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

27 December – 2 January

**Hebrew University-Bar Ilan University Set Theory seminar****Time:** Wednesday, 29 December, 14:00-16:00 Israel Time (13:00-15:00 CET)**Speaker:** Ur Yaar**Title:** Iterating the cofinality-\omega constructible model **Abstract:** Consider C* – the inner model constructed in an L-like fashion, but using first order logic augmented with the “cofinality \omega” quantifier. C* has some canonicity properties similar to L, but a notable difference is that C* does not necessarily satisfy the axiom V=C*, that is – constructing the C* of C* may result in a smaller inner model. If this happens, one can iterate this construction, taking intersections at limit stages and ask at what stage it stabilizes or “breaks” (i.e. that the result is no longer a model of ZFC). This type of construction was considered with regards to HOD, where it was shown by McAaloon, Harrington, Jech and Zadrozny that “everything is possible” – on one hand there are models with iterated HODs of any ordinal length (and even of length Ord), and on the other hand it is possible that after \omega many stages the intersection either satisfies ZF without AC or even doesn’t satisfy ZF at all. In this talk we will discuss iterating the C* construction, and show that under ZFC alone we can only reach finitely many steps, while a sequence of length $\omega$ is equiconsistent with an inner model with a measurable cardinal.**Information:** Contact Menachem Magidor or Omer Ben-Neria ahead of time for the seminar announcement and zoom link.

20-26 December

**Hebrew University-Bar Ilan University Set Theory seminar****Time:** Wednesday, 22 December, 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.

**Paris-Lyon Séminaire de Logique****Time:** Monday, 20 December,15:15 CET**Speaker:** Francesco Parente**Title:** Combinatorial properties of ultrafilters and their orderings on Boolean algebras**Abstract:** In this talk, I shall report on joint work with Jörg Brendle, focusing on the combinatorial properties of ultrafilters on Boolean algebras in relation to the Tukey and Rudin-Keisler orderings. First, I aim to introduce the framework of Tukey reducibility and discuss the existence of non-maximal ultrafilters. Furthermore, I shall connect this discussion with a cardinal invariant of Boolean algebras, the ultrafilter number, and sketch consistency results (and open questions) concerning its possible values on Cohen and random algebras. Finally, I will analyse and compare two generalizations of the Rudin-Keisler ordering to ultrafilters on complete Boolean algebras, introducing new techniques to construct incomparable ultrafilters in this setting.**Information:** Join via the link on the seminar webpage

13-19 December

**Online Logic Seminar****Time:** Thursday, 16 December, 01:00pm US central time (20:00 CET)**Speaker:** Todor Tsankov, Université Claude Bernard Lyon 1**Title:** tba **Abstract:** tba **Information:** See the seminar webpage.

**KGRC Logic Colloquium, ViennaTime:** Thursday, 16 December, 15:00 – 15:45 CET

**Speaker:**Martin Hils, University of Münster

**Title:**Classification of definable quotients

**Abstract:**In many areas of mathematics, quotient objects play an important role, and it is often useful to close a category under quotients. In the talk, we will discuss so-called imaginaries, i.e., definable quotients in first order logic. In algebraically closed and in real closed fields, imaginaries may be eliminated. In valued fields, the situation is more interesting, as there are definable quotients like the residue field and value group which may not be eliminated. In algebraically closed valued fields, the imaginaries were classified by Haskell-Hrushovski-Macpherson. We will discuss a recent generalization of their work to more general henselian valued fields, which is joint with Silvain Rideau-Kikuchi.

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

**Bristol Logic and Set Theory SeminarTime:** Wednesday, 15 December, 16:30-18:00 UK time (17:30-19:00 CEST)

**Speaker:**Juan Aguilera, Ghent University

**Title:**The Pi12 Consequences of a Theory

**Abstract:**We define the \Pi^1_2-norm |T| of a theory T, an analogue of the proof-theoretic ordinal of T for statements of complexity \Pi^1_2. We go over the basic theory of |T| and use it to define the \Pi^1_2-soundness ordinal of T. Then, we characterise the recursive and admissible ordinals which are \Pi^1_2-soundness ordinals of some recursively enumerable extension of Arithmetical Comprehension. This is joint work with Fedor Pakhomov.

