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

We announce talks from the following set theory related seminars and colloquia:

– **European Set Theory Society Panel Discussions**, Thursdays 17:00-19:00 CEST/CET, several times each year

– Bristol Logic and Set Theory Seminar, Tuesdays 12:30 local time (13:30 CEST/CET)

– Vienna Set Theory Seminar, Tuesdays 15:00-16:30 local time (15:00-16:00 CEST/CET)

– Baltic Set Theory Seminar, Tuesdays 15:00-16:30 local time (15:00-16:00 CEST/CET)

– Carnegie-Mellon University Pittsburgh Core Model Seminar, Tuesdays 1:30pm local time (19:30 CEST)

– Carnegie-Mellon University Pittsburgh Logic Seminar, Tuesdays 3:30pm local time (21:30 CEST)

– Helsinki Logic Group Logic Seminar, Wednesdays 12.15 local time (11:15 CEST/CET)

– Leeds Models and Sets Seminar, Wednesdays, 13:45-15:00 local time (14:45-16:00 CEST/CET)

– Barcelona Set Theory Seminar, Wednesdays 4pm local time (15:00-16:30 CEST/CET)

– Caltech: Logic seminar, Wednesdays 11am-12pm local time (20:00-21:00 CEST/CET except in the first week of November, then 20:00 CET)

– Vienna Logic Colloquium, Thursdays 15:00 – 16:30 local time (15:00 – 16:30 CEST/CET)

– Cross-Alps Logic Seminar, Fridays 16:30 local time (16:30 CEST/CET)

– CUNY Set Theory Seminar, Fridays 12:15pm-1:45pm local time (18:15-19:45 CEST/CET except in the first week of November, then 17:15-18:45 CET)

– University of Toronto Set Theory seminar, Fridays 1:30pm local time (19:30 CEST/CET except in the first week of November, then 18:30 CET)

– CUNY Logic Workshop, Fridays 2:00pm-3:30pm local time (20:00-21:30 CEST/CET except in the first week of November, then 19:00-20:30 CET)

26 December – 1 January

No announcements

19-25 December

No announcements

12-18 December

**Vienna Research Seminar in Set Theory****Time:** Tuesday, 13 December, 15:00-16:30 CET**Speaker:** W. Kubiś, Czech Academy of Sciences**Title:** The weak Ramsey property**Abstract:** The weak Ramsey property is a variant of the finite Ramsey property, used for characterizing extreme amenability of automorphism groups of ultra-homogeneous

structures, known as the Kechris – Pestov – Todorcevic correspondence. We shall describe a far reaching extension of the KPT correspondence, indicating also how it works in metric-enriched categories, capturing objects from topology and functional analysis.**Information:** This talk will be given in hybrid format. Please contact Richard Springer for information how to participate.

**Baltic Set Theory Seminar****Time:** Tuesday, 13 December, 15:00-16:30 CET**Speaker:** Several**Title:** Baltic Set Theory Seminar**Abstract:** This is a learning seminar, the goal is to actually go over proofs and more or less understand them. Discussions are encouraged. The topic of the seminar is the following:

1. Sandra Müller, Stationary-tower-free proof of Woodin’s Sealing Theorem.

2. Matteo Viale, Generic absoluteness theorem for the omega_1 Chang model conditioned to MM^{+++}.**Information:** Please see the seminar webpage.

**Helsinki Logic Seminar****Time:** Wednesday, 14 December, 12:00 – 14:00 Helsinki time (11:00-13:00 CET)**Speaker:** Sabrina Gaube**Title:** tba**Abstract:** tba**Information:** The talk will take place in hybrid mode. Please see the seminar webpage for the link.

**Leeds Models and Sets Seminar****Time:** Wednesday, 14 December, 13:45-15:00 local time (14:45-16:00 CET)**Speaker:** Jonathan Schilhan, University of Leeds**Title:** Wetzel’s problem and the continuum**Abstract:** In the early 60’s, John Wetzel came up with the following question in his PhD thesis on harmonic functions: If $\mathcal{F}$ is a family of entire functions (functions that are holomorphic on the complex plane) which at each point attains at most countably many values, is $\mathcal{F}$ itself necessarily countable? This question makes sense considering the quite restrictive nature of holomorphic functions. Not much thereafter, Erdős could show that a negative answer to Wetzel’s Problem is in fact equivalent to the continuum hypothesis. His argument shows that any family of entire functions, that attains at each point less values than elements of that family, must have size continuum. Recently Kumar and Shelah have shown that consistently such a family exists while the continuum has size $\aleph_{\omega_1}$. We answer their main open problem by showing that continuum $\aleph_2$ is possible as well. This is joint work with Thilo Weinert.**Information:** Please see the seminar webpage.

**Vienna Logic Colloquium****Time:** Thursday, 15 December, 15:00 – 16:30 CET

**Speaker:**W. Kubiś , Czech Academy of Sciences

**Title:**Abstract Evolution Systems

**Abstract:**An abstract evolution system is a category endowed with a fixed family of arrows (called transitions) and with a distinguished object, called the origin. An evolution is an infinite sequence of transitions starting from the origin. We will show that evolution systems provide a good framework for the study of highly symmetric mathematical structures, namely those having rich automorphism groups. On the other hand, evolution systems also describe terminating transition systems, leading to an extension of the celebrated Newman’s Lemma: A locally confluent terminating system is confluent.

**Information:**This talk will be given in hybrid format. Please contact Richard Springer for information how to participate.

**Vienna Logic Colloquium****Time:** Thursday, 15 December, 16:45 – 18:15 CET

**Speaker:**R. Schindler, University of Münster

**Title:**Martin’s Maximum, Woodin’s P_max axiom (*), and Cantor’s Continuum Problem

**Abstract:**In 2019, D. Asperó and the speaker showed that Martin’s Maximum++ implies the Pmax axiom (∗). This amalgamated two prominent maximality principles which before had often been considered as competitors. We provide some background and give an outline of the proof method. We also discuss to which extent our result has an impact on the question as to how many real numbers there are.

**Information:**This talk will be given in hybrid format. Please contact Richard Springer for information how to participate.

**Toronto Set Theory Seminar****Time:** Friday, 16 December, 1.30-3.00 Toronto time (19.30-21.00 CET)**Speaker:** tba**Title:** tba**Abstract:** tba**Information:** Please see http://gfs.fields.utoronto.ca/activities/22-23/set-theory-seminar.

5-11 December

**Vienna Research Seminar in Set Theory****Time:** Tuesday, 6 December, 15:00-16:30 CET**Speaker:** D. Chodounský, Czech Academy of Sciences**Title:** Introduction to big Ramsey degrees**Abstract:** I will give a quick introduction to big Ramsey degrees, sketch proofs of some basic results, and I will try to explain the ideas behind these proofs.

The talk is intended as an introduction to the topic, setting up the background for the talk of Jan Hubička. There will be a substantial overlap with the talk I gave at this seminar in June 2021.**Information:** This talk will be given in hybrid format. Please contact Richard Springer for information how to participate.

**Vienna Research Seminar in Set Theory****Time:** Tuesday, 6 December, 16:45-18:15 CET**Speaker:** J. Hubička, Charles University Prague**Title:** Big Ramsey degrees for structures with forbidden substructures**Abstract:** We discuss a new method used to prove that big Ramsey degrees of a given structure are finite. We start with a simple new proof of the theorem by Dobrinen showing the big Ramsey degrees of the homogeneous triangle free graphs are finite. This is based on an application of the Carlson-Simpson theorem. We outline how this proof can be carried to other structures including partial orders and metric spaces. Then we discuss a new theorem for trees with a successor operation that can be used to give bounds on big Ramsey degrees for structures with bigger forbidden configurations and in languages with higher arity.**Information:** This talk will be given in hybrid format. Please contact Richard Springer for information how to participate.

**Baltic Set Theory Seminar****Time:** Tuesday, 6 December, 15:00-16:30 CET**Speaker:** Several**Title:** Baltic Set Theory Seminar**Abstract:** This is a learning seminar, the goal is to actually go over proofs and more or less understand them. Discussions are encouraged. The topic of the seminar is the following:

1. Sandra Müller, Stationary-tower-free proof of Woodin’s Sealing Theorem.

2. Matteo Viale, Generic absoluteness theorem for the omega_1 Chang model conditioned to MM^{+++}.**Information:** Please see the seminar webpage.

**CMU Core Model Theory Seminar****Time:** Tuesday, 6 December, 1:30 – 3:00pm Pittsburgh time (19:30 – 21:00 CET) **Speaker:** Gabriel Goldberg, University of California, Berkeley**Title:** External ultrapowers of HOD in models of determinacy**Abstract: ** We show that in L(R) under determinacy, the external ultrapower

of HOD^L(R) by a countably complete ultrafilter on an ordinal less than

theta^L(R) is a normal iterate of HOD^L(R) via its iteration strategy.

This is joint work with Grigor Sargsyan.**Information:** Please contact Ernest Schimmerling in advance for the zoom link.

**Helsinki Logic Seminar****Time:** Wednesday, 7 December, 12:00 – 14:00 Helsinki time (11:00-13:00 CET)**Speaker:** Aleksi Anttila**Title:** A remark on the dual negation in propositional/modal team semantics**Abstract:** tba**Information:** The talk will take place in hybrid mode. Please see the seminar webpage for the link.

**Leeds Models and Sets Seminar****Time:** Wednesday, 7 December, 13:45-15:00 local time (14:45-16:00 CET)**Speaker:** Deirdre Haskell, McMaster University**Title:** Residue field domination in some theories of valued fields**Abstract:** A paraphrase of the Ax-Kochen-Ersov theorem for some theories of valued fields is that the elementary theory is determined by the theory of the value group and the residue field. At the level of types, the intuition is that a type should be controlled by its trace in each of the residue field and value group. In this talk, I will explore some ways in which this intuition can be made precise, and also some limitations to that preliminary intuition. I will try to give lots of examples to keep the discussion concrete.**Information:** Please see the seminar webpage.

**Caltech Logic Seminar****Time:** Wednesday, 7 December, 11:00am-12:00pm Pacific time (20:00-21:00 CET)**Speaker:** Aristotelis Panagiotopoulos, CMU**Title:** General relativity does not admit enough observables**Abstract:** One of the biggest open problems in mathematical physics has been the problem of formulating a complete and consistent theory of quantum gravity. Some of the core technical and epistemological difficulties come from the fact that General Relativity (GR) is fundamentally a geometric theory and, as such, it oughts to be “generally covariant”, i.e., invariant under change of coordinates by any element of the diffeomorphism group Diff(M)Diff(M) of the ambient manifold MM. The *Problem of Observables* is a famous instance of the difficulties associated with general covariance, and one directly related to ineffectiveness of classical quantization recipes when it comes to GR. In a nutshell, the problem of observables asks whether GR admits a complete set of smooth observables. That is, whether the family of all diffeomorphism-invariant, real-valued, smooth maps on the space Ein(M)Ein(M) of all Einstein metrics on MM is rich enough to separate all physical spacetimes. So far the only smooth observables known (when M=R4)M=R4) are the constant maps. In this talk, we will employ methods from descriptive set theory in order to answer the problem of observables in the negative. These results are inspired by old discussions with Marios Christodoulou and are based on recent work with George Sparling.**Information:** Please see the seminar webpage.

**CUNY Set Theory Seminar****Time:** Friday, 9 December, 12:30pm New York time (18:30 CET)**Speaker: **Vladimir Kanovei, Institute for Information Transmission Problems**Title:** On the significance of parameters in the comprehension and choice schemata in second-order arithmetic**Abstract:** Parameters are free variables in various axiom schemata in PA, ZFC, and other similar theories. Given an axiom schema S, we let S* be the parameter-free sub-schema. Kreisel (A survey of proof theory, JSL 1968) was one of those who paid attention to the comparison of some schemata in second-order PA and their parameter-free versions. In particular, Kreisel noted that […] if one is convinced of the significance of something like a given axiom schema, it is natural to study details, such as the effect of parameters. This talk is devoted to the effect of parameters in the schemata of Comprehension and Choice in second-order arithmetic.**Information:** The seminar will take place virtually. Please email Victoria Gitman (vgitman@nylogic.org) for the meeting id.

**Toronto Set Theory Seminar****Time:** Friday, 9 December, 1.30-3.00 Toronto time (19.30-21.00 CET)**Speaker:** Spencer Unger**Title:** Ramsey theory of trees**Abstract:** We make several observations about the infinite dimensional Ramsey theory of trees. To start we define a weakening of a Ramsey space and give a natural class of examples these spaces which do not satisfy Todorcevic’s axioms (in particular the amalgamation axiom fails). Motivated by these examples, we give a classical proof of a version of the Halpern-Lauchli theorem which allows us to analyze spaces of copies of the rationals and more generally SDAP classes of Coulson-Dobrinen-Patel. This is joint work with Stevo Todorcevic.**Information:** Please see http://gfs.fields.utoronto.ca/activities/22-23/set-theory-seminar.

**CUNY Logic Workshop****Time:** Friday, 9 December, 2:00 – 3:30 New York time (20:00-21:30 CET)**Speaker: **Daniel Turetsky, Victoria University of Wellington**Title: **Wadge degrees, games, and the separation and reduction properties**Abstract:** In this talk, I will give an overview of the picture of the Borel Wadge degrees. Our system of descriptions allows us to describe their Delta-classes, as well as specify which degrees have the separation or reduction properties. Part of our analysis is based on playing games along our descriptions, and so I will explain how these games are played and what they can tell us.**Information:** The talk will take place in hybrid mode. Please see the seminar webpage or email Victoria Gitman for the link.

28 November – 4 December

**Vienna Research Seminar in Set Theory****Time:** Tuesday, 29 November, 15:00-16:30 CET**Speaker:** Corey Switzer, Universität Wien **Title:** Subversion Forcing, part 2**Abstract:** In these two talks we will introduce Jensen’s classes of subcomplete and subproper forcing as well as discuss some applications due to the speaker and Fuchs, and the speaker and Sakai. An important feature of proper forcing is the countable covering property: every countable set of ordinals added by a proper forcing notion is contained in a ground model countable set of ordinals. This is important in iteration theorems. Subproper forcing is a weakening of proper forcing that is still iterable while including some well known forcing notions which do add countable sets of ordinals that are not covered by anything in the ground model including Namba forcing (under CH) and Prikry forcing. One can weaken other classes of forcing notions in a similar way and the “sub”version of the countably closed forcing, known as subcomplete forcing, is a particularly interesting subclass of subproper forcing that was used by Jensen in several applications including his solution to the extended Namba problem.

In the first of these talks I will introduce the classes subproper and subcomplete forcing as well as discuss simplifications of them due to Fuchs and myself. Time permitting I will discuss new iterations theorems for these classes reminiscent of similar theorems proved for proper forcing in the context of the reals and combinatorics on ω1 (ωω-bounding, preservation of Souslin trees etc). In the second talk I will discuss the forcing axioms for these classes including their applications and limitations. In particular, time permitting, I will discuss a recent result, joint with Hiroshi Sakai that the forcing axiom for subcomplete forcing is compatible with a □ω1-sequence. The take away is a class of strong forcing axioms that are compatible with a wide variety of behaviour on the level of the reals and combinatorics on cardinals below the continuum.**Information:** This talk will be given in hybrid format. Please contact Richard Springer for information how to participate.

**Baltic Set Theory Seminar****Time:** Tuesday, 29 November, 15:00-16:30 CET**Speaker:** Several**Title:** Baltic Set Theory Seminar**Abstract:** This is a learning seminar, the goal is to actually go over proofs and more or less understand them. Discussions are encouraged. The topic of the seminar is the following:

1. Sandra Müller, Stationary-tower-free proof of Woodin’s Sealing Theorem.

2. Matteo Viale, Generic absoluteness theorem for the omega_1 Chang model conditioned to MM^{+++}.**Information:** Please see the seminar webpage.

**CMU Core Model Theory Seminar****Time:** Tuesday, 29 November, 1:30 – 3:00pm Pittsburgh time (19:30 – 21:00 CET) **Speaker:** Benjamin Siskind, Carnegie Mellon University**Title:** Full normalization and HOD^L(R)**Abstract: **We’ll review some aspects of the HOD analysis of L(R) and

introduce the full normalization machinery developed by Steel and

Schlutzenberg. We’ll use this to show that HOD^L(R)|theta is actually a

normal iterate of M_omega|delta via M_omega’s canonical iteration strategy

(and a bit more), a result of Steel and Schlutzenberg. Time permitting, we

may discuss other applications of full normalization, due to Steel.**Information:** Please contact Ernest Schimmerling in advance for the zoom link.

**CMU Logic Seminar****Time:** Tuesday, 29 November, 3:30 – 4:30pm Pittsburgh time (21:30 – 22:30 CET) **Speaker:** Verónica Becher, University of Buenos Aires**Title:** Turing’s Normal Numbers**Abstract: **In a manuscript entitled “A note on normal numbers” and written

presumably in 1937 Alan Turing gave an algorithm that produces real

numbers normal to every integer base. This proves, for the first time, the

existence of computable instances and provides an answer to Borel’s

problem on giving examples of normality. Furthermore, with this work

Turing pioneers the theory of randomness and shows that he had the

insight, ahead of his time, that traditional mathematical concepts, like

measure or continuity, could be made computational. In this talk I will

highlight the ideas in these achievements of Turing, which are largely

unknown because his manuscript remained unpublished until it appeared in

his Collected Works in 1992.**Information:** See the seminar webpage.

**Hebrew University-Bar Ilan University Set Theory seminar****Time:** Wednesday, 30 November, 13:00-15:00 Israel Time (12:00-14:00 CET)**Speaker:** Grigor Sargsyan**Title:** An invitation to inner model theory**Abstract:** The talk will be aimed at non-experts and will outline some research directions pursued by inner model theorists. **Information:** Contact Menachem Magidor or Omer Ben-Neria ahead of time for the zoom link.

**Leeds Models and Sets Seminar****Time:** Wednesday, 30 November, 13:45-15:00 local time (14:45-16:00 CET)**Speaker:** Thilo Weinert, University of Vienna**Title:** Two New Inequalities for Cardinal Characteristics of the Continuum**Abstract:** Over the last decades the theory of cardinal characteristics of the continuum has emerged as one among several important subfields of set theory. Some of the classical results in it precede the invention of forcing and arguably the aforementioned emergence. Open problems in this field have inspired the invention over ever more versatile constructions of forcing notions and much of the progress has consisted of proving the values of cardinal characteristics not to be ZFC-provably related. A recent outlier has been the celebrated result by Malliaris and Shelah that p is equal to t. I had guessed that there might be more ZFC-provable relations between the hitherto defined characteristics and I am going to talk about what I found up to now. This is to say that I am going to present some ZFC-provable inequalities. In particular I am going to show that the evasion number is at most the subseries number. These cardinal characteristics have been introduced in work by Blass, Brendle, Brian, and Hamkins and originate from Algebra and Analysis, respectively. The proof interpolates via the pair-splitting number which is due to Minami.**Information:** Please see the seminar webpage.

**Caltech Logic Seminar****Time:** Wednesday, 30 November, 11:00am-12:00pm Pacific time (20:00-21:00 CET)**Speaker:** Sumun Iyer, Cornell**Title:** Dynamics of the Knaster continuum homeomorphism group**Abstract:** We use the projective Fraissé approach and Ramsey’s theorem to show that the universal minimal flow of the homeomorphism group of the universal Knaster continuum is homeomorphic to the universal minimal flow of the free abelian group on countably many generators. We will define a projective Fraissé class whose limit approximates the universal Knaster continuum in such a way that the group Aut(K)Aut(K) of automorphisms of the Fraissé limit is a dense subgroup of the group, Homeo(K)Homeo(K), of homeomorphisms of the universal Knaster continuum. The computation of the universal minimal flow involves modifying the Fraissé class in a natural way so that it approximates an open, normal, extremely amenable subgroup of Homeo(K)Homeo(K).**Information:** Please see the seminar webpage.

**Cross-Alps Logic Seminar****Time:** Friday, 2 December 16.00-18.00 CET**Speaker:** F. Parente, University of Turin**Title:** Good ultrafilters and universality properties of forcing**Abstract:** In 1964, Keisler introduced κ-good ultrafilters, which can be characterized as those ultrafilters which produce κ-saturated ultrapowers. The problem of finding an analogous characterization for ultrafilters on Boolean algebras has been considered by Mansfield (1971), Benda (1974), and Balcar and Franek (1982), who proposed and compared different notions of “goodness” for such ultrafilters. In the first part of my talk, I shall outline the different definitions introduced in the literature and show that they are in fact all equivalent, thus providing a complete characterization of those ultrafilters which produce κ-saturated Boolean ultrapowers. In the second part of the talk, I shall present a joint work with Matteo Viale, which started in 2015 during my Master’s thesis and was recently revived during the last few months in Torino. The aim of our project is to study the universality properties of forcing. More precisely, we shall prove that, for many interesting signatures, every model of the universal theory of an initial segment of the universe can be embedded into a model constructed by forcing. To achieve this goal, we build good ultrafilters on forcing notions such as the Lévy collapsing algebra and Woodin’s stationary tower.**Information:** The event will stream on the Webex platform. Please write to luca.mottoros [at] unito.it for the link to the event.

**Toronto Set Theory Seminar****Time:** Friday, 2 December, 1.30-3.00 Toronto time (19.30-21.00 CET)**Speaker:** Eduardo Duenez, University of Texas at San Antonio, and Jose Iovino, University of Texas at San Antonio**Title:** Integration and generalized functions in real-valued structures**Abstract:** We propose a formalism (joint with José Iovino) of measure and integration of functions in the framework of real-valued logic. In contrast to existing such frameworks that either rely on nonstandard analysis, impose strict pointwise bounds on functions, or abstract away the underlying measure theory, our setup is meant to be user-friendly for mathematicians working in (standard) analysis or probability. In our integration structures, classical functions peacefully coexist with function-like objects that, from a classical perspective, have “escaped off” to become supported at infinity, or have concentrated as point masses. As a case study, we present a simple proof (both conceptually and technically) of the stability of Orlicz function spaces. We also discuss the suitability of our framework as an alternate foundation to construct “randomized structures” à la Keisler.**Information:** Please see http://gfs.fields.utoronto.ca/activities/22-23/set-theory-seminar.

**CUNY Logic Workshop****Time:** Friday, 2 December, 2:00 – 3:30 New York time (20:00-21:30 CET)**Speaker: **Michał Tomasz Godziszewski, University of Vienna**Title: **Cardinal characteristics of the Calkin algebra and other interactions between logic and operator algebras**Abstract:** In recent years we have been witnessing a dynamic and fertile connection between logic and operator algebras. Many methods from set theory and model theory have been successfully applied to the investigations of C∗-algebras and other topics in abstract functional analysis (with a brilliant textbook on the ‘Combinatorial Set Theory of C∗-algebras’ by I. Farah presenting the current state of the art in this developing field). The purpose of this talk is to provide an introduction to this fruitful interplay with a focus on a certain set-theoretic problem concerning cardinal characteristics of the Calkin algebra which is a structure that may be thought of as a quantum counterpart of the Boolean algebra of subsets of natural numbers modulo finite sets.

Namely, I will present a result concerning possible sizes of families of projections (on a certain Hilbert space) that are mutually pairwise almost orthogonal, which informally means that they are orthogonal modulo ‘compact perturbation’. The aforementioned result is joint work with V. Fischer (Vienna).**Information:** The talk will take place in hybrid mode. Please see the seminar webpage or email Victoria Gitman for the link.

21-27 November

**Vienna Research Seminar in Set Theory****Time:** Tuesday, 22 November, 15:00-16:30 CET**Speaker:** Corey Switzer, Universität Wien **Title:** Subversion Forcing, part 1**Abstract:** In these two talks we will introduce Jensen’s classes of subcomplete and subproper forcing as well as discuss some applications due to the speaker and Fuchs, and the speaker and Sakai. An important feature of proper forcing is the countable covering property: every countable set of ordinals added by a proper forcing notion is contained in a ground model countable set of ordinals. This is important in iteration theorems. Subproper forcing is a weakening of proper forcing that is still iterable while including some well known forcing notions which do add countable sets of ordinals that are not covered by anything in the ground model including Namba forcing (under CH) and Prikry forcing. One can weaken other classes of forcing notions in a similar way and the “sub”version of the countably closed forcing, known as subcomplete forcing, is a particularly interesting subclass of subproper forcing that was used by Jensen in several applications including his solution to the extended Namba problem.

In the first of these talks I will introduce the classes subproper and subcomplete forcing as well as discuss simplifications of them due to Fuchs and myself. Time permitting I will discuss new iterations theorems for these classes reminiscent of similar theorems proved for proper forcing in the context of the reals and combinatorics on ω1 (ωω-bounding, preservation of Souslin trees etc). In the second talk I will discuss the forcing axioms for these classes including their applications and limitations. In particular, time permitting, I will discuss a recent result, joint with Hiroshi Sakai that the forcing axiom for subcomplete forcing is compatible with a □ω1-sequence. The take away is a class of strong forcing axioms that are compatible with a wide variety of behaviour on the level of the reals and combinatorics on cardinals below the continuum.**Information:** This talk will be given in hybrid format. Please contact Richard Springer for information how to participate.

**Baltic Set Theory Seminar****Time:** Tuesday, 22 November, 15:00-16:30 CET**Speaker:** Several**Title:** Baltic Set Theory Seminar**Abstract:** This is a learning seminar, the goal is to actually go over proofs and more or less understand them. Discussions are encouraged. The topic of the seminar is the following:

1. Sandra Müller, Stationary-tower-free proof of Woodin’s Sealing Theorem.

2. Matteo Viale, Generic absoluteness theorem for the omega_1 Chang model conditioned to MM^{+++}.**Information:** Please see the seminar webpage.

**Helsinki Logic Seminar****Time:** Wednesday, 23 November, 12:00 – 14:00 Helsinki time (11:00-13:00 CET)**Speaker:** Balthasar Grabmayr**Title:** Fixing Montague’s Problem**Abstract:** Turing machines only operate directly on strings. Turing computation over any other domain therefore requires a notation system for the domain. Ever since Montague’s (1960) observation that different notation systems in general yield different notions of Turing computability, the task of distinguishing those notation systems that are admissible for computation from those that are not continues to be a much debated and open problem in the philosophy of computation. In the first part of this talk, I will introduce a generalized version of Montague’s problem. In the second part, I will formulate and defend a solution to this problem.**Information:** The talk will take place in hybrid mode. Please see the seminar webpage for the link.

**Leeds Models and Sets Seminar****Time:** Wednesday, 23 November, 11:00-12:00 local time (12:00-13:00 CET) – note the taim change**Speaker:** Will Johnson, Fudan University**Title:** Around definable types in valued fields**Abstract:** Haskell, Hrushovski, and Macpherson showed that the theory ACVF of algebraically closed valued fields has elimination of imaginaries after adding the so-called “geometric sorts” to the language. The same result holds in $p$-adically closed fields ($p$CF) by work of Hrushovski, Martin, and Rideau. In the case of ACVF, one way to prove this is to encode imaginaries using definable types, and then encode definable types in the geometric sorts. While $p$CF does not have “enough” definable types to encode imaginaries, the encoding of definable types carries over. Surprisingly, the geometric sorts are unnecessary: any definable type in $p$CF has a code in the home sort (the field sort). This fact and its proof have some unexpected applications to definable groups and definable topological spaces in $p$CF. For example, certain quotient groups are definable rather than interpretable, and there is a unified notion of “definable compactness” for definable topological spaces. Parts of this talk are joint work with Pablo And\’ujar Guerrero.**Information:** Please see the seminar webpage.

**Barcelona Set Theory Seminar****Time:** Wednesday, 23 November, 16:00-17:30 CET**Speaker:** Jiachen Yuan**Title:** On the cofinality of the least omega_1-strongly compact cardinal**Abstract:** In [1,2], Bagaria and Magidor introduced the notion of l-strong

compactness, which generalized the well-known notion of strong compactness. Most

surprisingly, they proved, relative to a supercompact cardinal k with a measurable d

below k, that it is consistent that the least w1-strongly compact cardinal is singular, in

contrast to the fact that strongly compact cardinals are always inaccessible. In

particular, in their model they made k be the least w1-strongly compact cardinal, which

has cofinality exactly d.

In this talk, we explore all possible cofinalities of the least w1-strongly compact

cardinal. We’ll see that it is impossible to strengthen Bagaria-Magidor’s result under

their original assumption while we are able to prove under a slightly stronger (which is

showed to be necessary) that there is no non-trivial restrictions on the cofinality.

[1] Joan Bagaria, Menachem Magidor. On w1-strongly compact cardinals, Journal of

Symbolic Logic 79, 2014

[2] Joan Bagaria, Menachem Magidor. Group radicals and strongly compact cardinals,

Trans. Amer. Math. Soc. 366, 2014, No. 4, 1857-1877.**Information:** Online. If you wish to attend, please send an email to bagaria@ub.edu asking for the link.

**Vienna Logic Colloquium****Time:** Thursday, 24 November, 15:00 – 16:30 CET

**Speaker:**F.-V. Kuhlmann, University of Szczecin

**Title:**Nonstandard models of the reals and symmetrical completeness

**Abstract:**The notion of power series fields provides an easy method for the construction of nonstandard models of the ordered field of real numbers. I will define them, as well as Hahn products, which are their equivalent in the case of ordered abelian groups. The question arises whether these power series models can also have additional structures or properties that we know from the reals. For example, it was shown in joint work with Salma Kuhlmann and Saharon Shelah that they do not admit exponential functions which have the same elementary properties as the exponential function on the reals. In a different direction, the question came up whether they could support generalizations of Banach’s Fixed Point Theorem. I will introduce the notions of symmetrically complete ordered fields, abelian groups and sets and characterize those power series models of the reals that are symmetrically complete. They indeed admit a (nonarchimedean) generalization of Banach’s Fixed Point Theorem. Their construction is the result of joint work with Katarzyna Kuhlmann and Saharon Shelah. It heavily relies on the analysis of cuts in ordered power series fields and Hahn products.

**Information:**This talk will be given in hybrid format. Please contact Richard Springer for information how to participate.

**Toronto Set Theory Seminar****Time:** Friday, 25 November, 1.30-3.00 Toronto time (19.30-21.00 CET)**Speaker:** Cesar Corral, Universidad Nacional Autónoma de México**Title:** Convergence properties in the realm of Psi-spaces**Abstract:** We will deal with convergence properties, mainly the Frechet-Urysohn property and some strong versions of it, as well as some properties related to the preservation of Frechetness under products, like the [\alpha_i] -properties introduced by Arhangel’skii.

We will show that these classes of spaces are (consistently) different by constructing some counterexamples making use of almost disjoint families.**Information:** Please see http://gfs.fields.utoronto.ca/activities/22-23/set-theory-seminar.

14-20 November

**Bristol Logic and Set Theory SeminarTime:** Tuesday, 15 November, 12:30-13:30 UK time (13:30-14:30 CET)

**Speaker:**Sam Coskey, Boise State University

**Title:**Jumps in the Borel complexity hierarchy

**Abstract:**There are several well-studied “jumps” on the class of Borel equivalence relations under Borel reducibility, which carry an equivalence relation to one of greater complexity. Examples include the jump of Friedman–Stanley and the jumps of Louveau. In joint work with John Clemens, we defined a new (ish) family of jumps called Bernoulli jumps. In this talk I will introduce and describe Bernoulli jumps, and present an application to the classification of countable scattered orders. I will conclude by summarizing some more recent developments, due to others, relating to Bernoulli jumps.

**Information:**The login information is posted on the seminar webpage.

**Vienna Research Seminar in Set Theory****Time:** Tuesday, 15 November, 15:00-16:30 CET**Speaker:** Monroe Eskew, Universität Wien**Title:** Compactness versus hugeness at successor cardinals, part 2**Abstract:** There are several ways in which small cardinals can behave like large

ones. One variety is compactness phenomena, such as the tree property,

which characterize when inaccessible cardinals satisfy some strong large

cardinal notions, but can consistently hold at small cardinals such as

$\omega_2$. Another variety is generic embedding properties coming from

saturated ideals or Chang’s Conjecture that resemble embeddings associated

with huge cardinals. The known forcing strategies for obtaining

compactness and hugeness properties at small cardinals are very different.

Can they be made to hold simultaneously? In these talks, we present some

combinatorial barriers to combining them, and we show why several forcing

approaches will not work. Hopefully, by narrowing down the space of

possibilities, these negative results will point towards a path to

answering our question.**Information:** This talk will be given in hybrid format. Please contact Richard Springer for information how to participate.

**Baltic Set Theory Seminar****Time:** Tuesday, 15 November, 15:00-16:30 CET**Speaker:** Several**Title:** Baltic Set Theory Seminar**Abstract:** This is a learning seminar, the goal is to actually go over proofs and more or less understand them. Discussions are encouraged. The topic of the seminar is the following:

1. Sandra Müller, Stationary-tower-free proof of Woodin’s Sealing Theorem.

2. Matteo Viale, Generic absoluteness theorem for the omega_1 Chang model conditioned to MM^{+++}.**Information:** Please see the seminar webpage.

**CMU Core Model Theory Seminar****Time:** Tuesday, 15 November, 1:30 – 3:00pm Pittsburgh time (19:30 – 21:00 CET) **Speaker:** Sandra Müller, TU Wien**Title:** A stationary-tower-free proof of Sealing, part 2**Abstract: **Sealing is a generic absoluteness principle for the theory of

the universally Baire sets of reals introduced by Woodin. It is deeply

connected to the Inner Model Program and plays a prominent role in recent

advances in inner model theory. Woodin showed in his famous Sealing

Theorem that in the presence of a proper class of Woodin cardinals Sealing

holds after collapsing a supercompact cardinal. In the first talk, I will

outline the importance of Sealing and discuss a new and

stationary-tower-free proof of Woodin’s Sealing Theorem that is based on

Sargsyan’s and Trang’s proof of Sealing from iterability. In the second

talk, I will outline the proof of an extension of the Sealing Theorem that

gives models in which Theta is regular. This is joint work with Grigor

Sargsyan.**Information:** Please contact Ernest Schimmerling in advance for the zoom link.

