006.01103) Information: Zoom link https://cmu.zoom.us/j/621951121, meeting ID: 621 951 121 Week 5-11 October CUNY Set Theory Seminar Time: Friday, October 9, 11:00 New York time (17:00 CEST) Speaker: Heike Mildenberger, Albert-Ludwigs-Universität Freiburg Title: Forcing with variants of Miller trees Abstract: Guzmán and Kalajdzievski introduced a variant of Miller forcing P(F) that diagonalises a given filter F over ω and has Axiom A. We investigate the effect of P(F) for particularly chosen Canjar filters F. This is joint work with Christian Bräuninger. Information: The seminar will take place virtually. Please email Victoria Gitman (vgitman@nylogic.org) for the meeting id. Kurt Gödel Research Center Seminar Time: Thursday, October 8, 15:00 CEST Speaker: Colin Jahel, Claude Bernard University Lyon 1 Title: Actions of automorphism groups of Fraïssé limits on the space of linear orderings Abstract: In 2005, Kechris, Pestov and Todorčević 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: Talk via zoom. Caltech Logic Seminar Time: Wednesday, October 7, 12:00 – 1:00pm Pacific time (21:00 CEST) Speaker: Riley Thornton, UCLA Title: Factor of i.i.d. processes and Cayley diagrams Abstract: A Cayley diagram for a Cayley graph G=Cay(Γ,E) is an edge labelling of G with generators from E so that a path is labelled with a relation in Γ if and only if it is a cycle. I will show how Aut(G)-f.i.i.d. Cayley diagrams can be used to lift Γ-f.i.i.d. solutions of local combinatorial problems to Aut⁡(G)-f.i.i.d. solutions. And, I will investigate which graphs admit Aut⁡(G)-f.i.i.d. Cayley diagrams, answering a question of Weilacher on approximate chromatic numbers in the process. Information: Online talk https://caltech.zoom.us/j/99296122790?pwd=bUN4RS94RVYrTEtGTGhqTHRJbm9nZz09 Helsinki Logic Seminar Time: Wednesday, October 7, 12:15 Helsinki Time (11:15 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 -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 a byproduct of the groundbreaking result of Schindler and Asperò showing that Woodin’s axiom (*) can be forced by a stationary set preserving forcing. Information: Zoom link https://helsinki.zoom.us/j/61883091956?pwd=Yzl2NTE5c1pZdGhVR1d1N2x4Q1dwUT09 Logic Seminar, Carnegie Mellon University Time: Tuesday, October 6, 3:30 – 4:30pm Eastern Daylight Time (21:00 CEST) Speaker: Dima Sinapova, University of Illinois at Chicago Title: Iteration, reflection, and Prikry forcing Abstract: There is an inherent tension between stationary reflection and the failure of the singular cardinal hypothesis (SCH). The former is a compactness type principle that follows from large cardinals. Compactness is the phenomenon where if a certain property holds for every smaller substructure of an object, then it holds for the entire object. In contrast, failure of SCH is an instance of incompactness. It is 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$\aleph_\omega$, combining two classical results of Magidor. This is joint work with Alejandro Poveda and Assaf Rinot. Organizers’ note: This is a more general talk to be immediately followed by a more technical talk. Information: Zoom link https://cmu.zoom.us/j/621951121, meeting ID: 621 951 121 Week 27 September – 4 October CUNY Set Theory Seminar Time: Friday, October 2, 11:00 New York time (17:00 CEST) Speaker: David Aspero, University of East Anglia Title: Martin’s Maximum^++ implies the P_max axiom (*) (Part 2) Abstract: This will be a sequel to Ralf Schindler’s talk on 9/25. My plan is to give a reasonably detailed account of the proof of the result in the title. Information: The seminar will take place virtually. Please email Victoria Gitman (vgitman@nylogic.org) for the meeting id. Kurt Gödel Research Center Seminar Time: Thursday, October 1, 15:00 CEST Speaker: Jonathan Schilhan, KGRC Title: Definability of maximal families of reals in forcing extensions Abstract: The definability of combinatorial families of reals, such as mad families, has a long history. The constructible universe L is a good model for definability, for its nice structural properties. On the other hand, as a rule of thumb, the universe can’t be too far from L if it allows for low projective witnesses of such families. Thus it makes sense to look at forcing extensions of L. We show that after a countable support iteration of Sacks forcing or splitting forcing (or many others) over L, every analytic hypergraph on a Polish space has a Δ12 maximal independent set. This means that in the models obtained by these iterations, most types of interesting “maximal families” have Δ12 witnesses. In particular, this solves an open problem of Brendle, Fischer and Khomskii by providing a model with a Π11 mif (maximal independent family) while the independence number 𝔦i is bigger than ℵ. Information: Talk via zoom. CUNY Logic Seminar (MOPA) Time: Wednesday, September 30, 2pm New York time (20:00 CEST) – note the time Speaker: Leszek Kolodziejczyk, University of Warsaw Title: Ramsey’s Theorem over RCA^*_0: Part 2 Abstract: The usual base theory used in reverse mathematics, RCA0, is the fragment of second-order arithmetic axiomatized by Δ01 comprehension and Σ01 induction. The weaker base theory RCA∗0 is obtained by replacing Σ01 induction with Δ01 induction (and adding the well-known axiom exp in order to ensure totality of the exponential function). In first-order terms, RCA0 is conservative over IΣ1 and RCA∗0 is conservative over BΣ1+exp. Some of the most interesting open problems in reverse mathematics concern the first-order strength of statements from Ramsey Theory, in particular Ramsey’s Theorem for pairs and two colours. In this talk, I will discuss joint work with Kasia Kowalik, Tin Lok Wong, and Keita Yokoyama concerning the strength of Ramsey’s Theorem over RCA∗0.Given standard natural numbers n,k≥2, let RTnk stand for Ramsey’s Theorem for k-colourings of n-tuples. We first show that assuming the failure of Σ01 induction, RTnk is equivalent to its own relativization to an arbitrary Σ01-definable cut. Using this, we give a complete axiomatization of the first-order consequences of RCA∗0+RTnk for n≥3 (this turns out to be a rather peculiar fragment of PA) and obtain some nontrivial information about the first-order consequences of RT2k. Time permitting, we will also discuss the question whether our results have any relevance for the well-known open problem of characterizing the first-order consequences of RT22 over the traditional base theory RCA0. In the first part of the talk, we concentrated on Ramsey’s Theorem for n-tuples where n≥3. In this second part, the focus will be on RT22. Information: The seminar will take place virtually. Please email Victoria Gitman (vgitman@nylogic.org) for the meeting id. Paris-Lyon Séminaire de Logique Time: Wednesday, September 30, 16:00-17:15 CEST Speaker: Luca Motto Ros, University of Torino Title: Anti-classification results for Archimedean groups Abstract: We study the complexity of the isomorphism relation for countable Archimedean groups, both in terms of Borel reducibility and with respect to the theory of potential classes developed by Hjorth, Kechris and Louveau. This will lead to a number of anti-classification results for such groups. We will also present similar results concerning the bi-embeddability relation over countable Archimedean groups and, if time permits, we will speak about analogous problems for countable models of certain o-minimal theories (ordered divisible abelian groups, real closed fields). Joint work with F. Calderoni, D. Marker, and A. Shani. Information: Join via the link on the seminar webpage 10 minutes before the talk. Logic Seminar, Carnegie Mellon University Time: Tuesday, September 29, 3:30 – 4:30pm Eastern Daylight Time (21:00 CEST) Speaker: Gunter Fuchs, CUNY Title: The subcomplete fragment of SRP Abstract: I show how Todorcevic’s strong reflection principle SRP can be relativized to any forcing class. Of particular interest is the fragment of SRP that results from relativizing to the class of subcomplete forcing notions, a class introduced by Jensen that has very interesting features, such as being compatible with Jensen’s diamond principle. Like the original SRP, this fragment has a very natural characterization. I will describe some of the consequences of this fragment of SRP and explore its relationship to the subcomplete forcing axiom. Information: Zoom link https://cmu.zoom.us/j/621951121, meeting ID: 621 951 121 Week 21-27 September CUNY Set Theory Seminar Time: Friday, September 25, 11:00 New York time (17:00 CEST) Speaker: Ralf Schindler, University of Münster Title: Martin’s Maximum^++ implies the P_max axiom (*) Abstract: Forcing axioms spell out the dictum that if a statement can be forced, then it is already true. The P_max axiom (*) goes beyond that by claiming that if a statement is consistent, then it is already true. Here, the statement in question needs to come from a resticted class of statements, and ‘consistent’ needs to mean ‘consistent in a strong sense.’ It turns out that (*) is actually equivalent to a forcing axiom, and the proof is by showing that the (strong) consistency of certain theories gives rise to a corresponding notion of forcing producing a model of that theory. This is joint work with D. Asperó building upon earlier work of R. Jensen and (ultimately) Keisler’s ‘consistency properties’. Information: The seminar will take place virtually. Please email Victoria Gitman (vgitman@nylogic.org) for the meeting id. Caltech Logic Seminar Time: Wednesday, September 16, 12:00 – 1:00pm Pacific time (21:00 CEST) Speaker: Dana Bartošová, University of Florida Title: On phase spaces of universal minimal flows of groups with compact normal subgroups Abstract: For a topological group G, a G-flow is a continuous action of G on a compact Hausdorff space X; we call X the phase space of the G-flow. A G-flow on X is minimal if X has no closed non-trivial invariant subset. The universal minimal G-flow, M(G), has every minimal G-flow as a quotient and it is unique up to isomorphism. We show that whenever we have a short exact sequence 0→K→G→H→0 of topological groups with the image of K a compact normal subgroup of G, then the phase space of M(G) is homeomorphic to the product of the phase space of M(H) with K. For instance, if G is a Polish, non-Archimedean group, and the image of K is open in G, then H is a countable discrete group. The phase space of M(H) is homeomorphic to Gl⁡(22ℵ0), the Stone space of the completion of the free Boolean algebra on 2ℵ0 generators by Balcar-Błaszczyk and Glasner-Tsankov-Weiss-Zucker. Therefore, the phase space of M(G) is homeomorphic to K×Gl⁡(22ℵ0). When the sequence splits, that is, G≅H⋉K, then the homeomorphism witnesses an isomorphism of flows, recovering a result of Kechris and Sokić. Information: Online talk https://caltech.zoom.us/j/99296122790?pwd=bUN4RS94RVYrTEtGTGhqTHRJbm9nZz09 Southern Illinois University Logic Seminar Time: Thursday, 10 September, 1pm US Central Daylight Time (20:00 CEST) Speaker: Arno Pauly, Swansea University Title: How computability-theoretic degree structures