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

**Barcelona Set Theory Seminar****Time:** Wednesday, 15 December, 16:00-17:30 CET**Speaker:** Luca Motto-Ros, University of Torino**Title:** Generalized Polish spaces at regular uncountable cardinals**Abstract:** In the context of generalized descriptive set theory, we systematically compare and analyze various notions of Polish-like spaces and standard k-Borel spaces for k an uncountable (regular) cardinal satisfying k<k = k. As a result, we obtain a solid framework where one can develop the theory in full generality. We also provide natural characterizations of the generalized Cantor and Baire spaces. Some of the results obtained considerably extend previous work by Sikorski, Coskey-Schlicht, Galeotti, and Luecke-Schlicht, and answer some questions contained therein. Joint work with C. Agostini and P. Schlicht.**Information:** Online. If you wish to attend, please send an email to bagaria@ub.edu asking for the link.

**Leeds Models and Sets Seminar****Time:** Wednesday, 15 December, 13:45 UK time (14:45 CET)**Speaker: **Aris Papadopoulos, University of Leeds**Title:** Around Generalised Indiscernibles and Higher-arity Independence Properties**Abstract:** The machinery of generalised indiscernibles has played a key role in recent developments of stability theory. One of the most important applications of this machinery is characterising dividing lines by collapsing indiscernibles, a programme essentially tracing back to the early work of Shelah in the 1980s which has seen a resurgence lately, starting with the work of Scow.

In my talk, I will survey the main definitions and some important notions concerning these generalised indiscernibles and give some examples of characterising dividing lines by collapsing indiscernibles. Finally, if time permits, I will discuss an application of generalised indiscernibles to higher-arity independence properties, showing that IP_k can be witnessed by formulas in singleton variables if one allows parameters (from some model).**Information:** Please see the seminar webpage.

**Hebrew University-Bar Ilan University Set Theory seminar****Time:** Wednesday, 15 December, 14:00-16:00 Israel Time (13:00-15:00 CET)**Speaker:** Dominik Adolf **Title:** Cofinalities of Chang’s sets**Abstract:** Consider the property (ω4 ,ω3) –>> (ω2,ω1). A “typical” witness for this property is X a subset of ω4 containing ω1 such that the ordertype of (X ∩ ω4) is ω2 and the cofinality of (X ∩ ω3) is ω1. We are interested in what can be said about the cofinality of (X ∩ ω2). In this talk we will give some very basic results about this topic. We will then use these results to give an example of an unusual Supercompact Prikry-like forcing. If time permits we will discuss some interesting open problems. **Information:** Contact Menachem Magidor or Omer Ben-Neria ahead of time for the seminar announcement and zoom link.

**KGRC Set Theory Seminar, ViennaTime:** Tuesday, 14 December, 17:00-17:50 CET

**Speaker:**Yuxin Zhou, University of Florida

**Title:**Distinguish chromatic numbers for isosceles triangles in choiceless set theory

**Abstract:**For n a positive natural number, let Γn be the hypergraph of isosceles triangles on Rn. Under the axiom of choice, the existence of a countable coloring for Γn holds for every n. Without the axiom of choice, the chromatic numbers may or may not be countable. With an inaccessible cardinal assumption, there is a model of ZF + DC in which Γ2 has countable chromatic number while Γ3 has uncountable chromatic number. This result is obtained by a balanced forcing over the symmetric Solovay model.

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

**KGRC Set Theory Seminar, ViennaTime:** Tuesday, 14 December, 16:00-16:50 CET

**Speaker:**Marlene Koelbing, Uni Wien

**Title:**Special Aronszajn trees and Kurepa trees

**Abstract:**I will talk about special אn-Aronszajn trees and אn-Kurepa trees. The main result I want to present is the consistency of the statement that the following holds for every 0 < n < ω: all אn-Aronszajn trees are special, there are such, and there exists no אn-Kurepa tree. The proof needs ω-many supercompact cardinals. I will discuss the main ideas of the proof. This is joint work with Sy-David Friedman.

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

**KGRC Set Theory Seminar, ViennaTime:** Tuesday, 14 December, 15:30-15:55 CET

**Speaker:**Lukas Schembecker , Uni Wien

**Title:**Independence of the Whitehead Problem

**Abstract:**An abelian group A is called Whitehead i for every abelian group B we have that every short exact sequence of the form 0 → Z ι→ B π→ A → 0 splits, which means that there is a group homomorphism ρ : A → B such that π ◦ ρ = id A, i.e. it satises one of the equivalent conditions of the splitting lemma. Clearly, every free abelian group is Whitehead – conversely, we are interested in the Whitehead Problem: Is every Whitehead group free? We will see that for countable groups this is a theorem of ZFC. However, Shelah established the independence of the Whitehead Problem for groups of size א1 – a surprising result as it was the rst purely algebraic statement proven to be independent from ZFC. More specically, we show that the diamond principles in L give a positive answer and MA + ¬CH gives a negative answer to the Whitehead Problem.