**Helsinki Logic Seminar****Time:** Wednesday, 16 November, 12:00 – 14:00 Helsinki time (11:00-13:00 CET)**Speaker:** Tobias Boege**Title:** Incidence geometry, conditional independence and the existential theory of the reals**Abstract:** Deciding whether a system of polynomial equations and inequalities has a solution over the real numbers is a basic task in computational geometry, optimization and algebraic statistics. Under polynomial-time many-one reductions, this problem generates the complexity class $\exists\mathbb{R}$. I will briefly recall a geometric technique, due to von Staudt, for obtaining completeness results for this complexity class and apply it to the implication problem for conditional independence among jointly normal random variables.**Information:** The talk will take place in hybrid mode. Please see the seminar webpage for the link.

**Leeds Models and Sets Seminar****Time:** Wednesday, 16 November, 13:45-15:00 local time (14:45-16:00 CET)**Speaker:** Petra Staynova, University of Derby**Title:** Spotting rare Pokemon**Abstract:** Sometimes more abstract concepts in general topology are considered as having little relation with areas outside of topology. In this talk we will explore a beautiful construction that unexpectedly links the notion of n-Hausdorffness and a special topology in the dynamical systems setting.**Information:** Please see the seminar webpage.

**Barcelona Set Theory Seminar****Time:** Wednesday, 16 November, 16:00-17:30 CET**Speaker:** Jeffrey Bergfalk, University of Barcelona**Title:** The definable content of homological invariants**Abstract:** This talk is intended as an overview of the first two installments in a series of joint works with Martino Lupini and Aristotelis Panagiotopoulos (arXiv:2210.11098, arXiv:2008.08782). The theme of this series is the recognition that many classical homological and homotopical functors from the field of algebraic topology factor through “definable categories” such as what we term the category of groups with a

Polish cover; this recognition gives rise to stronger versions of those invariants, together with closer analyses of their classifying powers. Somewhat more informally, this series is about the topological and Borel structures underlying classical invariants like the Ext or lim1 or Cech cohomology groups; although these structures had, throughout these invariants’ history, periodically been considered, what distinguishes the works under discussion is their recognition of the remarkable utility of descriptive set theoretic frameworks for systematically and efficiently doing so.**Information:** Online. If you wish to attend, please send an email to bagaria@ub.edu asking for the link.

**Caltech Logic Seminar****Time:** Wednesday, 16 November, 11:00am-12:00pm Pacific time (20:00-21:00 CET)**Speaker:** Dexuan Hu, Cornell University**Title:** Polish modules over subrings of Q**Abstract:** We give a method of producing a Polish module over an arbitrary subring of Q from an ideal of subsets of N and a sequence in N. The method allows us to construct two Polish Q-vector spaces, U and V, such that

– both U and V embed into R, but

– U does not embed into V and V does not embed into U,

where by an embedding we understand a continuous Q-linear injection. This construction answers a question of Frisch and Shinko. In fact, our method produces a large number of incomparable with respect to embeddings Polish Q-vector spaces.

This is joint work with Slawomir Solecki.**Information:** Please see the seminar webpage.

**Vienna Logic Colloquium****Time:** Thursday, 17 November, 15:00 – 16:30 CET

**Speaker:**F. Schlutzenberg, University of Münster

**Title:**The Axiom of Choice and large cardinals

**Abstract:**The Axiom of Choice (AC) is mostly accepted by mathematicians, and is

essential in many proofs. However, it seems to be accepted with less

confidence than the other axioms of set theory, probably due to its

non-constructive nature and its various unexpected consequences. Large

cardinals are central axioms in set theory, with compelling consequences

for the universe of sets, not only for “large” sets but also for “small”

ones like real numbers and sets thereof. It turns out that the

relationship between AC and large cardinals is intricate, and not entirely

without conflict. The connections might even be taken to suggest that the

correct picture of the universe of sets is one in which very large

cardinals exist and the full Axiom of Choice must fail. I will survey some

of the recent work in this area. The talk will be aimed at a general logic

audience.

**Information:**This talk will be given in hybrid format. Please contact Richard Springer for information how to participate.

**Cross-Alps Logic Seminar****Time:** Friday, 18 November 16.00-18.00 CET**Speaker:** A. Conversano, Massey University**Title:** Tools of o-minimality in the study of groups**Abstract:** In this talk we will see how geometric invariants of definable sets in o-minimal structures can be used to understand the structure of groups in several categories.**Information:** The event will stream on the Webex platform. Please write to luca.mottoros [at] unito.it for the link to the event.

**CUNY Set Theory Seminar****Time:** Friday, 18 November, 12:30pm New York time (18:30 CET)**Speaker: **Brent Cody, Virginia Commonwealth University**Title:** Sparse analytic systems**Abstract:** Given a set S, an S-predictor P is a function that takes as inputs functions of the form f:(−∞,t)→S, where t∈R, and outputs a guess P(f) for what f(t) ‘should be.’ An S-predictor is good if for all total functions F:R→S the set of t∈R for which the guess P(F↾(−∞,t)) is not equal to F(t) has measure zero. Hardin and Taylor proved that every set S has a good S-predictor and they raised various questions asking about the extent to which the prediction P(f) made by a good predictor might be invariant after precomposing f with various well-behaved functions – this leads to the notion of ‘anonymity’ of good predictors under various classes of functions. Bajpai and Velleman answered several of Hardin and Taylor’s questions and asked: Does there exist, for every set S, a good S-predictor that is anonymous with respect to the strictly increasing analytic homeomorphisms of R? We provide a consistently negative answer to this question by strengthening a result of Erdős, which states that the Continuum Hypothesis is equivalent to the existence of an uncountable family F of (real or complex) analytic functions, such that {f(x):f∈F} is countable for every x. We strengthen Erdős’ result by proving that CH is equivalent to the existence of what we call *sparse analytic systems* of functions. This is joint work with Sean Cox and Kayla Lee.**Information:** The seminar will take place virtually. Please email Victoria Gitman (vgitman@nylogic.org) for the meeting id.

**Toronto Set Theory Seminar****Time:** Friday, 18 November, 1.30-3.00 Toronto time (19.30-21.00 CET)**Speaker:** Nishant Chandgotia, TIFR**Title:** About Borel and almost Borel embeddings for Zd actions**Abstract:** How can you code a system? This question has many perspectives depending on what we mean by code and by system. In this talk we will take the perspective of an ergodic theorist and look at encoding of free ergodic probability preserving transformations. A classical result here is due to Krieger who showed that this encoding can be done by bi-infinite sequence of unconstrained symbols from a finite alphabet. In this talk we will be talking about analogous theorems for encodings of Zd actions when these symbols have constraints (for instance when adjacent symbols are distinct or systems arising from tilings) which use some interesting combinatorial estimates of independent interest.**Information:** Please see http://gfs.fields.utoronto.ca/activities/22-23/set-theory-seminar.

**CUNY Logic Workshop****Time:** Friday, 18 November, 2:00 – 3:30 New York time (20:00-21:30 CET)**Speaker: **Dima Sinapova, Rutgers University**Title: **Prikry sequences and square properties at ℵω**Abstract:** It is well known that if an inaccessible cardinal κ is singularized to countable cofinality while preserving cardinals, then □κω holds in the outer model. Moreover, this remains true even when relaxing the cardinal preservation assumption a bit. In this talk we focus on when Prikry forcing adds weaker forms of square in a more general setting. We prove abstract theorems about when Prikry forcing with interleaved collapses to bring down the singularized cardinal to ℵω will add a weak square sequence. This can be viewed as a partial positive result to a question of Woodin about whether the failure of SCH at ℵω implies weak square.**Information:** The talk will take place in hybrid mode. Please see the seminar webpage or email Victoria Gitman for the link.

7-13 November

**Vienna Research Seminar in Set Theory****Time:** Tuesday, 8 November, 15:00-16:30 CEST**Speaker:** Monroe Eskew, Universität Wien**Title:** Compactness versus hugeness at successor cardinals, part 1**Abstract:** There are several ways in which small cardinals can behave like large

ones. One variety is compactness phenomena, such as the tree property,

which characterize when inaccessible cardinals satisfy some strong large

cardinal notions, but can consistently hold at small cardinals such as

$\omega_2$. Another variety is generic embedding properties coming from

saturated ideals or Chang’s Conjecture that resemble embeddings associated

with huge cardinals. The known forcing strategies for obtaining

compactness and hugeness properties at small cardinals are very different.

Can they be made to hold simultaneously? In these talks, we present some

combinatorial barriers to combining them, and we show why several forcing

approaches will not work. Hopefully, by narrowing down the space of

possibilities, these negative results will point towards a path to

answering our question.**Information:** This talk will be given in hybrid format. Please contact Richard Springer for information how to participate.

**Baltic Set Theory Seminar****Time:** Tuesday, 8 November, 15:00-16:30 CET**Speaker:** Several**Title:** Baltic Set Theory Seminar**Abstract:** This is a learning seminar, the goal is to actually go over proofs and more or less understand them. Discussions are encouraged. The topic of the seminar is the following:

1. Sandra Müller, Stationary-tower-free proof of Woodin’s Sealing Theorem.

2. Matteo Viale, Generic absoluteness theorem for the omega_1 Chang model conditioned to MM^{+++}.**Information:** Please see the seminar webpage.

**CMU Core Model Theory Seminar****Time:** Tuesday, 8 November, 1:30 – 3:00pm Pittsburgh time (19:30 – 21:00 CET) **Speaker:** Sandra Müller, TU Wien**Title:** A stationary-tower-free proof of Sealing, part 1**Abstract: **Sealing is a generic absoluteness principle for the theory of

the universally Baire sets of reals introduced by Woodin. It is deeply

connected to the Inner Model Program and plays a prominent role in recent

advances in inner model theory. Woodin showed in his famous Sealing

Theorem that in the presence of a proper class of Woodin cardinals Sealing

holds after collapsing a supercompact cardinal. In the first talk, I will

outline the importance of Sealing and discuss a new and

stationary-tower-free proof of Woodin’s Sealing Theorem that is based on

Sargsyan’s and Trang’s proof of Sealing from iterability. In the second

talk, I will outline the proof of an extension of the Sealing Theorem that

gives models in which Theta is regular. This is joint work with Grigor

Sargsyan.**Information:** Please contact Ernest Schimmerling in advance for the zoom link.

**CMU Logic Seminar****Time:** Tuesday, 8 November, 3:30 – 4:30pm Pittsburgh time (21:30 – 22:30 CEST) **Speaker:** Harrison-Trainor, University of Michigan**Title:** Computable Approximations and True Stages**Abstract: **The limit lemma says that a set is computable relative to the Halting problem if it can be approximated computably with finitely many mind changes. More complicated sets can also be approximated with more complicated approximation schemes, but these quickly become unwieldy. Many frameworks have been proposed for dealing with these approximations, such the $alpha$-systems of Ash and Knight or the true stages of Montalban. In the first half of the talk, I’ll describe the general ideas of some of these frameworks and some of their applications. In the second half of the talk, I will talk about a new framework of true stages (with Adam Day, Noam Greenberg, and Dan Turetsky) with connections to descriptive set theory. This framework can be thought of as a computability-theoretic change-of-topology with additional bonuses.**Information:** See the seminar webpage.

**Helsinki Logic Seminar****Time:** Wednesday, 9 November, 12:00 – 14:00 Helsinki time (11:00-13:00 CET)**Speaker:** Vadim Weinstein**Title:** Blurry filters and coassification by countable structures (Joint work with Martina Iannella) **Abstract:** The Stone duality gives a neat way to go back-and-forth between totally disconnected Polish spaces and countable Boolean algebras. The main ingredient is the Stone space of all ultrafilters on a Boolean algebra. In this talk we introduce a weaker concept which we call the “blurry filter”. Using blurry filters instead of ultrafilters enables one to extend the class of spaces under consideration from totally disconnected ones to a larger class. As an application of this method, we show that the following are completely classifiable by countable structures: the homeomorphism on 3-manifolds (also applicable to 2-manifolds; but this was known since 1971), and wild embeddings of Cantor sets in R³. By “classification” in this talk we mean classical Borel-reducibility. **Information:** The talk will take place in hybrid mode. Please see the seminar webpage for the link.

**Leeds Models and Sets Seminar****Time:** Wednesday, 9 November, 13:45-15:00 local time (14:45-16:00 CET)**Speaker:** Mariana Vicaria, UCLA**Title:** Elimination of imaginaries in ordered abelian groups of bounded regular rank**Abstract:** In this talk I will present some results about elimination of imaginaries in pure ordered abelian groups. For the class of ordered abelian groups with bounded regular rank (equivalently with finite spines) we obtain weak elimination of imaginaries once we add sorts for the quotient groups $\Gamma/ \Delta$ for each definable convex subgroup $\Delta$, and sorts for the quotient groups $\Gamma/(\Delta+ \ell\Gamma)$ where $\Delta$ is a definable convex subgroup and $\ell \in \mathbb{N}{\geq 2}$. We refer to these sorts as the \emph{quotient sorts}. For the dp-minimal case we obtain a complete elimination of imaginaries if we also add constants to distinguish the cosets of $\Delta+\ell\Gamma$ in $\Gamma$, where $\Delta$ is a definable convex subgroup and $\ell \in \mathbb{N}{\geq 2}$.**Information:** Please see the seminar webpage.

**Caltech Logic Seminar****Time:** Wednesday, 9 November, 11:00am-12:00pm Pacific time (20:00-21:00 CET)**Speaker:** Simon Thomas, Rutgers University**Title:** Some Open Problems On Invariant Random Subgroups**Abstract:** Let G be a countably infinite group and let SubG be the compact space of subgroups H⩽G. Then an *invariant random subgroup* (IRS) of G is a probability measure ν on SubG which is invariant under the conjugation action of G on SubG.

In this talk, after a brief introduction to the theory of invariant random subgroups, I will discuss some of the many basic questions in this relatively new area. For example, if ν is an ergodic IRS of a countable group G, then we obtain a corresponding *zero-one law* on SubG for the class of group-theoretic properties Φ such that the set {H∈SubG∣H has property Φ} is ν-measurable; and thus ν concentrates on a collection of subgroups which are quite difficult to distinguish between. Consequently, it is natural to ask whether there exists an ergodic IRS of a countable group G which does not concentrate on the subgroups H⩽G of a single isomorphism type.**Information:** Please see the seminar webpage.

**Vienna Logic Colloquium****Time:** Thursday, 10 November, 15:00 – 16:30 CET

**Speaker:**J. Schilhan, University of Leeds

**Title:**Entire functions and the continuum

**Abstract:**In the 60’s, Erdős showed that the continuum hypothesis is equivalent to the statement that there is an uncountable family of entire functions on the complex plane that attains only countably many values at each point. The argument in fact shows that any family of entire functions, that attains at each point less values than elements of that family, must have size continuum. Recently Kumar and Shelah have shown that consistently such a family exists while the continuum has size ℵω1. We answer their main open problem by showing that continuum ℵ2 is possible as well.

This is joint work with T. Weinert.

**Information:**This talk will be given in hybrid format. Please contact Richard Springer for information how to participate.

**CUNY Set Theory Seminar****Time:** Friday, 11 November, 12:30pm New York time (18:30 CET)**Speaker: **Peter Holy, Technical University of Vienna**Title:** Asymmetric Cut and Choose Games**Abstract:** We consider the following two player game of infinite length: We are given a starting set X, and the players go by the names ‘Cut’ and ‘Choose’. They take turns making moves, and in each step, Cut partitions a given set into two disjoint pieces, starting from the set X in their first move, and then Choose gets to pick one of the pieces, which is then partitioned into two pieces by Cut in their next move etc. In the end, Choose wins in case the intersection of all of their choices has at least two (distinct) elements.

We will investigate some of the properties of this game — in particular, we will discuss some classic results on when it is possible for one of the players to have a strategy for winning the game. We will then continue to discuss some variations of this game and their relevance to set theory — many central set theoretic notions, such as certain large cardinal properties, notions of distributivity, precipitousness and strategic closure were either known or turned out to be closely connected and often equivalent to the (non-)existence of winning strategies in certain cut and choose games.

This is joint work with Philipp Schlicht, Christopher Turner and Philip Welch (all University of Bristol).**Information:** The seminar will take place virtually. Please email Victoria Gitman (vgitman@nylogic.org) for the meeting id.

**Toronto Set Theory Seminar****Time:** Friday, 11 November, 1.30-3.00 Toronto time (19.30-21.00 CET)**Speaker:** David Schrittesser, University of Toronto**Title:** Nonstandard methods for statistics**Abstract:** I will discuss recent joint work with Haosui Duanmu and Daniel M. Roy, in which we give a precise characterization of admissibility in Bayesian terms, solving a long-standing problem in the field of statistical decision theory. This result uses so-called hyperpriors, which can give infinitesimal weight to events, to achieve this characterization, and also has interesting classical consequences (that is, not mentioning hyperpriors or infinitesimals). (I have already presented an early, weaker version of the present result in a previous lecture at the seminar; said result only held in problems with strictly convex loss functions. The present result holds without any assumptions on the loss function or the decision problem.)**Information:** Please see http://gfs.fields.utoronto.ca/activities/22-23/set-theory-seminar.

**Bristol Logic and Set Theory SeminarTime:** Tuesday, 1 November, 12:30-13:30 UK time (13:30-14:30 CET)

**Speaker:**Sean Cox, Virginia Commonwealth University

**Title:**Homological algebra, elementary submodels, and stationary logic

**Abstract:**The talk will focus on the use of set-theoretic elementary submodel techniques to solve, or partially solve, some problems from homological algebra (independence of Salce’s Problem about cotorsion pairs, and the precovering problem in Gorenstein Homological Algebra).

**Information:**The login information is posted on the seminar webpage.

**Baltic Set Theory Seminar****Time:** Tuesday, 1 November, 15:00-16:30 CET**Speaker:** Several**Title:** Baltic Set Theory Seminar**Abstract:** This is a learning seminar, the goal is to actually go over proofs and more or less understand them. Discussions are encouraged. The topic of the seminar is the following:

1. Sandra Müller, Stationary-tower-free proof of Woodin’s Sealing Theorem.

2. Matteo Viale, Generic absoluteness theorem for the omega_1 Chang model conditioned to MM^{+++}.**Information:** Please see the seminar webpage.

**CMU Core Model Theory Seminar****Time:** Tuesday, 1 November, 1:30 – 3:00pm Pittsburgh time (19:30 – 21:00 CET) **Speaker:** Derek Levinson**Title:** Unreachability of pointclasses in L(R), part 2**Abstract: **We present Hjorth’s proof that there is no sequence of distinct

$\Sigma^1_2$ sets of length $\delta^1_2$. Then we prove in $L(R)$ if

$\Gamma$ is an inductive-like pointclass then there is no sequence of

distinct $\Gamma$ sets of length $\delta_\Gamma^+$.**Information:** Please contact Ernest Schimmerling in advance for the zoom link.

**Helsinki Logic Seminar****Time:** Wednesday, 2 November, 12:00 – 14:00 Helsinki time (11:00-13:00 CET)**Speaker:** Miika Hannula**Title:** Dependencies and information inequalities**Abstract:** Dependence and independence can be interpreted as inequalities over Shannon entropies. In fact, basic principles about dependencies are already at work at the more abstract level of polymatroids, which encapsulate the elementary properties of the entropy function. In this talk we investigate these connections and survey some basic facts about information-theoretic inequalities.**Information:** The talk will take place in hybrid mode. Please see the seminar webpage for the link.

**Leeds Models and Sets Seminar****Time:** Wednesday, 2 November, 13:45-15:00 local time (14:45-16:00 CET)**Speaker:** Sebastian Eterovic, University of Leeds**Title:** Strong Existential Closedness**Abstract:** The strong existential closedness problem was introduced in Zilber’s work on pseudoexponentiation. Since then, it has been naturally adapted to many situations in arithmetic geometry. In this talk I will introduce the problem, review some important Diophantine questions that are connected to it, and discuss some of the known results.**Information:** Please see the seminar webpage.

**Barcelona Set Theory Seminar****Time:** Wednesday, 2 November, 16:00-17:30 CET**Speaker:** Philipp Lücke, University of Barcelona**Title:** Rowbottom cardinals and definability**Abstract:** If k is a Rowbottom cardinal, then a short argument shows that

no uncountable ordinal below k is definable by a S1-formula with parameter

k. In my talk, I want to present work that aims to expand this connection

and use undefinability consequences of partition properties to restrict the

class of possible models of set theory in which Àw is a Rowbottom cardinal.

As an example, these results provide a canonical argument that shows that

Àw is not Rowbottom in the standard models of strong forcing axioms.

This is joint work in progress with Omer Ben-Neria (Jerusalem).**Information:** Online. If you wish to attend, please send an email to bagaria@ub.edu asking for the link.

**Caltech Logic Seminar****Time:** Wednesday, 2 November, 11:00am-12:00pm Pacific time (20:00-21:00 CET)**Speaker:** Dino Rossegger, UC Berkeley**Title:** Analytic complete equivalence relations and their degree spectra**Abstract:** In this talk, we will show that elementary bi-embeddability is an analytic complete equivalence relation under Borel reducibility by giving a reduction from the bi-embeddability relation on graphs. We will then discuss the degree spectra realized by these relations. The degree spectrum of a countable structure with respect to an equivalence relation EE, a central notion in computable structure theory, is the set of Turing degrees of structures EE-equivalent to it. By analyzing the computability theoretic properties of our reduction from bi-embeddability to elementary bi-embeddability we show that the degree spectra of these two relations are related. Suppose a set of Turing degrees XX is the bi-embeddability spectrum of a graph. Then the set of degrees whose Turing jump is in XX is the elementary bi-embeddability spectrum of a graph. Combining results of Harrison-Trainor and the coauthor one sees that this is sharp: There is a bi-embedddability spectrum that is not an elementary bi-embeddability spectrum.**Information:** Please see the seminar webpage.

**Vienna Logic Colloquium****Time:** Thursday, 2 November, 15:00 – 16:30 CET

**Speaker:**W. Brian, University of North Carolina at Charlotte

**Title:**Covering versus partitioning with Polish spaces

**Abstract:**A topological space is Polish if it is second countable and completely metrizable. We may think of these as the small, or “essentially countable” members of the category of completely metrizable spaces. In this talk, we explore the question of whether, given a completely metrizable space X, it is possible to cover X with fewer Polish spaces than it can be partitioned into. Surprisingly, this question not only turns out to be independent of ZFC, but proving its independence requires large cardinal axioms. I will sketch some of the ideas that go into one direction of this independence proof. Specifically, I will describe how a version of the model-theoretic transfer principle called Chang’s Conjecture implies that there is a completely metrizable space that can be covered with fewer Polish spaces than it can be partitioned into.

**Information:**This talk will be given in hybrid format. Please contact Richard Springer for information how to participate.

**Vienna Research Seminar in Set Theory****Time:** Thursday, 3 November, 16:45-18:15 CET – note the time change**Speaker:** W. Brian, University of North Carolina at Charlotte**Title:** Partitioning the real line into Borel sets**Abstract:** I will sketch a proof that, assuming 0† does not exist, if there is a partition of the real line R into ℵω Borel sets, then there is also a partition of R into ℵω+1 Borel sets. (And the same is true for any singular cardinal of countable cofinality in place of ℵω.) This contrasts starkly with the situation for successor-of-successor cardinals, where the spectrum of possible sizes of partitions of Rinto Borel sets can seemingly be made completely arbitrary. For example, given any A⊆ω with 0,1∈A, there is a forcing extension in which A={n<ω: there is a partition of R into ℵn Borel sets}.**Information:** This talk will be given in hybrid format. Please contact Richard Springer for information how to participate.

**Cross-Alps Logic Seminar****Time:** Friday, 4 November, 16.00-17.00 CET **Speaker:** J. Emmenegger, University of Genoa**Title:** Quotients and equality, (co)algebraically**Abstract:** Doctrines were introduced by Lawvere as an algebraic tool to work with logical theories and their extensions. In fact, this algebraic character makes the theory of doctrines a suitable context where to address the question: “”What is the theory obtained by (co)freely adding logical structure?”” or the closely related question: “”How to express additional logical structure in terms of what is already available?””. More precisely, in the first case we ask whether a certain forgetful functor is adjoint and, in the second case, whether the adjunction obtained in this way is (co)monadic. After an introduction to doctrines and their connection to logic and type theory, I shall discuss the above questions in the case of two forgetful functors: the one from theories with conjunctions, equality and quotients to theories with conjunctions and equality, and the one that further forgets equality. Not surprisingly, the answers revolve around the concept of equivalence relation. I shall discuss applications to useful constructions in categorical logic and type theory, as well as to the theory of imaginary elements in the sense of Poizat. If time allows, I shall also describe how to lift this setting to Grothendieck fibrations (of which doctrines are a particular case) using groupoids instead of equivalence relations. **Information:** The event will stream on the Webex platform. Please write to luca.mottoros [at] unito.it for the link to the event.

**CUNY Set Theory Seminar****Time:** Friday, 4 November, 12:30pm New York time (18:30 CET)**Speaker: **Corey Switzer, University of Vienna**Title:** The Special Tree Number**Abstract:** A tree of height ω1 with no cofinal branch is called *special* if it can be decomposed into countably many antichains or, equivalently if it carries a specializing function: a function f:T→ω so that if f(s)=f(t) then s and t are incomparable in the tree ordering. It is known that there is always a non-special tree of size continuum, but the existence of a smaller one is independent of ZFC. Motivated by this we introduce the special tree number, st, the least size of a tree of height ω1which is neither non-special nor has a cofinal branch. Classical facts imply that st can be smaller than essentially all well studied cardinal characteristics. Conversely in this talk we will show that stcan be larger than a, g, and both the left hand side and bottom row of the Cichon diagram. Thus stis independent of many well known cardinal invariants. Central to this result is an in depth investigation of the types of reals added by the Baumgartner specialization poset which we will discuss as well.**Information:** The seminar will take place virtually. Please email Victoria Gitman (vgitman@nylogic.org) for the meeting id.

**Toronto Set Theory Seminar****Time:** Friday, 4 November, 1.30-3.00 Toronto time (19.30-21.00 CET)**Speaker:** Vinicius De Oliveira Rodrigues**Title:** Fin-intersecting MAD families and the pseudocompactness of hyperspaces**Abstract:** We study Ginsburg’s questions on the relations between the pseudocompactness of the Vietoris hyperspace of a given topological space and the pseudocompactness of its powers restricted to the context of Isbell-Mrówka space. An almost disjoint family is said to be pseudocompact if the Vietoris Hyperspace of its Isbell-Mrówka space is pseudocompact. In the context of Isbell-Mrówka spaces, Ginsburg’s questions become questions about the existence of pseudocompact MAD families. We will discuss what has been done around this problem and what remains open. To further study this problem we propose a new class of almost disjoint families which we call fin-intersecting almost disjoint families. Every fin intersecting MAD family is pseudocompact. Under p=c such families exist generically, but there is also a MAD pseudocompact family which is not fin-intersecting. Also, under CH, forcing extensions obtained by adding Cohen reals and Random reals contain such families.**Information:** Please see http://gfs.fields.utoronto.ca/activities/22-23/set-theory-seminar.

**CUNY Logic Workshop****Time:** Friday, 4 November, 2:00 – 3:30 New York time (20:00-21:30 CET)**Speaker: **Dave Marker, University of Illinois at Chicago**Title: **Automorphisms of differentially closed fields**Abstract:** Answering a question of Russell Miller, we show that there are differentially closed fields with no non-trivial automorphisms.**Information:** The talk will take place in hybrid mode. Please see the seminar webpage or email Victoria Gitman for the link.

24-30 October

**Vienna Research Seminar in Set Theory****Time:** Tuesday, 25 October, 15:00-16:30 CEST**Speaker:** Diana Carolina Montoya, University of Vienna **Title:** Maximal independence and singulars**Abstract:** In these talks, we will discuss the concept of maximal independent families for uncountable singular cardinals. In the first part, I will present the existing background results in the regular case from Kunen and Eskew-Fischer. In the second part, we will focus on the joint results obtained in joint work with Omer Ben-Neria: some on the existence of maximal independent families at a singular strong limit, and finally some on the possible sizes of such families.**Information:** This talk will be given in hybrid format. Please contact Richard Springer for information how to participate.

**Baltic Set Theory Seminar****Time:** Tuesday, 25 October, 15:00-16:30 CEST**Speaker:** Several**Title:** Baltic Set Theory Seminar**Abstract:** This is a learning seminar, the goal is to actually go over proofs and more or less understand them. Discussions are encouraged. The topic of the seminar is the following:

1. Sandra Müller, Stationary-tower-free proof of Woodin’s Sealing Theorem.

2. Matteo Viale, Generic absoluteness theorem for the omega_1 Chang model conditioned to MM^{+++}.**Information:** Please see the seminar webpage.

**CMU Core Model Theory Seminar****Time:** Tuesday, 25 October, 1:30 – 3:00pm Pittsburgh time (19:30 – 21:00 CEST) **Speaker:** Derek Levinson**Title:** Unreachability of pointclasses in L(R), part 1**Abstract: **We present Hjorth’s proof that there is no sequence of distinct

$\Sigma^1_2$ sets of length $\delta^1_2$. Then we prove in $L(R)$ if

$\Gamma$ is an inductive-like pointclass then there is no sequence of

distinct $\Gamma$ sets of length $\delta_\Gamma^+$. **Information:** Please contact Ernest Schimmerling in advance for the zoom link.

**CMU Logic Seminar****Time:** Tuesday, 25 October, 3:30 – 4:30pm Pittsburgh time (21:30 – 22:30 CEST) **Speaker:** Kristin Yvonne Rozier, Iowa State University **Title:** Proofs that Fly! Logic, Automata, and Set Theory in Air and Space**Abstract: **We are at the dawn of the age of unmanned aircraft, automated air traffic control, and autonomous spacecraft, where safe operation remains the primary consideration. Before we can build or deploy such safety-critical systems we must formally prove that they always satisfy safety requirements. Such certification critically depends on logic, automata, and set theory! We directly employ these techniques to formalize aerospace operational concepts: we must unambiguously specify safety requirements to produce automated, re-playable proofs that safety-critical systems behave the way we expect them to, before we allow them to interact with humans. Techniques from set theory and graph theory enable the scalability required to analyze a large (20K+) set of possible air traffic control designs from NASA. We address areas for future collaborations: what are the next big questions we must answer to proceed safely from here?**Information:** See the seminar webpage.

**Leeds Models and Sets Seminar****Time:** Wednesday, 26 October, 13:45-15:00 local time (14:45-16:00 CEST)**Speaker:** Andrew Brooke-Taylor, University of Leeds**Title:** Cardinal characteristics modulo nice ideals on omega**Abstract:** Many of the standard cardinal characteristics of the continuum are defined in terms of a relation holding almost everywhere, where “almost everywhere” means on all but a finite set. A very natural generalisation is to take “almost everywhere” to mean on all but a member of a given ideal. I will talk about what happens when we do this, with the density 0 ideal on omega as a focal example.**Information:** Please see the seminar webpage.

**Barcelona Set Theory Seminar****Time:** Wednesday, 26 October, 16:00-17:30 CEST**Speaker:** Juan P. Aguilera**Title:** Determinacy on the Edge of Second-Order Arithmetic**Abstract:** We calculate the exact strength of the strongest theories of determinacy which are provable in Second-Order Arithmetic, answering a question of Montalbán. This is joint work with Philip Welch. **Information:** Online. If you wish to attend, please send an email to bagaria@ub.edu asking for the link.

**Caltech Logic Seminar****Time:** Wednesday, 26 October, 11:00am-12:00pm Pacific time (20:00-21:00 CEST)**Speaker:** Andy Zucker, University of Waterloo**Title:** Big Ramsey degrees and Galvin-Prikry theorems for binary free-amalgamation classes**Abstract:** Given a finite relational language L and a (possibly infinite) set F of finite irreducible L-structures, the class Forb(F) describes those finite L-structures which do not embed any member of F. Classes of the form Forb(F) exactly describe those classes of finite L-structures with free amalgamation. In recent joint work with Balko, Chodounsky, Dobrinen, Hubicka, Konecny, and Vena, we exactly characterize big Ramsey degrees for those classes Forb(F) where the forbidden set F is finite. This characterization proceeds by defining tree-like objects called diagonal diaries, then showing that the big Ramsey degree of any A in Forb(F) is exactly the number of diagonal diaries which code the structure A. After giving a brief description of these objects, the talk will then consider those infinite diagonal diaries which code the Fraisse limit of Forb(F). In upcoming joint work with Dobrinen, we prove a Galvin-Prikry theorem for any such infinite diagonal diary, giving new examples of objects satisfying the Galvin-Prikry theorem which dramatically fail to satisfy Todorcevic’s Ramsey space axioms A1 through A4.**Information:** Please see the seminar webpage.