and topological spaces are related Abstract: We can generalize Turing reducibility to points in a large class of topological spaces. The point degree spectrum of a space is the collection of the degrees of its points. This is always a collection of Medvedev degrees, and it turns out that topological properties of the space are closely related to what degrees occur in it. For example, a Polish space has only Turing degrees iff it is countably dimensional. This connection can be used to bring topological techniques to bear on problems from computability theory and vice versa. The talk is based on joint work with Takayuki Kihara and Keng Meng Ng (https://arxiv.org/abs/1405.6866 and https://arxiv.org/abs/1904.04107). Information: The seminar will take place virtually via zoom. Week 14-20 September CUNY Set Theory Seminar Time: Friday, September 18, 14:00 New York time (20:00 CEST) Speaker: Arthur Apter, CUNY Title: UA and the Number of Normal Measures over ℵω+1 Abstract: The Ultrapower Axiom UA, introduced by Goldberg and Woodin, is known to have many striking consequences. In particular, Goldberg has shown that assuming UA, the Mitchell ordering of normal measures over a measurable cardinal is linear. I will discuss how this result may be used to construct choiceless models of ZF in which the number of normal measures at successors of singular cardinals can be precisely controlled. Information: The seminar will take place virtually. Please email Victoria Gitman (vgitman@nylogic.org) for the meeting id. CUNY Logic Seminar (MOPA) Time: Wednesday, September 16, 5pm New York time (23:00 CEST) – note the time Speaker: Sam Coskey, Boise State University Title: Classification of countable models of ZFC Abstract: In 2009 Roman Kossak and I showed that the classification of countable models of PA is Borel complete, which means it is as complex as possible. The proof is a straightforward application of Gaifman’s canonical I-models. In 2017 Sam Dworetzky, John Clemens, and I showed that the argument may also be used to show the classification of countable models of ZFC is Borel complete too. In this talk I’ll outline the original argument for models of PA, the adaptation for models of ZFC, and briefly consider several subclasses of countable models of ZFC. ​ Information: The seminar will take place virtually. Please email Victoria Gitman (vgitman@nylogic.org) for the meeting id. Caltech Logic Seminar Time: Wednesday, September 16, 12:00 – 1:00pm Pacific time (21:00 CEST) Speaker: Steve Jackson, University of North Texas Title: Some complexity results in dynamics and number theory Abstract: The Ki-Linton theorem asserts that the set of base b normal numbers is a Π03-complete set. The base bb normal numbers can be viewed as the set of generic points for an associated dynamical system. This leads to the question of the complexity of the set of generic points for other numeration/dynamical systems, for example continued fractions, β-expansions, Lüroth expansions to name a few. We prove a general result which covers all of these cases, and involves a well-known property in dynamics, a form of the specification property. We then consider differences of these sets. Motivated by the descriptive set theory arguments, we are able to show that the set of continued fraction normal but not base b normal numbers is a complete D2(Π30) set. Previously, the best known result was that this set was non-empty (due to Vandehey), and this assumed the generalized Riemann hypothesis. The first part of the work is joint with Mance and Kwietniak, and the second part with Mance and Vandehey. Information: Online talk https://caltech.zoom.us/j/95952118325?pwd=QzFPa3ZOeTJKWXJnSW5VbHhGOXJEZz09 Logic Seminar, Carnegie Mellon University Time: Tuesday, September 15, 3:30 – 4:30pm Eastern Daylight Time (21:00 CEST) Speaker: William Chan (Carnegie Mellon University) Title: A Survey of Combinatorics and Cardinality under Determinacy Abstract: We will survey some recent work with Jackson and Trang concerning combinatorics under the axiom of determinacy. We will be especially concerned with ultrapowers of the first uncountable cardinal by the partition measures and related questions concerning club uniformization and continuity of functions around the first uncountable cardinal. We will show that the cardinals below the power set of the first and second uncountable cardinals have a very complicated and rich structure under determinacy axioms. We will summarize our knowledge of this structure under AD, AD+, and the axiom of real determinacy. Information: Zoom link https://cmu.zoom.us/j/621951121, meeting ID: 621 951 121 Logic and Metaphysics Workshop, CUNY Time: Monday, September 14th, 4.15-6.15 pm (22.15 CEST) For zoom information, email Yale Weiss at: yweiss@gradcenter.cuny.edu Speaker: Chris Scambler (NYU) Title: Cantor’s Theorem, Modalized Abstract: I will present a modal axiom system for set theory that (I claim) reconciles mathematics after Cantor with the idea there is only one size of infinity. I’ll begin with some philosophical background on Cantor’s proof and its relation to Russell’s paradox. I’ll then show how techniques developed to treat Russell’s paradox in modal set theory can be generalized to produce set theories consistent with the idea that there’s only one size of infinity. Week 7-13 September Southern Illinois University Logic Seminar Time: Thursday, 10 September, 1pm US Central Daylight Time (20:00 CEST) Speaker: Mirna Džamonja (IHPST, CNRS-Université Panthéon-Sorbonne Paris, France) Title: On logics that make a bridge from the Discrete to the Continuous Abstract: We study logics which model the passage between an infinite sequence of finite models to an uncountable limiting object, such as is the case in the context of graphons. Of particular interest is the connection between the countable and the uncountable object that one obtains as the union versus the combinatorial limit of the same sequence. Information: The seminar will take place virtually via zoom. CUNY Logic Seminar (MOPA) Time: Wednesday, September 9, 3pm New York time (21:00 CEST) – note the time Speaker: Saeideh Bahrami, Institute for Research in Fundamental Sciences, Tehran Title: Fixed Points of Initial Self-Embeddings of Models of Arithmetic Abstract: In 1973, Harvey Friedman proved his striking result on initial self-embeddings of countable nonstandard models of set theory and Peano arithmetic. In this talk, I will discuss my joint work with Ali Enayat focused on the fixed point set of initial self-embeddings of countable nonstandard models of arithmetic. Especially, I will survey the proof of some generalizations of well-known results on the fixed point set of automorphisms of countable recursively saturated models of PA, to results about the fixed point set of initial self-embeddings of countable nonstandard models of IΣ1. Information: The seminar will take place virtually. Please email Victoria Gitman (vgitman@nylogic.org) for the meeting id. Genoa Logic Seminar Time: Tuesday, September 8, 17:00-18:00 CEST Speaker: Gianluca Basso, Université de Lausanne Title: Topological dynamics beyond Polish groups Abstract: When$G$is a Polish group, one way of knowing that it has “nice” dynamics is to show that$M(G)$, the universal minimal flow of$G$, is metrizable. For non-Polish groups, this is not the relevant dividing line: the universal minimal flow of$\mathrm{Sym}(\kappa)$is the space of linear orders on$\kappa$—not a metrizable space, but still “nice”—, for example. In this talk, we present a set of equivalent properties of topological groups which characterize having “nice” dynamics. We show that the class of groups satisfying such properties is closed under some topological operations and use this to compute the universal minimal flows of some concrete groups, like$\mathrm{Homeo}(\omega_{1})$. This is joint work with Andy Zucker. Information: The seminar will take place on Microsoft Teams, at the page of the Genoa logic group. The access code is fpedcxn. Alternatively, you can write to camerlo@dima.unige.it to have an access link. Further information on the activities of the Genoa logic group can be found at http://www.dima.unige.it/~camerlo/glhome.html Note: Due to the current emergency situation, the web page might not be updated. Week 31 August – 6 September CUNY Set Theory Seminar Time: Friday, September 4, 14:00 New York time (20:00 CEST) Speaker: Mirna Džamonja, IHPST, CNRS-Université Panthéon-Sorbonne Paris Title: On logics that make a bridge from the Discrete to the Continuous Abstract: We study logics which model the passage between an infinite sequence of finite models to an uncountable limiting object, such as is the case in the context of graphons. Of particular interest is the connection between the countable and the uncountable object that one obtains as the union versus the combinatorial limit of the same sequence. Information: The seminar will take place virtually. Please email Victoria Gitman (vgitman@nylogic.org) for the meeting id. CUNY Logic Seminar (MOPA) Time: Wednesday, September 2, 14:00 New York time (20:00 CEST) – note the time Speaker: Petr Glivický, Universität Salzburg Title: The ω-iterated nonstandard extension of N and Ramsey combinatorics Abstract: In the theory of nonstandard methods (traditionally known as nonstandard analysis), each mathematical object (a set) x has a uniquely determined so called nonstandard extension ∗x. In general, ∗x⊋{∗y;y∈x} – that is, besides the original ‘standard’ elements ∗y for y∈x, the set ∗x contains some new ‘nonstandard’ elements. For instance, some of the nonstandard elements of ∗R can be interpreted as infinitesimals (there is ε∈∗R such that 0<ε<1/n for all n∈N) allowing for nonstandard analysis to be developed in ∗R, while ∗N turns out to be an (at least ℵ1-saturated) nonstandard elementary extension of N (in the language of arithmetic). While the whole nonstandard real analysis is most naturally developed in ∗R (with just a few advanced topics where using the second extension ∗∗R is convenient, though far from necessary), recent successful applications of nonstandard methods in combinatorics on N have utilized also higher order extensions (n)∗N=∗∗∗⋯∗N with the chain ∗∗∗⋯∗ of length n>2. In this talk we are going to study the structure of the ω-iterated nonstandard extension ⋅N=⋃n∈ω(n)∗N of N and show how the obtained results shed new light on the complexities of Ramsey combinatorics on N and allow us to drastically simplify proofs of many advanced Ramsey type theorems such as Hindmann’s or Milliken’s and Taylor’s. Information: The seminar will take place virtually. Please email Victoria Gitman (vgitman@nylogic.org) for the meeting id. Paul Bernays Lectures 2020: Struggling with the size of infinity, Lecture 3 Time: Tuesday, September 1, 16:30 CEST Speaker: Prof. Saharon Shelah, Hebrew University Jerusalem Title: Cardinal invariants of the continuum: are they all independent? Abstract: Experience has shown that in almost all cases, if you define a bunch of cardinal invariants of the continuum, then modulo some easy inequalities, it follows by forcing (the method introduced by Cohen) that there are no more restrictions. Well, those independence results have been mostly for the case of the continuum being at most aleph2. But this seems to be just due to our lack of ability, as the problems are harder. However, this opinion ignores the positive side of having forcing, of us being able to prove independence results: clearing away the rubble of independence results, the cases where we fail may indicate that there are theorems there. We shall on the one hand deal with cases where this succeeds and on the other hand with cofinality arithmetic, and what was not covered in the first lecture. Information: Due to the unusual circumstances of the COVID-​​19 pandemia, the Paul Bernays Lectures 2020 will take place as a webinar: Link for this webinar. All lectures are given in English and are self-​contained. Lecture 1 is aimed at a general audience; lecture 2 and 3 address the scientific community. Paul Bernays Lectures 2020: Struggling with the size of infinity, Lecture 2 Time: Tuesday, September 1, 14:15 CEST Speaker: Prof. Saharon Shelah, Hebrew University Jerusalem Title: How large is the continuum? Abstract: Cantor discovered that in mathematics we can distinguish many infinities, called the aleph numbers. The works of Gödel and Cohen told us that we cannot decide what the value of the continuum is, that is, which of the aleph numbers is the answer to the question “How many real numbers are there?”