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

**KGRC Set Theory Seminar, ViennaTime:** Tuesday, 14 December, 15:00-15:25 CET

**Speaker:**Julia Millhouse, Uni Wien

**Title:**Ramsey uniformization and madness

**Abstract:**A classical result of Mathias states that there do not exist analytic innite maximal almost disjoint (mad) families. Moreover he proves that all innite A ⊆ [ω]^ω in Solovay’s model satisfy the Ramsey Property; Silver established the same property for analytic sets. Mathias posed the natural question if there exist innite mad families in Solovay’s model, to which Asger Tornquist responded negatively in 2014. A stronger result appeared in 2019, when Törnquist and Schrittesser proved that if all infinite A ⊆ [ω]^ω have the Ramsey property, then there do not exist infinite mad families. They work in ZF with the fragment of choice, DC. Their proof relies signicantly on the assumption of a certain uniformization property, as well as the topological properties of a rather canonical object the equivalence relation of eventual agreement on binary sequences. In this talk I will give an outline of this 2019 proof.

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

6-12 December

**CUNY Set Theory Seminar****Time:** Friday, 10 December, 2pm New York time (20:00 CET)**Speaker:** Eyal Kaplan, Tel Aviv University**Title:** Non-stationary support iterations of Prikry forcings and restrictions of ultrapower embeddings to the ground model, part II **Abstract:** Assume that P is a forcing notion, G is a generic set for it over the ground model V, and a cardinal κ is measurable in the generic extension. Let j be an ultrapower embedding, taken in V[G] with a normal measure on κ. We consider the following questions:

1. Is the restriction of j to V an iterated ultrapower of V (by its measures or extenders)?

2. Is the restriction of j to V definable in V?

By a work of Schindler [1], the answer to the first question is affirmative, assuming that there is no inner model with a Woodin Cardinal and V=K is the core model. By a work of Hamkins [2], the answer to the second question is positive for forcing notions which admit a Gap below κ.

We will address the above questions in the context of nonstationary-support iteration of Prikry forcings below a measurable cardinal κ. Assuming GCH only in the ground model, we provide a positive answer for the first question, and describe in detail the structure of j restricted to V as an iteration of V. The answer to the second question may go either way, depending on the choice of the measures used in the Prikry forcings along the iteration; we will provide a simple sufficient condition for the positive answer. This is a joint work with Moti Gitik.

[1] Ralf Schindler. Iterates of the core model. Journal of Symbolic Logic, pages 241–251, 2006.

[2] Joel David Hamkins. Gap forcing. Israel Journal of Mathematics, 125(1):237–252, 2001.**Information:** The seminar will take place virtually. Please email Victoria Gitman (vgitman@nylogic.org) for the meeting id.

**Toronto Set Theory Seminar****Time:** Friday, 10 December, 1.30-3.00pm Toronto time (19.30-21.00 CET)**Speaker:** Mirna Dzamonja, IRIF (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 the seminar webpage.

**CMU Model Theory Seminar****Time:** Thursday, 9 December, 11:00-12:30 Pittsburgh time (17:00-18:30 CET)**Speaker:** John T. Baldwin, University of Illinois at 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 mathematics 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:** Please see the seminar webpage.

**Caltech Logic Seminar****Time:** Wednesday, 8 December, 12:00-1:00pm Pacific time (21:00-22:00 CET)**Speaker:** Andrew Marks, UCLA**Title:** A dichotomy for σ-Baire class α functions**Abstract:** In the 1920s, Lusin asked whether every Borel function on a Polish space is a union of countably many partial continuous functions (i.e. whether every Borel function is σσ-continuous). This question has a negative answer; an example of a non-piecewise continuous Borel function is the Turing jump. A dichotomy of Solecki and Zapletal is that the Turing jump is the basis for every counterexample: every Borel function ff is either σσ-continuous, or the Turing jump continuously reduces to ff. We generalize the Solecki-Zapletal dichotomy throughout the Borel hierarchy. Recall that a Borel function is Baire class αα if and only if it is Σ0α+1Σα+10-measurable. We show that every Borel function ff is either σσ-Baire class αα, or a complete Baire class α+1α+1 function (an appropriate iterate of the Turing jump) continuously reduces to ff. Our proof uses an adaptation of Montalbán’s game metatheorem for priority arguments to boldface descriptive set theory. This is joint work with Antonio Montalbán.**Information:** Please see the seminar webpage.