**European Set Theory Society Panel Discussions****Time:** Thursday, 27 October, 17:00-19:00 CEST**Panelists:** Michael Hrusak, Universidad Nacional Autónoma de México

Juliette Kennedy, University of Helsinki

Menachem Magidor, Hebrew University of Jerusalem

Justin Tatch Moore, Cornell University**Title:** European Set Theory Society Panel Discussions**Abstract:** Four experts will be invited to describe the general area they represent, explain where the area is heading and discuss how it relates to other areas of set theory and mathematics.**Information:** Online. Zoom link for 27 October: https://univienna.zoom.us/j/62758810753?pwd=cFJkZjVWMGpmWmtKWUd4UE1EdXQ1dz09 , Meeting-ID: 627 5881 0753, Code: est22

**CUNY Set Theory Seminar****Time:** Friday, 28 October, 12:30pm New York time (18:30 CEST)**Speaker: **Andreas Lietz, University of Münster**Title:** Forcing ‘NSω1 is ω1-dense’ from Large Cardinals – A Journey guided by the Stars: Part II**Abstract:** An ideal I on ω1 is ω1-dense if (P(ω1)/I)+ has a dense subset of size ω1. We prove, assuming large cardinals, that there is a semiproper forcing P so that VP⊨‘NSω1 is ω1-dense’. This answers a question of Woodin positively. Our general strategy is based on the observation that replacing the role of Pmax in Woodin’s axiom (∗) by Qmax results in an axiom Qmax−(∗) which implies ‘NSω1 is ω1-dense’.

We proceed in three steps: First we define and motivate a new forcing axiom QM and then modify the Asperó-Schindler proof of ‘MM++⇒(∗)’ to show ‘QM⇒Qmax−(∗)’. Finally, assuming a supercompact limit of supercompact cardinals exists, we construct a semiproper partial order forcing QM. This last step involves proving two new iteration theorems both of which allow for forcings killing stationary sets.**Information:** The seminar will take place virtually. Please email Victoria Gitman (vgitman@nylogic.org) for the meeting id.

**Toronto Set Theory Seminar****Time:** Friday, 28 October, 1.30-3.00 Toronto time (19.30-21.00 CEST)**Speaker:** Andy Zucker, University of Waterloo**Title:** Galvin-Prikry theorems for big Ramsey structures**Abstract:** The Galvin-Prikry theorem, an infinite-dimensional generalization of the infinite Ramsey theorem, can be stated in the language of semigroups as follows: for any finite Borel partition of the semigroup of monotone injections from *ω* into itself, one piece contains a right ideal. This talk provides a generalization of this result to the semigroups of embeddings of certain more complicated first-order structures. The definition and construction of these structures arise from the study of big Ramsey degrees in binary free amalgamation classes. This is joint work with Natasha Dobrinen.**Information:** Please see http://gfs.fields.utoronto.ca/activities/22-23/set-theory-seminar.

**CUNY Logic Workshop****Time:** Friday, 28 October, 2:00 – 3:30 New York time (20:00-21:30 CEST)**Speaker: **Corey Switzer, University of Vienna**Title: **Ideal Independence, Filters and Maximal Sets of Reals**Abstract:** A family I⊆[ω]ω is called ideal independent if given any finite, distinct A,B0,…,Bn−1∈I, the set A∖⋃i<nBi is infinite. In other words, the ideal generated by I∖{A} does not contain A for any A∈I. The least size of a maximal (with respect to inclusion) ideal independent family is denoted smm and has recently been tied to several interesting questions in cardinal characteristics and Boolean algebra theory. In this talk we will sketch our new proof that this number is ZFC-provably greater than or equal to the ultrafilter number – the least size of a base for a non-principal ultrafilter on ω. The proof is entirely combinatorial and relies only on a knowledge of ultrafilters and their properties. Time permitting, we will also discuss some interesting new applications of ideal independent families to topology via a generalization of Mrowka spaces usually studied for almost disjoint families. This is joint work with Serhii Bardyla, Jonathan Cancino and Vera Fischer.**Information:** The talk will take place in hybrid mode. Please see the seminar webpage or email Victoria Gitman for the link.

17-23 October

**Vienna Research Seminar in Set Theory****Time:** Tuesday, 18 October, 15:00-16:30 CEST**Speaker:** Diana Carolina Montoya, University of Vienna **Title:** Maximal independence and singulars**Abstract:** In these talks, we will discuss the concept of maximal independent families for uncountable singular cardinals. In the first part, I will present the existing background results in the regular case from Kunen and Eskew-Fischer. In the second part, we will focus on the joint results obtained in joint work with Omer Ben-Neria: some on the existence of maximal independent families at a singular strong limit, and finally some on the possible sizes of such families.**Information:** This talk will be given in hybrid format. Please contact Richard Springer for information how to participate.

**Baltic Set Theory Seminar****Time:** Tuesday, 18 October, 15:00-16:30 CEST**Speaker:** Several**Title:** Baltic Set Theory Seminar**Abstract:** This is a learning seminar, the goal is to actually go over proofs and more or less understand them. Discussions are encouraged. The topic of the seminar is the following:

1. Sandra Müller, Stationary-tower-free proof of Woodin’s Sealing Theorem.

2. Matteo Viale, Generic absoluteness theorem for the omega_1 Chang model conditioned to MM^{+++}.**Information:** Please see the seminar webpage.

**Helsinki Logic Seminar****Time:** Wednesday, 19 October, 12:00 – 14:00 Helsinki time (11:00-13:00 CEST) **Speaker:** Davide Quadrellaro**Title:** Weak Dependence Logic**Abstract:** We introduce and study a fragment of dependence logic which we call weak dependence logic. The interest in this fragment lays in the fact that, although it is much less expressive than full dependence logic, it has three desirable features:

(i) a strong form of compactness;

(ii) the De Jongh property w.r.t. propositional dependence logic;

(iii) it admits a notion of model-theoretic type which gives rise to Esakia spaces.

We show these properties and establish connections to the algebraic and topological semantics of propositional dependence logic.**Information:** The talk will take place in hybrid mode. Please see the seminar webpage for the link.

**Leeds Models and Sets Seminar****Time:** Wednesday, 19 October, 13:45-15:00 local time (14:45-16:00 CEST)**Speaker:** Philipp Schlicht, University of Bristol**Title:** Interaction of determinacy and forcing**Abstract:** Determinacy principles provide a unified theory of definable sets of reals beyond Borel and analytic sets, while forcing is an important technique to study the independence of properties of sets of reals. This suggests studying the interaction of the two: how robust are determinacy principles under well behaved forcings? I will talk about the history of this problem as well as recent joint results with Jonathan Schilhan and Johannes Schürz on iterations of proper forcings. A sample application of our results is the following: starting from a model of analytic determinacy, one can construct a model of analytic determinacy and the Borel conjecture.**Information:** Please see the seminar webpage.

**Barcelona Set Theory Seminar****Time:** Wednesday, 19 October, 16:00-17:30 CEST**Speaker:** Christopher Turner, University of Bristol**Title:** Lowenheim-Skolem-Tarski Numbers for Regularity Quantifiers**Abstract:** Let Q1, …, Qn be quantifiers of second-order logic. The Lowenheim-Skolem-Tarski number LST(Q0, … ,Qn) is the smallest cardinal kappa such that for any first order structure A in a language of size less than kappa, there exists a substructure B<A of size less than kappa, which is elementary in the language L U {Q0, …, Qn}. In 2011, Magidor and Väänänen found exact lower bounds for two important second order quantifiers: the Härtig quantifier I, and the equal cofinality quantifier Qec. Informally, I tells us about the cardinals of V, and Qec tells us which of them are V-inaccessible. In this talk I will present a generalisation of these to the two schemes of intermediate quantifiers Qa and Ra. These two quantifiers (which are defined for any ordinal a) fit between I and Qec, telling us about precisely the inaccessibles with Cantor-Bendixson rank less than a. I will introduce the quantifiers, and examine how their LST numbers relate to each other and to the LST numbers for I and Qec. I will then give a lower bound for each of the LST numbers. Finally, I will sketch a proof that this bound is exact, assuming the consistency of supercompacts.**Information:** Online. If you wish to attend, please send an email to bagaria@ub.edu asking for the link.

**Bristol Logic and Set Theory SeminarTime:** Thursday, 20 October, 10:30-11:30 UK time (11:30-12:30 CEST) – note the time change

**Speaker:**Frank Stephan, National University of Singapore

**Title:**Initial Segment Complexity for Measures

**Abstract:**Based on a joint paper with Andre Nies (https://arxiv.org/abs/1902.07871), the speaker will present the background and selected results of the paper. Furthermore, the slides are available here: https://www.comp.nus.edu.sg/~fstephan/measurerandomtalkslides.pdf.

**Information:**The login information is posted on the seminar webpage.

**CUNY Set Theory Seminar****Time:** Friday, 21 October, 12:30pm New York time (18:30 CEST)**Speaker: **Andreas Lietz, University of Münster**Title:** Forcing ‘NSω1 is ω1-dense’ from Large Cardinals – A Journey guided by the Stars**Abstract:** An ideal I on ω1 is ω1-dense if (P(ω1)/I)+ has a dense subset of size ω1. We prove, assuming large cardinals, that there is a semiproper forcing P so that VP⊨‘NSω1 is ω1-dense’. This answers a question of Woodin positively. Our general strategy is based on the observation that replacing the role of Pmax in Woodin’s axiom (∗) by Qmax results in an axiom Qmax−(∗) which implies ‘NSω1 is ω1-dense’.

We proceed in three steps: First we define and motivate a new forcing axiom QM and then modify the Asperó-Schindler proof of ‘MM++⇒(∗)’ to show ‘QM⇒Qmax−(∗)’. Finally, assuming a supercompact limit of supercompact cardinals exists, we construct a semiproper partial order forcing QM. This last step involves proving two new iteration theorems both of which allow for forcings killing stationary sets.**Information:** The seminar will take place virtually. Please email Victoria Gitman (vgitman@nylogic.org) for the meeting id.

**Toronto Set Theory Seminar****Time:** Friday, 21 October, 1.30-3.00 Toronto time (19.30-21.00 CEST)**Speaker:** Shaun Allison, University of Toronto**Title:** The classification strength of Polish groups**Abstract:** We reframe much of the study of Borel reductions between orbit equivalence relations as the study of the classification strength of Polish groups. This is a partial order where we say G is stronger than H iff every orbit equivalence relation induced by a continuous action of H on a Polish space is Borel reducible to such an orbit equivalence relation induced by G. We discuss recent results pertaining to the non-Archimedean Polish groups with maximum classification strength, namely, the groups which involve S_infty. We give several surprising conditions which are equivalent to involving S_infty. Time permitting, we will discuss other work on other parts of the hierarchy of classification strength, including joint work with Aristotelis Panagiotopouloss.**Information:** Please see http://gfs.fields.utoronto.ca/activities/22-23/set-theory-seminar.

**CUNY Logic Workshop****Time:** Friday, 21 October, 2:00 – 3:30 New York time (20:00-21:30 CEST)**Speaker: **Philipp Rothmaler, CUNY**Title: **Generalized Bass modules**Abstract:** Over half a century ago Hyman Bass proved that all flat left modules are projective precisely when the underlying ring satisfies the descending chain condition on right principal ideals. He called such rings left perfect. Gena Puninski noticed that this can be given a model theoretic proof. Every infinite descending chain of principal right ideals gives rise to a descending chain of (pp) formulas which, in turn, gives rise to a direct limit of finitely generated projective modules that is not projective. Such a module is flat and not projective, and called a Bass module.

I demonstrate how this construction is elementary model theory and at the same time generalizes to other classes of (pp) formulas and modules, which, among other things, yields a new proof of the late Daniel Simson’s result that all left modules are Mittag-Leffler iff the ring is left pure-semisimple (which, to model theorists, means that all left modules are totally transcendental).

I will emphasize the model theoretic ideas and explain the connection with the algebraic concepts. This is part of ongoing work with Anand Pillay.**Information:** The talk will take place in hybrid mode. Please see the seminar webpage or email Victoria Gitman for the link.

10-16 October

**Vienna Research Seminar in Set Theory****Time:** Tuesday, 11 October, 15:00-16:30 CEST**Speaker:** L. Schembecker, University of Vienna**Title:** A Sacks-indestructible partion of Baire space into compact sets II**Abstract:** Remember that maximal almost disjoint families of finitely splitting trees (a.d.f.s. families) are in one-to-one correspondence with partitions of Baire space into compact sets. In part I we saw how to construct an a.d.f.s. family which is indestructible by the product of Sacks forcing of size ℵ0. In part II we strengthen the construction to get an a.d.f.s family which stays maximal after forcing with countably supported product or iteration of Sacks forcing of any length. The proof is an adaptation of the construction of a Sacks-indestructible maximal eventually different family by V. Fischer and D. Schrittesser. If time permits we give an idea how to generalize the construction to other combinatorial families, for example maximal cofinitary groups. **Information:** This talk will be given in hybrid format. Please contact Richard Springer for information how to participate.

**Baltic Set Theory Seminar****Time:** Tuesday, 11 October, 15:00-16:30 CEST**Speaker:** Several**Title:** Baltic Set Theory Seminar**Abstract:** This is a learning seminar, the goal is to actually go over proofs and more or less understand them. Discussions are encouraged. The topic of the seminar is the following:

1. Sandra Müller, Stationary-tower-free proof of Woodin’s Sealing Theorem.

2. Matteo Viale, Generic absoluteness theorem for the omega_1 Chang model conditioned to MM^{+++}.**Information:** Please see the seminar webpage.

**CMU Core Model Theory Seminar****Time:** Tuesday, 11 October, 1:30 – 3:00pm Pittsburgh time (19:30 – 21:00 CEST) **Speaker:** Sean Cody**Title:** Full Determinacy from Turing Determinacy over L(R)**Abstract: **The precise relationship between the Axiom of Determinacy and

Turing Determinacy has been a longstanding open problem. In this talk we

will look at a core model induction proof of Woodin’s result that, in

L(R), Turing Determinacy (+DC) implies the Axiom of Determinacy. This

example avoids several of the conceptual roadblocks present in more

difficult CMI arguments, so it will (hopefully) be presented in an

approachable manner.**Information:** Please contact Ernest Schimmerling in advance for the zoom link.

**CMU Logic Seminar****Time:** Tuesday, 11 October, 3:30 – 4:30pm Pittsburgh time (21:30 – 22:30 CEST) **Speaker:** Rehana Patel, Bentley University and Northeastern University**Title:** The number of ergodic models of an infinitary sentence**Abstract: **Given an $L_{\omega_1\omega}$-sentence $\varphi$ in a countable language, we call an ergodic $S_\infty$-invariant probability measure on the Borel space of countable models of $\varphi$ (having fixed underlying set) an \emph{ergodic model} of $\varphi$. I will discuss the number of ergodic models of such a sentence $\varphi$, including the case when $\varphi$ is a Scott sentence. This is joint work with N. Ackerman, C. Freer, A. Kruckman and A. Kwiatkowska.**Information:** See the seminar webpage.

**Helsinki Logic Seminar****Time:** Wednesday, 12 October, 14:00 – 16:00 Helsinki time (13:00-14:00 CEST) **Speaker:** Jouko Väänänen**Title:** Inner models from extended logics**Abstract:** If we replace first order logic by second order logic in the original definition of Gödel’s inner model L, we obtain HOD. In this talk we consider inner models that arise if we replace first order logic by a logic that has some, but not all, of the strength of second order logic. Typical examples are the extensions of first order logic by generalized quantifiers, such as the Magidor-Malitz quantifier, the cofinality quantifier, or stationary logic. It can be shown that both L and HOD manifest some amount of formalism freeness in the sense that they are not very sensitive to the choice of the underlying logic.

On the other hand, the cofinality quantifier gives rise to a new robust inner model between L and HOD. Assuming a proper class of Woodin cardinals the regular cardinals above aleph-1 of V are weakly compact in the inner model arising from the cofinality quantifier and the theory of that model is (set) forcing absolute and

independent of the cofinality in question. Assuming three Woodin cardinals and a measurable above them, if the construction is relativized to a real, then on a cone of reals the Continuum Hypothesis is true in the relativized model.

A potentially bigger inner model C(aa) arises from stationary logic. Assuming a proper class of Woodin cardinals, or alternatively MM- plus-plus, the regular uncountable cardinals of V are measurable in the inner model C(aa), the theory of C(aa) is (set) forcing absolute, and C(aa) satisfies CH. We introduce an auxiliary concept that we call club determinacy, which simplifies the construction of C(aa) greatly but may have also independent interest. Based on club determinacy, we introduce the concept of aa-mouse which we use to prove CH and other properties of the inner model C(aa). Finally, we discuss a delicate matter related to the Axiom of Choice in the inner model C(aa) and in inner models of the same kind.

This is joint work with Juliette Kennedy and Menachem Magidor.**Information:** The talk will take place in hybrid mode. Please see the seminar webpage for the link.

**Leeds Models and Sets Seminar****Time:** Wednesday, 12 October, 13:45-15:00 local time (14:45-16:00 CEST)**Speaker:** Assaf Rinot, Bar-Ilan University**Title:** The small Dowker space problem**Abstract:** It is well-known that the product of two normal topological spaces need not be normal, but what about the normality of the product of a normal space X with the unit interval [0,1]? A counterexample space X is called a “Dowker space”. In 1972, Rudin proved that such a space exists, but it remains open whether there must exist a Dowker space of size Aleph_1. In this talk, we shall report on a joint work with Shalev and Todorcevic in which we present a weak sufficient condition for the existence of a small Dowker space.**Information:** Please see the seminar webpage.

**Caltech Logic Seminar****Time:** Wednesday, 12 October, 11:00-12:00pm Pacific time (20:00-21:00 CEST)**Speaker:** Todor Tsankov, Université Lyon 1**Title:** Topological dynamics of kaleidoscopic groups**Abstract:** Kaleidoscopic groups are infinite permutation groups recently introduced by Duchesne, Monod, and Wesolek by analogy with a classical construction of Burger and Mozes of subgroups of automorphism groups of regular trees. In contrast with the Burger-Mozes groups, kaleidoscopic groups are never locally compact and they are realized as groups of homeomorphisms of Wazewski dendrites (tree-like, compact spaces whose branch points are dense). The input for the construction is a finite or infinite permutation group Γ and the output is the kaleidoscopic group K(Γ).

In this talk, I will discuss recent joint work with Gianluca Basso, in which we explain how these groups can be viewed as automorphism groups of homogeneous structures and characterize the universal minimal flow of K(Γ) in terms of the original group Γ.**Information:** Please see the seminar webpage.

**CUNY Set Theory Seminar****Time:** Friday, 14 October, 12:30pm New York time (18:30 CEST)**Speaker: **Philipp Lücke, University of Barcelona**Title:** Large cardinals, strong logics and reflection principles**Abstract:** Various results establish deep connections between the existence of large cardinals, regularity properties of strong logics and the validity of set-theoretic reflection principles. In particular, several compactness properties of strong logics were proven to be equivalent to large cardinal axioms. An important example of such an equivalence is given by a theorem of Makowsky that shows that *Vopěnka’s Principle* is equivalent to the existence of strong compactness cardinals for all abstract logics. Motivated by work of Boney, Dimopoulos, Gitman and Magidor, I recently proved an analogous combinatorial characterization of the existence of weak compactness cardinals for all abstract logics that is closely connected to the notion of *subtle cardinals*, introduced by Kunen and Jensen in their studies of strong diamond principles, and the concept of *shrewd cardinals*, defined by Rathjen in proof-theoretic work. In my talk, I want to first discuss the details of this characterization and then present connections to recent joint work with Joan Bagaria (Barcelona) on recurring patterns 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, 14 October, 1.30-3.00 Toronto time (19.30-21.00 CEST)**Speaker:** Shaun Allison, University of Toronto**Title:** The classification strength of Polish groups**Abstract:** We reframe much of the study of Borel reductions between orbit equivalence relations as the study of the classification strength of Polish groups. This is a partial order where we say G is stronger than H iff every orbit equivalence relation induced by a continuous action of H on a Polish space is Borel reducible to such an orbit equivalence relation induced by G. We discuss recent results pertaining to the non Archimedean Polish groups with maximum classification strength, namely, the groups which involve S_infty. We give several surprising conditions which are equivalent to involving S_infty. Time permitting, we will discuss other work on other parts of the hierarchy of classification strength, including joint work with Aristotelis Panagiotopoulos.**Information:** Please see http://gfs.fields.utoronto.ca/activities/22-23/set-theory-seminar.

**CUNY Logic Workshop****Time:** Friday, 14 October, 2:00 – 3:30 New York time (20:00-21:30 CEST)**Speaker:** Chris Conidis, CUNY**Title:** The computability of Artin-Rees and Krull Intersection**Abstract:** We will explore the computational content of two related algebraic theorems, namely the Artin-Rees Lemma and Krull Intersection Theorem. In particular we will show that, while the strengths of these theorems coincide for individual rings, they become distinct in the uniform context.**Information:** The talk will take place in hybrid mode. Please see the seminar webpage or email Victoria Gitman for the link.

3-9 October

**CMU Core Model Theory Seminar****Time:** Tuesday, 4 October, 1:30 – 3:00pm Pittsburgh time (19:30 – 21:00 CEST) **Speaker:** Nam Trang, University of North Texas**Title:** Core model induction toolbox and examples, part 4**Abstract: **This is the abstract for the first four lectures.

We discuss the relevant concepts and tools that go into a core model induction through L(R) (or more generally through models of the form Lp^\Sigma(R)). We provide examples (PFA, the existence of an \omega_1-dense ideal on \omega_1 etc) that illustrate how these concepts are used in practice.**Information:** Please contact Ernest Schimmerling in advance for the zoom link.

**Vienna Research Seminar in Set Theory****Time:** Tuesday, 4 October, 15:00-16:30 CEST**Speaker:** L. Schembecker, University of Vienna**Title:** Partitions of Baire space into compact sets**Abstract:** We follow up on the last talk by going through a detailed proof of the construction of an almost disjoint family of finitely splitting trees (a.d.f.s. family) which stays maximal after forcing with countably supported product or iteration of Sacks forcing of any length. Remember that maximal a.d.f.s. families are in one-to-one correspondence with partitions of Baire space into compact sets. To this end we first prove the main fusion lemma which lets us construct a maximal a.d.f.s. family which is indestructible by countably supported product of Sacks forcing of size ℵ0. We then adapt the construction of a Sacks-indestructible maximal eventually different family by V. Fischer and D. Schrittesser to show that this family already satisfies the indestructibility properties of our theorem. If time permits we give an idea how to generalize the construction to other combinatorial families, for example maximal cofinitary groups.**Information:** This talk will be given in hybrid format. Please contact Richard Springer for information how to participate.

**Baltic Set Theory Seminar****Time:** Tuesday, 4 October, 15:00-16:30 CEST**Speaker:** Several**Title:** Baltic Set Theory Seminar**Abstract:** This is a learning seminar, the goal is to actually go over proofs and more or less understand them. Discussions are encouraged. The topic of the seminar is the following:

1. Sandra Müller, Stationary-tower-free proof of Woodin’s Sealing Theorem.

2. Matteo Viale, Generic absoluteness theorem for the omega_1 Chang model conditioned to MM^{+++}.**Information:** Please see the seminar webpage.

**Helsinki Logic Seminar****Time:** Wednesday, 5 October, 12:00 – 14:00 Helsinki time (11:00-13:00 CEST)**Speaker:** Tuomas Hakoniemi**Title:** Ideal Refutations of Knapsack Contradictions**Abstract:** The Ideal Proof System (IPS), introduced by Grochow and Pitassi, is a strong algebraic proof system with close connections to central questions in algebraic circuit complexity. In this talk we present superpolynomial lower bounds in a constant-depth subsystem of IPS for a variant of Knapsack. Our argument builds on the recent breakthrough lower bounds for constant-depth algebraic circuits by Limaye, Srinivasan and Tavenas.

This talk is based on joint work with Nashlen Govindasamy and Iddo Tzameret. **Information:** The talk will take place in hybrid mode. Please see the seminar webpage for the link.**Leeds Models and Sets Seminar****Time:** Wednesday, 5 October, 13:45-15:00 local time (14:45-16:00 CEST)**Speaker:** Calliope Ryan-Smith, University of Leeds**Title:** Shattering Domination**Abstract:** The Erd\H{o}s-Rado arrow relation is to the order property as the shattering domination relation is to the independence property. In attempting to create a faithful translation of the independence property in a first-order theory in the setting of abstract elementary classes, the problem of having no compactness becomes clear immediately. In a logical setting involving the compactness theorem, it is easy to find `tree-indiscernible’ sequences with the same EM-type as any arbitrary tree (as in the same way one can find order-indiscernible sequences with the same EM-type as any arbitrary sequence). However, without such tools, we are left with a much more blunt weapon: taking large, extant trees and finding within them structures that just so happen to be indiscernible. The shattering domination relation (and its numerous derivatives) is an attempt to measure how blunt that weapon is, that is to say how large a tree has to be before we can find an indiscernible sub-tree of a given size. In the setting of ZFC+GCH, this is solved, but it seems likely that in ZFC+$\lnot$GCH, it is independent. **Information:** Please see the seminar webpage.

**Vienna Logic Colloquium****Time:** Thursday, 6 October, 15:00 – 16:30 CET

**Speaker:**A. Brooke-Taylor, University of Leeds

**Title:**Cardinal characteristics modulo nice ideals on omega

**Abstract:**Many of the standard cardinal characteristics of the continuum in terms of a relation holding almost everywhere, where “almost everywhere” means on all but a finite set. A very natural generalisation is to take “almost everywhere” to mean on all but a member of a given ideal. I will talk about what happens when we do this, with the density 0 ideal on ω as a focal example.

**Information:**This talk will be given in hybrid format. Please contact Richard Springer for information how to participate.

**Toronto Set Theory Seminar****Time:** Friday, 7 October, 1.30-3.00 Toronto time (19.30-21.00 CEST)**Speaker:** Keegan Dasilva Barbosa, The Fields Institute and University of Toronto**Title:** The weak Ramsey property and extreme amenability**Abstract:** In their highly influential work “Fra\”iss\’e Limits, Ramsey Theory, and Topological Dynamics of Automorphism Groups”, Kechris, Pestov, and Todorcevic characterized fixed point properties of non-archimedean Polish groups using structural Ramsey Theory. Category theoretic methods have demonstrated to be quite effective at both proving Ramsey theoretic results, and generalizing Fra\”iss\’e Theory as witnessed by works of Mašulović and Kubiś respectively. In fact, the Fra\”iss\’e sequence approach of Kubiś has been fruitful in producing a variety of generic structures. Our goal for this talk is to manufacture a correspondence in the vein of Kechris, Pestov, and Todorcevic in the categorical setting with weaker combinatorial hypotheses, but the same dynamical strength. Time permitted, we will also look at some examples. This is a joint work with Adam Bartoš, Tristan Bice, Wiesław Kubiś.**Information:** Please see http://gfs.fields.utoronto.ca/activities/22-23/set-theory-seminar.

**CUNY Set Theory Seminar****Time:** Friday, 7 October, 11:00pm New York time (17:00 CEST)**Speaker: **Sakae Fuchino, Kobe University**Title:** Definability of Laver-generic large cardinals and largeness of generic large cardinals with chain conditions**Abstract:** For a class P of posets, a cardinal κ is said to be generically supercompact by P (or P-gen. supercompact for short) if, for any λ≥κ, there are P∈P such that, for all (V,P)-generic G there are j, M⊆V[G] with j:V≺→κM, j(κ)>λ, and j”λ∈M.

A cardinal κ is Laver-generically supercompact for P (or P-Laver-gen. supercompact for short) if, for any λ≥κ, P∈P and (V,P)-generic G, there are P-name ˙Q with ∥–P“˙Q∈P” such that, for all (V,P∗˙Q)-generic H⊇G, there are j, M⊆V[H] such that j:V≺→κM, j(κ)>λ, and P∗˙Q, H, j”λ∈M.

Perhaps it is not apparent at first sight in the formulation the definitions above but these notions of generic large cardinals are first-order definable (S.F, and H. Sakai [1]).

While the generic supercompactness does not determine the size of the cardinal. Laver-generic supercompactness determines the size of the cardinal __and__ that of the continuum in most of the natural settings of P (see S.F., A.Ottenbreit Maschio Rodrigues, and H. Sakai [0] for a proof):

(A) If κ is P-Laver-gen. supercompact for a class P of posets such that (1) all P∈P are ω1preserving, (2) all P∈P do not add reals, and (3) there is a P1∈P which collapses ω2, __then__ κ=ℵ2 and CH holds.

(B) If κ is P-Laver-gen. supercompact for a class P of posets such that (1) all P∈P are ωpreserving, (2)’ there is an a P0∈P which add a real, and (3) there is a P1 which collapses ω2, __then__κ=ℵ2=2ℵ2.

(C) If κ is P-Laver-gen. supercompact for a class P of posets such that (1)’ all P∈P preserve cardinals, and (2)’ there is a P0∈P which adds a real, __then__ κ is very large and κ≤2ℵ0.

The case (C) can be still improved ([0]):

(C’) If κ is tightly P-Laver-gen. supercompact for a class P of posets such that (1)’ all P∈Ppreserve cardinals, and (2)’ there is a P0∈P which adds a real, __then__ κ is very large and κ=2ℵ0.* *

(A P-Laver-gen. supercompact cardinal κ is tightly P-Laver-gen. supercompact, if ˙Q in the definition of Laver-gen. supercompactness can always be chosen to be small enough — see [0] for a precise definition.)* *

In this talk, we are going to give a sketch of the proof of definability and discuss about a theorem which assesses the largeness of κ in (C) under the additional assumption that elements of P satisfy certain chain conditions.* *

[0] S.F., A.Ottenbreit Maschio Rodrigues, and H.Sakai, Strong downward Löwenheim-Skolem theorems for stationary logics, II — reflection down to the continuum, Archive for Mathematical Logic, Vol.60, 3-4, (2021), 495–523. https://fuchino.ddo.jp/papers/SDLS-x.pdf

[1] S.F., and H.Sakai, Generically supercompact cardinals by forcing with chain conditions RIMS Kôkûroku, No.2213 (2022). https://fuchino.ddo.jp/papers/RIMS2021-ccc-gen-supercompact-x.pdf

[2] S.F., and H.Sakai, The first-order definability of generic large cardinals, to appear. https://fuchino.ddo.jp/papers/definability-of-glc-x.pdf**Information:** The seminar will take place virtually. Please email Victoria Gitman (vgitman@nylogic.org) for the meeting id.

**CUNY Logic Workshop****Time:** Friday, 7 October, 2:00 – 3:30 New York time (20:00-21:30 CEST)**Speaker:** Krzysztof Krupiński, University of Wroclaw**Title:** Some Ramsey theory and topological dynamics for first order theories**Abstract:** I will discuss a theory developed in my joint paper with Junguk Lee and Slavko Moconja. One can view it as a variant of Kechris, Pestov, and Todorčević theory in the context of (complete first order) theories. I will discuss several ‘definable’ Ramsey-theoretic properties of first order theories and their dynamical characterizations. The point is that all the Ramsey-theoretic properties that we introduce involve ‘definable colorings’ and the dynamical characterizations are ‘dynamical properties of the theories’, i.e. they are expressed in terms of the action of the group of automorphisms of a monster (i.e. sufficiently saturated and homogeneous) model of the theory in question on the appropriate space of types. One of the basic results says that a theory has the definable Ramsey property iff it is extremely amenable (as defined by Hrushovski, Pillay and myself). But there are various other results, some of which are essentially new and may be surprising in comparison with the Kechris, Pestov, Todorčević theory. One of the motivations to study those properties was to understand better the so-called Ellis group of a theory (which was used by Pillay, Rzepecki, and myself to explain the nature of the Lascar Galois groups of first order theories and spaces of strong types, and led E. Hrushovski to some original development with striking applications to approximate subgroups). Using our dynamical characterizations, we obtain several criteria for profiniteness and for triviality of this Ellis group, with many examples where they apply. I will try to discuss it during my talk. If time permits, I may very briefly mention an abstract generalization of the above considerations and results, which also applies both to the context of definable groups as well as to the classical context of Kechris Pestov, Todorčević theory, leading to some new notions, results, and questions.**Information:** The talk will take place in hybrid mode. Please see the seminar webpage or email Victoria Gitman for the link.

26 September – 2 October

**CMU Core Model Theory Seminar****Time:** Tuesday, 27 September, 1:30 – 3:00pm Pittsburgh time (19:30 – 21:00 CEST) **Speaker:** Nam Trang, University of North Texas **Title:** Core model induction toolbox and examples, part 3**Abstract: **This is the abstract for the first three to four lectures.

We discuss the relevant concepts and tools that go into a core model induction through L(R) (or more generally through models of the form Lp^\Sigma(R)). We provide examples (PFA, the existence of an \omega_1-dense ideal on \omega_1 etc) that illustrate how these concepts are used in practice.**Information:** Please contact Ernest Schimmerling in advance for the zoom link.

**CMU Logic Seminar****Time:** Tuesday, 27 September, 3:30 – 4:30pm Pittsburgh time (21:30 – 22:30 CEST) **Speaker:** Riley Thornton, CMU**Title:** Complexity and dichotomy theorems**Abstract: **I will survey some recent results on projective complexity, and I will explain what they tell us about dichotomy theorems in descriptive set theory. In particular, I will explain how the (unfortunately named) CSP Dichotomy Theorem implies a formal connection between computational and projective complexity and how this tells us exactly which graph homomorphism problems admit descriptive set theoretic dichotomies.**Information:** See the seminar webpage.

**Caltech Logic Seminar****Time:** Wednesday, 28 September, 11:00am-12:00pm Pacific time (20:00-21:00 CEST)**Speaker:** Adrian Mathias, Université de la Réunion**Title:** Iteration problems in Symbolic Dynamics**Abstract:** In the decade 1994 – 2004 I wrote five papers applying techniques from descriptive set theory to a question posed by the dynamics group of Barcelona concerning the possible lengths of iterations. In August 2021 I gave two talks in the CUNY set theory Zoominar of Vika Gitman which were largely devoted to expounding the last and hardest of my constructions in this area. The present talk will be devoted to my earlier and more basic results, some recent work, and various open problems which I hope might attract logicians working in areas such as the descriptive set theory of group actions.**Information:** Please see the seminar webpage.

