May 26

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

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

May 27

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

**Paris-Lyon Séminaire de LogiqueTime:** Wednesday, May 27, 16:00-17:15 CEST

**Speaker:**Eliott Kaplan – University of Illinois at Urbana-Champaign

**Title:**Model completeness for the differential field of transseries with exponentiation

**Abstract:**I will discuss the expansion of the differential field of logarithmic-exponential transseries by its natural exponential function. This expansion is model complete and locally o-minimal. I give an axiomatization of the theory of this expansion that is effective relative to the theory of the real exponential field. These results build on Aschenbrenner, van den Dries, and van der Hoeven’s model completeness result for the differential field of transseries. My method can be adapted to show that the differential field of transseries with its restricted sine and cosine and its unrestricted exponential is also model complete and locally o-minimal.

**Information:**Join via the link on the seminar webpage 10 minutes before the talk.

**Bristol Logic and Set Theory SeminarTime:** Wednesday, May 27, 14:00-15:30 (UK time)

**Speaker:**Philip Welch, University of Bristol

**Title:**Higher type recursion for Infinite time Turing Machines IV

**Abstract:**This is part of a series informal working seminars on an extension of Kleene’s early 1960’s on recursion in higher types. (This formed a central theme on the borders of set theory and recursion theory in the 60’s and early 70’s, although now unfortunately not much discussed. Amongst the main names here were Gandy, Aczel, Moschovakis, Harrington, Normann.) We aim to present a coherent version of type-2 recursion for the infinite time Turing machine model. We aim to be somewhat (but not entirely) self-contained. Basic descriptive set theory, and recursion theory, together with admissibility theory will be assumed.

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

**CUNY Set Theory Seminar****Time:** Wednesday, May 27, 7pm New York time (1am May 14 CEST)**Speaker: **Bartosz Wcisło, University of Warsaw**Title:** Tarski boundary II**Abstract:** Truth theories investigate the notion of truth with axiomatic methods. To a fixed base theory (typically Peano Arithmetic PA) we add a unary predicate T(x) with the intended interpretation ‘x is a (code of a) true sentence.’ Then we analyse how adding various possible sets of axioms for that predicate affects its behaviour. One of the aspects we are trying to understand is which truth-theoretic principles make the added truth predicate ‘strong’ in that the resulting theory is not conservative over the base theory. Ali Enayat proposed to call this ‘demarcating line’ between conservative and non-conservative truth theories ‘the Tarski boundary.’ Research on Tarski boundary revealed that natural truth theoretic principles extending compositional axioms tend to be either conservative over PA or exactly equivalent to the principle of global reflection over PA. It says that sentences provable in PA are true in the sense of the predicate T. This in turn is equivalent to Δ0-induction for the compositional truth predicate which turns out to be a surprisingly robust theory.

In our talk, we will try to sketch proofs representative of research on Tarski boundary. We will present the proof by Enayat and Visser showing that the compositional truth predicate is conservative over PA. We will also try to discuss how this proof forms a robust basis for further conservativeness results.

On the non-conservative side of Tarski boundary, the picture seems less organised, since more arguments are based on *ad hoc* constructions. However, we will try to show some themes which occur rather repeatedly in these proofs: iterated truth predicates and the interplay between properties of good truth-theoretic behaviour and induction. To this end, we will present the argument that disjunctive correctness together with the internal induction principle for a compositional truth predicate yields the same consequences as Δ0-induction for the compositional truth predicate (as proved by Ali Enayat) and that it shares arithmetical consequences with global reflection. The presented results are currently known to be suboptimal.

This talk is intended as a continuation of ‘Tarski boundary’ presentation. However, we will try to avoid excessive assumptions on familiarity with the previous part.**Information:** The seminar will take place virtually. Please email Victoria Gitman (vgitman@nylogic.org) for the meeting id.

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

May 28

**Kurt Gödel Research Center Seminar** (organised by Ben Miller)**Time:** Thursday, May 28, 16:00 CEST

**Speaker:**Diego Mejía, Shizuoka University, Japan

**Title:**Preserving splitting families

**Abstract:**We present a method to force splitting families that can be preserved by a large class of finite support iterations of ccc posets. As an application, we show how to force several cardinal characteristics of the continuum to be pairwise different.

**Information:**Talk via zoom.

May 29

**CUNY Set Theory Seminar****Time:** Friday, May 29, 2pm New York time (8pm CEST)**Speaker:** Kameryn Williams University of Hawaii at Mānoa**Title:** The geology of inner mantles**Abstract:** An inner model is a ground if V is a set forcing extension of it. The intersection of the grounds is the mantle, an inner model of ZFC which enjoys many nice properties. Fuchs, Hamkins, and Reitz showed that the mantle is highly malleable. Namely, they showed that every model of set theory is the mantle of a bigger, better universe of sets. This then raises the possibility of iterating the definition of the mantle—the mantle, the mantle of the mantle, and so on, taking intersections at limit stages—to obtain even deeper inner models. Let’s call the inner models in this sequence the inner mantles.

In this talk I will present some results about the sequence of inner mantles, answering some questions of Fuchs, Hamkins, and Reitz. Specifically, I will present the following results, analogues of classic results about the sequence of iterated HODs.

1. (Joint with Reitz) Consider a model of set theory and consider an ordinal eta in that model. Then this model has a class forcing extension whose eta-th inner mantle is the model we started out with, where the sequence of inner mantles does not stabilize before eta.

2. It is consistent that the omega-th inner mantle is an inner model of ZF + ¬AC.

3. It is consistent that the omega-th inner mantle is not a definable class, and indeed fails to satisfy Collection.**Information:** The seminar will take place virtually. Please email Victoria Gitman (vgitman@nylogic.org) for the meeting id.

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

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