. This still does not stop people from having opinions and arguments. One may like to assume extra axioms which will decide the question (usually as aleph1 or aleph2), and argue that they should and eventually will be adopted. We feel that assuming the continuum is small makes us have equalities which are incidental. So if we can define 10 natural cardinals which are uncountable but at most the continuum, and the continuum is smaller than aleph10, at least two of them will be equal, without any inherent reasons. Such numbers are called cardinal invariants of the continuum, and they arise naturally from various perspectives. We would like to show that they are independent, that is, that there are no non-​trivial restrictions on their order. More specifically, we shall try to explain the Cichon diagram and what we cannot tell about it. Information: Due to the unusual circumstances of the COVID-​​19 pandemia, the Paul Bernays Lectures 2020 will take place as a webinar: Link for this webinar. All lectures are given in English and are self-​contained. Lecture 1 is aimed at a general audience; lecture 2 and 3 address the scientific community. Paul Bernays Lectures 2020: Struggling with the size of infinity, Lecture 1 Time: Monday, August 31, 17:00 CEST Speaker: Prof. Saharon Shelah, Hebrew University Jerusalem Title: Cardinal arithmetic: Cantor’s paradise Abstract: We will explain Hilbert’s first problem. Specifically, this asks what the value of the continuum is: Is the number of real numbers equal to aleph1 — the first infinite cardinal above aleph0=the number of natural numbers? Recall that Cantor introduces infinite numbers just as equivalence classes of sets under “there is a bijection”. The problem of the size of the continuum really means: ”What are the laws of cardinal arithmetic, i.e. the arithmetic of infinite numbers”. We will review its history, (including Gödel and Cohen), mention different approaches, explain what is undecidable, and mainly present some positive answers which we have nowadays. Those will be mainly about cofinality arithmetic, the so called pcf theory; but we will also mention cardinal invariants of the continuum. Information: Due to the unusual circumstances of the COVID-​​19 pandemia, the Paul Bernays Lectures 2020 will take place as a webinar: Link for this webinar. All lectures are given in English and are self-​contained. Lecture 1 is aimed at a general audience; lecture 2 and 3 address the scientific community. Week 24-30 August CUNY Set Theory Seminar Time: Friday, August 28, 14:00 New York time (20:00 CEST) Speaker: Miha Habic, Bard College at Simon’s Rock Title: Normal ultrapowers with many sets of ordinals Abstract: Any ultrapower M of the universe by a normal measure on a cardinal κ is quite far from V in the sense that it computes V_κ+2 incorrectly. If GCH holds, this amounts to saying that M is missing a subset of κ+. Steel asked whether, even in the absence of GCH, normal ultrapowers at κ must miss a subset of κ+. In the early 90s Cummings gave a negative answer, building a model with a normal measure on κ whose ultrapower captures the entire powerset of κ+. I will present some joint work with Radek Honzík in which we improved Cummings’ result to get this capturing property to hold at the least measurable cardinal. Information: Please email Victoria Gitman (vgitman@nylogic.org) for meeting id (this talk will have a different meeting ID!). CUNY Logic Seminar (MOPA) Time: Wednesday, August 26, 12:00 New York time (18:00 CEST) – note the time Speaker: Emil Jeřábek, Czech Academy of Sciences Title: Feasible reasoning with arithmetic operations Abstract: In bounded arithmetic, we study weak fragments of arithmetic that often correspond in a certain sense to computational complexity classes (e.g., polynomial time). Questions about provability in such theories can be thought of as a form of feasible reasoning: considering a natural object of interest from a complexity class C, can we prove its fundamental properties using only concepts from C? Our objects of interest in this talk will be the elementary integer arithmetic operations +,−,×,/, whose complexity class is (uniform) TC0, a small subclass of P. The corresponding arithmetical theory is VTC0. Since we do not know yet if the theory can prove the totality of division and iterated multiplication ∏i<nXi which are in TC0 by an intricate result of Hesse, Allender, and Barrington, we will also consider an extension of the theory VTC0+IMUL. Our main question is what can VTC0±IMUL prove about the elementary arithmetic operations. The answer is that more than one might expect: VTC0+IMUL proves induction for quantifier-free formulas in the basic language of arithmetic (IOpen), and even induction and minimization for Σb0 (sharply bounded) formulas in Buss’s language. This result is connected to the existence of TC0 constant-degree root-finding algorithms; the proof relies on a formalization of a form of the Lagrange inversion formula in VTC0+IMUL, and on model-theoretic abstract nonsense involving valued fields. The remaining problem is if VTC0 proves IMUL. We will discuss issues with formalization of the Hesse–Allender–Barrington construction in VTC0, and some partial results (this is a work in progress). Information: The seminar will take place virtually. Please email Victoria Gitman (vgitman@nylogic.org) for the meeting id. Bar-Ilan-Jerusalem Set Theory Seminar Time: Wednesday, August 26, 11:00am Israel Time (10:00 CEST) Speaker: Menachem Magidor, Hebrew University Title: Around weak diamond and uniformization Abstract: I’ll present some old results about weak diamond, uniformization and maybe some connections to Whitehead problem. In particular I’ll present Woodin’s elegant proof to the Devlin-Shelah equivalence of Weak diamond with 2^\aleph_0<2^\aleph_1. Information: Contact Menachem Magidor, Asaf Rinot or Omer Ben-Neria ahead of time for the zoom link. ALGOS 2020 – ALgebras, Graphs and Ordered Sets – August 26th to 28th (Online) Time: Wednesday, August 26, 9:30am – Friday, August 29, 19:30pm, CEST Information: Please see here for the program and advance registration. Week 17-23 August CUNY Set Theory Seminar Time: Friday, August 21, 14:00 New York time (20:00 CEST) Speaker: Dan Hathaway University of Vermont Title: A relative of ZF+DC+‘ω1 is measurable’ Abstract: Let Φ be the statement that for any function f:ω1×ω1→ω, there are functions g1,g2:ω1→ω such that for all (x,y)∈ω1×ω1, we have f(x,y)≤max {g1(x),g2(y)}. We will show that Φ follows from ZF+DC+‘ω1 is measurable’. On the other hand using core models, we will show that Φ+‘the club filter on ω1 is normal’ implies there are inner models with many measurable cardinals. We conjecture that Φ and ZF+DC+‘ω1 is measurable’ have the same consistency strength. The research is joint with Francois Dorais at the University of Vermont. Information: Please email Victoria Gitman (vgitman@nylogic.org) for meeting id (this talk will have a different meeting ID!). CUNY Logic Seminar (MOPA) Time: Wednesday, August 19, 14:00 New York time (20:00 CEST) – note the time Speaker: Leszek Kołodziejczyk, University of Warsaw Title: Ramsey’s Theorem over RCA_0* Abstract: The usual base theory used in reverse mathematics, RCA0, is the fragment of second-order arithmetic axiomatized by Δ01 comprehension and Σ01 induction. The weaker base theory RCA∗0 is obtained by replacing Σ01 induction with Δ01 induction (and adding the well-known axiom exp in order to ensure totality of the exponential function). In first-order terms, RCA0 is conservative over IΣ1 and RCA∗0 is conservative over BΣ1+exp. Some of the most interesting open problems in reverse mathematics concern the first-order strength of statements from Ramsey Theory, in particular Ramsey’s Theorem for pairs and two colours. In this talk, I will discuss joint work with Kasia Kowalik, Tin Lok Wong, and Keita Yokoyama concerning the strength of Ramsey’s Theorem over RCA∗0.Given standard natural numbers n,k≥2, let RTnk stand for Ramsey’s Theorem for k-colourings of n-tuples. We first show that assuming the failure of Σ01 induction, RTnk is equivalent to its own relativization to an arbitrary Σ01-definable cut. Using this, we give a complete axiomatization of the first-order consequences of RCA∗0+RTnk for n≥3 (this turns out to be a rather peculiar fragment of PA) and obtain some nontrivial information about the first-order consequences of RT2k. Time permitting, we will also discuss the question whether our results have any relevance for the well-known open problem of characterizing the first-order consequences of RT22 over the traditional base theory RCA0. Information: The seminar will take place virtually. Please email Victoria Gitman (vgitman@nylogic.org) for the meeting id. Bar-Ilan-Jerusalem Set Theory Seminar Time: Wednesday, August 19, 11:00am Israel Time (10:00 CEST) Speaker: Uri Abraham, Ben-Gurion University Title: Coding well ordering of the reals with ladders, part 5 Abstract: Results from the 2002 paper “Coding with Ladders a Well Ordering of the Reals” by Abraham and Shelah. Information: contact Menachem Magidor, Asaf Rinot or Omer Ben-Neria ahead of time for the zoom link. Week 10-16 August CUNY Set Theory Seminar Time: Friday, August 14, 14:00 New York time (20:00 CEST) Speaker: Gunter Fuchs CUNY Title: Canonical fragments of the strong reflection principle Abstract: I have been working over the past few years on the project of trying to improve our understanding of the forcing axiom for subcomplete forcing. The most compelling feature of this axiom is its consistency with the continuum hypothesis. On the other hand, it captures many of