**Bristol Logic and Set Theory SeminarTime:** Thursday, 29 September, 12:30-13:30 UK time (13:30-14:30 CEST) – note the time change

**Speaker:**Will Stafford, University of Bristol

**Title:**Is the Proof-Theoretically Valid Logic Intuitionistic?

**Abstract:**Several recent results bring into focus the superintuitionistic nature of most notions of proof-theoretic validity, but little work has been done evaluating the consequences of these results. Proof-theoretic validity claims to offer a formal explication of how inferences follow from the definitions of logic connectives (which are defined by their introduction rules). This paper explores whether the new results undermine this claim. It is argued that, while the formal results are worrying, superintuitionistic inferences are valid because the treatments of atomic formulas are insufficiently general, and a resolution to this issue is proposed.

**Information:**For the zoom access code, please contact Philip Welch in advance.

**CUNY Set Theory Seminar****Time:** Friday, 30 September, 12:30pm New York time (18:30 CEST)**Speaker: **Victoria Gitman, CUNY**Title:** Jensen’s forcing at an inaccessible**Abstract:** Jensen constructed in L, using ♢, a subposet of the Sacks forcing with the ccc and the property that it adds a unique generic real over L (in contrast to, say, Cohen forcing which adds continuum many generic reals). He used what came to be known as Jensen’s forcing to show that, consistently, there can be a Π12-definable non-constructible real. The ‘uniqueness of generic reals’ property of Jensen’s forcing extends to products of Jensen’s forcing and to finite iterations, when forcing over L. Indeed, a Jensen-like forcing with the same uniqueness properties can be constructed in any universe with a ♢-sequence. In a joint work with Friedman and Kanovei, we used a tree iteration of Jensen’s forcing to construct (in a symmetric submodel of the forcing extension) a model of full second-order arithmetic Z2 with the choice scheme in which the dependent choice scheme failed for a Π12-assertion (this is optimal because Z2 with the choice scheme implies dependent choice for Σ12-assertions). In this talk, I will present a generalization of Jensen’s forcing to forcing with perfect κ-trees for an inaccessible cardinal κ. I will show that Jensen’s forcing at an inaccessible has the same ‘uniqueness of generics’ properties as Jensen’s forcing. One of the goals of this work is to prove an analogue of the second-order arithmetic result for second-order set theory by showing that the dependent choice scheme is independent of the second-order Kelley-Morse set theory with the choice scheme. This is joint work with Sy-David Friedman.**Information:** The seminar will take place virtually. Please email Victoria Gitman (vgitman@nylogic.org) for the meeting id.

**Toronto Set Theory Seminar****Time:** Friday, 30 September, 1.30-3.00 Toronto time (19.30-21.00 CEST)**Speaker:** Jing Zhang**Title:** Diamonds at Large Cardinals, part 2**Abstract:** It is well-known that large cardinal properties can give rise to guessing principles. For example, Jensen and Kunen showed that if kappa is a subtle cardinal, then the diamond principle holds at kappa and Laver showed that if kappa is a supercompact cardinal, then a stronger guessing principle, nowadays known as the Laver diamond, holds at kappa. The following two questions are long-standing and they motivate much of the research in the area: 1) can diamond fail at a weakly compact cardinal, 2) does GCH imply that diamond holds at an inaccessible cardinal. We will introduce weakenings of the diamond principle, which follow from weak compactness. Using these principles, we demonstrate a few ways to make diamond fail at an inaccessible cardinal as well as obstacles to make diamond fail at a weakly compact cardinal. Joint work with Omer Ben-Neria.**Information:** Please see http://gfs.fields.utoronto.ca/activities/22-23/set-theory-seminar.

**CUNY Logic Workshop****Time:** Friday, 30 September, 2:00 – 3:30 New York time (20:00-21:30 CEST)**Speaker:** Hans Schoutens, CUNY**Title:** The model-theory of categories**Abstract:** One could make the claim that category theory is as foundational as set-theory or model-theory. So, we should be able to transfer from one perspective to the other. In this talk, I will consider one aspect of this meta-equivalence, by introducing a theory in a very simple, one-sorted(!) language, whose models are all categories admitting a terminal object (many categories do). Many categorical constructions then turn out to be first-order. But something even more strange happens: standard categories (like the category of Abelian groups) become actually universal models! I’ll explain this apparent contradiction.

In the second part of the talk, I will concentrate on one particularly interesting category: that of compact Hausdorff spaces. I will show that we can recover the natural numbers N and the reals R, or rather, (the isomorphism classes of) their compactifications ¯N and ¯R, by parameter-free definitions, including their order relation, addition and multiplication. Moreover, in any category that is elementary equivalent to the category of compact Hausdorff spaces, the resulting objects are then a model of PA and a real closed field respectively. Full disclosure: while I have a complete proof for the first assertion, the second is still conjectural.

Apart from some basic model-theory, category theory and topology, everything else will be explained in the talk and so it should be accessible to many.**Information:** The talk will take place in hybrid mode. Please see the seminar webpage or email Victoria Gitman for the link.

19-25 September

**CMU Core Model Theory Seminar****Time:** Tuesday, 20 September, 1:30 – 3:00pm Pittsburgh time (19:30 – 21:00 CEST) **Speaker:** Nam Trang, University of North Texas **Title:** Core model induction toolbox and examples, part 2**Abstract: **This is the abstract for the first three to four lectures.

We discuss the relevant concepts and tools that go into a core model induction through L(R) (or more generally through models of the form Lp^\Sigma(R)). We provide examples (PFA, the existence of an \omega_1-dense ideal on \omega_1 etc) that illustrate how these concepts are used in practice.**Information:** Please contact Ernest Schimmerling in advance for the zoom link.

**CMU Logic Seminar****Time:** Tuesday, 20 September, 3:30 – 4:30pm Pittsburgh time (21:30 – 22:30 CEST) **Speaker:** James Cummings, Carnegie Mellon University**Title:** Extender based forcing and cardinal invariants of singular cardinals**Abstract: **(Joint work with Charles Morgan) Many of the cardinal invariants of the continuum (EG the bounding number) generalise in a reasonable way to uncountable regular cardinals, but not to singular cardinals. I will discuss a cardinal invariant (the ultrafilter character spectrum) which does generalise, and how to control its value by forcing.**Information:** See the seminar webpage.

**Helsinki Logic Seminar****Time:** Wednesday, 21 September, 12:00 – 14:00 Helsinki time (11:00-13:00 CEST) **Speaker:** Matilda Häggblom **Title:** Axiomatizing modal inclusion logic**Abstract:** Modal inclusion logic is team-based modal logic extended with inclusion atoms. The talk will cover the main result of my master’s thesis: A complete axiomatization of modal inclusion logic. We begin by recalling that modal inclusion logic is expressively complete for classes that have the empty team property, are closed under unions and closed under k-bisimulation for some k. Through the expressive completeness proof, we obtain a normal form for the logic. In the talk, we suggest a simplified version of the normal form compered to the one currently in the literature. We then introduce a natural deduction proof system and show completeness using the normal form.**Information:** The talk will take place in hybrid mode. Please see the seminar webpage for the link.

**Toronto Set Theory Seminar****Time:** Friday, 23 September, 1.30-3.00 Toronto time (19.30-21.00 CEST)**Speaker:** Jing Zhang**Title:** Diamonds at Large Cardinals**Abstract:** It is well-known that large cardinal properties can give rise to guessing principles. For example, Jensen and Kunen showed that if kappa is a subtle cardinal, then the diamond principle holds at kappa and Laver showed that if kappa is a supercompact cardinal, then a stronger guessing principle, nowadays known as the Laver diamond, holds at kappa. The following two questions are long-standing and they motivate much of the research in the area: 1) can diamond fail at a weakly compact cardinal, 2) does GCH imply that diamond holds at an inaccessible cardinal. We will introduce weakenings of the diamond principle, which follow from weak compactness. Using these principles, we demonstrate a few ways to make diamond fail at an inaccessible cardinal as well as obstacles to make diamond fail at a weakly compact cardinal. Joint work with Omer Ben-Neria.**Information:** Please see http://gfs.fields.utoronto.ca/activities/22-23/set-theory-seminar.

**CUNY Logic Workshop****Time:** Friday, 23 September, 2:00 – 3:30 New York time (20:00-21:30 CEST)**Speaker:** Russell Miller, CUNY**Title:** Interpreting a field in its Heisenberg group**Abstract:** The Heisenberg group G(F) of a field F is the group of upper triangular matrices in GL_3(F), with 1’s along the diagonal and 0’s below it. This group is obviously interpretable (indeed definable) in the field F. Mal’cev showed that one can recover F from G(F), and indeed that there is an interpretation of F in G(F) using two parameters. Any two noncommuting elements of G(F) can serve as the parameters, but Mal’cev was unable to produce an interpretation without parameters.

After introducing the notions of a *computable functor* and an *effective interpretation*, we will present joint work showing that there is an effective interpretation of each countable field in its Heisenberg group, without parameters, uniformly in F. (That is, the same formulas give the interpretation, no matter which field F we consider.) Moreover, from the effective interpretation we will then extract a traditional interpretation without parameters, in the usual model-theoretic sense. Finally we will note that, whereas Mal’cev’s result actually gives a definition of F in G(F), there is no parameter-free definition of F there.

This work is joint with Rachael Alvir, Wesley Calvert, Grant Goodman, Valentina Harizanov, Julia Knight, Andrey Morozov, Alexandra Soskova, and Rose Weisshaar.**Information:** The talk will take place in hybrid mode. Please see the seminar webpage or email Victoria Gitman for the link.

12-18 September

**CMU Core Model Theory Seminar****Time:** Tuesday, 13 September, 1:30 – 3:00pm Pittsburgh time (19:30 – 21:00 CEST) **Speaker:** Nam Trang, University of North Texas **Title:** Core model induction toolbox and examples, part 1**Abstract: **This is the abstract for the first three to four lectures.

We discuss the relevant concepts and tools that go into a core model induction through L(R) (or more generally through models of the form Lp^\Sigma(R)). We provide examples (PFA, the existence of an \omega_1-dense ideal on \omega_1 etc) that illustrate how these concepts are used in practice.**Information:** Please contact Ernest Schimmerling in advance for the zoom link.

**CMU Logic Seminar****Time:** Tuesday, 13 September, 3:30 – 4:30pm Pittsburgh time (21:30 – 22:30 CEST) **Speaker:** Rick Statman, Carnegie Mellon University**Title:** Cayley Monoids**Abstract: **Everyone is familiar with Cayley’s regular representation of groups in the symmetric group. And, if you think about monoids at all, you noted that it applies in a limited way to monoids. The notion of a Cayley monoid is just an internalization of this type of representation. There are many examples of Cayley monoids. Some of these are obtained from the group of a monoid acting on the monoid itself. There are some interesting special cases of Cayley monoids with strong <_{L} commutativity properties . An example is provided by the monoid Z*Q where Q is the quaternion group. In these cases we say that the Cayley monoid has an autonomous commutator. Here will will show that every monoid can be extended to a Cayley monoid with an autonomous commutator.**Information:** See the seminar webpage.

**Helsinki Logic Seminar****Time:** Wednesday, 14 September, 14:00 – 16:00 Helsinki time (13:00-15:00 CEST) – note the time change!**Speaker:** Andres Villaveces, Universidad Nacional de Colombia, Bogotá**Title:** Around Definability in AECs: Logics and Patterns**Abstract:** Definability in Abstract Elementary Classes has been both central and elusive. The possibility itself of the development of a good stability theory (now solidly established) was made possible early by the Shelah Presentation Theorem. Yet, in spite of the centrality of this theorem for the start of the theory, definability questions were never at the center of the model theory of AECs, at least not in a direct way. Later, results by Kueker et al. started capturing better definitions (not second order) of some AECs. In 2020, with Saharon Shelah, we provided a way to define arbitrary AECs with no expansion of the vocabulary. Leung improved the bound of the logic, at the expense of an infinitary game quantifier. We later improved Leung’s bound, removing the use of that game quantifier. More recently, in joint work with Nájar, we have started doing classification theory using our newly gained definability; recasting older results and capturing issues around definability of types. I will discuss this line of research, as well as possible connections with Hrushovski’s recent work on “definability patterns” aimed at balancing the Galois theory of (first order) model theory.**Information:** The talk will take plce in hybrid mode. Please see the seminar webpage for the link.

**Toronto Set Theory Seminar****Time:** Friday, 16 September, 1.30-3.00 Toronto time (19.30-21.00 CEST)**Speaker:** Ilijas Farah**Title:** Hilbert spaces with no choice**Abstract:** Every Hilbert space H has an orthonormal basis. This basis is a Hamel basis if and only if H is finite-dimensional. The algebra of bounded linear operators on H, B(H), is quite complicated, but its quotient over the ideal of compact operators (the Calkin algebra) is even worse (or better, depending on one’s taste) than P(N)/Fin. All of these statements fail in some models of ZF in which the Axiom of Choice fails. This is a report on joint work in progress with Bruce Blackadar and Asaf Karagila.**Information:** Please see http://gfs.fields.utoronto.ca/activities/22-23/set-theory-seminar.

**CUNY Logic Workshop****Time:** Friday, 16 September, 2:00 – 2:30 New York time (20:00-21:30 CEST)**Speaker:** Gunter Fuchs, CUNY**Title:** The blurry HOD hierarchy**Abstract:** Classically, an object is ordinal definable if it is the unique one satisfying a formula with ordinal parameters. Generalizing this concept, given a cardinal κ, I call an object <κ-blurrily ordinal definable if it belongs to an ordinal definable set with fewer than κ elements (called a <κ-blurry definition). By considering the hereditary versions of this notion, one arrives at a hierarchy of inner models, indexed by cardinals κ: the collection of all hereditarily <κ-blurrily ordinal definable sets, which I call <κ-HOD. In a ZFC-model, this hierarchy spans the entire spectrum from HOD to V.

The special cases κ=ω and κ=ω1 have been previously considered, but no systematic study of the general setting has been carried out, it seems. One main aspect of the analysis is the notion of a leap, that is, a cardinal at which a new object becomes hereditarily blurrily definable.

In this talk, I will focus on the ZFC-provable structural properties of the blurry HOD hierarchy, which turn out to be surprisingly plentiful. So for the most part, the talk will be forcing-free. Time permitting, I may hint at the result of the equiconsistency between the least leap being the successor of a singular strong limit cardinal and the existence of a measurable cardinal, for which, admittedly, forcing is used in one direction.**Information:** The talk will take place in hybrid mode. Please see the seminar webpage or email Victoria Gitman for the link.

5-11 September

**Helsinki Logic Seminar****Time:** Wednesday, 7 September, 12:00 – 14:00 Helsinki time (11:00-13:00 CEST)**Speaker:** Miguel Moreno, University of Vienna**Title:** Indestructibility and characterization of filter reflection**Abstract:** Filter reflection is a natural generalization of the stationary reflection. It was introduced by Fernandes-Moreno-Rinot when studying the generalized-Borel-reducibility. It was shown that it has deep implications in the generalized Baire space and Cantor space. It is known that filter reflection is independent from ZFC, it doesn’t require large cardinals, and it is consistent that filter reflection holds at any uncountable regular cardinal. In this talk we will go through some recent results about filter reflection, including indestructibility results. We will also show that filter reflection is equivalent to the existence of continuous reductions in the generalized Baire spaces.**Information:** The talk will take plce in hybrid mode. Please see the seminar webpage for the link.

29 August – 4 September

No seminars announced.

22-28 August

No seminars announced.

15-21 August

No seminars announced.

8-14 August

No seminars announced.

1-7 August

No seminars announced.

18-24 July

**Barcelona Set Theory Seminar****Time:** Wednesday, 20 July, 16:00-17:30 CEST**Speaker:** Omer Ben-Neria**Title:** Diamond, Compactness, and product approximations**Abstract:** It is well known that certain compactness principles imply the existence of diamonds. A long-standing open problem in the area asks if a weakly compact cardinal must carry a diamond sequence. We introduce a weak form of the diamond principle given in terms of function estimates on products of cardinals. We use the weaker principle to find new methods for forcing the failure of diamonds at inaccessible, Mahlo, and stationary reflecting cardinals and show that the weaker principle must hold at a weakly compact cardinal. This is joint work with Jing Zhang.**Information:** Online. If you wish to attend, please send an email to bagaria@ub.edu asking for the link.

11-17 July

No seminars announced.

4-10 July

No seminars announced.

27 June – 3 July

**KGRC Logic Colloquium, ViennaTime:** Thursday, 30 June, 15:00 – 15:45 CET

**Speaker:**A. Vignati, University of Paris

**Title:**Set theory and coronas of C*-algebras

**Abstract:**As abelian C*-algebras correspond functorially to locally compact Hausdorff space, studying C*-algebras is often viewed as noncommutative topology. A locally compact Hausdorff space X can be embedded densely in its Čech-Stone compactification βX, the ‘largest compact space in which X sits densely’. Similarly, to every nonunital C*-algebras A one can associate ‘the largest unital C*-algebra in which A sit densely’, the multiplier algebra M(A). Corona C*-algebras, quotients of the form M(A)/A, correspond to Čech-Stone remainders (space of the form βX∖X). Čech-Stone remainders have been studied with set theoretical methods since the ’80s. The work of Rudin, Shelah, Steprans, Velickovic, and Farah among others, showed that the structure of the space βX∖X, and its autohomeomorphisms, often depend on the set theoretic axioms in play. Similar phenomenons appear when studying corona C*-algebras, as Farah’s work on the Calkin algebra (the corona algebra of the compact operators) witnesses. This talk is dedicated to overview how different axioms in set theory impact the structure of automorphisms of corona C*-algebras.

**Information:**This talk will be given in hybrid format. Please contact Richard Springer for information how to participate.

**KGRC Set Theory Seminar, Vienna****Time:** Tuesday, 28 June, 15:00-16:30 CEST**Speaker:** F. Maesano, L. Schembecker, M. Koelbing, Ö. F. Bag, J. Millhouse, R. Doerner**Title:** Various topics**Abstract:** Doctoral and master students specialising in set theory will speak on selected topics from their work during the semester. For the detailed program with titles and abstracts, please see the seminar webpage. **Information:** This talk will be given via Zoom. Please contact Richard Springer for information how to participate.

20-26 June

**Barcelona Set Theory Seminar****Time:** Wednesday, 22 June, 16:00-17:30 CEST**Speaker:** Jeffrey Bergfalk**Title:** Higher derived limits and higher dimensional partitions of partial orders **Abstract:** This talk will primarily survey the recent work “A descriptive approach to higher derived limits”, joint with Nathaniel Bannister, Justin Moore, and Stevo Todorcevic (arXiv:2203.00165). At the heart of this work is a new family of partition principles which synthesize several recent advances in the study of higher derived limits, rendering those results far more amenable to combinatorial analyses. These principles admit formulation on any directed quasi-order, and are of particular, and interrelated, interest on the quasi-orders (ww, £*) and the ordinals wn. We’ll begin with a brief review of higher derived limits, then turn our attention to these partition principles, closing with a selection of open questions.**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, 22 June, 14:00-16:00 Israel Time (13:00-15:00 CEST)**Speaker:** tba**Title:** tba**Abstract:** tba**Information:** Contact Menachem Magidor or Omer Ben-Neria ahead of time for the seminar announcement and zoom link.

**KGRC Set Theory Seminar, Vienna****Time:** Tuesday, 21 June, 15:00-16:30 CEST**Speaker:** David Schrittesser, University of Toronto**Title:** The Ramsey property, MAD families, and their higher dimensional relatives**Abstract:** Infinite maximal almost disjoint families, dubbed “MAD families” by A.R.D. Mathias, have long been an object of interest in set theory, topology, and other areas. A question which has been tossed around for quite a while was whether there can exist an analytic Fin^{2}-MAD family – that is, a two-dimensional variant of the usual notion of MAD family. Analytic (one-dimensional) MAD families cannot exist, so the conjecture has always been “no” – and indeed the answer is “no”, as was shown in joint work with Törnquist and Bakke Haga in 2016.

In 1969, Mathias asked whether Ramsey regularity rules out the existence of (one dimensional) MAD families. This question was answered positively in joint work with Törnquist in 2019.

But now that we know that Fin^{2} MAD families behave like MAD families in some respects, and that Ramsey regularity rules out (one dimensional) MAD families, does Ramsey regularity also rule out the two dimensional variant?

Yes, Ramsey regularity rules out the existence of Fin^{2} MAD families (also joint work with Asger Törnquist). The result even holds for J-MAD families, where J is an ideal in the smallest class containing the ideal of finite sets and closed under Fubini products. I will also report on work in progress with our student Severin Mejak.**Information:** This talk will be given via Zoom. Please contact Richard Springer for information how to participate.

13-19 June

**Bristol Logic and Set Theory SeminarTime:** Thursday, 16 June, 15:00-16:30 UK time (16:00-17:30 CEST)

**Speaker:**Tanmay Inamdar

**Title:**From Sierpinski-type colourings to Ulam-type matrices

**Abstract:**Ulam matrices were introduced by Ulam in his study of the measure problem. Ulam’s construction applies to all successor cardinals Kappa, and later Hajnal extended the construction to apply to some limit cardinals as well. In my talk I will show how a colouring principle introduced by Sierpinski can be used to construct matrices with similar applications as the matrices of Ulam and Hajnal. I will also show how such colouring principles can be obtained from the existence of a non-trivial C-sequence on Kappa using walks on ordinals. As a consequence, the resulting matrices are more readily available than the matrices of Ulam and Hajnal. The results I present are joint work with Assaf Rinot.

**Information:**For the zoom access code, please contact Philip Welch in advance.

**Barcelona Set Theory Seminar****Time:** Wednesday, 15 June, 16:00-17:30 CEST**Speaker:** Kameryn Williams**Title:** Inner mantles: the good, bad, and ugly**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 class forcing extension. 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, call these 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:** 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, 15 June, 14:00-16:00 Israel Time (13:00-15:00 CEST)**Speaker:** Shaun Allison**Title:** Polish groups with the pinned property, continued**Abstract:** We will discuss a property of Polish groups called the “pinned property” which means that every orbit equivalence relation they generate is “pinned”, a metamathematical notion which is used to separate the complexity of different equivalence relations up to Borel reducibility. We will discuss the subtle way that the amount of choice assumed influences the pinned property. In particular, we will discuss results of Su Gao and Alex Thompson which imply that in a mode of ZFC, a Polish group has the pinned property if and only if it has a complete compatible left-invariant metric. We will also present a new result which, along with a result of Larson-Zapletal, implies that in the Solovay model derived from a measurable, a Polish group has the pinned property if and only if it involves S_\infty (caveat: for the special case of non-Archimedian groups). Time permitting, we will discuss Larson-Zapletal’s result as well. This is part of a larger project to measure and categorize the “classification strength” of Polish groups.**Information:** Contact Menachem Magidor or Omer Ben-Neria ahead of time for the seminar announcement and zoom link.

**KGRC Set Theory Seminar, Vienna****Time:** Tuesday, 14 June, 15:00-16:30 CEST**Speaker:** G. Fuchs, CUNY**Title:** Another look at bounded forcing axioms**Abstract:** Bounded forcing axioms were introduced by Goldstern and Shelah as weak versions of the usual forcing axioms, and were subsequently proved to be equivalent to a form of generic absoluteness. I will describe a characterization of these bounded forcing axioms that is more closely related to Woodin’s characterization of the regular forcing axioms, and that can be seen more directly to entail Bagaria’s generic absoluteness property. This new way of stating the bounded forcing axioms is very natural and provides a different perspective. It grew out of work with Ben Goodman.**Information:** This talk will be given in hybrid format. Please contact Richard Springer for information how to participate.

**Leeds Models and Sets Seminar****Time:** Tuesday, 14 June, 13:45-14:55 UK time (14:45-15:55 CEST)**Speaker:** Daisuke Ikegami, Shibaura Institute of Technology**Title:** On preserving AD via forcings**Abstract:** It is well-known that forcings preserve ZFC, i.e., any set generic extension of any model of ZFC is again a model of ZFC. How about the Axiom of Determinacy (AD) under ZF? It is not difficult to see that Cohen forcing always destroys AD, i.e., any set generic extension of a model of ZF + AD via Cohen forcing is not a model of AD. Actually it is open whether there is a forcing which adds a new real while preserving AD. In this talk, we present some results on preservation & non-preservation of AD via forcings, whose details are as follows:

1. Starting with a model of ZF + AD^+ + V = L(P(R)), any forcing increasing \Theta destroys AD.

2. It is consistent relative to ZF + AD_R that ZF + AD^+ + There is a forcing which increases \Theta while preserving AD.

3. In ZF, no forcings on the reals preserve AD. (This is an improvement of the result of Chan and Jackson where they additionally assumed \Theta is regular.)

4. In ZF + AD^+ + V = L(P(R)) + \Theta is regular, there is a forcing on \Theta which adds a new subset of \Theta while preserving AD.

This is joint work with Nam Trang.**Information:** Please see the seminar webpage.

6-12 June

**Cross-Alps Logic Seminar****Time:** Friday, 10 June, 16.00-18.00 CEST **Speaker:** S. L’Innocente, University of Camerino**Title:** A factorisation theory for generalised power series**Abstract:** A classical tool in the study of real closed fields are the fields K((G)) of generalised power series (i.e., formal sums with well-ordered support) with coefficients in a field K of characteristic 0 and exponents in an ordered abelian group G. A fundamental result of Berarducci ensures the existence of irreducible series in the subring of generalised power series with non-positive exponents. This report aims at describing a factorisation theory in this context: a joint work with Vincenzo Mantova proves that every series admits a factorisation into a bounded number of irreducibles and a unique product, up to multiplication by a unit, of factors whose supports are finite and generate rational linear spaces of dimension one. Analogous results are deduced for the ring of omnific integers within Conway’s surreal numbers, using a suitable notion of infinite product. In turn, Gonshor’s conjecture is solved: the omnific integer omega 2 + omega + 1 is prime. Other possible generalizations will also be sketched.**Information:** The event will stream on the Webex platform. Please write to luca.mottoros [at] unito.it for the link to the event.

**Hebrew University-Bar Ilan University Set Theory seminar****Time:** Wednesday, 8 June, 14:00-16:00 Israel Time (13:00-15:00 CEST)**Speaker:** Shaun Allison**Title:** Polish groups with the pinned property**Abstract:** We will discuss a property of Polish groups called the “pinned property” which means that every orbit equivalence relation they generate is “pinned”, a metamathematical notion which is used to separate the complexity of different equivalence relations up to Borel reducibility. We will discuss the subtle way that the amount of choice assumed influences the pinned property. In particular, we will discuss results of Su Gao and Alex Thompson which imply that in a mode of ZFC, a Polish group has the pinned property if and only if it has a complete compatible left-invariant metric. We will also present a new result which, along with a result of Larson-Zapletal, implies that in the Solovay model derived from a measurable, a Polish group has the pinned property if and only if it involves S_\infty (caveat: for the special case of non-Archimedian groups). Time permitting, we will discuss Larson-Zapletal’s result as well. This is part of a larger project to measure and categorize the “classification strength” of Polish groups.**Information:** Contact Menachem Magidor or Omer Ben-Neria ahead of time for the seminar announcement and zoom link.

**Leeds Models and Sets Seminar****Time:** Tuesday, 7 June, 13:45-14:55 UK time (14:45-15:55 CEST)**Speaker:** Lorna Gregory, Università degli Studi della Campania Luigi**Title:** Decidability of Theories of Modules of Prüfer domains**Abstract:** An integral domain is Prüfer if its localisation at each maximal ideal is a valuation domain. Many classically important rings are Prüfer domains. For instance, they include Dedekind domains and hence rings of integers of number fields; Bézout domains and hence the ring of complex entire functions and the ring of algebraic integers; the ring of integer valued polynomials with rational coefficients and the real holomorphy rings of formally real fields.

Over the last 15 years, efforts have been made to characterise when the theory of modules of (particular types of) Prüfer domains are decidable. I will give an overview of such decidability results culminating in recently obtained elementary conditions completely characterising when the theory of modules of an arbitrary Prüfer domain is decidable.**Information:** Please see the seminar webpage.

30 May – 5 June

**KGRC Logic Colloquium, ViennaTime:** Thursday, 2 June, 15:00 – 15:45 CET

**Speaker:**J. Aguilera, TU Wien

**Title:**The metamathematics of \Pi^1_2 sentences

**Abstract:**We will survey some recent results on the metamathematics of Π12 sentences. Most of the work involves a kind of Proof Theory analogous to classical ordinal analysis, but focused on a Π12 notion instead. The talk will be aimed at a general logic audience. Topics will include: proof-theoretic Π12-norms, a characterization of the Π12 consequences of arithmetical comprehension and related systems, Π12-soundness ordinals, and the Π12-Spectrum Conjecture. This is joint work with F. Pakhomov.

**Information:**This talk will be given in hybrid format. Please contact Richard Springer for information how to participate.

**Caltech Logic Seminar****Time:** Wednesday, 1 June, 12:00-1:00pm Pacific time (21:00-22:00 CEST)**Speaker:** Adrian Mathias (Université de la Réunion)**Title:** Iteration problems in Symbolic Dynamics**Abstract:** In the decade 1994-2004, I wrote five papers applying techniques from descriptive set theory to a question posed by the dynamics group of Barcelona concerning the possible lengths of iterations. Last August, I gave two talks in the CUNY set theory Zoominar of Vika Gitman which were largely devoted to expounding the last and hardest of my constructions in this area. The present talk will be devoted to my earlier and more basic results, some recent work, and various open problems which I hope might attract logicians working in areas such as the descriptive set theory of group actions.**Information:** Please see the seminar webpage.

**Barcelona Set Theory Seminar****Time:** Wednesday, 1 June, 16:00-17:30 CEST**Speaker:** William Chan **Title:** Almost Disjoint Families under Determinacy**Abstract:** In this talk, we will consider almost disjoint families on singular and regular

cardinals with respect to the ideal of small cardinality (which is the classical almost

disjoint family) and the ideal of bounded sets under the axiom of determinacy. We will

show that under certain determinacy assumptions, all almost disjoint families for either

ideal on certain cardinals of uncountable cofinality must be wellorderable. When

combined with suitable instances of boldface GCH, this will imply there are no maximal

almost disjoint families for either ideal for suitable cardinals of uncountable cofinality

which do not inject into its own cofinality. Every infinite maximal almost disjoint family

for either ideal on cardinals of countable cofinality (if they exists) must be

nonwellorderable. Extending arguments of Schritteser and Tornquist, suitable

determinacy assumptions will imply there are no infinite maximal almost disjoint

families for the ideal of bounded sets for any cardinal of countable cofinality. In L(R)

satisfying AD, we will be able to give a fairly complete answer to the maximal almost

disjoint family problem for the ideal of bounded sets for all cardinals. This is joint work

with Jackson and Trang.**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, 1 June, 14:00-16:00 Israel Time (13:00-15:00 CEST)**Speaker:** Ur Yaar**Title:** Inner models from extended logics**Abstract:** Given some logic L* extending first-order logic, we can construct an inner model C(L*) in an L-like fashion – at each level of the construction, we add all the subsets of the current level, which are L*-defineable instead of merely first order definable. This gives us a plethora of inner models, with various properties. In my research I focused on models constructed from the logics obtained by adding cofinality-quantifiers, and the so-called stationary-logic. In this talk I will review some of the results concerning models from cofinality-quantifiers, and then focus on the question of iterating the construction: unlike the case of L, it may be the case that C(L*) does not satisfy V=C(L*), in which case we can repeat the construction inside C(L*), obtaining a descending sequence of models, and at some cases even proceed transfinitely by taking intersections at limit stages. We will show, as time permits, how we can obtain models with descending sequences of the stationary-logic constructable model C(aa). For that purpose we will present shortly the mechanism of iterating club shooting forcings of specific kinds, which we use as a coding mechanism.**Information:** Contact Menachem Magidor or Omer Ben-Neria ahead of time for the seminar announcement and zoom link.

**Leeds Models and Sets Seminar****Time:** Tuesday, 31 May, 13:45-14:55 UK time (14:45-15:55 CEST)**Speaker:** Omer Ben-Neria, Hebrew University of Jerusalem**Title:** Diamonds, Compactness, and Global Scales**Abstract:** In pursuit of an understanding of the relations between compactness and approximation principles we address the question: To what extent do compactness principles assert the existence of a diamond sequence?It is well known that a cardinal kappa that satisfies a sufficiently strong compactness assumption must also carry a diamond sequence. However, other results have shown that certain weak large cardinal assumptions are consistent with the failure of the full diamond principle. We will discuss this gap and describe recent results with Jing Zhang which connect this problem to the existence of a certain global notion of cardinal arithmetic scales.**Information:** Please see the seminar webpage.