**Barcelona Set Theory Seminar****Time:** Wednesday, 8 December, 16:00-17:30 CET**Speaker:** Philipp Schlicht, University of Bristol **Title:** Forcing axioms via ground model interpretations **Abstract:** In a joint project with Christopher Turner, we study principles of the form: if a name σ is forced to have a certain property φ, then there is a ground model filter g such that φ(σ^g) holds. Such principles are implicit in strong forcing axioms such as PFA+. We prove a general correspondence connecting these name principles to forcing axioms. One can use this to obtain characterisations of forcing axioms and their λ-bounded versions. A further goal is to systematically study name principles where φ is a notion of largeness for subsets of ω1 (such as being unbounded, stationary or in the club filter) and corresponding forcing axioms. The preprint is available on the arxiv.**Information:** Online. If you wish to attend, please send an email to bagaria@ub.edu asking for the link.

**Leeds Models and Sets Seminar****Time:** Wednesday, 8 December, 13:45 UK time (14:45 CET)**Speaker: **Anush Tserunyan, McGill University**Title:** Tba**Abstract:** Tba**Information:** Please see the seminar webpage.

**Hebrew University-Bar Ilan University Set Theory seminar****Time:** Wednesday, 8 December, 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.

**KGRC Set Theory Seminar, ViennaTime:** Tuesday, 7 December, 15:00-16:30 CET

**Speaker:**S. Bardyla, Universität Wien

**Title:**On regular countably compact \mathbb{R}-rigid spaces

**Abstract:**A regular separable first-countable countably compact space is called a Nyikos space. A space X is called R-rigid if any continuous real-valued function on X is constant. Under MA we construct an R-rigid Nyikos space. This way we answer a few questions of Tzannes and extend results of Ciesielski and Wojciechowski. This is a joint work with Zdomskyy.

**Information:**This talk will be given via Zoom. Please note that the Zoom meeting ID and passcode change for the Set Theory Seminar starting with this talk (but remain unchanged for other KGRC seminars).

29 November – 5 December

**CUNY Set Theory Seminar****Time:** Friday, 3 December, 2pm New York time (20:00 CET)**Speaker:** Eyal Kaplan, Tel Aviv University**Title:** Non-stationary support iterations of Prikry forcings and restrictions of ultrapower embeddings to the ground model**Abstract:** Assume that P is a forcing notion, G is a generic set for it over the ground model V, and a cardinal κ is measurable in the generic extension. Let j be an ultrapower embedding, taken in V[G] with a normal measure on κ. We consider the following questions:

1. Is the restriction of j to V an iterated ultrapower of V (by its measures or extenders)?

2. Is the restriction of j to V definable in V?

By a work of Schindler [1], the answer to the first question is affirmative, assuming that there is no inner model with a Woodin Cardinal and V=K is the core model. By a work of Hamkins [2], the answer to the second question is positive for forcing notions which admit a Gap below κ.

We will address the above questions in the context of nonstationary-support iteration of Prikry forcings below a measurable cardinal κ. Assuming GCH only in the ground model, we provide a positive answer for the first question, and describe in detail the structure of j restricted to V as an iteration of V. The answer to the second question may go either way, depending on the choice of the measures used in the Prikry forcings along the iteration; we will provide a simple sufficient condition for the positive answer. This is a joint work with Moti Gitik.

[1] Ralf Schindler. Iterates of the core model. Journal of Symbolic Logic, pages 241–251, 2006.

[2] Joel David Hamkins. Gap forcing. Israel Journal of Mathematics, 125(1):237–252, 2001.**Information:** The seminar will take place virtually. Please email Victoria Gitman (vgitman@nylogic.org) for the meeting id.

**Cross-Alps Logic Seminar****Time:** Friday, 3 December, 16.00-18.00 CET **Speaker:** Gabriel Goldberg, UC Berkeley**Title:** tba **Abstract:** tba **Information:** Online on WebEx. Please see the seminar webpage.

**Toronto Set Theory Seminar****Time:** Friday, 3 December, 10.00-11.30pm Toronto time (16.00-17.30 CET)**Speaker:** Mariam Beriashvili, Iv. Javakhishvili Tbilisi State University, Tbilisi **Title:** On two-point sets and other nontrivial point sets**Abstract:** We consider certain pathological point sets from the general measure theoretical point of view. Namely, we discuss Mazurkiewicz sets, also called two-point sets, which have difficult and interesting descriptive as well as measure theoretic properties. Moreover, we will discuss also uniform subsets of the Euclidean space and their connections to the Mazurkiewizc sets. Also, in the talk will be considered Bernstein sets and Hamel bases.**Information:** Please see the seminar webpage.