the major consequences of Martin’s Maximum. It is a compelling feature of Martin’s Maximum that many of its consequences filter through Todorcevic’s Strong Reflection Principle SRP. SRP has some consequences that the subcomplete forcing axiom does not have, like the failure of CH and the saturation of the nonstationary ideal. It has been unclear until recently whether there is a version of SRP that relates to the subcomplete forcing axiom as the full SRP relates to Martin’s Maximum, but it turned out that there is: I will detail how to associate in a canonical way to an arbitrary forcing class its corresponding fragment of SRP in such a way that (1) the forcing axiom for the forcing class implies its fragment of SRP, (2) the stationary set preserving fragment of SRP is the full principle SRP, and (3) the subcomplete fragment of SRP implies the major consequences of the subcomplete forcing axiom. I will describe how this association works, describe some hitherto unknown effects of (the subcomplete fragment of) SRP on mutual stationarity, and say a little more about the extent of (3). Information: Please email Victoria Gitman (vgitman@nylogic.org) for meeting id (this talk will have a different meeting ID!). CUNY Logic Seminar (MOPA) Time: Wednesday, August 12, 19:00 New York time (1:00am July 30 CEST) – note the time Speaker: Athar Abdul-Quader Purchase College Title: CP-genericity and neutrality Abstract: In a paper with Kossak in 2018, we studied the notion of neutrality: a subset X of a model M of PA is called neutral if the definable closure relation in (M, X) coincides with that in M. This notion was suggested by Dolich. motivated by work by Chatzidakis-Pillay on generic expansions of theories. In this talk, we will look at a more direct translation of the Chatzidakis-Pillay notion of genericity, which we call ‘CP-genericity’, and discuss its relation to neutrality. The main result shows that for recursively saturated models, CP-generics are always neutral; previously we had known that not all neutral sets are CP-generic. ​ Information: The seminar will take place virtually. Please email Victoria Gitman (vgitman@nylogic.org) for the meeting id. Bar-Ilan-Jerusalem Set Theory Seminar Time: Wednesday, August 12, 11:00am Israel Time (10:00 CEST) Speaker: Uri Abraham, Ben-Gurion University Title: Coding well ordering of the reals with ladders, part 4 Abstract: Results from the 2002 paper “Coding with Ladders a Well Ordering of the Reals” by Abraham and Shelah. Information: contact Menachem Magidor, Asaf Rinot or Omer Ben-Neria ahead of time for the zoom link. Week 3-9 August CUNY Set Theory Seminar Time: Friday, August 7, 14:00 New York time (20:00 CEST) Speaker: Brent Cody, Virginia Commonwealth University Title: Higher indescribability Abstract: To what extent can formulas from infinitary logics be used in set-theoretic reflection arguments? If κ is a measurable cardinal, any Lκ,κ sentence which is true in (κ,∈), must be true about some strictly smaller cardinal. Whereas, there are Lκ+,κ+ sentences of length κ which are true in (κ,∈)and which are not true about any smaller cardinal. However, if κ is a measurable cardinal and some Lκ+,κ+ sentence φ is true in (κ,∈), then there must be some strictly smaller cardinal α<κsuch that a canonically restricted version of φ holds about α. Building on work of Bagaria and Sharpe-Welch, we use canonical restriction of formulas to define notions of Π1ξ-indescribability of a cardinal κ for all ξ<κ+. In this context we show that such higher indescribability hypotheses are strictly weaker than measurability, we prove the existence of universal Π1ξ-formulas, study the associated normal ideals and notions of ξ-clubs and prove a hierarchy result. Time permitting we will discuss some applications. Information: Please email Victoria Gitman (vgitman@nylogic.org) for meeting id (this talk will have a different meeting ID!). Bar-Ilan-Jerusalem Set Theory Seminar Time: Wednesday, August 5, 11:00am Israel Time (10:00 CEST) Speaker: Uri Abraham, Ben-Gurion University Title: Coding well ordering of the reals with ladders, part 3 Abstract: Results from the 2002 paper “Coding with Ladders a Well Ordering of the Reals” by Abraham and Shelah. Information: contact Menachem Magidor, Asaf Rinot or Omer Ben-Neria ahead of time for the zoom link. Week 27 July – 2 August CUNY Set Theory Seminar Time: Friday, July 31, 12:00 New York time (18:00 CEST) Speaker: Corey Switzer, CUNY Title: Dissertation defense: Alternative Cichoń diagrams and forcing axioms compatible with CH Abstract: This dissertation surveys several topics in the general areas of iterated forcing, infinite combinatorics and set theory of the reals. There are two parts. In the first half I consider alternative versions of the Cichoń diagram. First I show that for a wide variety of reduction concepts there is a Cichoń diagram for effective cardinal characteristics relativized to that reduction. As an application I investigate in detail the Cichoń diagram for degrees of constructibility relative to a fixed inner model of ZFC. Then I study generalizations of cardinal characteristics to the space of functions from ωω to ωω. I prove that these cardinals can be organized into two diagrams analogous to the standard Cichoń diagram show several independence results and investigate their relation to cardinal invariants on omega. In the second half of the thesis I look at forcing axioms compatible with CH. First I consider Jensen’s subcomplete and subproper forcing. I generalize these notions to larger classes which are (apparently) much more nicely behaved structurally. I prove iteration and preservation theorems for both classes and use these to produce many new models of the subcomplete forcing axiom. Finally I deal with dee-complete forcing and its associated axiom DCFA. Extending a well-known result of Shelah, I show that if a tree of height ω1 with no branch can be embedded into an ω1 tree, possibly with uncountable branches, then it can be specialized without adding reals. As a consequence I show that DCFA implies there are no Kurepa trees, even if CH fails. Information: Note the different time! The seminar will take place virtually at 12pm US Eastern Standard Time. Please email Victoria Gitman (vgitman@nylogic.org) for meeting id (this talk will have a different meeting ID!). CUNY Logic Seminar (MOPA) Time: Wednesday, July 29, 14:00 New York time (20:00 CEST) Speaker: Kameryn Williams, University of Hawai‘i at Mānoa Title: End-extensions of models of set theory and the Σ1 universal finite sequence Abstract: Recall that if M⊆N are models of set theory then N end-extends M if N does not have new elements for sets in M. In this talk I will discuss a Σ1-definable finite sequence which is universal for end extensions in the following sense. Consider a computably axiomatizable extension ¯ZF of ZF. There is a Σ1-definable finite sequencea0,a1,…,anwith the following properties. * ZF proves that the sequence is finite. * In any transitive model of ¯ZF the sequence is empty. * If M is a countable model of ¯ZF in which the sequence is s and t∈M is a finite sequence extending s then there is an end-extension N⊨¯ZF of M in which the sequence is exactly t. * Indeed, for the previous statements it suffices that M⊨ZF and end-extends a submodel W⊨¯ZF of height at least (ωL1)M. This universal finite sequence can be used to determine the modal validities of end-extensional set-theoretic potentialism, namely to be exactly the modal theory S4. The sequence can also be used to show that every countable model of set theory extends to a model satisfying the end-extensional maximality principle, asserting that any possibly necessary sentence is already true. This talk is about joint work with Joel David Hamkins. The Σ1 universal finite sequence is a sister to the Σ2 universal finite sequence for rank-extensions of Hamkins and Woodin, and both are cousins of Woodin’s universal algorithm for arithmetic. Information: The seminar will take place virtually. Please email Victoria Gitman (vgitman@nylogic.org) for the meeting id. Bar-Ilan-Jerusalem Set Theory Seminar – cancelled Time: Wednesday, July 29, 11:00am Israel Time (10:00 CEST) – cancelled Speaker: Uri Abraham, Ben-Gurion University Title: Coding well ordering of the reals with ladders, part 3 Abstract: Results from the 2002 paper “Coding with Ladders a Well Ordering of the Reals” by Abraham and Shelah. Information: contact Menachem Magidor, Asaf Rinot or Omer Ben-Neria ahead of time for the zoom link. Week 20-26 July July 24 CUNY Set Theory Seminar Time: Friday, July 24, 14:00 New York time (20:00 CEST) Speaker: Andrew Brooke-Taylor, University of Leeds Title: Measurable cardinals and limits in the category of sets Abstract: An old result of Isbell characterises measurable cardinals in terms of certain canonical limits in the category of sets. After introducing this characterisation, I will talk about recent work with Adamek, Campion, Positselski and Rosicky teasing out the importance of the canonicity for this and related results. The language will be category-theoretic but the proofs will be quite hands-on combinatorial constructions with sets. Information: The seminar will take place virtually. Please email Victoria Gitman (vgitman@nylogic.org) for the meeting id. July 23 Southern Illinois University Logic Seminar Time: Thursday, 23 July, 1pm US Central Daylight Time (20:00 CEST) Speaker: Dana Bartošová, University of Florida Title: Dynamics of finite products of groups and of group extensions Abstract: tba Information: The seminar will take place virtually via zoom. July 22 CUNY Logic Seminar (MOPA) Time: Wednesday, July 22, 14:00 New York time (20:00 CEST) Speaker: Tin Lok Wong, National University of Singapore Title: Properties preserved in cofinal extensions Abstract: Cofinal extensions generally preserve many more properties of a model of arithmetic than their sisters, end extensions. Exactly how much must or can they preserve? The answer is intimately related to how much arithmetic the model can do. I will survey what is known and what is not known about this question, and report on some recent work on this line. Information: The seminar will take place virtually. Please email Victoria Gitman (vgitman@nylogic.org) for the meeting id. Bar-IlanJerusalem Set Theory Seminar Time: Wednesday, July 22, 11:00am (Israel Time) Speaker: Uri Abraham, Ben-Gurion University Title: Coding well ordering of the reals with ladders, part 2 Abstract: Results from the 2002 paper “Coding with Ladders a Well Ordering of the Reals” by Abraham and Shelah. Information: contact Menachem Magidor, Asaf Rinot or Omer Ben-Neria ahead of time for the zoom link. Week 13-19 July July 17 CUNY Set Theory Seminar Time: Friday, July 17, 14:00 New York time (20:00 CEST) Speaker: Kaethe Minden, Bard College at Simon’s Rock Title: Maximality and Resurrection Abstract: The maximality principle (MP) is the assertion that any sentence which can be forced in such a way that after any further forcing the sentence remains true, must already be true. In modal terms, MPstates that forceably