23 – 29 May

**Toronto Set Theory Seminar****Time:** Friday, 27 May, 13.30-15.00 Toronto time (19.30-21.00 CEST)**Speaker:** Juris Steprans, York University**Title:** Selective ultrafilters do not imply the existence of Milliken-Taylor ultrafilters**Abstract:** What is now known as Hindman’s Theorem (Theorem 3.1 in [3]) was proved to establish the truth of a conjecture of Graham and Rothschild [2]. It says that if the positive integers are partitioned into finitely many cells, then there is an infinite set of integers all of whose non-empty finite subsets have their sum in the same cell. In [3] van Douwen is credited with realizing that, assuming the Continuum Hypothesis, it is possible to construct an ultrafilter U such that if the positive integers are partitioned into finitely many cells, then there is X ∈ U such that all of the non-empty finite subsets of X have a sum belonging to the same cell. It was noticed by van Douwen that certain ultrafilters had an even stronger property, in that they had a base consisting of all of the finite sums of some set of positive integers. Such ultrafilters are now known as strongly summable ultrafilters. The strongly summable ultrafilters play a significant role in the theory of the semigroup (βN, +). Closely related to Hindman’s Theorem is Theorem 3.3 from [3], a result about the semigroup obtained by replacing addition on N with the union operation on the finite subsets of the positive integers. The corresponding ultrafilters, denoted as stable, ordered, union ultrafilters by Blass in [1] are now sometimes called Milliken-Taylor ultrafilters, since Blass showed that they satisfy the Milliken-Taylor Theorem.

Blass also showed that, associated with every stable, ordered, union ultrafilter there is a pair of RK- inequivalent selective ultrafilters and, assuming the Continuum Hypothesis, the correspondence can be reversed. At the end of [1] Blass asks whether his result using the Continuum Hypothesis can be improved by using only the usual axioms of set theory, conjecturing a negative answer. A proof of his conjecture will be presented by constructing a model with two RK-inequivalent selective ultrafilters, but no Milliken-Taylor ultrafilters.

References:

[1] Andreas Blass. Ultrafilters related to Hindman’s finite-unions theorem and its extensions. In Logic and combinatorics (Arcata, Calif., 1985), volume 65 of Contemp. Math., pages 89–124. Amer. Math. Soc., Providence, RI, 1987.

[2] R. L. Graham and B. L. Rothschild. Ramsey’s theorem for n-parameter sets. Trans. Amer. Math. Soc., 159:257–292, 1971.

[3] Neil Hindman. Finite sums from sequences within cells of a partition of N. J. Combinatorial Theory Ser. A, 17:1–11, 1974.**Information:** Please see http://gfs.fields.utoronto.ca/activities/21-22/set-theory-seminar.

**CUNY Set Theory Seminar****Time:** Friday, 27 May, 12:30pm New York time (18:30 CEST)**Speaker: **William Chan, Carnegie Mellon University**Title:** Determinacy and Partition Properties: Part II**Abstract:** In this talk, we will review some basic properties of partition cardinals under the axiom of determinacy. We will be particularly interested with the strong partition property of the first uncountable cardinal and the good coding system used to derive these partition properties. We will discuss almost everywhere behavior of functions on partition spaces of cardinals with respect to the partition measures including various almost everywhere continuity and monotonicity properties. These continuity results will be used to distinguish some cardinalities below the power set of partition cardinals. We will also use these continuity results to produce upper bounds on the ultrapower of the first uncountable cardinal by each of its partition measures, which addresses a question of Goldberg. Portions of the talk are joint work with Jackson and Trang.**Information:** The seminar will take place virtually. Please email Victoria Gitman (vgitman@nylogic.org) for the meeting id.

**Cross-Alps Logic Seminar****Time:** Friday, 27 May, 16.00-18.00 CEST **Speaker:** T. Nemoto, Japan Advanced Institute of Science and Technology**Title:** Determinacy of infinite games and reverse mathematics**Abstract:** Reverse mathematics is a program to classify mathematical theorems by set comprehension axioms in second order arithmetic [1]. In this program, it is presented that most of the theorems from undergraduate mathematics are equivalent to set comprehension axioms characterizing systems called “Big Five”. Comparing to the systems of set theory, second order arithmetic is a rather weak system, which enables the classification of weak determinacy schemata for the classes in the very low level of the Wadge hierarchy. In this talk, we will see that determinacy of infinite games up to the difference hierarchy over \Sigma^0_3 makes a fine hierarchy in second order arithmetic. References [1] S. G. Simpson, Subsystems of second order arithmetic (2nd edition), Cambridge University Press, 2010 [2] T. Nemoto, Determinacy of Wadge classes and subsystems of second order arithmetic, Mathematical Logic Quarterly, Volume 55, Issue 2, February 2009, pp. 154 – 176. [3] A. Montalbán and R. A. Shore, The limits of determinacy in second order arithmetic: consistency and complexity strength, Israel J. Math., 204 (2014), 477–508.**Information:** The event will stream on the Webex platform. Please write to luca.mottoros [at] unito.it for the link to the event.

**Caltech Logic Seminar****Time:** Wednesday, 25 May, 12:00-1:00pm Pacific time (21:00-22:00 CEST)**Speaker:** Asaf Shani, Harvard**Title:** Classifying invariants for E1**Abstract:** We introduce a framework for studying “reasonable” classifying invariants, more permitting than “classification by countable structures”. This framework respects the intuitions and results about classifications by countable structures, and allows for equivalence relations such as E1 and E1+ to be “reasonably classifiable” as well. In this framework we show that E1 has classifying invariants which are κκ-sequences of E0-classes for κ=b, and it does not have such classifying invariants if κ<add(B).

The result relies on analysing the tail intersection model ⋂n<ωV[cn,cn+1,…], where ⟨c0,c1,…⟩ is a generic sequence of Cohen reals.**Information:** Please see the seminar webpage.

**Barcelona Set Theory Seminar****Time:** Wednesday, 25 May, 16:00-17:30 CEST**Speaker:** Peter Holy**Title:** Asymmetric Cut and Choose Games**Abstract:** Joint with Philipp Schlicht, Christopher Turner and Philip Welch

(University of Bristol). We consider two player games in which the first

player cuts a given set into pieces, and the second player picks one of them.

This goes on for some infinite number of steps, and the second player wins

if the intersection of their choices is (in some sense) large. We consider

properties of these games, and their relationship to (generic) large

cardinals, notions of distributivity, strategic closure and precipitousness.**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, 25 May, 14:00-16:00 Israel Time (13:00-15:00 CEST)**Speaker:** Ur Yaar**Title:** Inner models from extended logics**Abstract:** Given some logic L* extending first-order logic, we can construct an inner model C(L*) in an L-like fashion – at each level of the construction, we add all the subsets of the current level, which are L*-defineable instead of merely first order definable. This gives us a plethora of inner models, with various properties. In my research I focused on models constructed from the logics obtained by adding cofinality-quantifiers, and the so-called stationary-logic. In this talk I will review some of the results concerning models from cofinality-quantifiers, and then focus on the question of iterating the construction: unlike the case of L, it may be the case that C(L*) does not satisfy V=C(L*), in which case we can repeat the construction inside C(L*), obtaining a descending sequence of models, and at some cases even proceed transfinitely by taking intersections at limit stages. We will show, as time permits, how we can obtain models with descending sequences of the stationary-logic constructable model C(aa). For that purpose we will present shortly the mechanism of iterating club shooting forcings of specific kinds, which we use as a coding mechanism.**Information:** Contact Menachem Magidor or Omer Ben-Neria ahead of time for the seminar announcement and zoom link.

**KGRC Set Theory Seminar, Vienna****Time:** Wednesday, 25 May, 11:30 CEST**Speaker:** Luke Elliott, University of St Andrews**Title:** Unindexed subshifts of finite type and a connecton to Thompsons groups**Abstract:** I will give a brief introduction to subshifts of finite type defined by

finite directed graphs. In particular I will mention a category of

“digaphs and foldings” introduced by Jim Belk, Collin Bleak, and Peter

Cameron which is useful for studying these systems. I will then discuss my

recent work in building an analogous category in which isomorphisms don’t

necessarily preserve indexing and path length. This category gives us both

more flexible notions of (strong) shift equivalence and a connection to

automorphisms of Thompsons groups.**Information:** This talk will be given via Zoom. Please contact Richard Springer for information how to participate.

**Helsinki Logic Seminar**, Special lecture**Time:** Wednesday, 25 May, 12:00 – 14:00 Helsinki time (11:00-13:00 CEST)**Speaker:** Philip Welch, University of Bristol**Title:** Reworking Kleene’s Higher Type Recursions**Abstract:** In the late 50’s and early 60’s Kleene wrote four papers on recursion in finite types. In two of them he gave an equation theoretic basis for such recursions, making it look rather like extensions of Goedel-Herbrand recursion. These two papers were very influential. In the other two papers he gave an alternative definition using Turing machines and showed their equivalence. The latter papers seem not to have had any substantial consequences, or indeed readership. Nevertheless they produce in type 2, a notion of computation whereby the ‘halting problem’ in this context is a complete Pi^1_1 set of integers, or equivalently a complete GSigma^0_1 set. By taking Kleene’s idea and reworking it for infinite time Turing machines (ITTM’s) one has a halting problem that is a complete GSigma^0_3 set, thereby neatly tying in ITTM’s with classical descriptive set theory.**Information:** The talk will take place in hybrid format. Please see the seminar webpage for the login information.

**KGRC Set Theory Seminar, Vienna****Time:** Tuesday, 24 May, 15:00-16:30 CEST**Speaker:** Sandra Müller, TU Wien**Title:** Inner Models, Determinacy, and Sealing**Abstract:** Inner model theory has been very successful in connecting determinacy

axioms to the existence of inner models with large cardinals and other

natural hypotheses. Recent results of Larson, Sargsyan, and Trang suggest

that a Woodin limit of Woodin cardinals is a natural barrier for our

current methods to prove these connections. One reason for this comes from

Sealing, a generic absoluteness principle for the theory of the

universally Baire sets of reals introduced by Woodin. Woodin showed in his

famous Sealing Theorem that in the presence of a proper class of Woodin

cardinals Sealing holds after collapsing a supercompact cardinal. I will

outline the importance of Sealing and discuss a new and

stationary-tower-free proof of Woodin’s Sealing Theorem that is based on

Sargsyan’s and Trang’s proof of Sealing from iterability. This is joint

work with Grigor Sargsyan and Bartosz Wcisło.**Information:** This talk will be given via Zoom. Please contact Richard Springer for information how to participate.

**Leeds Models and Sets Seminar****Time:** Tuesday, 24 May, 13:45-14:55 UK time (14:45-15:55 CEST)**Speaker:** Juvenal Murwanashyaka, University of Oslo**Title:** Weak Essentially Undecidable Theories of Concatenation**Abstract:** We sketch a proof of mutual interpretability of Robinson arithmetic and a weak finitely axiomatized theory of concatenation.**Information:** Please see the seminar webpage.

16 – 22 May

**CUNY Set Theory Seminar****Time:** Friday, 20 May, 12:30pm New York time (18:30 CEST)**Speaker: **William Chan, Carnegie Mellon University**Title:** Determinacy and Partition Properties**Abstract:** In this talk, we will review some basic properties of partition cardinals under the axiom of determinacy. We will be particularly interested with the strong partition property of the first uncountable cardinal and the good coding system used to derive these partition properties. We will discuss almost everywhere behavior of functions on partition spaces of cardinals with respect to the partition measures including various almost everywhere continuity and monotonicity properties. These continuity results will be used to distinguish some cardinalities below the power set of partition cardinals. We will also use these continuity results to produce upper bounds on the ultrapower of the first uncountable cardinal by each of its partition measures, which addresses a question of Goldberg. Portions of the talk are joint work with Jackson and Trang.**Information:** The seminar will take place virtually. Please email Victoria Gitman (vgitman@nylogic.org) for the meeting id.

**Cross-Alps Logic Seminar****Time:** Friday, 20 May, 16.00-18.00 CEST **Speaker:** A. Marcone, University of Udine**Title:** The transfinite Ramsey theorem**Abstract:** In this talk I discuss generalizations of the classic finite Ramsey theorem that substitute “set of cardinality n” with the notion of alpha-large set, where alpha is a countable ordinal. The prototype of these results is the statement that Paris and Harrington showed unprovable in PA in 1977. Since then several extensions were proved, typically for ordinals up to epsilon_0. Our results extend this approach by dealing with ordinals (at least) up to Gamma_0 and using simultaneously alpha-large sets (almost) everywhere in the statements. Quite surprisingly, in many cases we obtain tight bounds on the generalized Ramsey numbers, in contrast with the classical finite case where tight bounds are known only for very few cases involving very small numbers. This is joint work with Antonio Montalbán.**Information:** The event will stream on the Webex platform. Please write to luca.mottoros [at] unito.it for the link to the event.

**KGRC Logic Colloquium, ViennaTime:** Thursday, 19 May, 15:00 – 15:45 CET

**Speaker:**Corey Switzer, University of Vienna

**Title:**Axiomatizing Kaufmann Models of Arithmetic in Strong Logics

**Abstract:**A

*Kaufmann model*of PA is an ω1-like, recursively saturated, rather classless model (these terms will be defined in the talk). Such models have been an important object of study in model theory of arithmetic and its environs since the 70’s. Kaufmann models are natural counterexamples to several theorems about countable models of PA holding at the uncountable. Moreover they are a witness to incompactness at ω1 similar to an Aronszajn tree. The proof that Kaufmann models exist lies along a somewhat twisted road. Kaufmann showed that there are Kaufmann models under the combinatorial principle ♢ω1 and, later, Shelah eliminated the use of ♢ω1 by appealing to a forcing absoluteness argument involving the strong logic Lω1,ω(Q) where Q is the quantifier “there exists uncountably many”. It remains an extremely interesting, if somewhat vague, question, attributed to Hodges, whether one can build a Kaufmann model “by hand” in ZFC without appealing to generic absoluteness.

In this talk we will report on our recent progress in this area. Specifically we will consider the role that the strong logic Lω1,ω(Q)plays in Kaufmann models and show that the statement “Kaufmann models can be axiomatized by Lω1,ω(Q)” is independent of ZFC. Along the way we will discuss how Kaufmann models are affected by forcing and in particular show that it is independent of ZFC whether or not there is a Kaufmann model which can be “killed” by forcing without collapsing ω1.

**Information:**This talk will be given via Zoom. Please contact Richard Springer for information how to participate.

**Caltech Logic Seminar****Time:** Wednesday, 18 May, 12:00-1:00pm Pacific time (21:00-22:00 CEST)**Speaker:** Aristotelis Panagiotopoulos, CMU**Title:** Strong ergodicity phenomena for Bernoulli shifts of bounded algebraic dimension**Abstract:** For every Polish permutation group P≤Sym(N), let A↦[A]P be the assignment which maps every A⊆N to the set of all k∈N whose orbit under the action of the stabilizer PF of some finite F⊆A is finite. Then A↦[A]P is a closure operator and hence it endows P with a natural notion of dimension dim(P)dim(P). This notion of dimension has been extensively studied in model theory when A↦[A]P satisfies additionally the exchange principle, that is, when A↦[A]P forms a pregeometry. However, under the exchange principle, every Polish permutation group P with dim(P)<∞ is locally compact and therefore unable to generate any “wild” dynamics. In this talk, we will discuss the relationship between dim(P) and certain strong ergodicity phenomena in the absence of the exchange principle. In particular, for every n∈N, we will provide a Polish permutation group P with dim(P)=n whose Bernoulli shift P↷RN is generically ergodic relative to the injective part of the Bernoulli shift of any permutation group QQ with dim(Q)<n. We will use this to exhibit an equivalence relation of pinned cardinal ℵ1 which strongly resembles Zapletal’s counterexample to a question of Kechris, but which does not Borel reduce to the latter. Our proofs rely on the theory of symmetric models of choiceless set theory and in the process we establish that a vast collection of symmetric models admit a theory of supports similar to the basic Cohen model. This is joint work with Assaf Shani. **Information:** Please see the seminar webpage.

**Barcelona Set Theory Seminar****Time:** Wednesday, 18 May, 16:00-17:30 CEST**Speaker:** Laura Fontanella, Creteil University**Title:** Representing ordinals in classical realizability**Abstract:** Realizability aims at extracting the computational content of mathematical

proofs. Introduced in 1945 by Kleene as part of a broader program in constructive

mathematics, realizability has later evolved to include classical logic and even set theory.

Krivine’s work led to define realizability models for the theory ZF following a technique

that generalizes the method of Forcing. After a brief presentation of this technique, we

will discuss the problem of representing ordinals in realizability models for set theory,

thus we will present the solution proposed in a joint work with Guillaume Geoffroy that

led to realize uncountable versions of the Axiom of Dependent Choice.**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, 18 May, 14:00-16:00 Israel Time (13:00-15:00 CEST)**Speaker:** Shaun Allison**Title:** Polish groups with the pinned property, part 2**Abstract:** We will discuss a property of Polish groups called the “pinned property” which means that every orbit equivalence relation they generate is “pinned”, a metamathematical notion which is used to separate the complexity of different equivalence relations up to Borel reducibility. We will discuss the subtle way that the amount of choice assumed influences the pinned property. In particular, we will discuss results of Su Gao and Alex Thompson which imply that in a mode of ZFC, a Polish group has the pinned property if and only if it has a complete compatible left-invariant metric. We will also present a new result which, along with a result of Larson-Zapletal, implies that in the Solovay model derived from a measurable, a Polish group has the pinned property if and only if it involves S_\infty (caveat: for the special case of non-Archimedian groups). Time permitting, we will discuss Larson-Zapletal’s result as well. This is part of a larger project to measure and categorize the “classification strength” of Polish groups.**Information:** Contact Menachem Magidor or Omer Ben-Neria ahead of time for the seminar announcement and zoom link.

**KGRC Set Theory Seminar, Vienna****Time:** Tuesday, 17 May, 15:00-16:30 CEST**Speaker:** Philipp Lücke, University of Barcelona**Title:** Patterns in the large cardinal hierarchy**Abstract:** In my talk, I will present results showing that the existence of various well-known large cardinals can be characterized through the validity of strong extensions of the downward Löwenheim-Skolem theorem.

These equivalences show that certain patterns recur throughout the large cardinal hierarchy.

In particular, they show that strongly unfoldable cardinals, introduced by Villaveces in his model-theoretic investigations of models of set theory, relate to subtle cardinals, introduced by Kunen and Jensen in their studies of strong diamond principles, in the same way as supercompact cardinals relate to Vopěnka cardinals and strong cardinals relate to Woodin cardinals.

This is joint work in progress with Joan Bagaria (Barcelona).**Information:** This talk will be given via Zoom. Please contact Richard Springer for information how to participate.

**Leeds Models and Sets Seminar****Time:** Tuesday, 17 May, 13:45-14:55 UK time (14:45-15:55 CEST)**Speaker:** Julia Knight, University of Notre Dame**Title:** Freeness and typical behavior for algebraic structures**Abstract:** The talk is on joint work with Johanna Franklin and Turbo Ho. Gromov asked “What is a typical group?” He was thinking of finitely presented groups. He proposed an approach involving limiting density. In 2013, I conjectured that for elementary first order sentences $\varphi$, and for group presentations with n generators ($n\geq 2$) and a single relator, the limiting density for groups satisfying $\varphi$ always exists, with value 0 or 1, and the value is 1 iff $\varphi$ is true in the non-Abelian free groups. The conjecture is still open, but there are positive partial results by Kharlampovich and Sklinos, and by Coulon, Ho, and Logan. We ask Gromov’s question about structures in other equational classes, or \emph{algebraic varieties} in the sense of universal algebra. We give examples illustrating different possible behaviors. Focusing on languages with just finitely many unary function symbols, we prove a result with conditions sufficient to guarantee that the analogue of the conjecture holds. The proof uses a version of Gaifman’s Locality Theorem, plus ideas from random group theory and probability. **Information:** Please see the seminar webpage.

**Helsinki Logic Seminar**, Special lecture**Time:** Monday, 16 May, 12:00 – 14:00 Helsinki time (11:00-13:00 CEST)**Speaker:** Philip Welch, University of Bristol**Title:** Quasi-Inductive Definitions**Abstract:** Such definitions extend the well researched theory of monotone inductive definitions by allowing non-monotone processes that are structured by liminf rules at limits rather than simple unions. Much of the Moschovakian theory of induction over abstract structures can be performed in this context, resulting in certain Spector classes of sets. Whereas the theory of inductive definitions leads to the idea of the least admissible set over a structure A, here one constructs the least ‘strongly Sigma_2-admissible set’ over A. Just as Sigma^0_1-Determinacy is associated with HYP(N), and Kleene’s higher type recursion, so there are connections to be explored here with a higher type form of quasi-inductive recursion for a q-HYP(N).**Information:** The talk will take place in hybrid format. Please see the seminar webpage for login information.

9 – 15 May

**Toronto Set Theory Seminar****Time:** Friday, 13 May, 13.30-15.00 Toronto time (19.30-21.00 CEST)**Speaker:** Vladimir Tkachuk, Universidad Autónoma Metropolitana**Title:** Lindelöf Σ-spaces in 2022**Abstract:** This talk is a survey and an advertisement of the theory of Lindelöf Σ-spaces. We will present ten equivalent definitions of the Lindelöf Σ-property and a selection of results that have numerous applications in General Topology, Topological Algebra and Cp-theory.**Information:** Please see http://gfs.fields.utoronto.ca/activities/21-22/set-theory-seminar.

**CUNY Set Theory Seminar****Time:** Friday, 13 May, 12:30pm New York time (18:30 CEST)**Speaker: **Andrew Brooke-Taylor, University of Leeds**Title:** Categorifying Borel reducibility**Abstract:** The theory of Borel reducibility has had great success in ruling out proposed classifications in various areas of mathematics. However, this framework doesn’t account for an important feature of such classifications – they are often expected to be functorial, not just respecting isomorphism but taking any homomorphism between the objects in question to a homomorphism of the invariants. I will talk about some work in progress with Filippo Calderoni, extending the framework to include functoriality and noting some differences this immediately introduces from the standard framework.**Information:** The seminar will take place virtually. Please email Victoria Gitman (vgitman@nylogic.org) for the meeting id.

**Cross-Alps Logic Seminar****Time:** Friday, 13 May, 16.00-18.00 CEST **Speaker:** U. Darji, University of Louisville**Title:** Descriptive complexity and local entropy **Abstract:** Blanchard introduced the concepts of Uniform Positive Entropy (UPE) and Complete Positive Entropy (CPE) as topological analogues of K-automorphism. He showed that UPE implies CPE, and that the converse is false. A flurry of recent activity studies the relationship between these two notions. For example, one can assign a countable ordinal which measures how complicated a CPE system is. Recently, Barbieri and García-Ramos constructed Cantor CPE systems at every level of CPE. Westrick showed that natural rank associated to CPE systems is actually a \Pi^1_1-rank. More importantly, she showed that the collection of CPE Z2-SFT’s is a \Pi^1_1-complete set. In this talk, we discuss some results, where UPE and CPE coincide and others where we show that the complexity of certain classes of CPE systems is \Pi^1_1-complete. This is joint work with García-Ramos.**Information:** The event will stream on the Webex platform. Please write to luca.mottoros [at] unito.it for the link to the event.

**Caltech Logic Seminar****Time:** Wednesday, 11 May, 12:00-1:00pm Pacific time (21:00-22:00 CEST)**Speaker:** Rehana Patel, African Institute for Mathematical Sciences**Title:** The number of ergodic models of an infinitary sentence**Abstract:** Given an Lω1ω-sentence φ in a countable language, we call an ergodic S∞-invariant probability measure on the Borel space of countable models of φ (having fixed underlying set) an **ergodic model** of φ. I will discuss the number of ergodic models of such a sentence φ, including the case when φ is a Scott sentence. This is joint work with N. Ackerman, C. Freer, A. Kruckman and A. Kwiatkowska.**Information:** Please see the seminar webpage.

**Barcelona Set Theory Seminar****Time:** Wednesday, 11 May, 16:00-17:30 CEST**Speaker:** Sandra Müller, TU Wien**Title:** Inner Models, Determinacy, and Sealing**Abstract:** Inner model theory has been very successful in connecting determinacy axioms to the existence of inner models with large cardinals and other natural hypotheses. Recent results of Larson, Sargsyan, and Trang suggest that a Woodin limit of Woodin cardinals is a natural barrier for our current methods to prove these connections. One reason for this comes from Sealing, a generic absoluteness principle for the theory of the universally Baire sets of reals introduced by Woodin. Woodin showed in his famous Sealing Theorem that in the presence of a proper class of Woodin cardinals Sealing holds after collapsing a supercompact cardinal. I will outline the importance of Sealing and discuss a new and stationary-tower-free proof of Woodin’s Sealing Theorem that is based on Sargsyan’s and Trang’s proof of Sealing from iterability. This is joint work with Grigor Sargsyan and Bartosz Wcisło.**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, 11 May, 14:00-16:00 Israel Time (13:00-15:00 CEST)**Speaker:** Shaun Allison**Title:** Polish groups with the pinned property**Abstract:** We will discuss a property of Polish groups called the “pinned property” which means that every orbit equivalence relation they generate is “pinned”, a metamathematical notion which is used to separate the complexity of different equivalence relations up to Borel reducibility. We will discuss the subtle way that the amount of choice assumed influences the pinned property. In particular, we will discuss results of Su Gao and Alex Thompson which imply that in a mode of ZFC, a Polish group has the pinned property if and only if it has a complete compatible left-invariant metric. We will also present a new result which, along with a result of Larson-Zapletal, implies that in the Solovay model derived from a measurable, a Polish group has the pinned property if and only if it involves S_\infty (caveat: for the special case of non-Archimedian groups). Time permitting, we will discuss Larson-Zapletal’s result as well. This is part of a larger project to measure and categorize the “classification strength” of Polish groups. **Information:** Contact Menachem Magidor or Omer Ben-Neria ahead of time for the seminar announcement and zoom link.

**KGRC Set Theory Seminar, Vienna****Time:** Tuesday, 10 May, 15:00-16:30 CEST**Speaker:** Ö. Bag, Universität Wien**Title:** On higher Baire spaces combinatorics**Abstract:** In this talk we will consider some recent results regarding the higher Baire spaces analogues of the almost disjointness, bounding, dominating and splitting numbers. In particular, we will discuss the recently established relative consistency of s(κ)=κ+<b(κ) for κ-strongly unfoldable, the relative consistency of κ+<a(κ)=b(κ)<d(κ)<c(κ) for κ regular uncountable, as well as the global evaluation of the spectrum of κ-mad families.

The results build to a great extent on the method of non-linear iterations of Cumming-Shelah, the method of matrix iterations as originally appearing in the work on the ultrafilter and dominating numbers of Blass-Shelah and its subsequent developments, and give (among others) an interesting application of Kunen’s isomorphism of names argument in the context of Easton forcing. We will conclude the talk with a brief discussion of remaining open questions.**Information:** This talk will be given via Zoom. Please contact Richard Springer for information how to participate.

**Leeds Models and Sets Seminar****Time:** Tuesday, 10 May, 13:45-14:55 UK time (14:45-15:55 CEST)**Speaker:** Diana Carolina Montoya, University of Vienna**Title:** tba**Abstract:** tba**Information:** Please see the seminar webpage.

2 – 8 May

**Toronto Set Theory Seminar****Time:** Friday, 6 May, 13.30-15.00 Toronto time (19.30-21.00 CEST)**Speaker:** Mirna Dzamonja, IRIF – Centre national de la recherche scientifique (CNRS) – Université deParis**Title:** Morass-generic structures**Abstract:** We discuss a joint work with Wiesław Kubiś on a specific way of constructing structures of size ℵ1 using finite approximations, namely by organising the approximations along a simplified morass. We demonstrate a connection with Fraïssé limits and show that the naturally obtained structure of size ℵ1 is homogeneous. Moreover, this is preserved under expansions, which leads us to a partial answer to a question of Bassi and Zucker. We give some examples of interesting structures constructed, such as the antimetric space of size ℵ1. Finally, we comment on the situation when one Cohen real is added.**Information:** Please see http://gfs.fields.utoronto.ca/activities/21-22/set-theory-seminar.

**CUNY Set Theory Seminar****Time:** Friday, 6 May, 12:30pm New York time (18:30 CEST)**Speaker:** James Holland, Rutgers University**Title:** Weak Indestructibility and Reflection**Abstract:** Assuming multiple of strong cardinals, there are lots of cardinals with small degrees of strength (i.e. κ that are κ+2-strong). We can calculate the consistency strength of these all cardinal’s small degrees of strength being weakly indestructible using forcing and core model techniques in a way similar to Apter and Sargsyan’s previous work. This yields some easy relations between indestructibility and Woodin cardinals, and also generalizes easily to supercompacts. I will give a proof sketches of these results.**Information:** The seminar will take place virtually. Please email Victoria Gitman (vgitman@nylogic.org) for the meeting id.

**Caltech Logic Seminar****Time:** Wednesday, 4 May, 12:00-1:00pm Pacific time (21:00-22:00 CEST)**Speaker:** Garrett Ervin, CMU**Title:** Filter flows**Abstract:** A directed hypergraph G consists of a vertex set V along with a collection of directed hyperedges (A,B), where A and B are finite subsets of V. Given a set of vertices X, we think of the edge (A,B) as being on the boundary of X if X intersects A and does not completely contain B.

We can generalize the notion of directed hypergraph as follows. A *filter graph* G consists of an infinite vertex set VV along with a collection of edges (F,G), where F and G are filters on VV. Given a set of vertices X, we think of the edge (F,G) as being on the boundary of X if X is F-positive and the complement of X is G-positive.

Filter graphs seem to be surprisingly graph-like. We’ll show that filter graphs satisfy the natural generalization of the max-flow/min-cut theorem, where point masses flowing along directed edges in the usual hypergraph setting are replaced by ultrafilters flowing along filter-edges.**Information:** Please see the seminar webpage.

**Barcelona Set Theory Seminar****Time:** Wednesday, 4 May, 16:00-17:30 CET**Speaker:** Will Boney**Title:** Compactness of strong logics and large cardinals**Abstract:** Connections between large cardinals in set theory and

compactness principles of strong logics in model theory have a long

history, going back to Tarski (compact cardinals), Magidor,

(extendibles), and Benda (supercompacts). We discuss several

recent advances, including connecting omitting types and normal

ultrafilters; sort logic and C(n)-cardinals; abstract Henkin models and

Woodin cardinals; and virtual logic and virtual large cardinals.

Additionally, this work has connections to category theory.**Information:** Online. If you wish to attend, please send an email to bagaria@ub.edu asking for the link.

**Helsinki Logic Seminar****Time:** Wednesday, 4 May, 12:00 – 14:00 Helsinki time (11:00-13:00 CEST)**Speaker:** Philip Welch, University of Bristol **Title:** Free subsets of structures**Abstract:** If A is a first order structure, a subset X of dom(A) is *free* if no element of X can be defined in A from other elements of X. In general, finding infinite free subsets of infinite structures, requires large cardinals. We survey the field here, and examine extensions of this property due to Pereira, distilled from his work on the pcf conjecture, and recently another due to Adolf and Ben Neria. Work of the latter now establishes, with older work of the speaker, an equiconsistency between inner models with sequences of measures and their extension of Pereira’s “Approachable Free Subset Property”.**Information:** The talk will take plce in hybrid mode. Please see the seminar webpage for the link.

**KGRC Set Theory Seminar, Vienna****Time:** Tuesday, 3 May, 15:00-16:30 CEST**Speaker:** M. A. Cardona Montoya, TU Wien**Title:** On the cardinal characteristics associated with \varepsilon**Abstract:** Let ε be the σ-ideal generated by closed measure zero sets of reals. We prove that, for ε, their associated cardinal characteristics (i.e. additivity, covering, uniformity and cofinality) are pairwise different.**Information:** This talk will be given via Zoom. Please contact Richard Springer for information how to participate.

**Leeds Models and Sets Seminar****Time:** Tuesday, 3 May, 13:45-14:55 UK time (14:45-15:55 CEST)**Speaker:** Noa Lavi, Hebrew University of Jerusalem**Title:** tba**Abstract:** tba**Information:** Please see the seminar webpage.

25 April – 1 May

**Toronto Set Theory Seminar****Time:** Friday, 29 April, 13.30-15.00 Toronto time (19.30-21.00 CEST)**Speaker:** Matteo Viale, University of Turin**Title:** The (absolute) model companionship spectrum of a mathematical theory and the continuum problem**Abstract:** We introduce a classification tool for mathematical theories based on Robinson’s notion of model companionship; roughly the idea is to attach to a mathematical theory *T* those signatures *L* such that *T* as axiomatized in *L* admits a(n absolute) model companion. To do so we also introduce a slight strengthening of model companionship (absolute model companionship – AMC) which characterize those model companionable *L*-theories *T* whose model companion is axiomatized by the Π2 sentences for L which are consistent with the universal theory of any *L*-model of *T*.

We use the above to analyze set theory, and we show that the above classification tools can be used to extract (surprising?) information on the continuum problem.**Information:** Please see http://gfs.fields.utoronto.ca/activities/21-22/set-theory-seminar.

**CUNY Set Theory Seminar****Time:** Friday, 29 April, 12:30pm New York time (18:30 CEST)**Speaker: **Andreas Blass, University of Michigan**Title:** Do these ultrafilters exist, II: not Tukey top**Abstract:** This is the second of two talks devoted to two properties of ultrafilters (non-principal, on omega) for which the question ‘Do such ultrafilters exist?’ is open. In this talk, I’ll discuss the property of not being at the top of the Tukey ordering (of ultrafilters on omega). I’ll start with the definition of the Tukey ordering, and I’ll give an example of an ultrafilter that is ‘Tukey top’. It’s consistent with ZFC that some ultrafilters are not Tukey top. The examples and the combinatorial characterizations involved here are remarkably similar but not identical to examples and the characterization from the previous talk. That observation suggests some conjectures, one of which I’ll disprove if there’s enough time.**Information:** The seminar will take place virtually. Please email Victoria Gitman (vgitman@nylogic.org) for the meeting id.