**CMU Model Theory Seminar****Time:** Thursday, 2 December, 11:00-12:30 Pittsburgh time (17:00-18:30 CET)**Speaker:** Samson Leung, CMU**Title:** Axiomatizing AECs and applications, part 3**Abstract:** Let $K$ be an abstract elementary class and $\lambda=LS(K)$. $K$ can be axiomatized by a sentence in $L_{(2^\lambda)^+,\lambda^+}$, allowing game quantification. This extends Kueker’s result which assumes finite character and countable LS(K). It is also a parallel to Shelah-Villaveces result which demands a much higher complexity of junctions but without game quantification. Shelah’s presentation theorem gives $K=PC(T,\Gamma,L(K))$ where $T$ is a first-order theory in an expansion of $L(K)$ and $\Gamma$ is a set of $T$-types. We provide a better bound of $|\Gamma|$ in terms of $I_2(\lambda,K)$. We also give conditions under which the categoricity in two successive cardinals implies the existence of models in the next cardinal. This improves the result of Shelah and as a corollary we extend Shelah-Vasey’s result.**Information:** Please see the seminar webpage.

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

**Speaker:**Martin Goldstern, TU Wien

**Title:**The world between aleph1 and continuum: from Martin’s Axiom to Cichoń’s Maximum

**Abstract:**Georg Cantor’s “Continuum Hypothesis” (CH) postulates that the continuum (the cardinality of the set of real numbers) is equal to ℵ1, the smallest uncountable cardinal. Martin’s Axiom (MA) is a weakening of CH; it implies that all infinite cardinals below the continuum are similar to ℵ0, the cardinality of a countable set. For example, MA implies that not only every countable union of null (measure zero) sets is still null, but even every union of fewer than continuum many such sets. This motivates the definition of a so-called cardinal characteristic, the additivity number of the measure zero sets – the answer to the question “how many null sets do we have to join together to get a non-null set”. There is a whole zoo of such cardinal characteristics (some of them defined long before the advent of forcing); whenever you know that any countable set of objects with property X will never have property Y, you may ask how many such objects you need to get to Y.

Accepting CH or just MA as an axiom gives a picture that is on the one hand very clean, but on the other hand also rather poor: most cardinal characteristics can then be shown to be equal to the continuum.

In my talk I will discuss – or at least hint at – some recent (and some old) techniques for constructing “anti-MA” universes, where many cardinals between ω1 and the continuum appear as cardinal characteristics (defined by some natural properties X and Y).

I will try to hide all technical details, so that my talk will hopefully be understandable also for non-set-theorists.

**Information:**This talk will be given in mixed mode, in person as well as via Zoom.

**Caltech Logic Seminar****Time:** Wednesday, 1 December, 12:00-1:00pm Pacific time (21:00-22:00 CET)**Speaker:** Lauren Wickman, University of Florida**Title:** On the universal minimal flow of the homeomorphism group of a Knaster continuum**Abstract:** The correspondence between Ramsey theory, Fraïssé theory, and dynamics established in 2003 by Kechris, Pestov, and Todorčević has had far-reaching consequences in the study of automorphism groups of discrete first-order structures. The dual notion, projective Fraïssé theory (developed by Irwin and Solecki in 2006), considers classes of topological structures and in 2017, Panagiotopoulos proved that every compact, metrizable space can be obtained as a quotient of a projective Fraïssé limit. Therefore, projective Fraïssé theory provides an ideal framework to consider the homeomorphism groups of compact spaces. In this talk, for each Knaster continuum KK, we will give a projective Fraïssé class of finite objects that approximates KK (up to homeomorphism) and use the dual of the KPT correspondence (proved by Bartošová and Kwiatkowska, 2018) to determine if the homeomorphism group of KK is extremely amenable.**Information:** Please see the seminar webpage.

**Barcelona Set Theory Seminar****Time:** Wednesday, 1 December, 16:00-17:30 CET**Speaker:** Menachem Magidor, Jerusalem**Title:** Borel canonization of analytic and universally Baire relations**Abstract:** The basic question is due to Kanovie,Sabok and Zapletal: Given an analytic equivalence relation over the Polish space E \subseteq X x X. To what extent E can be “Borelized”? Namely , can one find a ”sizable” Borel set B \subseteq X such that E|X is Borel? ”Sizable” here means positive with respect to some sigma-ideal of Borel sets, I. Results of Kanovei-Sabok-Zapletal shows that one needs to make some assumptions about the equivalence relation and about the ideal. Natural assumptions are that every E equivalence class is Borel and that the natural forcing notion associated with I, P_I, is a proper forcing notion. The following theorem is due independently to Drucker and Chan:

Theorem 0.1. Assume that there is a measurable cardinal (actually (P(R))# is enough ) then the answer is ”yes” for E with Borel equivalence classes and P_I proper. The proof yields the following theorem:

Theorem 0.2. Assume that there is a measurable cardinal. Let E \subseteq X x Y be an analytic relation such that for every x in X the vertical section E_x is Borel . Let I be a sigma-ideal of Borel subsets of X such that P_I is a proper forcing notion. Then for every B in I+ there is C \subseteq B, C in I+ such that E \ (C x Y ) is Borel.