necessary sentences are true. The resurrection axiom (RA) asserts that the ground model is as existentially closed in its forcing extensions as possible. In particular, RArelative to Hc states that for every forcing Q there is a further forcing R such that HVc≺HV[G][H]c, for G∗H⊆Q∗˙R generic. It is reasonable to ask whether MP and RA can consistently both hold. I showed that indeed they can, and that RA+MP is equiconsistent with a strongly uplifting fully reflecting cardinal, which is a combination of the large cardinals used to force the principles separately. In this talk I give a sketch of the equiconsistency result. Information: The seminar will take place virtually. Please email Victoria Gitman (vgitman@nylogic.org) for the meeting id. Week 6-12 July July 10 CUNY Set Theory Seminar Time: Friday, July 10, 14:00 New York time (20:00 CEST) Speaker: Peter Holy, University of Udine Title: Uniform large cardinal characterizations and ideals up to measurability Abstract: Many prominent large cardinal notions up to measurability can be characterized by the existence of certain ultrafilters for small models of set theory. Most prominently, this includes weakly compact, ineffable, Ramsey and completely ineffable cardinals, but there are many more, and our characterization schemes also give rise to many new natural large cardinal concepts. Moreover, these characterizations allow for the uniform definition of ideals associated to these large cardinals, which agree with the ideals from the set-theoretic literature (for example, the weakly compact, the ineffable, the Ramsey or the completely ineffable ideal) whenever such had been previously established. For many large cardinal notions, we can show that their ordering with respect to direct implication, but also with respect to consistency strength corresponds in a very canonical way to certain relations between their corresponding large cardinal ideals. This is all material from a fairly extensive joint paper with Philipp Luecke, and I will try to provide an overview as well as present some particular results from this paper. Information: The seminar will take place virtually. Please email Victoria Gitman (vgitman@nylogic.org) for the meeting id. July 8 CUNY Logic Seminar (MOPA) Time: Wednesday, July 8, 14:00 New York time (20:00 CEST) Speaker: Corey Switzer, CUNY Title: Axiomatizing Kaufmann models in strong logics Abstract: A Kaufmann model is an ω1-like, recursively saturated, rather classless model of PA. Such models were constructed by Kaufmann under the ♢ assumption and then shown to exist in ZFC by Shelah using an absoluteness argument involving the logic Lω1,ω(Q) where Q is the quantifier ‘there exists uncountably many…’. It remains an intriguing, if vague, open problem whether one can construct a Kaufmann model in ZFC ‘by hand’ i.e. without appealing to some form of absoluteness or other very non-constructive methods. In this talk I consider the related problem of axiomatizing Kaufmann models in Lω1,ω(Q) and show that this is independent of ZFC. Along the way we’ll see that it is also independent of ZFC whether there is an ω1-preserving forcing notion adding a truth predicate to a Kaufmann model. Information: The seminar will take place virtually. Please email Victoria Gitman (vgitman@nylogic.org) for the meeting id. Bristol Logic and Set Theory Seminar (recurring lecture series – last lecture!) Time: Wednesday, July 8, 13:30-15:00 UK time (14:30-16:00 CEST) Speaker: Philip Welch, University of Bristol Title: Higher type recursion for Infinite time Turing Machines XI 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: zoom via https://zoom.us/j/96803195711 (open 30 minutes before). July 7 Münster Set Theory Seminar Time: Tuesday, June 23, 4:15pm CEST Speaker: Ralf Schindler, University of Münster Title: MM and (∗)++ Abstract: The axiom (∗)++ is a strengthening of (∗) which was also introduced by Woodin. (∗)++ says that the set of sets of reals, or equivalently, Hc+, is contained in a Pmax extension of a determinacy model. Woodin showed that (∗)++ is false in all the known models of Martin’s Maximum. We will give a proof of this result. It remains open if MM++ refutes (∗)++. Information: contact rds@wwu.de ahead of time in order to participate. Week 29 June-5 July July 3 CUNY Set Theory Seminar Time: Friday, July 3, 14:00 New York time (20:00 CEST) Speaker: Vera Fischer, University of Vienna Title: More ZFC inequalities between cardinal invariants Abstract: We will discuss some recent ZFC results concerning the generalized Baire spaces, and more specifically the generalized bounding number, relatives of the generalized almost disjointness number, as well as generalized reaping and domination. Information: The seminar will take place virtually. Please email Victoria Gitman (vgitman@nylogic.org) for the meeting id. July 1 CUNY Logic Seminar (MOPA) Time: Wednesday, July 1, 14:00 New York time (20:00 CEST) Speaker: Zachiri McKenzie Title: Initial self-embeddings of models of set theory: Part II 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. Bristol Logic and Set Theory Seminar (recurring lecture series) Time: Wednesday, July 1, 13:30-15:00 UK time (14:30-16:00 CEST) Speaker: Philip Welch, University of Bristol Title: Higher type recursion for Infinite time Turing Machines X 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: zoom via https://zoom.us/j/96803195711 (open 30 minutes before). Week 22-28 June June 26 CUNY Set Theory Seminar Time: Friday, June 26, 14:00 New York time (20:00 CEST) Speaker: Joel David Hamkins, Oxford University Title: Categorical cardinals Abstract: Zermelo famously characterized the models of second-order Zermelo-Fraenkel set theory ZFC2 in his 1930 quasi-categoricity result asserting that the models of ZFC2 are precisely those isomorphic to a rank-initial segment Vκ of the cumulative set-theoretic universe V cut off at an inaccessible cardinal κ. I shall discuss the extent to which Zermelo’s quasi-categoricity analysis can rise fully to the level of categoricity, in light of the observation that many of the Vκ universes are categorically characterized by their sentences or theories. For example, if κ is the smallest inaccessible cardinal, then up to isomorphism Vκ is the unique model of ZFC2 plus the sentence ‘there are no inaccessible cardinals.’ This cardinal κ is therefore an instance of what we call a first-order sententially categorical cardinal. Similarly, many of the other inaccessible universes satisfy categorical extensions of ZFC2 by a sentence or theory, either in first or second order. I shall thus introduce and investigate the categorical cardinals, a new kind of large cardinal. This is joint work with Robin Solberg (Oxford). 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 26, 1.30pm Toronto time (7.30pm CEST) Speaker: Will Brian, UNC Charlotte Title: Limited-information strategies in Banach-Mazur games. Abstract: The Banach-Mazur game is an infinite-length game played on a topological space X, in which two players take turns choosing members of an infinite decreasing sequence of open sets, the first player trying to ensure that the intersection of this sequence is empty, and the second that it is not. A limited-information strategy for one of the players is a game plan that, on any given move, depends on only a small part of the game’s history. In this talk we will discuss Telgársky’s conjecture, which asserts roughly that there must be topological spaces where winning strategies for the Banach-Mazur game cannot be too limited, but must rely on large parts of the game’s history in a significant way. Recently, it was shown that this conjecture fails in models of set theory satisfying GCH + square. In such models it is always possible for one player to code all information concerning a game’s history into a small piece of it. We will discuss these so-called coding strategies, why assuming GCH + square makes them work so well, and what can go wrong in other models of set theory. Information: The seminar will take place virtually. Follow the link below: https://yorku.zoom.us/j/96087161597 June 25 Kurt Gödel Research Center Seminar (organised by Ben Miller) Time: Thursday, June 25, 16:00 CEST Speaker: Victoria Gitman, CUNY Title: Class forcing in its rightful setting Abstract: The use of class forcing in set theoretic constructions goes back to the proof Easton’s Theorem that GCH can fail at all regular cardinals. Class forcing extensions are ubiquitous in modern set theory, particularly in the emerging field of set-theoretic geology. Yet, besides the pioneering work by Friedman and Stanley concerning pretame and tame class forcing, the general theory of class forcing has not really been developed until recently. A revival of interest in second-order set theory has set the stage for understanding the properties of class forcing in its natural setting. Class forcing makes a fundamental use of class objects, which in the first-order setting can only be studied in the meta-theory. Not surprisingly it has turned out that properties of class forcing notions are fundamentally determined by which other classes exist around them. In this talk, I will survey recent results (of myself, Antos, Friedman, Hamkins, Holy, Krapf, Schlicht, Williams and others) regarding the general theory of class forcing, the effects of the second-order set theoretic background on the behavior of class forcing notions and the numerous ways in which familiar properties of set forcing can fail for class forcing even in strong second-order set theories. Information: Talk via zoom. June 24 CUNY Logic Seminar (MOPA) Time: Wednesday, June 24, 14:00 New York time (20:00 CEST) Speaker: Bartosz Wcisło, Polish Academy of Sciences Title: Tarski boundary III Abstract: Truth theories investigate the notion of truth using 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 which 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 A. It says that sentences provable in PA are true in the sense of the predicate T. This in turn is equivalent to Δ_0-induction for the compositional truth predicate which turns out to be a surprisingly robust theory. The equivalences between nonconservative truth theories are typically proved by relatively direct ad hoc arguments. However, certain patterns seem common to these proofs. The first one is construction of various arithmetical partial truth predicates which provably in a given theory have better properties than the original truth predicate. The second one is deriving induction for these truth predicates from internal induction, a principle which says that for any arithmetical formula, the set of those elements for which that formula is satisfied under the truth predicate satisfies the usual induction axioms. As an example of this phenomenon, we will present two proofs. First, we will show that global reflection principle is equivalent to local induction. Global reflection expresses that any sentence provable in PA is true. Local induction says that any predicate obtained by restricting truth predicate to sentences of a fixed syntactic complexity satisfies full induction. This is an observation due to Mateusz Łełyk and the author of this presentation. The second example is a result by Ali Enayat who showed that CT_0, a theory compositional truth with Δ_0-induction, is arithmetically equivalent to the theory of compositional truth together with internal induction and disjunctive correctness.This talk is intended as a continuation of ‘Tarski boundary II’ presentation at the same seminar. 