**KGRC Logic Colloquium, ViennaTime:** Thursday, 28 April, 15:00 – 15:45 CET

**Speaker:**Tobias Kaiser, University of Passau

**Title:**tba

**Abstract:**tba

**Information:**This talk will be given via Zoom. Please contact Richard Springer for information how to participate.

**Caltech Logic Seminar****Time:** Wednesday, 27 April, 12:00-1:00pm Pacific time (21:00-22:00 CEST)**Speaker:** Lionel Nguyen Van Thé, Aix-Marseille Université**Title:** Revisiting the Erdős-Rado canonical partition theorem**Abstract:** One of the numerous strengthenings of Ramsey’s theorem is due to Erdős and Rado, who analyzed what partition properties can be obtained on mm-subsets of the naturals when colorings are not necessarily finite. Large monochromatic sets may not appear in that case, but there is a finite list of behaviors, called “canonical”, to which every coloring reduces. The purpose of this talk will be to remind certain not so well-known analogous theorems of the same flavor that were obtained by Prömel in the eighties for various classes of structures (like graphs or hypergraphs), and to show how such theorems can in fact be deduced in the more general setting of Fraïssé classes.**Information:** Please see the seminar webpage.

**KGRC Set Theory Seminar, Vienna****Time:** Tuesday, 26 April, 15:00-16:30 CEST**Speaker:** Miguel Moreno, Universität Wien**Title:** The isomorphism relation of unsuperstable theories in the generalized Borel-reducibility hierarchy**Abstract:** One of the most important questions in generalized descriptive set theory is whether there is a generalized Borel-reducibility counterpart of Shelah’s main gap theorem? (i.e. for any classifiable theory T and nonclassifiable theory T′, is the isomorphism relation of T Borel reducible to the isomorphism relation of T′?) In this talk we will study the case of unsuperstable theories. By introducing the notion of K-colorable linear orders we can construct generalized Ehrenfeucht-Mostowski models and a Borel reduction from the isomorphism relation of any classifiable theory to the isomorphism relation of any unsuperstable theory.**Information:** This talk will be given via Zoom. Please contact Richard Springer for information how to participate.

**Leeds Models and Sets Seminar****Time:** Tuesday, 26 April, 13:45-14:55 UK time (14:45-15:55 CEST) **Speaker:** Tamara Servi, IMJ-PRG & Fields Institute**Title:** Interdefinability and compatibility in certain o-minimal expansions of the real field**Abstract:** tba**Information:** Please see the seminar webpage.

18 – 24 April

**CUNY Set Theory Seminar****Time:** Friday, 22 April, 2:00pm New York time (20:00 CEST)**Speaker: **Jouko Väänänen, University of Helsinki**Title:** Stationary logic and set theory**Abstract:** Stationary logic was introduced in the 1970’s. It allows the quantifier ‘for almost all countable subsets s…’. Although it is undoubtedly a kind of second order logic, it is completely axiomatizable, countably compact and satisfies a kind of Downward Lowenheim-Skolem theorem. In this talk I give first a general introduction to the extension of first order logic by this ‘almost all’-quantifier. As ‘almost all’ is interpreted as ‘for a club of’, the theory of this logic is entangled with properties of stationary sets. I will give some examples of this. The main reason to focus on this logic in my talk is to use it to build an inner model of set theory. I will give a general introduction to this inner model, called C(aa), or the aa-model, and sketch a proof of CH in the model. My work on the aa-model is joint work with Juliette Kennedy and Menachem Magidor.**Information:** The seminar will take place virtually. Please email Victoria Gitman (vgitman@nylogic.org) for the meeting id.

**Toronto Set Theory Seminar****Time:** Friday, 22 April, 13.30-15.00 Toronto time (19.30-21.00 CEST)**Speaker:** Asger Törnquist, University of Copenhagen**Title:** The mathematics of a model of the mind in psychology**Abstract:** Jens Mammen, a psychologist, has proposed a model of the human mind based on the idea that the brain organizes objects in the world into two kinds of general categories: Broad categories, which he called “sense categories”, and categories of special, distinguished objects (or people), which he called “choice categories”.

From a mathematical point of view, it is interesting that Mammen formulated his model of the mind axiomatically, based on the notion of a topological space. The objects in the universe are modelled by the points in a topological space (U,S), where the (broad) sense categories are modelled by open sets in the topology S. The choice categories forms an additional collection of subsets of the universe, C, that together with the topology must adhere to certain axioms. The triple (U,S,C) is called a “Mammen space” (a term that I introduced).

Several mathematical questions arise out Mammen’s theory. For instance, if we want Mammen’s model to be able to account for all possible subsets of the universe (a property Mammen called “completeness”), then the Axiom of Choice, or at least some non-trivial consequences thereof, seem to play a role. There are also several interesting questions related to cardinal invariants, such as the “weight” of the underlying topological space of a complete Mammen space.

I will give an overview of the mathematics of Mammen spaces and known results, and also discuss the numerous unsolved problems that remain.**Information:** Please see http://gfs.fields.utoronto.ca/activities/21-22/set-theory-seminar.

**CUNY Set Theory Seminar****Time:** Friday, 22 April, 12:15pm New York time (18:30 CEST)**Speaker: **Andreas Blass, University of Michigan**Title:** Do these ultrafilters exist, I: preservation by forcing**Abstract:** This is the first of two talks devoted to two properties of ultrafilters (non-principal, on omega) for which the question ‘Do such ultrafilters exist?’ is open. In this talk, I’ll discuss the property of being preserved by some forcing that adds new reals. Some forcings destroy all ultrafilters, and some (in fact many) ultrafilters are destroyed whenever new reals are added, but it is consistent with ZFC that some ultrafilters are preserved when some kinds of reals are added. I plan to prove some of these things and describe the rest. I’ll also describe a combinatorial characterization, due to Arnie Miller, of preservable ultrafilters.**Information:** The seminar will take place virtually. Please email Victoria Gitman (vgitman@nylogic.org) for the meeting id.

**Cross-Alps Logic Seminar****Time:** Friday, 22 April, 16.00-18.00 CEST **Speaker:** S. Thei, University of Udine **Title:** The geology of pseudo-grounds**Abstract:** Four decades after the invention of forcing, Laver and independently Woodin answered one of the most natural questions regarding forcing. Is the ground model definable in its forcing extensions? Surprisingly, it turns out that the ground models of a given set-theoretic universe are uniformly definable. Fuchs, Hamkins and Reitz used this result to establish the formal foundations for set-theoretic geology that reverses the forcing construction by studying what remains from a model of set theory once the layers created by forcing are removed. Such a switch in perspective leads to another interesting question. Is the universe itself a nontrivial forcing extension of a smaller model? Reitz addressed the issue and introduced the Ground Axiom (the precursor to set-theoretic geology) which asserts that the universe is not obtained by forcing over any strictly smaller model.

This talk is about some types of inner models which are defined following the paradigm of “undoing” forcing. For example, a bedrock is a ground satisfying the Ground Axiom and the mantle is the intersection of all grounds. Once the main geological notions are in place, we will introduce inner models with the cover and approximation properties called pseudo-grounds. In particular, we will consider some generalizations of classical results to the context of class forcing and pseudo-grounds.**Information:** The event will stream on the Webex platform. Please write to luca.mottoros [at] unito.it for the link to the event.

**Caltech Logic Seminar****Time:** Wednesday, 20 April, 12:00-1:00pm Pacific time (21:00-22:00 CEST)**Speaker:** Maciej Malicki, IMPAN**Title:** Isomorphism of locally compact Polish metric structures**Abstract:** The talk will be devoted to the isomorphism relations on Borel classes of locally compact Polish metric structures. Using continuous logic, one can prove that they are always Borel reducible to graph isomorphism, which implies, in particular, that isometry of locally compact Polish metric spaces is reducible to graph isomorphism. This answers a question of Gao and Kechris. As a matter of fact, locally compact Polish metric structures behave very much like countable ones. For example, Hjorth, Kechris and Louveau proved that isomorphism of countable structures that is potentially of rank α+1 multiplicative Borel class is reducible to equality on hereditarily countable sets of rank α, and the same turns out to be true about locally compact Polish metric structures. Time permitting, I will also discuss certain variants of the Hjorth-isomorphism game, recently introduced by Lupini and Panagiotopoulos.**Information:** Please see the seminar webpage.

**CMU Set Theory Seminar****Time:** Tuesday, 19 April, 4:30 – 5:45pm Pittsburgh time (22:30 – 23:45 CEST) **Speaker:** Samson Leung, Carnegie Mellon University**Title:** Categoricity results of abstract elementary classes (Part II)**Abstract: **We will look at the main tools used in the proof of our

categoricity transfer: good frames, multidimensional diagrams and primes.

It is known that our assumptions allow a set-theoretic argument to

transfer categoricity down to $\beth_{(2^{LS(K)})^+}$. We will discuss

examples that encode the cumulative hierarchy, which have the first

categoricity cardinals up to $\beth_{(2^{LS(K)})^+}$, but fail

amalgamation. We conjecture that a more refined set-theoretic construction

might provide such examples that also satisfy amalgamation, which will

imply the above threshold is tight.**Information:** See the seminar webpage.

**CMU Logic Seminar****Time:** Tuesday, 19 April, 3:30 – 4:30pm Pittsburgh time (21:30 – 22:30 CEST) **Speaker:** Samson Leung, Carnegie Mellon University**Title:** Categoricity results of abstract elementary classes (Part I)**Abstract: **The notion of abstract elementary classes (AECs) is an axiomatic framework developed by Shelah to generalize classification theory beyond the first-order context. One central test question is the categoricity conjecture: if an AEC K is categorical in some $\mu\geq\beth_{(2^{LS(K)})^+}$, then it is categorical in all $\mu\geq\beth_{(2^{LS(K)})^+}$. After going through the axioms of AECs, we will overview some partial results in the literature, in particular those assuming tameness, type-shortness and the amalgamation property. We show that: assuming type-shortness and amalgamation over sets, the categoricity conjecture is true. Our result also provides an alternative proof to the upward categoricity transfer in first-order theories.**Information:** See the seminar webpage.

11 – 17 April

**CUNY Set Theory Seminar****Time:** Friday, 15 April, 12:15pm New York time (18:15 CEST)**Speaker:** Joel David Hamkins, Notre Dame University**Title:** The surprising strength of reflection in second-order set theory with abundant urelements**Abstract:** I shall give a general introduction to urelement set theory and the role of the second-order reflection principle in second-order urelement set theory GBCU and KMU. With the *abundant atom axiom*, asserting that the class of urelements greatly exceeds the class of pure sets, the second-order reflection principle implies the existence of a supercompact cardinal in an interpreted model of ZFC. The proof uses a reflection characterization of supercompactness: a cardinal κ is supercompact if and only if for every second-order sentence ψ true in some structure M (of any size) in a language of size less than κ is also true in a first-order elementary substructure m≺M of size less than κ. This is joint work with Bokai Yao.**Information:** The seminar will take place virtually. Please email Victoria Gitman (vgitman@nylogic.org) for the meeting id.

**Caltech Logic Seminar****Time:** Wednesday, 13 April, 12:00-1:00pm Pacific time (21:00-22:00 CEST)**Speaker:** Jack Lutz, Iowa State**Title:** Extending the Reach of the Point-to-Set Principle**Abstract:** The *point-to-set principle* has recently enabled the theory of computing to be used to answer open questions about fractal geometry in Euclidean spaces RnRn. These are classical questions, meaning that their statements do not involve computation or related aspects of logic.

In this talk I will describe the extension of two algorithmic fractal dimensions — computability-theoretic versions of classical Hausdorff and packing dimensions that assign dimensions dim(x)dim(x) and Dim(x)Dim(x) to *individual points* x∈Xx∈X — to arbitrary separable metric spaces and to arbitrary gauge families. I will then discuss the extension of the point-to-set principle to arbitrary separable metric spaces and to a large class of gauge families. Finally, I will indicate how the extended point-to-set principle can be used to prove new theorems about classical fractal dimensions in hyperspaces.

This is joint work with Neil Lutz and Elvira Mayordomo.**Information:** Please see the seminar webpage.

**Hebrew University-Bar Ilan University Set Theory seminar****Time:** Wednesday, 13 April, 14:00-16:00 Israel Time (13:00-15:00 CEST)**Speaker:** tba **Title:** tba **Abstract:** tba **Information:** Contact Menachem Magidor or Omer Ben-Neria ahead of time for the seminar announcement and zoom link.

**Hebrew University-Bar Ilan University – Set theory lecture series****Time:** Wednesday, 13 April, 12:00-13:30 Israel Time (11:00-12:30 CEST)**Speaker:** tba **Title:** tba **Abstract:** tba **Information:** Contact Menachem Magidor or Omer Ben-Neria ahead of time for the seminar announcement and zoom link.

**Singapore Logic Seminar****Time:** Wednesday, 13 April, 16:00-17:00 Singapore time (10:00-11:00 CEST)**Speaker:** Wang Wei **Title:** Ackermann, Ramsey and Trees **Abstract:** Recently, Chong, Yang and I prove that a version of Pigeonholes Principle for trees (**TT ^{1}**) is

**Π**-conservative over

^{0}_{3}**RCA**. So,

_{0}**TT**does not imply the totality of the Ackermann function over

^{1}**RCA**, like the instance of Ramsey’s Theorem for

_{0}**2**-colorings of pairs. To fit the trend of logic talks, I am not going to present many details. Instead, I will try to recall some stories about the Ackermann function and its appearance in reverse mathematics.

**Information:**See the seminar webpage.

**CMU Logic Seminar****Time:** Tuesday, 12 April, 3:30 – 4:30pm Pittsburgh time (21:30 – 22:30 CEST) **Speaker:** Garrett Ervin, Carnegie Mellon University**Title:** Flowing through networks of generalized hyperarcs**Abstract: **There are many versions of the max-flow/min-cut theorem for graphs. All assert something like “the maximum flow that can travel from a set of source vertices S to a set of sinks T is equal to the minimum boundary size of a set X containing S and disjoint from T.” König’s lemma, Hall’s matching theorem, Menger’s theorem, and the max-flow/min-cut theorem itself can each be viewed as instances of this principle.

The proofs of these theorems rely crucially on the fact that the function f that measures the boundary of a given set of vertices is submodular. This means that for any finite sets of vertices X, Y we have

f(X \cap Y) + f(X \cup Y) <= f(X) + f(Y)

Here, f might measure the vertex boundary, the edge boundary, or some capacitated version of one of these.

Can the max-flow/min-cut theorem can be generalized to an arbitrary submodular function f, even one that doesn’t arise as the boundary function of a graph? At first this question seems meaningless, since the definition of a flow depends on an underlying graph. But it turns out that certain simple submodular functions can be viewed as analogues of directed arcs in a hypergraph, and sums of these functions can be viewed as networks on which flows can be defined. We show there is a class of these networks generalizing hypergraphs for which the max-flow/min-cut theorem holds.**Information:** See the seminar webpage.

4 – 10 April

**Toronto Set Theory Seminar****Time:** Friday, 8 April, 13.30-15.00 Toronto time (19.30-21.00 CEST)**Speaker:** Lyumbomyr Zdomskyy, University of Vienna**Title:** Combinatorial covering properties of co-ideals**Abstract:** We shall discuss consistent examples of coideals (=complements of ideals in P(w)) distinguishing between covering properties of Menger, Scheepers, and variations thereof, thus answering some questions posed in an earlier work with Bella and Tokgoz. These examples could be obtained both by recursive constructions using certain equalities between cardinal characteristics, as well as by forcing. We also plan to mention potential applications in functional analysis as well to the problem of existence of non-meager P-filters. **Information:** Please see http://gfs.fields.utoronto.ca/activities/21-22/set-theory-seminar.

**Cross-Alps Logic Seminar****Time:** Friday, 8 April, 16.00-17.00 CEST **Speaker:** A. Kechris, Caltech**Title:** Countable sections for actions of locally compact groups**Abstract:** A Borel action of a Polish locally compact group on a standard Borel space admits a countable Borel section, i.e., a Borel set that meets every orbit in a countable nonempty set. It is a long standing open problem whether this property characterizes locally compact groups. I will discuss the history of this problem and some recent progress in joint work with M. Malicki, A. Panagiotopoulos and J. Zielinski. **Information:** The event will stream on the Webex platform. Please write to luca.mottoros [at] unito.it for the link to the event.

**Caltech Logic Seminar****Time:** Wednesday, 6 April, 12:00-1:00pm Pacific time (21:00-22:00 CEST)**Speaker:** Raphaël Carroy, University of Turin**Title:** Constructing Wadge classes and describing Wadge quasi-orders**Abstract:** Let X and Y be two topological spaces. Given a subset A of X and a subset B of Y we say that A is Wadge-reducible (or sometimes continuously reducible) to B if there is a continuous function f from X to Y that satisfies f−1(B)=A. Wadge reducibility is a particularly nice quasi-order on subsets of Polish zero-dimensional spaces: Wadge’s Lemma guarantees indeed that its antichains are of size at most two, while Martin and Monk have proven that it is well-founded. This gives an ordinal ranking to every equivalence class for Wadge reducibility, thus generating various questions. I will talk about two of these questions.

First, given a Wadge equivalence class, can we build it using classes of lower ordinal rank, and how?

Second, given any Polish zero-dimensional space X, can we decide if there is an antichain of two classes or just one class of some specific ordinal rank of the Wadge quasi-order of X? For which ranks?**Information:** Please see the seminar webpage.

**Hebrew University-Bar Ilan University – Set theory lecture series****Time:** Wednesday, 6 April, 12:00-13:30 Israel Time (11:00-12:30 CET)**Speaker:** Uri Abraham**Title:** Open colouring axioms and their relatives, part 5 **Abstract:** Baumgartner proved in 1970 the consistency of the statement that every two \aleph_1 sets of reals (with no end-points) are order-isomorphic. The lectures will cover this result as well as some other results such as the open coloring axioms that were motivated by this work of Baumgartner. We will describe results and problems from the paper by Rubin, Shelah and the lecturer and perhaps other articles as well.

The lectures are intended to be accessible to any student that is familiar with the basic technique of forcing iterations. Familiarity with the standard proof of the consistency of Martin Axiom should be enough. **Information:** Contact Menachem Magidor or Omer Ben-Neria ahead of time for the seminar announcement and zoom link.

**Singapore Logic Seminar****Time:** Wednesday, 6 April, 16:00-17:00 Singapore time (10:00-11:00 CEST)**Speaker:** Frank Stephan, National University of Singapore**Title:** Matching regular pumping lemmas and automaticity**Abstract:** A pumping lemma for regular languages is matching iff it is satisfied exactly by the regular languages. Furthermore, one distinghises for pumping lemmas one-sided versions where only words in the language L are pumped and two-sided versions where all words of sufficiently long length are pumped.

Jaffe as well as Ehrenfeucht, Parikh and Rozenberg gave examples of matching two-sided pumping lemmas. The goal of the present talk is to show that if one requires that an automatic function (equivalently a transducer) marks of the pump in the input word, then most two-sided pumping lemmas including the traditional pumping lemma are matching. Furthermore, it is shown that in contrast to this, most one-sided pumping lemmas with an automatic function to compute the pump are not matching. This is only left open for the case of the one-sided automatic block pumping lemma.

This is joint work of Karen Celine, Ryan Chew, Sanjay Jain and the speaker.**Information:** See the seminar webpage.

**CMU Logic Seminar****Time:** Tuesday, 5 April, 3:30 – 4:30pm Pittsburgh time (21:30 – 22:30 CEST) **Speaker:** Benjamin Siskind, Carnegie Mellon University**Title:** An update on order-preserving Martin’s Conjecture **Abstract: **Martin’s Conjecture is a way of codifying a phenomenon observed in computability theory: the only natural functions on the Turing degrees seem to be the constant functions, the identity, and the transfinite iterates of the Turing jump. While the full conjecture is wide open, there has been significant progress on order-preserving Martin’s Conjecture–that is, Martin’s Conjecture restricted to the functions which preserve Turing-reducibility. In particular, the order-preserving version has been settled positively for Borel functions whereas Martin’s Conjecture for even low-level Borel functions is open. In this talk, we’ll discuss a plan for pushing order-preserving Martin’s Conjecture beyond Borel functions involving some AD combinatorics, higher recursion theory, and forcing. This is joint, in-progress work with Patrick Lutz.**Information:** See the seminar webpage.

**KGRC Set Theory Seminar, Vienna****Time:** Tuesday, 5 April, 15:00-16:30 CEST**Speaker:** W. Wohofsky, U Wien**Title:** Fresh function spectra**Abstract:** My talk will be about the notion of fresh function and I will discuss the corresponding spectrum. A function with domain λ is fresh if it is new but all its initial segments are in the ground model. I will give general facts how to compute the fresh function spectrum, also discussing what sets are realizable as a fresh function spectrum of a forcing. Moreover, I will provide several examples, including well-known tree forcings on ω such as Sacks, Laver, Miller, and Mathias forcing, as well as Prikry and Namba forcing to illustrate the difference between fresh functions and fresh subsets.

This is joint work with Vera Fischer and Marlene Koelbing.**Information:** This talk will be given via Zoom. Please contact Richard Springer for information how to participate.

**Paris-Lyon Séminaire de Logique****Time:** Monday, 4 April, 15:15 CEST**Speaker:** Alexis Chevalier**Title:** Piecewise Interpretable Hilbert Spaces (I) **Abstract:** We introduce piecewise interpretable Hilbert spaces and show their relevance to model theory and representation theory. Piecewise interpretable Hilbert spaces are direct limits of imaginary sorts of a continuous logic structure which carry definable Hilbert space operations. We will show that they offer an interesting unified framework for studying definable measures and Shelah-Galois groups, and that they offer an interesting point of contact between model theory and the theory of unitary group representations. We will briefly discuss a structure theorem for scattered piecewise interpretable Hilbert spaces and we will explain various applications of this theorem. This is joint work with Ehud Hrushovski.

In this talk we will focus on giving an overview of results and on discussing examples. More detailed results will be discussed on Tuesday in the Théorie des Modèles et Groupes seminar.**Information:** See the seminar webpage.

28 March – 3 April

**Toronto Set Theory Seminar****Time:** Friday, 1 April, 13.30-15.00 Toronto time (19.30-21.00 CEST)**Speaker:** Philipp Schlicht, University of Bristol**Title:** Dichotomies for open directed hypergraphs on generalised Baire spaces**Abstract:** The open graph dichotomy for a subset X of the Baire space states that any open graph on X either contains a large complete subgraph or admits a countable colouring. It is a definable version of the open colouring axiom for X and generalises the perfect set property. Recently, this was generalised to infinite dimensions by Miller, Carroy and Soukup. I will discuss extensions of this result to generalised Baire spaces and a number of applications such as variants of the Hurewicz dichotomy, the determinacy of Väänänen’s game and the asymmetric Baire property. This is a joint project with Dorottya Sziraki. **Information:** Please see http://gfs.fields.utoronto.ca/activities/21-22/set-theory-seminar.

**CUNY Set Theory Seminar****Time:** Friday, 1 April, 12:30pm New York time (18:30 CEST)**Speaker:** Vera Fischer, University of Vienna**Title:** Independent families, Spectra and Indestructibility: Part II**Abstract:** Independent families are families of infinite sets of integers with the property that for any two disjoint, non-empty, finite subfamilies A and B of the given family, the set ⋂A∖⋃B is infinite. Of particular interest are the sets of the possible cardinalities of maximal independent families, as well as their indestructibility by various forcing notions. In this talk, we will consider some recent advances in the area and point out to remaining open questions.**Information:** The seminar will take place virtually. Please email Victoria Gitman (vgitman@nylogic.org) for the meeting id.

**Cross-Alps Logic Seminar****Time:** Friday, 1 April, 16.00-18.00 CEST **Speaker:** David Evans, Imperial College London**Title:** Amalgamation properties in measured structures**Abstract:** In a paper published in 2008, Macpherson and Steinhorn introduced and studied structures in which each every definable set carries a well behaved dimension and measure: we refer to these as MS-measurable structures. Examples include totally categorical structures, pseudofinite fields and the random graph. MS-measurable structures are supersimple of finite SU-rank and we discuss some amalgamation properties which hold in MS-measurable structures, but not in all supersimple finite rank structures. We are interested in the question of whether every omega-categorical, MS-measurable structure is one-based. A construction of Hrushovski can be used to produce omega-categorical structures which are supersimple of finite SU-rank and not one-based: indeed, this construction is essentially the only known way to produce such structures. It is still an open question whether any of these Hrushovski constructions can be MS-measurable. However, I will discuss some work of myself and of my PhD student Paolo Marimon which uses the amalgamation results and other methods to show that at least some of the Hrushovski constructions are not MS-measurable.**Information:** The event will stream on the Webex platform. Please write to luca.mottoros [at] unito.it for the link to the event.

**Hebrew University-Bar Ilan University Set Theory seminar****Time:** Wednesday, 30 March, 14:00-16:00 Israel Time (13:00-15:00 CEST)**Speaker:** Alejandro Poveda **Title:** The failure of Galvin’s property at successors of singulars**Abstract:** Galvin’s property fails at the successor of every singular cardinal, simultaneously. In the upcoming session, we plan to revisit the results thus far presented by S. Garti and describe the challenges involved in producing failures of Galvin’s property at accessible cardinals; such as \aleph_{\omega+1}. Once this is accomplished we will move forward and comment on the main ideas behind the proof of the theorem. Our main forcing-theoretic tool will be the so-called Radin forcing with interleaved collapses. A more sophisticated version of this forcing appeared in work by Foreman & Woodin; instead, our approach will follow Cummings’ later exposition. We will describe the forcing taking for granted the existence of guiding generics. Once the exposition of the poset is accomplished we shall show how to produce such guiding generics in the absence of GCH. We plan to conclude our talks by proving the theorem and (if time permits) presenting some relevant open problems. **Information:** Contact Menachem Magidor or Omer Ben-Neria ahead of time for the seminar announcement and zoom link.

**Singapore Logic Seminar****Time:** Wednesday, 30 March, 16:00-17:00 Singapore time (10:00-11:00 CEST)**Speaker:** Liuzhen Wu**Title:** Continuum function and strongly compact cardinal**Abstract:** The continuum function is a key and long-studied object inside set theory. We will survey the study on the behavior of continuum function in presence of strongly compact cardinals. We will also introduce some major research problems in this area. Finally, We discuss our recent work on forcing continuum function of some special pattern. **Information:** See the seminar webpage.

**CMU Logic Seminar****Time:** Tuesday, 29 March, 3:30 – 4:30pm Pittsburgh time (21:30 – 22:30 CEST) **Speaker:** Itay Neeman, UCLA**Title:** Restrictions of OCA_T with large continuum**Abstract: **Todorcevic’s Open Coloring Axiom (OCA_T) states that any open graph on a separable metric space is either countably chromatic, or admits an uncountable clique. OCA_T has many interesting and important applications. Its known consistency proofs all lead to models where the continuum is $\aleph_2$. It is therefore natural to ask whether it implies that the continuum is $\aleph_2$, or whether there are other consistency proofs leading to models with larger continuum. (OCA_T negates the CH.) This question is still open. However we show that the restriction of OCA_T to spaces of size less than the continuum is consistent with arbitrarily large values of the continuum. Earlier work by Farah obtained this for the restriction to spaces of size $\aleph_1$**Information:** See the seminar webpage.

**Leeds Models and Sets Seminar****Time:** Tuesday, 29 March, 15:45-16:55 UK time (16:45-17:55 CEST) – changed time **Speaker:** Kameryn J Williams, Sam Houston State University**Title:** The potentialist multiverse of classes**Abstract:** Set-theoretic potentialism is the view that the universe of sets is never fully completed but is only given potentially. Tools from modal logic have been applied to understand the mathematics of potentialism. In recent work, Neil Barton and I extended this analysis to class-theoretic potentialism, the view that proper classes are given potentially (while the sets may or may not be fixed).In this talk, I will survey some results from set-theoretic potentialism. After seeing how the tools apply in that context I will then discuss our work in the class-theoretic context.**Information:** Please see the seminar webpage.

**KGRC Set Theory Seminar, Vienna****Time:** Tuesday, 29 March, 15:00-16:30 CEST**Speaker:** D. Sobota , University of Viena**Title:** P-measures in the random model **Abstract:** Let μ be a finitely additive probability measure on ω which vanishes on points, that is, μ({n})=0 for every n. It follows immediately that μ is not σ-additive, however it may be almost σ-additive in the following weak sense. We say that μ is a *P‑measure* if for every decreasing sequence (An) of subsets of ω there is a subset A such that A∖An is finite for every n and μ(A)=limnμ(An). It follows immediately that, e.g., an ultrafilter U on ω is a P‑point if and only if the one-point measure δU is a P‑measure. And similarly as in the case of P‑points the existence of P‑measures is independent of ZFC.

During my talk I will discuss basic properties of P‑measures and show, at least briefly, that using old ideas of Solovay and Kunen one can obtain a non-atomic P‑measure in the random model. The latter result implies that in this model there exists a nowhere dense ccc P‑set in ω∗, which may be treated as a (weak) partial answer to the question asking whether there are P‑points in the random model.

This is a joint work with Piotr Borodulin-Nadzieja.**Information:** This talk will be given via Zoom. Please contact Richard Springer for information how to participate.

21 – 27 March

**Toronto Set Theory Seminar****Time:** Friday, 25 March, 13.30-15.00 Toronto time (19.30-21.00 CET)**Speaker:** Asger Törnquist**Title:** tba**Abstract:** tba**Information:** Please see https://homepage.univie.ac.at/david.schrittesser/world-logic-day-2022.html.

**CUNY Set Theory Seminar****Time:** Friday, 25 March, 12:30pm New York time (17:30 CET)**Speaker:** Vera Fischer, 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.**Cross-Alps Logic Seminar****Time:** Friday, 25 March, 16.00-18.00 CET **Speaker:** Omer Ben-Neria, Hebrew University of Jerusalem**Title:** tba**Abstract:** tba**Information:** The event will stream on the Webex platform. Please write to luca.mottoros [at] unito.it for the link to the event.

**Online Logic Seminar****Time:** Thursday, 24 March, 1pm US central time (19:00 CET)**Speaker:** Riley Thornton, UCLA**Title:** An algebraic approach to Borel CSPs**Abstract:** I will explain how some of the algebraic tools behind the CSP dichotomy theorem in computer science can be adapted to answer questions in Borel combinatorics.**Information:** See the seminar webpage.

**CMU Model Theory Seminar****Time:** Thursday, 24 March, 11:00-12:30 Pittsburgh time (16:00-17:30 CET)**Speaker:** Marcos Mazari-Armida, University of Colorado Boulder**Title:** Characterizing categoricity in several classes of modules, part 1**Abstract:** The birth of modern model theory can be traced back to Morley’s Categoricity Theorem which asserts how to transfer categoricity in elementary classes. Currently, Shelah’s Categoricity Conjecture, which is a far-reaching generalization to Morley’s Categoricity Theorem, is the main test question in the development of non-elementary model theory. In this talk, we will show that the condition of being categorical in a tail of cardinals is a natural algebraic property by characterizing it algebraically for several classes of modules. As an application, we will provide rings such that the class of flat modules is categorical in a tail of cardinals but it is not first-order axiomatizable.**Information:** Please see the seminar webpage.

**Leeds-Ghent Virtual Logic SeminarTime:** Wednesday, 23 March, 3pm UK time (16:00 CET)

**Speaker:**John Truss, University of Leeds

**Title:**The small index property for superatomic Boolean algebras

**Abstract:**A Boolean algebra is said to be superatomic if all its quotients are atomic. These were studied originally by Mostowski and Tarski, and later by Day, and several equivalent conditions were given for superatomicity. I just consider the countable case, where the algebras are best viewed as the algebra of clopen subsets of a countable successor ordinal under the order topology. There is a natural notion of rank (how many times one can factor by the Frechet ideal). For instance, [0, omega^alpha] for a countable ordinal alpha has rank alpha. The main result is the small index property for superatomic Boolean algebras of this form for finite alpha (the case for general countable alpha is conjectured, but this is still work in progress).

Monk studied the automorphism groups of countable superatomic Boolean algebras and showed that for the finite rank n case there are at least n normal subgroups. Jacob Hilton has shown that for n >= 2 there are actually exactly 2^(2^(aleph_0)) normal subgroups and has classified all the ones which fix pointwise the set of limit points. It is hoped to extend this to arbitrary countable rank and also to classify all the normal subgroups.

**Information:**Please contact Paul Shafer in advance to participate.

**Hebrew University-Bar Ilan University Set Theory seminar****Time:** Wednesday, 23 March, 14:00-16:00 Israel Time (13:00-15:00 CET)**Speaker:** Alejandro Poveda**Title:** The failure of Galvin’s property at successors of singulars**Abstract:** The primary goal of these lectures is to present the proof of the following result: Assuming the existence of a supercompact cardinal there is a model of ZFC where Galvin’s property fails at the successor of every singular cardinal, simultaneously.

In the upcoming session, we plan to revisit the results thus far presented by S. Garti and describe the challenges involved in producing failures of Galvin’s property at accessible cardinals; such as \aleph_{\omega+1}. Once this is accomplished we will move forward and comment on the main ideas behind the proof of the theorem. Our main forcing-theoretic tool will be the so-called Radin forcing with interleaved collapses. A more sophisticated version of this forcing appeared in work by Foreman & Woodin; instead, our approach will follow Cummings’ later exposition. We will describe the forcing taking for granted the existence of guiding generics. Once the exposition of the poset is accomplished we shall show how to produce such guiding generics in the absence of GCH. We plan to conclude our talks by proving the theorem and (if time permits) presenting some relevant open problems. **Information:** Contact Menachem Magidor or Omer Ben-Neria ahead of time for the seminar announcement and zoom link.