Further results (jointly with W. Chan) show that, under the appropriate large cardinals, these results can be generalized to universally Baire relation E. In the talk we shall survey these results and discuss the extent of the large cardinals needed for these results. **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 December, 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.

**KGRC Set Theory Seminar, ViennaTime:** Tuesday, 30 November, 15:00-16:30 CET

**Speaker:**David Chodounský, TU Wien

**Title:**Big Ramsey degrees of 3-uniform hypergraphs are finite, part 2

**Abstract:**This is a continuation of the KGRC Set Theory seminar talk I gave in June 2021. I will quickly repeat the content of the first talk and focus on things I did not cover then. It is well known that the (universal countable) Rado graph has finite big Ramsey degrees. I.e., given a finite colouring of n-tuples of its vertices there is a copy of the Rado graph such that its n-tuples have at most D(n)-many colours. The proof of this fact uses a theorem of Milliken for trees. I will talk about the extension of this argument which works also for universal structures with higher arities, in particular 3-uniform hypergraphs.

Joint work with M. Balko, J. Hubička, M. Konečný, and L. Vena, see https://arxiv.org/abs/2008.00268

**Information:**This talk will be given via Zoom. Please note that the Zoom meeting ID and passcode change for the Set Theory Seminar starting with this talk (but remain unchanged for other KGRC seminars).

22-28 November

**Toronto Set Theory Seminar****Time:** Friday, 26 November, 1.30-2.30pm Toronto time (19.30-20.30 CET)**Speaker:** David Aspero, University of East Anglia**Title:** (*) and surrounding issues**Abstract:** I aim to present the proof that the ℙmax axiom (*) is implied by Martin’s Maximum++, as well as some further work related to this result and its proof.**Information:** Please see the seminar webpage.

**Barcelona Set Theory Seminar****Time:** Wednesday, 24 November, 16:00-17:30 CET**Speaker:** Chris Scambler, New York University**Title:** Axiomatic Potentialism**Abstract:** I’ll present some axiomatic systems for potentialism in modal logic, and some new results on their consistency strength. In particular I’ll show how a natural system combining height and width potentialism is (close to) bi-interpretable with with second order arithmetic + Pi_1^1 perfect set property, and (hence) equiconsistent with ZFC. **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, 24 November, 14:00-16:00 Israel Time (13:00-15:00 CET)**Speaker:** Moti Gitik **Title:** On definability of elementary embeddings **Abstract:** We will construct a generic extension V^* of V with a normal ultrafilter W over a cardinal \kappa such that:

W\cap V \in V, j_W(V) is definable in V, but j_W|V is not definable, where j_W is the ultrapower embedding of V^* by W.**Information:** Contact Menachem Magidor or Omer Ben-Neria ahead of time for the seminar announcement and zoom link.

**KGRC Set Theory Seminar, ViennaTime:** Tuesday, 23 November, 15:00 CET

**Speaker:**Rahman Mohammadpour, TU Wien

**Title:**Specializing Triples and Weak Embeddability

**Abstract:**A weak embedding between trees is a function that preserves the strict order. A class U of trees is said to be universal for a class C of trees if every tree in C weakly embeds in an element of U. It turns out that the pre-ordered structure induced by weak embeddability on a class C of trees is a plausible tool for the study of the elements of C. One can ask e.g., what is the universality number of a class of trees (the size of the smallest subclass which is universal)? can it be 1? whether a subclass is cofinal? etc. If CH holds, then the class of ℵ1-wide Aronszan trees (trees of height and size ℵ1 without cofinal branches) does not have a maximal tree under weak embeddability (this follows from Kurepa’s works). Todorcevic has proved, among other things, that under MAℵ1, the class of Aronszajn trees has no maximal object. In their joint work on wide Aronszajn trees under MAℵ1, Dzamonja and Shelah introduced the notion of a specializing triple that connects weak embeddings to the specialization of trees. In particular, they reproved Todorcevic’s result using specializing triples. In this talk, we shall focus on a variant of this notion in a general setting and demonstrate the main aspects of it. We shall then discuss some negative results on the universality problem for Aronszajn trees whose height is the successor of a regular cardinal, and hopefully, we shall finish the talk with some open problems.