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. Bristol Logic and Set Theory Seminar Time: Wednesday, June 24, 15:00-16:30 UK time (16:00-17:30 CEST) Speaker: Alessandro Andretta, University of Torino Title: Generalised Iteration Trees Abstract: A theorem of Gaifman states that any internal linear iteration whose length belongs to the model it is applied to has a well-founded direct limit.We have isolated a notion of “generalized iteration trees” for which a similar result is possible, at least if the length of the tree is$\omega$. These iterations are more general than the objects introduced by Martin and Steel over three decades ago in that the extender$E_n$used to construct$M_{n+1}$need not to belong to the last model$M_n$. In other words$E_n \in M_{d(n+1)}$, with$d(n+1) \leq n$. We isolate a simple property of the function$d$characterizing continuous ill-foundedness of generalized iteration trees.Any generalized iteration trees satisfying this property is not continuously ill-founded. Conversely, any tree order with a$d$function failing such property can be realized as a continuously ill-founded iteration tree on V. This is joint work with John Steel. Information: Via zoom: https://zoom.us/j/97281665521 (open 15 minutes before) Paris-Lyon Séminaire de Logique Time: Wednesday, June 24, 16:00-17:15 CEST Speaker: Ludovic Patey, University of Lyon Title: The computability-theoretic aspects of Milliken’s tree theorem and applications Abstract: Milliken’s tree theorem states that for every countable, finitely branching tree T with no leaves, and every finite coloring f of the strong subtrees of height n, there is an infinite strong subtree over which the strong subtrees of height n are monochromatic. This theorem has several applications, among which Devlin’s theorem about finite coloring of the rationals, and a theorem about the Rado graph. In this talk, we give a survey of the computability-theoretic aspects of these statements seen as mathematical problems, in terms of instances and solutions. Our main motivation is reverse mathematics. This is a joint work with Paul-Elliot Anglès d’Auriac, Peter Cholak and Damir Dzhafarov. Information: Join via the link on the seminar webpage 10 minutes before the talk. Bristol Logic and Set Theory Seminar (recurring lecture series) Time: Wednesday, June 24, 13:30-15:00 UK time (14:30-16:00 CEST) Speaker: Philip Welch, University of Bristol Title: Higher type recursion for Infinite time Turing Machines IX 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: zoom via https://zoom.us/j/96803195711 (open 30 minutes before). Bar-Ilan University and Hebrew University Set Theory Seminar Time: Wednesday, June 24, 11:00-13:00 Israel time (10:00-13:00 CEST) Speaker: Istvan Juhasz Title: On the free subset number of a topological space and their G_\delta modification Abstract: pdf available on seminar website. Information: Contact Assaf Rinot for the zoom id. June 23 Münster Set Theory Seminar Time: Tuesday, June 23, 4:15pm CEST Speaker: Farmer Schlutzenberg Title: Remarks on rank-into-rank embeddings part III Abstract: Recall that Woodin’s large cardinal axiom I0 gives an ordinal λ and an elementary embedding j:L(V_{λ+1})→L(V_{λ+1}) with critical point <λ. Using methods due to Woodin, we show that if ZFC+I0 is consistent then so is ZF+DC(λ)+ there is an ordinal λ and an elementary j:V_{λ+2}→V_{λ+2}”. (A version with the added assumption that V_{λ+1}^sharp exists is due to the author, and Goldberg observed that the appeal to V_{λ+1}^sharp could actually be replaced by some further calculations of Woodin’s.) Reference: https://arxiv.org/abs/2006.01077, “On the consistency of ZF with an elementary embedding from V_{λ+2} into V_{λ+2}”. Information: contact rds@wwu.de ahead of time in order to participate. Week 15-21 June June 19 CUNY Set Theory Seminar Time: Friday, June 19, 14:00 New York time (20:00 CEST) Speaker: Boban Velickovic, University of Paris 7 Title: Strong guessing models Abstract: The notion of a guessing model introduced by Viale and Weiss. The principle GM(ω2,ω1) asserts that there are stationary many guessing models of size ℵ1 in Hθ, for all large enough regular θ. It follows from PFA and implies many of its structural consequences, however it does not settle the value of the continuum. In search of higher of forcing axioms it is therefore natural to look for extensions and higher versions of this principle. We formulate and prove the consistency of one such statement that we call SGM+(ω3,ω1). It has a number of important structural consequences: • the tree property at ℵ2 and ℵ3 • the failure of various weak square principles • the Singular Cardinal Hypothesis • Mitchell’s Principle: the approachability ideal agrees with the non stationary ideal on the set of cof(ω1) ordinals in ω2 • Souslin’s Hypothesis • The negation of the weak Kurepa Hypothesis • Abraham’s Principles: every forcing which adds a subset of ω2 either adds a real or collapses some cardinals, etc. The results are joint with my PhD students Rahman Mohammadpour. 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 19, 1.30pm Toronto time (7.30pm CEST) Speaker: David Schrittesser, University of Vienna Title: Higher degrees of madness Abstract: The notion of mad family can be generalized by replacing the finite ideal by an iterated Fubini product of the finite ideal. While these ideals are more complicated both combinatorially and in terms of Borel complexity, it turns out that the same assumptions of Ramsey theoretic regularity can rule out their existence. We sketch a proof of this and some related results. This talk is a sequel to my last talk at the Fields Institute Seminar. Information: The seminar will take place virtually. ZOOM ID: https://yorku.zoom.us/j/96087161597 Udine graduate course Time: Friday, June 19, 10:00-12:00 CEST Speaker: Vincenzo Dimonte, University of Udine Title: Generalized Descriptive Set Theory II, Lecture 5 Abstract: The objective of the course is to prove an analogue of Silver’s Theorem for the space$2^\lambda$, where$\lambda$is an uncountable cardinal of cofinality$\omega$, using some large cardinal strength (the proof is still unpublished). This result has been chosen as an example to show, more in general, how to generalize a deep classical theorem in this setting, which properties of singular cardinals are useful in that respect, and what are the main obstacles of the generalization. The proof will use some peculiarities of singular cardinal combinatorics and some large cardinal strength, and everything will be introduced in the first three lessons. The course is self-contained (despite the name), the only prerequisite is to know basic set theory (the theory of forcing, the most basic descriptive set theory, maybe inaccessible cardinals). The following is a tentative schedule:Tuesday 10.00-12.00 CEST, Friday 10.00-12.00 CEST, from 5 June 2020, for 5 lessons. Lesson 1: Measurable cardinals Lesson 2: Prikry forcing, diagonal Prikry forcing Lesson 3: Strong Prikry condition, “double” diagonal Prikry forcing Lesson 4: generalized G_0 dichotomy Lesson 5: generalized Silver Theorem Information: Via Microsoft Teams, to participate contact vincenzo.dimonte@uniud.it. June 18 Kurt Gödel Research Center Seminar (organised by Ben Miller) Time: Thursday, June 18, 16:00 CEST Speaker: Anush Tserunyan, University of Illinois at Urbana-Champaign Title: Hyperfinite subequivalence relations of treed equivalence relations Abstract: A large part of measured group theory studies structural properties of countable groups that hold “on average”. This is made precise by studying the orbit equivalence relations induced by free Borel actions of these groups on a standard probability space. In this vein, the amenable groups correspond to hyperfinite equivalence relations, and the free groups to the treeable ones. In joint work with R. Tucker-Drob, we give a detailed analysis of the structure of hyperfinite subequivalence relations of a treed equivalence relation on a standard probability space, deriving the analogues of structural properties of amenable subgroups (copies of ℤZ) of a free group. Most importantly, just like every such subgroup is contained in a unique maximal one, we show that even in the non-pmp setting, every hyperfinite subequivalence relation is contained in a unique maximal one. Information: Talk via zoom. June 17 New York Logic Seminar (MOPA) Time: Wednesday, June 17, 14:00 New York time (20:00 CEST) Speaker: Mateusz Łełyk, University of Warsaw Title: Partial Reflection over Uniform Disquotational Truth Abstract: In the context of arithmetic, a reflection principle for a theory Th is a formal way of expressing that all theorems of Th are true. In the presence of a truth predicate for the language of Th this principle can be expressed as a single sentence (called the Global Reflection principle over Th) but most often is met in the form of a scheme consisting of all sentences of the form ∀x(ProvTh(ϕ(x˙))→ϕ(x)). Obviously such a scheme is not provable in a consistent theory Th. Nevertheless, such soundness assertions are said to provide a natural and justified way of extending ones initial theory. This perspective is nowadays very fruitfully exploited in the context of formal theories of truth. One of the most basic observations is that strong axioms for the notions of truth follow from formally weak types of axiomatizations modulo reflection principles. In such a way compositional axioms are consequences of the uniform disquotational scheme for for the truth predicate, which is ∀xT(ϕ(x˙))≡ϕ(x). The above observation is also used in the recent approach to ordinal analysis of theories of predicative strength by Lev Beklemishev and Fedor Pakhomov. The assignment of ordinal notations to theories proceeds via partial reflection principles (for formulae of a fixed Σn-complexity) over (iterated) disquotational scheme. It becomes important to relate theories of this form to fragments of standard theories of truth, in particular the ones based on induction for restricted classes of formulae such as CT0 (the theory of compositional truth with Δ0-induction for the extended language. The theory was discussed at length in Bartek Wcisło’s talk). Beklemishev and Pakhomov leave the following open question: Is Σ1-reflection principle over the uniform disquotational scheme provable in CT0? The main goal of our talk is to present the proof of the affirmative answer to this question. The result significantly improves the known fact on the provability of Global Reflection over PA in CT0. During the talk, we explain the theoretical context described above including the information on how the result fits into Beklemishev-Pakhomov project. In the meantime we give a different proof of their characterisation of Δ_0-reflection over the disquotational scheme.Despite the proof-theoretical flavour of these results, our proofs rests on essentially model-theoretical techniques. The important ingredient is the Arithmetized Completeness Theorem. 