**Hebrew University-Bar Ilan University – Set theory lecture series****Time:** Wednesday, 23 March, 12:00-13:30 Israel Time (11:00-12:30 CET)**Speaker:** Uri Abraham**Title:** Open colouring axioms and their relatives, part 4 **Abstract:** Baumgartner proved in 1970 the consistency of the statement that every two \aleph_1 sets of reals (with no end-points) are order-isomorphic. The lectures will cover this result as well as some other results such as the open coloring axioms that were motivated by this work of Baumgartner. We will describe results and problems from the paper by Rubin, Shelah and the lecturer and perhaps other articles as well.

The lectures are intended to be accessible to any student that is familiar with the basic technique of forcing iterations. Familiarity with the standard proof of the consistency of Martin Axiom should be enough. **Information:** Contact Menachem Magidor or Omer Ben-Neria ahead of time for the seminar announcement and zoom link.

**Singapore Logic Seminar****Time:** Wednesday, 23 March, 16:00-17:00 Singapore time (9:00-10:00 CET)**Speaker:** Wu Guohua**Title:** tba **Abstract:** tba **Information:** See the seminar webpage.

**CMU Logic Seminar****Time:** Tuesday, 22 March, 3:30 – 4:30pm Eastern Standard Time (20:30 – 21:30 CET) **Speaker:** Colin Jahel, Carnegie Mellon University**Title:** Asymptotic theories and homomorphically-avoided structures**Abstract: **Given a class of finite structures, one can consider μ_{n} the uniform measure on structures in said class of size n. We study the asymptotic behavior, when n goes to infinity, of the family (μ_{n})_{n}. In particular, one can ask: which sentences have converging probability, and when is this limit non-zero? I will present our results for classes of graphs and digraphs, in particular classes not containing any homorphic copies of certain sets of finite structures. Joint work with Manuel Bodirsky.**Information:** See the seminar webpage.

**MOPA (Models of Peano Arithmetic), CUNY****Time:** Tuesday, 22 March, 2pm New York time (19:00 CET)**Speaker: **Ermek Nurkhaidarov, Penn State Mont Alto**Title:** Generic Automorphisms**Abstract:** In this talk we investigate generic automorphisms of countable models. Hodges-Hodkinson-Lascar- Shelah 93 introduces the notion of SI (small index) generic automorphisms which are used to show the small index property. Truss 92 defines the notion of Truss generic automorphisms. We study the relationship between these two types of generic automorphisms.**Information:** The seminar will take place virtually. Please email Victoria Gitman (vgitman@nylogic.org) for the meeting id.**Leeds Models and Sets Seminar****Time:** Tuesday, 22 March, 13:45 UK time (14:45 CET)**Speaker:** Asaf Karagila, University of East Anglia**Title:** Ccc without C, si? Si.**Abstract:** What does the countable chain condition mean without the axiom of choice? We will discuss several possible definitions, all equivalent in ZFC, none equivalent in ZF(+DC). We will also present two “external” definitions (due to Bukovský and to Mekler) and see how they fit into this picture.

We will show that a ccc forcing can collapse ω_{1}, and quite possibly be countably closed while doing so. On the other hand, with the “correct definition” of ccc, no cofinalities or cardinals are changed above ω_{1}. Whether or not ω_{1} can be collapsed is open, but we know that would require it to be singular.

This is a joint work with Noah Schweber.**Information:** Please see the seminar webpage.

**KGRC Set Theory Seminar, Vienna****Time:** Tuesday, 22 March, 15:00-16:30 CET**Speaker:** Jeffrey Bergfalk, University of Viena**Title:** A family of higher dimensional partition principles**Abstract:** This talk will be an exposition of the recent work *A descriptive approach to higher derived limits*, joint with Nathaniel Bannister, Justin Moore, and Stevo Todorcevic (arXiv:2203.00165). The material of this paper is somewhat more ranging than its title would suggest.

At its heart is a new family of partition principles which synthesize several recent advances in the study of higher derived limits, rendering those results far more amenable to combinatorial analyses. These principles admit formulation on any directed quasi-order, and are of particular, and interrelated, interest on the quasi-orders (ωω,≤∗) and the ordinals ωn

A main implication of these principles in any case is the triviality of (higher dimensionally) coherent families of functions; we’ll use any remaining time to note ways that such objects, and even higher derived limits, are closer to classical set theoretic concerns than perhaps tends to be realized.**Information:** This talk will be given via Zoom. Please contact Richard Springer for information how to participate.

**Paris-Lyon Séminaire de Logique****Time:** Monday, 21 March, 15:15 CET**Speaker:** Joshua Frisch**Title:** Embedding Theorems for Polish Modules**Abstract:** A Polish module is a topological module whose underlying topology is Polish (separable and completely metrizable). In this talk I will discuss some results (joint with Forte Shinko) about when Polish modules continuously inject into one another and the pre-order induced by these injections. In particular we will show that, for a wide class of rings, there are countably many minimal elements in this pre-order. As an application we will construct a countable family of uncountable abelian Polish groups at least one of which embeds into any other uncountable abelian Polish group.**Information:** See the seminar webpage.

14 – 20 March

**Online Logic Seminar****Time:** Thursday, 17 March, 01:00pm US central time (19:00 CET)**Speaker:** John Baldwin, University of Illinois Chicago**Title:** Category Theory and Model Theory: Symbiotic Scaffolds**Abstract:** A *scaffold* for mathematics includes both *local* foundations for various areas of mathematics and productive guidance in how to unify them. In a scaffold the unification does not take place by a common axiomatic basis but consists of a systematic ways of connecting results and proofs in various areas of mathematics. Two scaffolds, model theory and category theory, provide local foundations for many areas of mathematic including two flavors (material and structural) of set theory and different approaches to unification. We will discuss salient features of the two scaffolds including their contrasting but bi-interpretable set theories. We focus on the contrasting treatments of `size’ in each scaffold and the advantages/disadvantages of each for different problems.**Information:** See the seminar webpage.

**CMU Model Theory Seminar****Time:** Thursday, 17 March, 11:00-12:30 Pittsburgh time (16:00-17:30 CET)**Speaker:** Wentao Yang, CMU**Title:** Shelah’s eventual categoricity conjecture in universal classes, part 3**Abstract:** We present Sebastien Vasey’s proof of Shelah’s categoricity conjecture in universal classes. First we change the substructure relation of the given class to obtain an AEC with better properties, except that the union axiom might not hold. To prove that the union axiom holds, we use the independence framework AxFr developed by Shelah, build an “independent” tree assuming the failure of union, and contradict stability.**Information:** Please see the seminar webpage.

**KGRC Logic Colloquium, ViennaTime:** Thursday, 17 March, 15:00 – 15:45 CET

**Speaker:**Stefan Hoffelner, University of Münster

**Title:**Forcing and the Separation, the Reduction and the Uniformization

Property

**Abstract:**The Separation property, the Reduction property and the Uniformization

property, introduced in the 1920’s and 1930’s are three classical

regularity properties of pointclasses of the reals.

The celebrated results of Y. Moschovakis on the one hand and D. Martin, J.

Steel and H. Woodin on the other, yield a global description of the

behaviour of these regularity properties for projective pointclasses under

the assumption of large cardinals. These results, impressive as they are,

still leave open a lot of natural questions. To name a few we mention:

Do we need large cardinals to obtain their effects on the behaviour of

these regularity property?

Is the $\Sigma^1_{2n+1}$-separation property actually consistent for n >1?

More generally: to what extent can we produce set theoretic universes

which display a different behaviour of these regularity properties?

Are the separation the reduction and the uniformization property different

notions at all?

The goal of this talk to introduce the three mentioned regularity

properties, present a couple of these natural problems and discuss new

results, utilising a novel forcing technique, which answer some of them.

**Information:**This talk will be given via Zoom. Please contact Richard Springer for information how to participate.

**Hebrew University-Bar Ilan University Set Theory seminar****Time:** Wednesday, 16 March, 14:00-16:00 Israel Time (13:00-15:00 CET)**Speaker:** Alejandro Poveda**Title:** The failure of the Galvin property at successors of singulars**Abstract:** The primary goal of these lectures is to present the proof of the following result: Assuming the existence of a supercompact cardinal there is a model of ZFC where Galvin’s property fails at the successor of every singular cardinal, simultaneously.

In the upcoming session, we plan to revisit the results thus far presented by S. Garti and describe the challenges involved in producing failures of Galvin’s property at accessible cardinals; such as \aleph_{\omega+1}. Once this is accomplished we will move forward and comment on the main ideas behind the proof of the theorem. Our main forcing-theoretic tool will be the so-called Radin forcing with interleaved collapses. A more sophisticated version of this forcing appeared in work by Foreman & Woodin; instead, our approach will follow Cummings’ later exposition. We will describe the forcing taking for granted the existence of guiding generics. Once the exposition of the poset is accomplished we shall show how to produce such guiding generics in the absence of GCH. We plan to conclude our talks by proving the theorem and (if time permits) presenting some relevant open problems. **Information:** Contact Menachem Magidor or Omer Ben-Neria ahead of time for the seminar announcement and zoom link.

**Hebrew University-Bar Ilan University – Set theory lecture series****Time:** Wednesday, 16 March, 12:00-13:30 Israel Time (11:00-12:30 CET)**Speaker:** Uri Abraham**Title:** Open colouring axioms and their relatives, part 2 **Abstract:** Baumgartner proved in 1970 the consistency of the statement that every two \aleph_1 sets of reals (with no end-points) are order-isomorphic. The lectures will cover this result as well as some other results such as the open coloring axioms that were motivated by this work of Baumgartner. We will describe results and problems from the paper by Rubin, Shelah and the lecturer and perhaps other articles as well.

The lectures are intended to be accessible to any student that is familiar with the basic technique of forcing iterations. Familiarity with the standard proof of the consistency of Martin Axiom should be enough. **Information:** Contact Menachem Magidor or Omer Ben-Neria ahead of time for the seminar announcement and zoom link.

**CMU Logic Seminar****Time:** Tuesday, 15 March, 3:30 – 4:30pm Eastern Standard Time (20:30 – 21:30 CET) **Speaker:** Alex Kruckman, Wesleyan University**Title:** Properly ergodic structures**Abstract: **One natural notion of “random (countably infinite) L-structure” is a probability measure on the space of L-structures with domain omega which is invariant and ergodic for the natural action of the symmetric group Sym(omega) on this space. We call such a measure an ergodic structure. The most famous example of an ergodic structure is the Erdős–Rényi random graph model on domain omega, which gives measure 1 to the isomorphism type of the Rado graph. Ergodic structures also arise naturally as limits of sequences of finite structures which are convergent in the appropriate sense, generalizing the graph limits of Lovász and Szegedy.

Some ergodic structures (like the Erdős–Rényi random graph model) are almost surely isomorphic to a single countable structure (like the Rado graph), and the countable structures which arise in this way have been completely characterized by Ackerman, Freer, and Patel. In this talk, we will consider properly ergodic structures, those which do not give measure 1 to any single isomorphism type. What do properly ergodic models “look like”? To address this question, we develop an analogue of the Scott rank for ergodic structures, which leads to a precise characterization of those first-order theories (and, more generally, those sentences of the infinitary logic L_{omega_1,omega}) which admit properly ergodic models. This is joint work with Ackerman, Freer, and Patel.**Information:** See the seminar webpage.

**KGRC Set Theory Seminar, Vienna****Time:** Tuesday, 15 March, 15:00-16:30 CET**Speaker:** Stefan Hoffelner, University of Münster**Title:** Forcing the \Pi^1_n-uniformization property**Abstract:** The uniformization property, introduced by N. Lusin in 1930, is an

extensively studied notion in descriptive set theory. For a given

projective pointclass $\Gamma$ it says that every subset of the plane

which belongs to $\Gamma$ has a uniformizing function whose graph is an

element of $\Gamma$ as well. The celebrated results of Y. Moschovakis on

the one hand and D. Martin, J. Steel and H. Woodin on the other, yield a

natural and global description of the behaviour of the uniformization

property for projective pointclasses under the assumption of large

cardinals. In particular, under PD, for every natural number n,

$\Pi^1_{2n+1}$-sets and hence $\Sigma^1_{2n+2}$-sets do have the

uniformization property.

Yet the question of universes which display an alternative behaviour of

theses regularity properties has remained in large parts a complete

mystery, mostly due to the absence of forcing techniques to produce such

models. Consequentially, a lot of very natural problems have remained wide

open ever since.

In my talk, I want to outline some recently obtained tools, which turn the

question of forcing a universe with the $\Pi^1_n$-uniformization property

into a fixed point problem for certain sets of forcing notions. This fixed

point problem can be solved, yielding a specific set of forcing notions

which in turn can be used to force the Uniformization property (for n>2)

over fine structural inner models with large cardinals (for n=3, the inner

model is just L). For even n, these universes witness for the first time

the consistency (relative to the existence of n-3 many Woodin cardinals)

of the $\Pi^1_{n}$-uniformization property, and, for odd n, give new lower

bounds in terms of consistency strength.**Information:** This talk will be given via Zoom. Please contact Richard Springer for information how to participate.

**Paris-Lyon Séminaire de Logique****Time:** Monday, 14 March, 14:15 CET**Speaker:** Ben De Bondt **Title:** On a playfully defined family of countable elementary submodels

of H_\theta**Abstract:** In order to make this talk accessible for a general audience

of logicians, we will start off softly by first introducing certain open

two-player games and then use these to isolate a special family of

countable elementary submodels of the structure H_\theta. We will then

analyse the existence of (projective stationary) many such models and

discuss the connection with precipitousness of the nonstationary ideal

on \omega_1. Next, we will discuss a particular forcing P that consists

of finite conditions in which these special models feature as side

conditions. Depending on the time, we might go on to survey some

interesting properties that this forcing shares with an L-forcing

defined by Claverie Schindler and a Namba-like forcing defined by

Ketchersid-Larson-Zapletal, to both of which it shows resemblance. It

will follow that this side condition approach using special models gives

yet another way to increase the second uniform indiscernible in a

stationary set preserving way beyond some arbitrary prespecified

ordinal. This is all part of joint ongoing work with my thesis

supervisor Boban Velickovic.**Information:** Join via the link on the seminar webpage

7 – 13 March

**CUNY Set Theory Seminar****Time:** Friday, 11 March, 2:00pm New York time (20:00 CET)**Speaker:** Joel David Hamkins, Notre Dame University**Title:** Infinite wordle and the mastermind numbers**Abstract:** I shall introduce and consider the natural infinitary variations of Wordle, Absurdle, and Mastermind. Infinite Wordle extends the familiar finite game to infinite words and transfinite play—the code-breaker aims to discover a hidden codeword selected from a dictionary Δ⊆Σω of infinite words over a countable alphabet Σ by making a sequence of successive guesswords, receiving feedback after each guess concerning its accuracy. For any dictionary using the usual 26-letter alphabet, for example, the code-breaker can win in at most 26 guesses, and more generally in n guesses for alphabets of finite size n. Meanwhile, for some dictionaries on an infinite alphabet, infinite play is required, but the code-breaker can always win by stage ω on a countable alphabet, for any fixed dictionary. Infinite Mastermind, in contrast, is a subtler game than Wordle because only the number and not the position of correct bits is given. When duplication of colors is allowed, nevertheless, then the code-breaker can still always win by stage ω, but in the no-duplication variation, no countable number of guesses (even transfinite) is sufficient for the code-breaker to win. I therefore introduce the *mastermind number*, denoted mm, to be the size of the smallest winning no-duplication Mastermind guessing set, a new cardinal characteristic of the continuum, which I prove is bounded below by the additivity number add(M) of the meager ideal and bounded above by the covering number cov(M). In particular, the precise value of the mastermind number is independent of ZFC and can consistently be strictly between ℵ1 and the continuum 2ℵ0. In *simplified Mastermind*, where the feedback given at each stage includes only the numbers of correct and incorrect bits (omitting information about rearrangements), then the corresponding simplified mastermind number is exactly the eventually different number d(≠∗). http://jdh.hamkins.org/infinite-wordle-and-the-mastermind-numbers-cuny-logic-workshop-march-2022/**Information:** The seminar will take place virtually. Please email Victoria Gitman (vgitman@nylogic.org) for the meeting id.

**CUNY Set Theory Seminar****Time:** Friday, 11 March, 12:30pm New York time (18:30 CET)**Speaker:** William Chan, Carnegie Mellon University**Title:** tba**Abstract:** tba**Information:** The seminar will take place virtually. Please email Victoria Gitman (vgitman@nylogic.org) for the meeting id.

**Cross-Alps Logic Seminar****Time:** Friday, 11 March, 16.00-17.00 CET **Speaker:** A. Andretta, University of Turin**Title:** Sierpinski’s partitions with Sigma^1_2 pieces**Abstract:** There are several statements in elementary geometry that depend on the size of the continuum, and most of them are modelled on the proof of a theorem of Sierpinski’s. In the first part of the talk I will survey a few of these geometric statements and show how these are related to each other. In the second part I will show how imposing a definability condition on the pieces of Sierpinski’s theorem yields a better bound on the size of the continuum.**Information:** The event will stream on the Webex platform. Please write to luca.mottoros [at] unito.it for the link to the event.

**Hebrew University-Bar Ilan University Set Theory seminar****Time:** Wednesday, 9 March, 14:00-16:00 Israel Time (13:00-15:00 CET)**Speaker:** Shimon Garti **Title:** The Galvin property**Abstract:** In this lecture we will discuss the consistency of the failure of the Galvin property at successors of regular cardinals (the first Abraham-Shelah model), followed by a similar statement at successors of strong limit singular cardinals (from a supercompact cardinal in the ground model). If time permits, we will say a few things about the strong vs. weak failure of the Galvin property. **Information:** Contact Menachem Magidor or Omer Ben-Neria ahead of time for the seminar announcement and zoom link.

**Hebrew University-Bar Ilan University – Set theory lecture series****Time:** Wednesday, 9 March, 12:00-13:30 Israel Time (11:00-12:30 CET)**Speaker:** Uri Abraham**Title:** Open colouring axioms and their relatives, part 1 **Abstract:** Baumgartner proved in 1970 the consistency of the statement that every two \aleph_1 sets of reals (with no end-points) are order-isomorphic. The lectures will cover this result as well as some other results such as the open coloring axioms that were motivated by this work of Baumgartner. We will describe results and problems from the paper by Rubin, Shelah and the lecturer and perhaps other articles as well.

The lectures are intended to be accessible to any student that is familiar with the basic technique of forcing iterations. Familiarity with the standard proof of the consistency of Martin Axiom should be enough. **Information:** Contact Menachem Magidor or Omer Ben-Neria ahead of time for the seminar announcement and zoom link.

**Leeds Models and Sets Seminar****Time:** Tuesday, 8 March, 13:45 UK time (14:45 CET)**Speaker:** Sandra Müller, Technische Universität Wien**Title:** The Interplay of Determinacy, Large Cardinals, and Inner Models**Abstract:** The standard axioms of set theory, Zermelo-Fraenkel set theory with Choice (ZFC), do not suffice to answer all questions in mathematics. While this follows abstractly from Kurt Gödel’s famous incompleteness theorems, we nowadays know numerous concrete examples for such questions. In addition to a large number of problems in set theory, even many problems outside of set theory have been showed to be unsolvable, meaning neither their truth nor their failure can be proven from ZFC. A major part of set theory is devoted to attacking this problem by studying various extensions of ZFC and their properties with the overall goal to identify the “right” axioms for mathematics that settle these problems.

Determinacy assumptions are canonical extensions of ZFC that postulate the existence of winning strategies in natural infinite two-player games. Such assumptions are known to enhance sets of real numbers with a great deal of canonical structure. Other natural and well-studied extensions of ZFC are given by the hierarchy of large cardinal axioms. Inner model theory provides canonical models for many large cardinal axioms. Determinacy assumptions, large cardinal axioms, and their consequences are widely used and have many fruitful implications in set theory and even in other areas of mathematics. Many applications, in particular, proofs of consistency strength lower bounds, exploit the interplay of determinacy axioms, large cardinals, and inner models. In this talk I will survey recent developments as well as my contribution to this flourishing area.**Information:** Please see the seminar webpage.

28 February – 6 March

**CUNY Set Theory Seminar****Time:** Friday, 4 March, 12:30pm New York time (18:30 CET)**Speaker:** Tom Benhamou, Tel Aviv University**Title:** Subforcings of the Tree-Prikry Forcing**Abstract:** We investigate which forcing notions can be embedded into a Tree-Prikry forcing. It turns out that the answer changes drastically under different large cardinal assumptions. We will focus on the class of κ-strategically closed forcings of cardinality κ, <κ-strategically closed forcings of cardinality κ and the κ-distributive forcing notions of cardinality κ. Then we will examine distributive subforcings of the Prikry forcing of cardinality larger than κ. This is a joint work with Moti Gitik and Yair Hayut.**Information:** The seminar will take place virtually. Please email Victoria Gitman (vgitman@nylogic.org) for the meeting id.

**Cross-Alps Logic Seminar****Time:** Friday, 4 March, 16.00-17.00 CET **Speaker:** M. Skrzypczak, University of Warsaw**Title:** The infinite tree – from Kolmogorov, Rabin, and Shelah to modern Theoretical Computer Science**Abstract:** The infinite binary tree (i.e. the free structure of two successors, aka S2S) seems to be a very simple and natural object. Nevertheless, due to its branching structure, it has rich abilities of modelling complex processes including e.g. nondeterminism, perfect information games, combinatorics of P(N), etc The fundamental result of Rabin from late 60’s (sometimes called “”the mother of all decidability results””) proves that the Monadic Second-Order (MSO) theory of S2S is decidable. Since then, the structure of properties expressible in MSO over S2S has been intensively studied. Many of these studies were related to and/or motivated by descriptive set theory. During the talk I would like to make a broad overview of these relations, including issues of Wadge degrees, measurability (with relations to Kolmogorov), and uniformisability (Gurevich-Shelah). Although a lot of questions have been already answered, there still remain important and natural open problems in all three mentioned directions of research.**Information:** The event will stream on the Webex platform. Please write to luca.mottoros [at] unito.it for the link to the event.

**CMU Model Theory Seminar****Time:** Thursday, 3 March, 11:00-12:30 Pittsburgh time (17:00-18:30 CET)**Speaker:** Wentao Yang, CMU**Title:** Shelah’s eventual categoricity conjecture in universal classes, part 2**Abstract:** We present Sebastien Vasey’s proof of Shelah’s categoricity conjecture in universal classes. First we change the substructure relation of the given class to obtain an AEC with better properties, except that the union axiom might not hold. To prove that the union axiom holds, we use the independence framework AxFr developed by Shelah, build an “independent” tree assuming the failure of union, and contradict stability.**Information:** Please see the seminar webpage.

**Caltech Logic Seminar****Time:** Wednesday, 2 March, 12:00-1:00pm Pacific time (21:00-22:00 CET)**Speaker:** Zoltán Vidnyánszky, Caltech**Title:** Ramsey and hypersmoothness**Abstract:** I will talk about an ongoing work in which new reasons of non-hypersmoothness are investigated. I will present an FσFσ equivalence relation, which is not hypersmooth because of a Ramsey-theoretic argument.**Information:** Please see the seminar webpage.

**CMU Logic Seminar****Time:** Tuesday, 1 March, 15:30-16:30 Pittsburgh time (21:30-12:30 CET)**Speaker:** Aristotelis Panagiotopoulos, Carnegie Mellon University**Title:** Every CBER is smooth below a Milliken-generic strong subtree (Part I)**Abstract:** The theory of Borel reductions becomes a very delicate subject of study when one restricts attention to the class of Countable Borel Equivalence Relations (CBERs). Indeed, no matter how complex a CBER is, its complexity tends to reside on a “small” piece of its domain. For example a result of Hjorth and Kechris states that every CBER on a Polish space is hyperfinite when restricted to some comeager set. Similarly, classical results of Mathias about forcing extensions by Mathias reals imply that every CBER on the Ellentuck Ramsey space is hyperfinite when restricted to some pure Ellentuck cube. In this talk we show that the Milliken space M of strong trees satisfies a much stronger canonization property: if E is CBER on M then for the Milliken-generic T in M we have that E and = agree on the pure Milliken cube [T].

This is joint work with Allison Wang.**Information:** Please see the seminar webpage.

**KGRC Set Theory Seminar, Vienna****Time:** Tuesday, 1 March, 15:00-16:30 CET**Speaker:** Thilo Weinert, University of Vienna**Title:** Ramsey Theory of Ordinals and Finite Combinatorics**Abstract:** The Ramsey Theory of Ordinals has been investigated over the last decades and a large variety of results have been attained. The talk is going to focus on the Ramsey Theory of finite multiples both of infinite cardinals and, in some cases products of two infinite cardinals. This leads to problems in finite combinatorics similar to the calculation of finite Ramsey numbers. On the one hand, exact results are usually only obtainable if the natural numbers involved remain somewhat small. On the other hand, sometimes asymptotic results can be attained.

More concretely, for any ordinal α and β, let r(α,β) denote the least ordinal γ such that any colouring of the pairs in γ in black and white either allows for a homogeneously white subset of order-type α or a homogeneously black subset of order-type β. Since the nineties it is known that the growth of r(n,3) is of order n2/log(n). It turns out that for any infinite cardinal λ, we have r(λ∗n,3)=λ∗r(In,L3) where the growth of r(In,L3) is of order n2/log(n) as well. Similarly, if κ>λ is weakly compact, we have r(κ∗λ∗n,3)=κ∗λ∗r(In,S3) where, again, the growth of r(In,L3) is of order n2/log(n). Finally there is a finitary characterisation of the Ramsey numbers r(ω2∗n,k) for natural numbers n and k. However the growth behaviour of r(ω2∗n,3) is still unknown.

This is partly joint work with Ferdinand Ihringer and Deepak Rajendraprasad.**Information:** This talk will be given via Zoom. Please contact Richard Springer for information how to participate.

21-27 February

**CUNY Set Theory Seminar****Time:** Friday, 25 February, 12:30pm New York time (18:30 CET)**Speaker:** Richard Matthews, University of Leeds**Title:** Big classes and the respected model**Abstract:** In standard (ZFC) set theory, proper classes are not sets because they are too ‘big’ or, to put it in a formal way, because they surject onto any non-zero ordinal. We shall study this notion of ‘bigness’ in weaker systems of set theory, in particular those in which the Power Set Axiom fails. We will observe that in many such theories it is possible to have proper classes which are not big.

As part of this, we shall see a failed attempt to find a proper class which is not big in the theory ZF without Power Set but with Collection – which is by taking a certain symmetric submodel of a class forcing. It will turn out that this approach fails because, unlike in the set forcing case, the symmetric submodel of a class forcing need not exhibit many of the nice properties that we would expect. Notably, Collection may fail and, in fact, it is unclear which axioms need necessarily hold.

This will lead to the definition of the ‘Respected Model’, an alternative approach to defining a submodel of a class forcing in which Choice fails. We will investigate the properties of this new model and compare it to the symmetric version.**Information:** The seminar will take place virtually. Please email Victoria Gitman (vgitman@nylogic.org) for the meeting id.

**Cross-Alps Logic Seminar****Time:** Friday, 25 February, 16.00-18.00 CET **Speaker:** V. Cipriani, University of Udine **Title:** tba**Abstract:** tba**Information:** The event will stream on the Webex platform. Please write to luca.mottoros [at] unito.it for the link to the event.

**CMU Model Theory Seminar****Time:** Thursday, 24 February, 11:00-12:30 Pittsburgh time (17:00-18:30 CET)**Speaker:** Wentao Yang, CMU**Title:** Shelah’s eventual categoricity conjecture in universal classes, part 1**Abstract:** We present Sebastien Vasey’s proof of Shelah’s categoricity conjecture in universal classes. First we change the substructure relation of the given class to obtain an AEC with better properties, except that the union axiom might not hold. To prove that the union axiom holds, we use the independence framework AxFr developed by Shelah, build an “independent” tree assuming the failure of union, and contradict stability.**Information:** Please see the seminar webpage.

**Caltech Logic Seminar****Time:** Wednesday, 23 February, 12:00-1:00pm Pacific time (21:00-22:00 CET)**Speaker:** Chris Laskowski, University of Maryland**Title:** Most(?) theories have Borel complete reducts and expansions**Abstract:** The talk is a meld of two papers, both joint with Douglas Ulrich.

We characterize which Φ∈Lω1,ωΦ∈Lω1,ω have Borel complete expansions and give several sufficient conditions for a theory to have a Borel complete reduct, e.g., having uncountably many complete 11-types.

We illustrate these general behaviors by considering a family ThTh of first-order theories, indexed by functions h:ω→ω∖{0,1}h:ω→ω∖{0,1} in a language L={En:n∈ω}L={En:n∈ω} asserting that each EnEn is an equivalence relation with h(n)h(n) classes, and that the classes cross-cut. The key to showing that many theories have Borel complete reducts follows from the fact that ThTh is Borel complete whenever hh is unbounded. We deduce our equivalents of having a Borel complete expansion in order to show the converse: if hh is bounded, then ThTh is not Borel complete.**Information:** Please see the seminar webpage.

**Helsinki Logic Seminar****Time:** Wednesday, 23 February, 12:00 – 14:00 Helsinki time (11:00-13:00 CET)**Speaker:** Gabriel Goldberg**Title:** tba **Abstract:** tba**Information:** Please see the seminar webpage. The talk will take place in person.

14-20 February

**CUNY Set Theory Seminar****Time:** Friday, 18 February, 12:30pm New York time (18:30 CET)**Speaker:** Sittinon Jirattikansakul,Tel Aviv University**Title:** Forcing with overlapping supercompact extenders, part 2**Abstract:** In the paper ‘Blowing up the power of a singular cardinal of uncountable cofinality’, Gitik introduced the forcing which can violate the SCH at singular cardinals of any cofinalities, assuming that the singular cardinals are also singular in the ground model. The forcing is built up from a Mitchell increasing sequence of strong extenders, and it preserves all cardinals and cofinalities in the generic extension. In this talk, we will discuss a forcing which is built from a Mitchell increasing sequence of supercompact extenders. The forcing also violates the SCH at singular cardinals of any cofinalities which are singular in the ground model. An important feature of this forcing is that it is possible to collapse the successor of a singular cardinal, while preserving cardinals above it.**Information:** The seminar will take place virtually. Please email Victoria Gitman (vgitman@nylogic.org) for the meeting id.

**Cross-Alps Logic Seminar****Time:** Friday, 18 February, 16.00-18.00 CET **Speaker:** T. Marinov, University of Turin **Title:** tba**Abstract:** tba**Information:** The event will stream on the Webex platform. Please write to luca.mottoros [at] unito.it for the link to the event.

**CMU Model Theory Seminar****Time:** Thursday, 17 February, 11:00-12:30 Pittsburgh time (17:00-18:30 CET)**Speaker:** Samson Leung, CMU**Title:** Stability results assuming tameness, monster model and continuity of nonsplitting, part 4**Abstract:** Assuming the existence of a monster model, tameness and continuity of nonsplitting in an abstract elementary class, we extend known superstability results: let $\mu>LS(K)$ be a regular stability cardinal and $\chi$ be the local character of $\mu$-nonsplitting, we have the following: (1) when $\mu$-nonforking is restricted to $(\mu,\geq\chi)$-limit models ordered by universal extensions, then it has invariance, monotonicity, uniqueness, existence, extension and continuity. Moreover it has a local character $\chi$. This generalizes Vasey’s result to the strictly stable setting. (2) There is a cardinal $\lambda<h(\mu)$ such that if $K$ is stable in every cardinal between $\mu$ and $\lambda$, then K has $\mu$-symmetry, (a) uniqueness of $(\mu,\geq\chi)$-limit models and (b) that any increasing chain of $\mu^+$-saturated models of length $\geq\chi$ has a $\mu^+$-saturated union. These generalize Boney, VanDieren and Vasey’s results. Furthermore, we show that the conclusions of (1), (2)(a)(b) are equivalent to $\chi$ being the local character of $\mu$-nonsplitting. These generalize Grossberg-Vasey’s superstability criteria and improve the cardinal threshold of equivalent criteria.**Information:** Please see the seminar webpage.

**Caltech Logic Seminar****Time:** Wednesday, 16 February, 12:00-1:00pm Pacific time (21:00-22:00 CET)**Speaker:** Konstantin Slutsky, Iowa State**Title:** Orbit equivalences of multidimensional Borel flows**Abstract:** An orbit equivalence between Borel actions of Polish groups is a Borel bijection between phase spaces that preserves orbit partitions. We are interested in free Rm-actions which are known as multidimensional flows. In this case, orbit equivalence is a coarse invariant collapsing all non-trivial flows into one class. Since any translation-invariant structure can be transferred from the acting group onto individual orbits, it is natural to consider strengthenings of orbit equivalence that respect these structures. Notable examples of such structures include measure, topology, and metric.

We will concentrate on two instances of this paradigm and discuss Borel versions of two ergodic-theoretical results: Katok’s representation theorem and Rudolph’s result on smooth orbit equivalence. The latter shows that any non-trivial free Rm-flow can be transformed into any other Rm-flow via an orbit equivalence that is a smooth orientation-preserving diffeomorphism on each orbit. Katok’s theorem provides a multidimensional generalization of the suspension flow construction and shows that all free Rm-flows emerge as special flows over Zm-actions.**Information:** Please see the seminar webpage.

7-13 February

**CUNY Set Theory Seminar****Time:** Friday, 11 February, 12:30pm New York time (18:30 CET)**Speaker:** Sittinon Jirattikansakul,Tel Aviv University**Title:** Forcing with overlapping supercompact extenders**Abstract:** In the paper ‘Blowing up the power of a singular cardinal of uncountable cofinality’, Gitik introduced the forcing which can violate the SCH at singular cardinals of any cofinalities, assuming that the singular cardinals are also singular in the ground model. The forcing is built up from a Mitchell increasing sequence of strong extenders, and it preserves all cardinals and cofinalities in the generic extension. In this talk, we will discuss a forcing which is built from a Mitchell increasing sequence of supercompact extenders. The forcing also violates the SCH at singular cardinals of any cofinalities which are singular in the ground model. An important feature of this forcing is that it is possible to collapse the successor of a singular cardinal, while preserving cardinals above it.**Information:** The seminar will take place virtually. Please email Victoria Gitman (vgitman@nylogic.org) for the meeting id.