The results have been obtained in a collaboration with Mirna Dzamonja.

**Information:**This talk will be given in mixed mode, in person as well as via Zoom.

**CUNY Models of Peano Arithmetic Seminar****Time:** Monday, 22 November, 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**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.

15-21 November

**CUNY Set Theory Seminar****Time:** Friday, 19 November, 1pm New York time (19:00 CET)**Speaker:** Corey Switzer, University of Vienna**Title:** Definable Well Orders and Other Beautiful Pathologies**Abstract:** Many sets of reals – well orders of the reals, MAD families, ultrafilters on omega etc – only necessarily exist under the axiom of choice. As such, it has been a perennial topic in descriptive set theory to try to understand when, if ever, such sets can be of low definitional complexity. Large cardinals rule out such the existence of projective well orders, MAD families etc while it’s known that if V=L (or even just ‘every real is constructible’) then there is a Δ12 well order of the reals and Π11 witnesses to many other extremal sets of reals such as MAD families and ultrafilter bases. Recently a lot of work on the border of combinatorial and descriptive set theory has focused on considering what happens to the definitional complexity of such sets in models in which the reals have a richer structure – for instance when CH fails and various inequalities between cardinal characteristics is achieved. In this talk I will present a recent advance in this area by exhibiting a model where the continuum is ℵ2, there is a Δ13 well order of the reals, and a Π11 MAD family, a Π11ultrafilter base for a P-point, and a Π11 maximal independent family, all of size ℵ1. These complexities are best possible for both the type of object and the cardinality hence this may be seen as a maximal model of ‘minimal complexity witnesses’. This is joint work with Jeffrey Bergfalk and Vera Fischer.**Information:** The seminar will take place virtually. Please email Victoria Gitman (vgitman@nylogic.org) for the meeting id.

**Toronto Set Theory Seminar****Time:** Friday, 19 November, 10.30am-12.00pm Toronto time (16.30-18.00 CET)**Speaker:** Mohammad Golshani, IPM**Title:** Ultraproducts and the continuum hypothesis**Abstract:** I will discuss some recent joint projects with Saharon Shelah about the relation between ultraproducts and the continuum hypothesis. In particular, we show that the Keisler’s isomorphism theorem implies the continuum hypothesis, and then prove some consistency results in the absence of the continuum hypothesis.**Information:** Please see the semianr webpage.

**Cross-Alps Logic Seminar****Time:** Friday, 19 November, 16.00-17.00 CET **Speaker:** Asger Törnquist, University of Copenhagen**Title:** tba **Abstract:** tba **Information:** Online on WebEx. Please see the seminar webpage.

**KGRC Logic Colloquium, ViennaTime:** Thursday, 18 November, 15:00 – 15:45 CET

**Speaker:**Radek Honzik, Charles University, Prague

**Title:**Compactness at small uncountable cardinals

**Abstract:**We will discuss various compactness principles such as stationary reflection, the tree property or Rado conjecture at small cardinals (for instance ω2). We will give context and motivation for the principles and discuss and compare the main sources of these principles: large cardinals and consequences of forcing axioms. We will focus on indestructibility of these principles with respect to classes of forcing notions, and give some examples (for instance we show that stationary reflection at ω2 cannot be destroyed by a ccc forcing). Indestructibility is important for investigating connections between compactness and other areas of set theory such as generalized cardinal invariants, and we will mention some applications.

This talk will be given in mixed mode, in person as well as via Zoom.

If you want to attend in person, please be aware of the fact that you will be required to show proof of your COVID-19 “2.5G” status (vaccinated, recovered, PCR tested) upon entry of the buildings, or during sporadic random checks in the seminar rooms. During the lectures we will also pass around an attendance sheet to facilitate contact tracing. (According to the regulations, this form will be kept for 28 days and destroyed thereafter.)