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 17, 16:00-17:30 UK time (17:00-18:30 CEST) Speaker: Corey Bacal Switzer, City University of New York Title: Some Set Theory of Kaufmann Models Abstract: A Kaufmann model is an ω1-like, recursively saturated, rather classless model of PA. Such models were shown to exist by Kaufmann under the assumption that ♢ holds, and in ZFC by Shelah via an absoluteness argument involving strong logics. They are important in the theory of models of arithmetic notably because they show that many classic results about countable, recursively saturated models of arithmetic cannot be extended to uncountable models. They are also a particularly interesting example of set theoretic incompactness at ω1, similar to an Aronszajn tree. In this talk we’ll look at several set theoretic issues relating to this class of models motivated by the seemingly naïve question of whether or not such models can be killed by forcing without collapsing ω1. Surprisingly the answer to this question turns out to be independent: under MAℵ1 no ω1-preserving forcing can destroy Kaufmann-ness whereas under ♢ there is a Kaufmann model M and a Souslin tree S so that forcing with S adds a satisfaction class to M (thus killing rather classlessness). The techniques involved in these proofs also yield another surprising side of Kaufmann models: it is independent of ZFC whether the class of Kaufmann models can be axiomatized in the logic Lω1,ω(Q) where Q is the quantifier “there exists uncountably many”. This is the logic used in Shelah’s aforementioned result, hence the interest in this level of expressive power. Information: For the Zoom access code, contact Samuel Adam-Day: me@samadamday.com. Paris-Lyon Séminaire de Logique Time: Wednesday, June 17, 16:00-17:15 CEST Speaker: Michał Skrzypczak – Université de Varsovie Title: Measure theory and Monadic Second-order logic over infinite trees Abstract: Monadic Second-order (MSO) logic is a well-studied formalism featuring many decision procedures and effective transformations. It is the fundamental logic considered in automata theory, equivalent to various other ways of defining sets of objects. In this talk, I will speak about the expressive power of MSO over infinite binary trees (i.e. free structures of two successors) – the theory from the famous Rabin’s decidability result. The goal of the talk is to survey recent results about measure properties of MSO-definable sets of infinite trees. First, I will argue that these sets are indeed measurable (which is not obvious, as there exist non-Borel sets definable in MSO). Then I will move to the question of our ability to compute the measure of the set defined by a given formula. Although the general question is still open (and seems to be demanding), I will speak about decidability results for fragments of MSO, focusing on the so-called weak-MSO. Information: Join via the link on the seminar webpage 10 minutes before the talk. Bristol Logic and Set Theory Seminar (recurring lecture series) Time: Wednesday, June 17, 13:30-15:00 UK time (14:30-16:00 CEST) Speaker: Philip Welch, University of Bristol Title: Higher type recursion for Infinite time Turing Machines VIII 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: zoom via https://zoom.us/j/96803195711 (open 30 minutes before). Bar-Ilan University and Hebrew University Set Theory Seminar Time: Wednesday, April 17, 11:00-13:00 Israel time (10:00-13:00 CEST) Speaker: Mirna Dzamonja (University of East Anglia) Title: Wide Aronszajn trees Abstract: A wide Aronszajn tree is a tree is size and height omega_1 but with no uncountable branch. Such trees arise naturally in the study of model-theoretic notions on models of size aleph_1 as well as in generalised descriptive set theory. In their 1994 paper devoted to various aspects of such trees, Mekler and Väänänen studied the so called weak embeddings between such trees, which are simply defined as strict-order preserving functions. Their work raised the question if under MA there exists a universal wide Aronszajn tree under such embeddings. We present a negative solution to this question, obtained in a paper to appear, joint with Shelah. We also discuss various connected notions and the history of the problem. Information: Contact Assaf Rinot for the zoom id. June 16 Münster Set Theory Seminar Time: Tuesday, June 16, 4:15pm CEST Speaker: Farmer Schlutzenberg Title: Remarks on rank-into-rank embeddings part II Abstract: Recall that Woodin’s large cardinal axiom I0 gives an ordinal λ and an elementary embedding j:L(V_{λ+1})→L(V_{λ+1}) with critical point <λ. Using methods due to Woodin, we show that if ZFC+I0 is consistent then so is ZF+DC(λ)+ there is an ordinal λ and an elementary j:V_{λ+2}→V_{λ+2}”. (A version with the added assumption that V_{λ+1}^sharp exists is due to the author, and Goldberg observed that the appeal to V_{λ+1}^sharp could actually be replaced by some further calculations of Woodin’s.) Reference: https://arxiv.org/abs/2006.01077, “On the consistency of ZF with an elementary embedding from V_{λ+2} into V_{λ+2}”. Information: contact rds@wwu.de ahead of time in order to participate. Udine graduate course Time: Tuesday, June 16, 10:00-12:00 CEST Speaker: Vincenzo Dimonte, University of Udine Title: Generalized Descriptive Set Theory II, Lecture 4 Abstract: The objective of the course is to prove an analogue of Silver’s Theorem for the space$2^\lambda$, where$\lambda$is an uncountable cardinal of cofinality$\omega$, using some large cardinal strength (the proof is still unpublished). This result has been chosen as an example to show, more in general, how to generalize a deep classical theorem in this setting, which properties of singular cardinals are useful in that respect, and what are the main obstacles of the generalization. The proof will use some peculiarities of singular cardinal combinatorics and some large cardinal strength, and everything will be introduced in the first three lessons. The course is self-contained (despite the name), the only prerequisite is to know basic set theory (the theory of forcing, the most basic descriptive set theory, maybe inaccessible cardinals). The following is a tentative schedule:Tuesday 10.00-12.00 CEST, Friday 10.00-12.00 CEST, from 5 June 2020, for 5 lessons. Lesson 1: Measurable cardinals Lesson 2: Prikry forcing, diagonal Prikry forcing Lesson 3: Strong Prikry condition, “double” diagonal Prikry forcing Lesson 4: generalized G_0 dichotomy Lesson 5: generalized Silver Theorem Information: Via Microsoft Teams, to participate contact vincenzo.dimonte@uniud.it. Week 8-14 June June 12 Udine graduate course Time: Friday, June 12, 10:00-12:00 CEST Speaker: Vincenzo Dimonte, University of Udine Title: Generalized Descriptive Set Theory II, Lecture 1 Abstract: The objective of the course is to prove an analogue of Silver’s Theorem for the space$2^\lambda$, where$\lambda$is an uncountable cardinal of cofinality$\omega$, using some large cardinal strength (the proof is still unpublished). This result has been chosen as an example to show, more in general, how to generalize a deep classical theorem in this setting, which properties of singular cardinals are useful in that respect, and what are the main obstacles of the generalization. The proof will use some peculiarities of singular cardinal combinatorics and some large cardinal strength, and everything will be introduced in the first three lessons. The course is self-contained (despite the name), the only prerequisite is to know basic set theory (the theory of forcing, the most basic descriptive set theory, maybe inaccessible cardinals). The following is a tentative schedule:Tuesday 10.00-12.00 CEST, Friday 10.00-12.00 CEST, from 5 June 2020, for 5 lessons. Lesson 1: Measurable cardinals Lesson 2: Prikry forcing, diagonal Prikry forcing Lesson 3: Strong Prikry condition, “double” diagonal Prikry forcing Lesson 4: generalized G_0 dichotomy Lesson 5: generalized Silver Theorem Information: Via Microsoft Teams, to participate contact vincenzo.dimonte@uniud.it. June 10 CUNY Set Theory Seminar Time: Wednesday, June 10, 7pm New York time (June 11, 1am CEST) Speaker: Zachiri McKenzie Title: Initial self-embeddings of models of set theory: Part II 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. Bristol Logic and Set Theory Seminar Time: Wednesday, June 10, 16:15-17:45 UK time (17:15-18:45 CEST) Speaker: Peter Holy, University of Udine Title: Ideal and Tree Forcing Topologies Abstract: While the usual topology on the kappa-reals is based on the bounded ideal, in the sense that the basic open sets are generated by bounded partial functions from kappa to 2, we consider generalized topologies based on arbitrary ideals on kappa, in particular on the nonstationary ideal on regular and uncountable cardinals kappa. We will illustrate the connections of these topologies with certain tree forcing topologies, and in particular the connection of the nonstationary topology with the topology generated by kappa-Silver forcing. We will show how properties of this forcing notion carry over to properties of the nonstationary topology, and we will also generalize results on kappa-Silver forcing of Friedman, Khomskii and Kulikov. This is joint work with Marlene Koelbing, Philipp Schlicht and Wolfgang Wohofsky. Information: Please login to the zoom meeting https://zoom.us/j/95916684321 a few minutes before the talk. Paris-Lyon Séminaire de Logique Time: Wednesday, June 10, 16:00-17:15 CEST Speaker: Assaf Rinot Title: Transformations of the transfinite plane Abstract: We study the existence of transformations of the transfinite plane that allow to reduce Ramsey-theoretic statements concerning uncountable Abelian groups into classic partition relations for uncountable cardinals. This is joint work with Jing Zhang. Information: Join via the link on the seminar webpage 10 minutes before the talk. Bristol Logic and Set Theory Seminar Time: Wednesday, June 10, 14:00-15:30 UK time (15:00-16:30 CEST) Speaker: Philip Welch, University of Bristol Title: Higher type recursion for Infinite time Turing Machines VII 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. June 9 Münster Set Theory Seminar Time: Tuesday, June 9, 4:15pm CEST Speaker: Farmer Schlutzenberg Title: Remarks on rank-into-rank embeddings Abstract: Recall that Woodin’s large cardinal axiom I0 gives an ordinal λ and an elementary embedding j:L(V_{λ+1})→L(V_{λ+1}) with critical point <λ. Using methods due to Woodin, we show that if ZFC+I0 is consistent then so is ZF+DC(λ)+ there is an ordinal λ and an elementary j:V_{λ+2}→V_{λ+2}”. (A version with the added assumption that V_{λ+1}^sharp exists is due to the author, and Goldberg observed that the appeal to V_{λ+1}^sharp could actually be replaced