**CMU Model Theory Seminar****Time:** Thursday, 10 February, 11:00-12:30 Pittsburgh time (17:00-18:30 CET)**Speaker:** Samson Leung, CMU**Title:** Stability results assuming tameness, monster model and continuity of nonsplitting, part 3**Abstract:** Assuming the existence of a monster model, tameness and continuity of nonsplitting in an abstract elementary class, we extend known superstability results: let $\mu>LS(K)$ be a regular stability cardinal and $\chi$ be the local character of $\mu$-nonsplitting, we have the following: (1) when $\mu$-nonforking is restricted to $(\mu,\geq\chi)$-limit models ordered by universal extensions, then it has invariance, monotonicity, uniqueness, existence, extension and continuity. Moreover it has a local character $\chi$. This generalizes Vasey’s result to the strictly stable setting. (2) There is a cardinal $\lambda<h(\mu)$ such that if $K$ is stable in every cardinal between $\mu$ and $\lambda$, then K has $\mu$-symmetry, (a) uniqueness of $(\mu,\geq\chi)$-limit models and (b) that any increasing chain of $\mu^+$-saturated models of length $\geq\chi$ has a $\mu^+$-saturated union. These generalize Boney, VanDieren and Vasey’s results. Furthermore, we show that the conclusions of (1), (2)(a)(b) are equivalent to $\chi$ being the local character of $\mu$-nonsplitting. These generalize Grossberg-Vasey’s superstability criteria and improve the cardinal threshold of equivalent criteria.**Information:** Please see the seminar webpage.

**Caltech Logic Seminar****Time:** Wednesday, 9 February, 12:00-1:00pm Pacific time (21:00-22:00 CET)**Speaker:** Matthew Bowen, McGill University**Title:** Measurable combinatorics in hyperfinite graphs**Abstract:** We discuss a few new results concerning the descriptive combinatorics of bounded degree hyperfinite Borel graphs. In particular, we show that the Baire measurable edge chromatic number of GG is at most ⌈32Δ(G)⌉+6⌈32Δ(G)⌉+6 when GG is a multigraph, and for bipartite graphs we improve this bound to Δ(G)+1Δ(G)+1 and show that degree regular one-ended bipartite graphs have Borel perfect matchings generically. Similar results hold in the measure setting assuming some hyperfiniteness conditions. This talk is based on joint work with Kun and Sabok, Weilacher, and upcoming work with Poulin and Zomback.**Information:** Please see the seminar webpage.

**Hebrew University-Bar Ilan University Set Theory seminar****Time:** Wednesday, 9 February, 14:00-16:00 Israel Time (13:00-15:00 CET)**Speaker:** tba**Title:** tba**Abstract:** tba**Information:** Contact Menachem Magidor or Omer Ben-Neria ahead of time for the seminar announcement and zoom link.

**Helsinki Logic Seminar****Time:** Wednesday, 9 February, 12:00 – 14:00 Helsinki time (11:00-13:00 CET)**Speaker:** Menachem Magidor **Title:** Large cardinals and properties of generalized logics **Abstract:** This is a joint work with Will Boney, Stamis Dimipoulos and Victoria Gitman.

It is well known that many large cardinals properties and schemata can be phrased as regularities properties of generalized logics. (Typical examples are Skolem-Lōwenheim and compactness properties ). In this talk we shall present some additional such characterizations.

An example is the connection between subtle cardinals and the existence of weak compactness for every abstract logic or the characterization of various virtual cardinals in terms of variations of compactness properties appropriate for the context of virtual large cardinals.

The basic definitions will be provided, so the talk should be accessible to a large group of logicians.**Information:** Please see the seminar webpage.

**Leeds Models and Sets Seminar****Time:** Tuesday, 8 February, 13:45 UK time (14:45 CET)**Speaker:** Vera Fischer, Universität Wien**Title:** Spectra and definability**Abstract:** In this talk, we will consider two aspects in the study of extremal sets of reals, sets like maximal families of eventually different functions, maximal cofinitary groups, or maximal independent families. On one side, we will discuss their spectrum, defined as the set of cardinalities of such families and on the other, the existence of witnesses of optimal projective complexity. We will emphasize recent developments in the area and indicate interesting remaining open questions. **Information:** Please see the seminar webpage.

31 January – 6 February

**CMU Model Theory Seminar****Time:** Thursday, 3 February, 11:00-12:30 Pittsburgh time (17:00-18:30 CET)**Speaker:** Samson Leung, CMU**Title:** Stability results assuming tameness, monster model and continuity of nonsplitting, part 2**Abstract:** Assuming the existence of a monster model, tameness and continuity of nonsplitting in an abstract elementary class, we extend known superstability results: let $\mu>LS(K)$ be a regular stability cardinal and $\chi$ be the local character of $\mu$-nonsplitting, we have the following: (1) when $\mu$-nonforking is restricted to $(\mu,\geq\chi)$-limit models ordered by universal extensions, then it has invariance, monotonicity, uniqueness, existence, extension and continuity. Moreover it has a local character $\chi$. This generalizes Vasey’s result to the strictly stable setting. (2) There is a cardinal $\lambda<h(\mu)$ such that if $K$ is stable in every cardinal between $\mu$ and $\lambda$, then K has $\mu$-symmetry, (a) uniqueness of $(\mu,\geq\chi)$-limit models and (b) that any increasing chain of $\mu^+$-saturated models of length $\geq\chi$ has a $\mu^+$-saturated union. These generalize Boney, VanDieren and Vasey’s results. Furthermore, we show that the conclusions of (1), (2)(a)(b) are equivalent to $\chi$ being the local character of $\mu$-nonsplitting. These generalize Grossberg-Vasey’s superstability criteria and improve the cardinal threshold of equivalent criteria.**Information:** Please see the seminar webpage.

**Caltech Logic Seminar****Time:** Wednesday, 2 February, 12:00-1:00pm Pacific time (21:00-22:00 CET)**Speaker:** Gábor Elek, Lancaster University**Title:** Uniform amenability and uniform hyperfiniteness**Abstract:** According to the Connes-Feldman-Weiss theorem, measurable amenability is equivalent to measurable hyperfiniteness for countable group actions It is easy to see that Borel hyperfiniteness implies Borel amenability, but the opposite direction is a very hard open problem (Jackson-Kechris-Louveau).

The graph-theoretical analogue of amenability is clearly Property A. One can view the so-called uniform local amenability (ULA) property (or local Følner-property) as the graph-theoretical analogue of hyperfiniteness. Brodzky et al. proved that Property A implies ULA. Two years ago I was able to prove that, in fact, ULA is equivalent to Property A. This result has an interesting application in distributed computing.

We say that an action is Uniformly (measurably) Hyperfinite, if all the positive measure subgraphings of the graphing of the action is hyperfinite with the same structure constants. I recently proved that Uniform Hyperfiniteness is equivalent to Uniform Amenability (the measurable version of Property A). Note that UA ⟹⟹ UH is very similar to the Connes-Feldman-Weiss proof, UH ⟹⟹ UA requires some new ideas. **Information:** Please see the seminar webpage.

**Barcelona Set Theory Seminar****Time:** Wednesday, 2 February, 16:00-17:30 CET**Speaker:** Rupert McCallum**Title:** Intrinsic Justifications in Set theory**Abstract:** tba **Information:** Online. If you wish to attend, please send an email to bagaria@ub.edu asking for the link.

**Leeds-Ghent Virtual Logic SeminarTime:** Wednesday, 2 February, 3pm UK time (16:00 CET)

**Speaker:**Richard Matthews, University of Leeds

**Title:**Big classes and the respected model

**Abstract:**In standard (ZFC) set theory, proper classes are not sets because they are too “big” or, to put it in a formal way, because they surject onto any non-zero ordinal. We shall study this notion of “bigness” in weaker systems of set theory, in particular those in which the Power Set Axiom fails. We will observe that in many such theories it is possible to have proper classes which are not big. As part of this, we shall see a failed attempt to find a proper class which is not big in the theory ZF without Power Set but with Collection – which is by taking a certain symmetric submodel of a class forcing. It will turn out that this approach fails because, unlike in the set forcing case, the symmetric submodel of a class forcing need not exhibit many of the nice properties that we would expect. Notably, Collection may fail and, in fact, it is unclear which axioms need necessarily hold. This will lead to the definition of the “Respected Model,” an alternative approach to defining a submodel of a class forcing in which Choice fails. We will investigate the properties of this new model and compare it to the symmetric version.

**Information:**Please contact Paul Shafer in advance to participate.

**Hebrew University-Bar Ilan University Set Theory seminar****Time:** Wednesday, 2 February, 14:00-16:00 Israel Time (13:00-15:00 CET)**Speaker:** tba**Title:** tba**Abstract:** tba**Information:** Contact Menachem Magidor or Omer Ben-Neria ahead of time for the seminar announcement and zoom link.

**Singapore Logic Seminar****Time:** Wednesday, 2 February, 16:00-17:00 Singapore time (9:00-10:00 CET)**Speaker:** Andre Nies, University of Auckland**Title:** The structure of the class of K-trivial sets**Abstract:** The K-trivial sets are antirandom in the sense that the initial segment complexity in terms of prefix-free Kolmogorov complexity K grows as slowly as possible. Chaitin introduced this notion in about 1975, and showed that each K-trivial is Turing below the halting set. Shortly after, Solovay proved that a K-trivial set can be noncomputable. In the past two decades, many alternative characterisations of this class have been found: properties such as being low for K, low for Martin-Loef (ML) randomness, and a basis for ML randomness, which state in one way or the other that the set is close to computable. Initially, the class of noncomputable K-trivials appeared to be amorphous. More recently, an internal structure has been found. Most of these results can be phrased in the language of a reducibility on the K-trivials which is weaker than Turing’s: A is ML-below B if each ML-random computing B also computes A.

Bienvenu, Greenberg, Kucera, Nies and Turetsky (JEMS 2016) showed that there an ML complete K-trivial set. Greenberg, Miller and Nies (JML, 2019) established a dense hierarchy of subclasses of the K-trivials based on fragments of Omega computing the set, and each such subclass is an initial segment for ML. More recent results (see arxiv.org/abs/1707.00258, updated and submitted Dec 2020) generalise these approaches using cost functions. They also show that each K-trivial set is ML-equivalent to a c.e. K-trivial. The extreme lowness of K-trivials, far from being an obstacle, allows for methods which don’t work in a wider setting. The talk provides an overview and discusses open questions. For instance, is ML-completeness an arithmetical property of K-trivials? This is joint work with Noam Greenberg, Joseph Miller and Dan Turetsky. **Information:** See the seminar webpage.

24-30 January

**Online Logic Seminar****Time:** Thursday, 27 January, 01:00pm US central time (20:00 CET)**Speaker:** Lauren Wickmann, University of Florida**Title:** tba **Abstract:** tba **Information:** See the seminar webpage.

**KGRC Logic Colloquium, ViennaTime:** Thursday, 27 January, 15:00 – 15:45 CET

**Speaker:**Yash Lodha, Uni Wien

**Title:**The Banach-Tarski paradox, monsters and their gentler cousins

**Abstract:**The Banach-Tarski paradox is one of the most striking and counterintuitive

phenomena in all of mathematics. Von Neumann’s study of the paradox led to

the formulation of the so called von Neumann-Day problem, which has been

attributed to Mahlon Day from the 1950s. The problem was solved in the

late 70s by the construction of various finitely generated “monster”

groups. However, I will explain how elementary tools from descriptive set

theory recently led to the construction of considerably “less scary” new

solutions, some of which are finitely presented (and even type

F_{\infty}).

**Information:**This talk will be given via Zoom. Please contact Richard Springer for information how to participate.

**Caltech Logic Seminar****Time:** Wednesday, 26 January, 12:00-1:00pm Pacific time (21:00-22:00 CET)**Speaker:** Severin Mejak, University of Copenhagen**Title:** An effective strengthening of Mathias’ theorem**Abstract:** A family of infinite subsets of natural numbers is almost disjoint if for any two distinct members of the family, their intersection is finite. The classical theorem of Mathias from the 70s states that there is no infinite analytic maximal almost disjoint (mad) family. The original proof used forcing and it wasn’t until almost four decades later that Asger Törnquist found a proof which used a derivative process on a tree.

In joint work with Asger Törnquist, we managed to simplify the derivative process so that it can be carried out (and it terminates) inLω1CK, thus proving that for any infinite Σ11 almost disjoint family, there is a Δ11witness to non-maximality (where both classes are lightface). The ongoing work in progress attempts to generalise the result to the ideals FinαFinα for α<ω1CK.

This is joint work with Asger Törnquist.**Information:** Please see the seminar webpage.

**Hebrew University-Bar Ilan University Set Theory seminar****Time:** Wednesday, 26 January, 14:00-16:00 Israel Time (13:00-15:00 CET)**Speaker:** tba**Title:** tba**Abstract:** tba**Information:** Contact Menachem Magidor or Omer Ben-Neria ahead of time for the seminar announcement and zoom link.

**Singapore Logic Seminar****Time:** Wednesday, 26 January, 16:00-17:00 Singapore time (9:00-10:00 CET)**Speaker:** Zhang Jing**Title:** Ramsey-type theorems on large structures**Abstract:** Structures like trees, linear orders, partial orders, graphs have been widely studied in different areas of mathematics. The Ramsey-type theorems we will discuss usually take the following form: for any given coloring of the structure, there exists a “large sub-structure” that intersects “very few” color classes. One example is a theorem of Laver that states for any finite coloring of Q x Q (ordered pairs of rationals), there exists X, Y order isomorphic to Q, such that X x Y intersects at most 2 color classes. We will discuss the uncountable analogues of these statements and their consistency. In particular, a diagonal version of the Halpern-Lauchli theorem plays a key role. The differences between countable combinatorics and uncountable combinatorics will be highlighted. **Information:** See the seminar webpage.

17-23 January

**CUNY Set Theory Seminar****Time:** Friday, 21 January, 2pm New York time (20:00 CET)**Speaker:** Wolfgang Wohofsky, University of Vienna**Title:** Distributivity spectrum and fresh functions, part 2**Abstract:** We discuss different notions of a distributivity spectrum of forcings.

In the first talk, I will mainly focus on the notion of fresh functions and the corresponding spectrum. A function with domain lambda is fresh if it is new but all its initial segments are in the ground model. I will give general facts how to compute the fresh function spectrum, also discussing what sets are realizable as a fresh function spectrum of a forcing. Moreover, I will provide several examples, including well-known tree forcings on omega such as Sacks and Mathias forcing, as well as Prikry and Namba forcing to illustrate the difference between fresh functions and fresh subsets.

In the second talk, I will also discuss another (‘combinatorial’) distributivity spectrum; most importantly, analyzing this notion for the forcing P(omega)/fin.

This is joint work with Vera Fischer and Marlene Koelbing.**Information:** The seminar will take place virtually. Please email Victoria Gitman (vgitman@nylogic.org) for the meeting id.

**Cross-Alps Logic Seminar****Time:** Friday, 21 January, 16.00-18.00 CET**Speaker:** L. Notaro, University of Turin**Title:** Tree representations for Borel functions**Abstract:** In 2009 Brian Semmes, in his PhD thesis, provided a characterization of Borel measurable functions from and into the Baire space using a reduction game called the Borel game. Around the same year, Alain Louveau wrote some (still unpublished) notes in which he provided a characterization of Baire class α functions (again from and into the Baire space), for all fixed α and, importantly, Σ0λ-measurable functions for λ countable limit, using tree-representations instead of games. In this talk, we present Louveau’s characterization, comparing it with Semmes’ one, and see that if we modify a bit the Borel game we end up characterizing functions having a Gδ graph. Then we notice that under AC there are functions for which the Borel game is undetermined, thus opening questions regarding the consistency strength of the general determinacy of this game.**Information:** The event will stream on the Webex platform. Please write to luca.mottoros [at] unito.it for the link to the event.

**Online Logic Seminar****Time:** Thursday, 20 January, 01:00pm US central time (20:00 CET)**Speaker:** Daniel Turetsky, Victoria University of Wellington**Title:** True Stages — From Priority Arguments to Descriptive Set Theory **Abstract:** The true stages machinery was conceived as a technique for organizing complex priority constructions in computability theory, much like Ash’s metatheorem. With a little modification, however, it can prove remarkably useful in descriptive set theory. Using this machinery, we can obtain nice proofs of results of Wadge, Hausdorff and Kuratowski, and Louveau, sometimes strengthening the result in the process. Without getting too deep into the details, I will give the ideas of the machinery and how it applies to descriptive set theory.**Information:** See the seminar webpage.

**Caltech Logic Seminar****Time:** Wednesday, 19 January, 12:00-1:00pm Pacific time (21:00-22:00 CET)**Speaker:** Asger Törnquist, University of Copenhagen**Title:** Set theory and a proposed model of the mind in psychology**Abstract:** Jens Mammen (Professor Emeritus of psychology at Aarhus and Aalborg University) has developed a theory in psychology, which aims to provide a model for the interface between a human being (and mind), and the real world.

This theory is formalized in a very mathematical way: Indeed, it is described through a mathematical axiom system. Realizations (“models”) of this axiom system consist of a non-empty set UU (the universe of objects), as well as a perfect Hausdorff topology SS on UU, and a family CC of subsets of UU which must satisfy certain axioms in relation to SS. The topology SS is used to model broad categories that we sense in the world (e.g., all the stones on a beach) and the CC is used to model the process of selecting an object in a category that we sense (e.g., a specific stone on the beach that we pick up). The most desirable kind of model of Mammen’s theory is one in which every subset of UU is the union of an open set in SS and a set in CC. Such a model is called “complete”.

The harder mathematical aspects of Mammen’s theory were first studied in detail by J. Hoffmann-Joergensen in the 1990s. Hoffmann-Joergensen used the Axiom of Choice (AC) to show that a complete model of Mammen’s axiom system, in which the universe UU is infinite, does exist. Hoffmann-Joergensen conjectured at the time that the existence of a complete model of Mammen’s axioms would imply the Axiom of Choice.

I will discuss the set-theoretic aspects of complete Mammen models. First of all, the question of “how much” AC is needed to obtain a complete Mammen model; secondly, I will introduce some cardinal invariants related to complete Mammen models and establish elementary ZFCZFC bounds for them, as well as some consistency results.

This is joint work with Jens Mammen.**Information:** Please see the seminar webpage.

**Hebrew University-Bar Ilan University Set Theory seminar****Time:** Wednesday, 18 January, 14:00-16:00 Israel Time (13:00-15:00 CET)**Speaker:** tba**Title:** tba**Abstract:** tba**Information:** Contact Menachem Magidor or Omer Ben-Neria ahead of time for the seminar announcement and zoom link.

**University of Vienna Set Theory Seminar****Time:** Tuesday, 18 January, 15:00-16:30 CET**Speaker:** Gunter Fuchs, City College of New York **Title:** Blurry definability**Abstract:** In this talk on ongoing research, I analyze blurry forms of ordinal definability and their hereditary versions which generalize ideas due to Hamkins/Leahy and Tzouvaras. Classically, a set is ordinal definable if it is the unique object satisfying some first order property in which ordinal parameters may occur. Given a cardinal kappa, I define that a set is <κ-blurrily ordinal definable if it belongs to an OD set of cardinality less than kappa. So in this case, the set is one of fewer than κ many objects with a certain property using ordinal parameters, not necessarily the unique such set. I will present some results on the class of hereditarily <κ-blurrily definable sets and the structure of leaps, that is, stages at which new sets become blurrily definable. There are some ZF(C) results and some relative consistency results using forcing.**Information:** This talk will be given via Zoom. Please contact Richard Springer for information how to participate.

**MOPA (Models of Peano Arithmetic), CUNY****Time:** Monday, 17 January, 2pm New York time (20:00 CET)**Speaker: **Mauro di Nasso, Università di Pisa**Title:** Nonstandard natural numbers in arithmetic Ramsey Theory and topological dynamics: Part II**Abstract:** The use of nonstandard models *N of the natural numbers has recently found several applications in arithmetic Ramsey theory. The basic observation is that every infinite number in *N corresponds to an ultrafilter on N, and the algebra of ultrafilters is a really powerful tool in this field. Note that this notion also makes sense in any model of PA, where one can consider the 1-type of any infinite number.

Furthermore, nonstandard natural numbers are endowed with a natural compact topology, and one can apply the methods of topological dynamics considering the shift operator x↦x+1 . This very peculiar dynamic has interesting characteristics.

In this talk I will also present a new result in the style of Hindman’s Theorem about the existence of infinite monochromatic configurations in any finite coloring of the natural numbers. A typical example is the following monochromatic pattern:

a, b, c, … , a+b+ab, a+c+ac, b+c+bc, … , a+b+c+ab+ac+bc+abc.**Information:** The seminar will take place virtually. Please email Victoria Gitman (vgitman@nylogic.org) for the meeting id.

10-16 January

**CUNY Set Theory Seminar****Time:** Friday, 14 January, 2pm New York time (20:00 CET)**Speaker:** Wolfgang Wohofsky, University of Vienna**Title:** Distributivity spectrum and fresh functions**Abstract:** We discuss different notions of a distributivity spectrum of forcings.

In the first talk, I will mainly focus on the notion of fresh functions and the corresponding spectrum. A function with domain lambda is fresh if it is new but all its initial segments are in the ground model. I will give general facts how to compute the fresh function spectrum, also discussing what sets are realizable as a fresh function spectrum of a forcing. Moreover, I will provide several examples, including well-known tree forcings on omega such as Sacks and Mathias forcing, as well as Prikry and Namba forcing to illustrate the difference between fresh functions and fresh subsets.

In the second talk, I will also discuss another (‘combinatorial’) distributivity spectrum; most importantly, analyzing this notion for the forcing P(omega)/fin.

This is joint work with Vera Fischer and Marlene Koelbing.**Information:** The seminar will take place virtually. Please email Victoria Gitman (vgitman@nylogic.org) for the meeting id.

**Toronto Set Theory Seminar****Time:** Friday, 14 January, 13.30-15.10 Toronto time (19.30-21.10 CET)**Speaker:** On this day the Toronto Set Theory Seminar at Fields Institute will hold

a special session with 4 short talks (20 minutes each), showcasing recent

research by faculty and postdocs at University of Toronto. Talks by Franklin D. Tall, Ivan Ongay Valverde, Clovis Hamel, Spencer Unger and David Schrittesser. **Information:** Please see https://homepage.univie.ac.at/david.schrittesser/world-logic-day-2022.html.

**Conference:** **Model theoretic logics and their frontiers ****Time:** Friday, 14 January, 9:30- Saturday 16 January 16:20 CET **Title:** Model theoretic logics and their frontiers**Abstract:** See the meeting’s website**Information:** See the meeting’s webpage.

**Conference: Still Stuk@Home ****Time:** Friday, 14 January, 9:30-17:00 UK time (10:00-18:00 CET) **Title:** Set Theory in the UK 7**Abstract:** See the meeting’s website**Information:** See the meeting’s webpage.

**Conference: **World Logic Day in Nigeria**Time:** Friday, 14 January, 10:00-13:10 CET**Title:** Logic – a world of interdiciplinary science 2**Abstract:** See the seminar webpage.**Information:** See the seminar webpage.

**Cross-Alps Logic Seminar****Time:** Thursday, 17 January, 17.00 CET **Speaker:** Menachem Magidor, Jerusalem**Title:** Sets of reals are not created equal: regularity properties of

subsets of the reals and other Polish spaces.**Abstract:** A ?pathological set? can be a non measurable set, a set which

does not have the property of Baire (namely it is not a Borel set modulo a

first category set). A subset $A \subseteq P^{\omega}(\mathbb{N})$ (= the

infinite subsets of natural numbers) can be considered to be ?pathological?

if it is a counterexample to infinitary Ramsey theorem. Namely there does

not exist an infinite set of natural numbers such that all its subsets are

in our sets or all its infinite subsets are not in the set. A subset of the

Baire space $A\subseteq \mathbb{N}^{\mathbb{N}}$ can be considered to be

?pathological? if the infinite game $G_A$ is not determined. The game $G_A$

is an infinite game where two players alternate picking natural numbers,

forming an infinite sequence, namely a member of $\mathbb{N}^{\mathbb{N}}$.

The first player wins the round if the resulting sequence is in $A$. The

game is determined if one of the players has a winning strategy.

A prevailing paradigm in Descriptive Set Theory is that sets that have a

?simple description? should not be pathological. Evidence for this maxim is

the fact that Borel sets are not pathological in any of the senses

described above.In these talks we shall present a notion of ?super

regularity? for subsets of a Polish space, the family of universally Baire

sets. This family of sets generalizes the family of Borel sets and forms a

$\sigma$-algebra. We shall survey some regularity properties of universally

Baire sets , such as their measurability with respect to any regular Borel

measure, the fact that they have an infinitary Ramsey property etc. Some of

these theorems will require assuming some strong axioms of infinity. Most

of the talk should be accessible to a general Mathematical audience, but in

the second part we shall survey some newer results.**Information:** The event will stream on the Webex platform. Please write to luca.mottoros [at] unito.it for the link to the event.

**KGRC Logic Colloquium, ViennaTime:** Thursday, 13 January, 11:30 – 12:15 CET

**Speaker:**Victor Torres-Pérez, TU Wien

**Title:**Worlds without Martin’s Axiom

**Abstract:**The first of Hilbert’s famous list of problems at the beginning of the

20th century was to establish Cantor’s Continuum Hypothesis (CH), i.e. if

there is no uncountable subset of the reals with cardinality strictly less

than the continuum. After the works of Gödel and Cohen, it was concluded

that the traditional axioms of Set Theory (ZFC) cannot decide CH.

Since then, new axioms have emerged. Prominently we have Forcing Axioms.

One of the first Forcing Axioms ever considered was Martin’s Axiom (MA).

While MA implies the negation of the CH, it does not decide the exact

value of the continuum. However, generalizations of MA like the Proper

Forcing Axiom (PFA) or Martin’s Maximum (MM) do imply that the continuum

is the second uncountable cardinal. Besides, PFA or MM imply the negation

of certain square principles or tree properties (among a very large number

of interesting consequences). This means in particular that these axioms

require the existence of large cardinals.

There are other relatively new principles, which have strong consequences

similar to the ones from PFA or MM, but they can coexist consistently with

the absence of MA or even imply this absence. A couple of these principles

are, for example, Rado’s Conjecture (RC) and the P-Ideal Dichotomy (PID).

We will give a general review of results involving these kinds of

principles, including some of ours obtained along the previous years.

There, it is possible to observe that even if they can avoid MA, they are

still quite powerful like these traditional Forcing Axioms. We will expose

one of our last results, where we prove (with L. Wu) that PID implies the

negation of a certain type of two-cardinals square principle.

**Information:**This talk will be given via Zoom. Please contact Richard Springer for information how to participate.

**Singapore Logic Seminar****Time:** Wednesday, 12 January, 16:00-17:00 Singapore time (9:00-10:00 CET)**Title:** Logic Day Special**Abstract:** This week’s Logic Seminar on Wed 12 January 2022 at 16:00 hrs is an open session where, in light of the World Logic Day on Friday, everyone is encouraged to give about 10 minutes presentation about his favourate result or results of his own work.**Information:** See the seminar webpage.

**KGRC Set Theory Seminar, Vienna****Time:** Tuesday, 11 January, 18:00-18:30 CET**Speaker:** Lorenzo Sauras, Technical University of Vienna**Title:** Generalization of proofs of universal sentences**Abstract:** In 1999, Baaz established a systematic way of generalizing proofs of universal

sentences that has led to certain progress on the challenging problem of the factorization of Fermat numbers (i.e., numbers of the form 22n 1, where n is a natural number). This talk will explain, in a concise way, how such generalization algorithm works, by inputting an ingenious calculation (i.e., a proof of a quantier-free formula without variables) of 641 divides 2251 that was discovered by Bennet and Kraïtchik. In addition, if time permits, some other of its outputs (remarkably, sucient conditions for a given value to be a divisor of a Fermat number), as well as related results from an ongoing research, will be stated.**Information:** This talk will be given via Zoom. Please contact Richard Springer for information how to participate.

**KGRC Set Theory Seminar, Vienna****Time:** Tuesday, 11 January, 17:15-17:45 CET**Speaker:** Michal Tomasz Godziszewski, University of Warsaw & University of Lódz**Title:** Spectra of maximal almost orthogonal families of projections in the Calkin algebra**Abstract:** Let H be an innite dimensional separable complex Hilbert space with inner

product h·|·i. Let B(H) be a Banach space of bounded linear operators on H with the

operator norm. In case when H = `2(ω), we can distinguish a particular subalgebra of the

Banach space B(H): we dene K(H) as the smallest Banach subalgebra of B(H) containing

all nite-dimensional operators, and we call its elements compact operators. So, T ∈ B(H)

is compact if it is a limit of nite-rank operators. The collection K(H) has the structure of a C∗-algebra and is a ring-theoretical ideal in B(H).**Information:** This talk will be given via Zoom. Please contact Richard Springer for information how to participate.

**KGRC Set Theory Seminar, Vienna****Time:** Tuesday, 11 January, 16:30-17:00 CET**Speaker:** Fabian Kaak, University of Vienna**Title:** Forcing Indestructibility of MAD families**Abstract:** An innite family A of innite subsets of natural numbers is called almost

disjoint if any two members of it have nite intersection, furthermore we call it a maximal almost disjoint (short: MAD) family if it is maximal with respect to inclusion. After forcing a MAD family will stay almost disjoint, but the maximality could be destroyed. For a forcing notion P we say that A is P-indestructible if it stays maximal in every forcing extension via P. In this talk I will present a property for forcings, which gives rise to a combinatorial characterization of indestructibility. Using this we can proof implications between indestructibility for dierent forcing notions. For example if a MAD family is indestructible for some forcing adding a real, then it is Sacks indestructible. The main focus will be the construction of a Sacks indestructible MAD family.**Information:** This talk will be given via Zoom. Please contact Richard Springer for information how to participate.

**KGRC Set Theory Seminar, Vienna****Time:** Tuesday, 11 January, 15:45-16:15 CET**Speaker:** Valentin Haberl, University of Vienna & Technical University of Graz**Title:** Tukey order on hyperspaces of compact subspaces and the Menger covering property**Abstract:** For some topological space X the hyperspace K(X) is the space of compact

subspaces of X equipped with the Vietoris topology. By considering the Tukey order for

the poset K(X) ordered by inclusion, it is possible to characterize topological properties of X. For separable metrizable spaces there are the important results K(X) ≤T ω^ω iff X is

Polish and K(Q) ≤T K(X) iff X is not hereditarily Baire. We look at the characterization K(X) hereditarily Baire i X is co-Menger in the special case for subsets of 2 ω. As an application we are able to prove the existence of X ⊆ 2 ω in ZFC such that K(X) T ω ω and K(X) T K(Q). Furthermore, we discuss Menger subspaces in the context of forcing,

for example proving that there are c many Menger subspaces of 2 ω in the iterated Sacks model.**Information:** This talk will be given via Zoom. Please contact Richard Springer for information how to participate.

**KGRC Set Theory Seminar, ViennaTime:** Tuesday, 11 January, 15:00-15:30 CET

**Speaker:**Lukas Koschat, University of Vienna

**Title:**A short intro duction to countable support iterations and proper forcing

**Abstract:**In this talk I want to present the basic ideas behind countable support itera-

tions and explain why we care about properness. I will give an overview of the fundamental theorems about properness and countable support iterations. In the end I will explain how these tools come together in a proof of the consistency of certain cardinal invariants constellations. The talk is aimed at anyone who is familiar with forcing but not yet familiar with countable support iterations and properness. The content of the talk is mainly extracted from my master thesis, and hence will be accessible to anyone who has completed an introductory course on forcing.

**Information:**This talk will be given via Zoom. Please contact Richard Springer for information how to participate.

3-9 January

**Hebrew University-Bar Ilan University Set Theory seminar****Time:** Wednesday, 5 January, 14:00-16:00 Israel Time (13:00-15:00 CET)**Speaker:** Yair Hayut**Title:** Stationary Reflection and the Successors of Singular cardinals**Abstract:** In this series of talks I’m going to present a few old and new results concerning the consistency of special assertions at successors of singular cardinals (i.e. $\aleph_{\omega + 1}$) – the reflection principles. The so called “reflection principle” are properties of the form: Let X be a subset of $\lambda$. such that X has some property. Then there is some $M$ subset of $\lambda$ of small cardinality, such that X \cap M has the same property.We will start with a very gentle introduction to Prikry forcing, showing its basic properties. Then, we will focus on stationary reflection and prove Magidor’s theorem on the consistency of stationary reflection at $\aleph_{\omega+1}$, starting with supercompact cardinals. Then, we will show how to get stationary reflection except one bad set, using Prikry forcing. After that, we will work towards the stronger result, getting full stationary reflection at $\aleph_{\omega+1}$, using a variant of the Prikry forcing (Unger and H.). We will introduce the Extender Based Prikry forcing, and prove the consistency of stationary reflection with the negation of SCH, using a partial supercompact cardinal (Ben-Neria, Unger and H.).In the last part, I will talk about a recent project with Magidor, improving the upper bound of the Delta reflection. **Information:** Contact Menachem Magidor or Omer Ben-Neria ahead of time for the seminar announcement and zoom link.