by some further calculations of Woodin’s.) Reference: https://arxiv.org/abs/2006.01077, “On the consistency of ZF with an elementary embedding from V_{λ+2} into V_{λ+2}”. Information: contact rds@wwu.de ahead of time in order to participate. Udine graduate course Time: Tuesday, June 9, 10:00-12:00 CEST Speaker: Vincenzo Dimonte, University of Udine Title: Generalized Descriptive Set Theory II, Lecture 1 Abstract: The objective of the course is to prove an analogue of Silver’s Theorem for the space$2^\lambda$, where$\lambda$is an uncountable cardinal of cofinality$\omega$, using some large cardinal strength (the proof is still unpublished). This result has been chosen as an example to show, more in general, how to generalize a deep classical theorem in this setting, which properties of singular cardinals are useful in that respect, and what are the main obstacles of the generalization. The proof will use some peculiarities of singular cardinal combinatorics and some large cardinal strength, and everything will be introduced in the first three lessons. The course is self-contained (despite the name), the only prerequisite is to know basic set theory (the theory of forcing, the most basic descriptive set theory, maybe inaccessible cardinals). The following is a tentative schedule:Tuesday 10.00-12.00 CEST, Friday 10.00-12.00 CEST, from 5 June 2020, for 5 lessons. Lesson 1: Measurable cardinals Lesson 2: Prikry forcing, diagonal Prikry forcing Lesson 3: Strong Prikry condition, “double” diagonal Prikry forcing Lesson 4: generalized G_0 dichotomy Lesson 5: generalized Silver Theorem Information: Via Microsoft Teams, to participate contact vincenzo.dimonte@uniud.it. Week 1-7 June June 5 Udine graduate course Time: Friday, June 5, 10:00-12:00 CEST Speaker: Vincenzo Dimonte, University of Udine Title: Generalized Descriptive Set Theory II, Lecture 1 Abstract: The objective of the course is to prove an analogue of Silver’s Theorem for the space$2^\lambda$, where$\lambda$is an uncountable cardinal of cofinality$\omega$, using some large cardinal strength (the proof is still unpublished). This result has been chosen as an example to show, more in general, how to generalize a deep classical theorem in this setting, which properties of singular cardinals are useful in that respect, and what are the main obstacles of the generalization. The proof will use some peculiarities of singular cardinal combinatorics and some large cardinal strength, and everything will be introduced in the first three lessons. The course is self-contained (despite the name), the only prerequisite is to know basic set theory (the theory of forcing, the most basic descriptive set theory, maybe inaccessible cardinals). The following is a tentative schedule:Tuesday 10.00-12.00 CEST, Friday 10.00-12.00 CEST, from 5 June 2020, for 5 lessons. Lesson 1: Measurable cardinals Lesson 2: Prikry forcing, diagonal Prikry forcing Lesson 3: Strong Prikry condition, “double” diagonal Prikry forcing Lesson 4: generalized G_0 dichotomy Lesson 5: generalized Silver Theorem Information: Via Microsoft Teams, to participate contact vincenzo.dimonte@uniud.it. June 4 Kurt Gödel Research Center Seminar (organised by Ben Miller) Time: Thursday, June 4, 16:00 CEST Speaker: Stefan Hoffelner, Universität Münster 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 Delta-1-2-determinacy implies the Sigma-1-3 separation property; yet Detla-1-2-determinacy implies an inner model with a Woodin cardinal. The question whether the Sigma-1-3-separation property is consistent relative to jsut ZFC remained open however since Mathias “Surrealist Landscape”-paper. We show that one can force it over L. Information: Talk via zoom. June 3 Bristol Logic and Set Theory Seminar Time: Wednesday, June 3, 14:00-15:30 UK time (15:00-16:30 CEST) Speaker: Philip Welch, University of Bristol Title: Higher type recursion for Infinite time Turing Machines VI 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. Week 25-31 May 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. 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 Toronto Set Theory Seminar Time: Friday, May 29, 1.30pm Toronto time (7.30pm CEST) Speaker: Michael Hrusak (UNAM) Title: Invariant Ideal Axiom Abstract: We introduce the Invariant Ideal Axiom and discuss its impact on the structure of countable topological groups. (joint work with Alexander Shibakov) Information: The seminar will take place virtually. ZOOM ID: https://yorku.zoom.us/j/96087161597 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 27 Oxford Set Theory Seminar Time: Wednesday, May 27, 16:00-17:30 UK time (17:00-18:30 CEST) 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 investihttp://jdh.hamkins.org/oxford-set-theory-seminar/gated 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. Information: For the Zoom access code, contact Samuel Adam-Day: me@samadamday.com. 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. Bristol Logic and Set Theory Seminar Time: Wednesday, May 27, 14:00-15:30 UK time (15:00-16:30 CEST) Speaker: Philip Welch, University of Bristol Title: Higher type recursion for Infinite time Turing Machines V 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. 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. 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. May 26 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. 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. Week 18-24 May 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. May 20 Oxford Set Theory Seminar Time: Wednesday, May 20, 16:00-17:30 UK time (17:00-18:30 CEST) Speaker: Joel David Hamkins, University of 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 ⟨Hω1,∈⟩ and ⟨Hω2,∈⟩ 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: For the Zoom access code, contact Samuel Adam-Day: me@samadamday.com. Paris-Lyon Séminaire de Logique Time: Wednesday, May 6, 16:00-17:15 CEST Speaker: Michale Hrusak, University of Morelia Title: Strong measure zero in Polish groups 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. (Joint with W. Wohofsky, J. Zapletal and/or O. Zindulka.) 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 (15:00-16:30 CEST) 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. Week 11-17 May 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. 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 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 13 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. Bristol Logic and Set Theory Seminar Time: Wednesday, May 13, 14:00-15:30 UK time (15:00-16:30 CEST) 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. Information: Please contact Philip Welch (p.welch@bristol.ac.uk) ahead of time to participate. 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 (joint work with Chris Laskowski.) 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. May 12 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. 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: Please contact Ralf Schindler (rds@wwu.de ) ahead of time to participate. Week 4-10 May May 8 CUNY Set Theory Seminar Time: Friday, May 8, 2pm New York time (8pm CEST) Speaker: Sandra Müller, Universität Wien 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. 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. May 7 Kurt Gödel Research Center Seminar (organised by Ben Miller) Time: Thursday, May 7, 16:00 CEST Speaker: Stefan Hoffelner, University of Münster Title: Tba Abstract: Tba Information: Talk via zoom. May 6 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. Information: For the Zoom access code, contact Samuel Adam-Day: me@samadamday.com. 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. Bristol Logic and Set Theory Seminar Time: Wednesday, May 6, 14:00-15:30 UK time (15:00-16:30 CEST) Speaker: Philip Welch, University of Bristol Title: Higher type recursion for Infinite time Turing Machines II 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. May 5 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 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. Information: The seminar will be held remotely via zoom. Please contact rds@wwu.de ahead of time in order to participate. Week 27 April-3 May May 1 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. 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. 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. 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 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. Information: The seminar will be held remotely via zoom. Please contact rds@wwu.de ahead of time in order to participate. Week 20-26 April 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. Toronto Set Theory Seminar Time: Friday, April 24, 1:30-3:00pm EDT (19:30-21:00 CEST) Speaker: Todd Eisworth, Ohio University Title: Representability and pseudopowers. Abstract: We will prove some basic facts about Shelah’s pseudopower function, and derive some new (?) ZFC results in cardinal arithmetic using basic topological ideas. This talk is designed to be an introduction to this part of pcf theory. Information: The talk will take place via zoom: https://yorku.zoom.us/j/925557716. 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 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 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\$.

Week 13-19 April

April 17

CUNY Set Theory Seminar
Time: Friday, April 17, 2pm New York time (8pm CEST)
Speaker: Corey Switzer, CUNY
Title: Specializing Wide Trees Without Adding Reals
Abstract: An important advancement in iterated forcing was Jensen’s proof that CH does not imply ♢ by iteratively specializing Aronszajn trees with countable levels without adding reals thus producing a model of CH plus ‘all Aronszajn trees are special’. This proof was improved by Shelah who developed a general method around the notion of dee-complete forcing. This class (under certain circumstances) can be iterated with countable support and does not add reals. However, neither Jensen’s nor Shelah’s posets will specialize trees of uncountable width and it remains unclear when one can iteratively specialize wider trees. Indeed a very intriguing example, due to Todorčević, shows that there is always a wide Aronszajn tree which cannot be specialized without adding reals. By contrast the ccc forcing for specializing Aronszajn trees makes no distinction between trees of different widths (but may add many reals). In this talk we will provide a general criteria a wide trees Aronszajn tree can have that implies the existence of a dee-complete poset specializing it. Time permitting we will discuss applications of this forcing to forcing axioms compatible with CH and some open questions related to set theory of the reals.
Information: The seminar will take place virtually. Please email Victoria Gitman (vgitman@nylogic.org) for the meeting id.

Toronto Set Theory Seminar
Time: Friday, April 17, 1:30-3:00pm EDT (19:30-21:00 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.
Information: The talk will take place via zoom: https://yorku.zoom.us/j/925557716.