**Information:**This talk will be given in mixed mode, in person as well as via Zoom.

**CMU Model Theory Seminar****Time:** Thursday, 18 November, 11:00-12:30 Pittsburgh time (17:00-18:30 CET)**Speaker:** Samson Leung, CMU**Title:** Axiomatizing AECs and applications, part 2**Abstract:** Let $K$ be an abstract elementary class and $\lambda=LS(K)$. $K$ can be axiomatized by a sentence in $L_{(2^\lambda)^+,\lambda^+}$, allowing game quantification. This extends Kueker’s result which assumes finite character and countable LS(K). It is also a parallel to Shelah-Villaveces result which demands a much higher complexity of junctions but without game quantification. Shelah’s presentation theorem gives $K=PC(T,\Gamma,L(K))$ where $T$ is a first-order theory in an expansion of $L(K)$ and $\Gamma$ is a set of $T$-types. We provide a better bound of $|\Gamma|$ in terms of $I_2(\lambda,K)$. We also give conditions under which the categoricity in two successive cardinals implies the existence of models in the next cardinal. This improves the result of Shelah and as a corollary we extend Shelah-Vasey’s result.**Information:** Please see the seminar webpage

**Caltech Logic Seminar****Time:** Wednesday, 17 November, 12:00 – 1:00pm Pacific time (21:00 CET)**Speaker:** Michal Doucha, Czech Academy of Sciences **Title:** Descriptive complexity of Banach spaces and generic objects**Abstract:** The talk will consist of two parts. In the first one, we introduce a certain natural Polish space of all separable Banach spaces. We compare it with the recent different approach to topologizing the space of separable Banach spaces, by Godefroy and Saint-Raymond.

Our main interest will be in the descriptive complexity of classical Banach spaces with respect to this Polish topology. We show that the separable infinite-dimensional Hilbert space is characterized as the unique Banach space whose isometry class is closed, and also as the unique Banach space whose isomorphism class is FσFσ, where the former employs the Dvoretzky theorem and the latter the solution to the homogeneous subspace problem. For pp in [1,∞)−{2}[1,∞)−{2}, we mention that the isometry class of Lp[0,1]Lp[0,1] is GδGδ-complete and the class of ℓpℓp is Fσ,δFσ,δ-complete.

In the second part, we connect it with the recent study of Fraïssé Banach spaces, initiated by Ferenczi, López-Abad, Mbombo, and Todorčević. We show that a Banach space has a comeager isometry class in its closure if and only if it is the unique limit of a ‘weak Fraïssé class’ of finite-dimensional spaces. While it is open whether there are other Fraïssé Banach spaces besides the Gurariĭ space and the LpLp’s, we show there are more examples of generic Banach spaces in the weaker sense above.

The first part will be based on joint work with M. Doležal, M. Cúth, and O. Kurka; the second is a work in progress jointly with M. Cúth and N. de Rancourt.**Information:** Please see the seminar webpage.

**Hebrew University-Bar Ilan University Set Theory seminar****Time:** Wednesday, 17 November, 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.

**Logic Seminar, Carnegie Mellon University****Time:** Tuesday, 16 November, 3:30 – 4:30pm Eastern Standard Time (21:30 – 22:30 CET) **Speaker:** William Chan, Carnegie Mellon University**Title:** Almost Disjoint Families under Determinacy, part 1**Abstract: **We will investigate some properties of almost disjoint families and the maximal almost disjoint (MAD) family problem on cardinals (regular and singular) within determinacy settings. We will show under suitable assumptions that every almost disjoint family on a cardinal of uncountable cofinality must be wellorderable. This will show under suitable assumptions (which includes the boldface GCH) that there are no MAD families on a regular cardinal kappa so that the family does not strictly inject into kappa. (This answers a question of Muller concerning uncountable MAD families on omega_1 under AD.) We will show under AD that every wellorderable almost disjoint family on a cardinal below Theta of countable cofinality is not maximal. This result may help explain why the Schrittesser-Tornquist or Neeman-Norwood arguments excluding a MAD family on omega has quite a different flavor than the MAD family question for cardinals of uncountable cofinality. We will review the ultrapower representation and measure analysis of Jackson below omega_omega. This will be used to investigate the MAD family question surrounding omega_1, omega_2, and the singular cardinals omega_n for n between 3 and omega. This is joint work with Stephen Jackson and Nam Trang.**Information:** Zoom link https://cmu.zoom.us/j/621951121, meeting ID: 621 951 121

**KGRC Set Theory Seminar, ViennaTime:** Tuesday, 16 November, 15:00 CET

**Speaker:**Radek Honzik, Charles University, Prague

**Title:**Indestructibility of some compactness principles over models of PFA

**Abstract:**Recall that the tree property at a regular cardinal κ says that every κ-tree has a cofinal branch, and the weak Kurepa hypothesis at κ says that there exists a tree of size and height κ which has at least κ+cofinal branches. We will prove that over any transitive model of PFA, the tree property at ω2 cannot be destroyed by the single Cohen forcing Add(ω,1) and the negation of the weak Kurepa hypothesis at ω1cannot be destroyed by a σ-centered forcing.

We will observe that a model-theoretic principle, Guessing model property (GMP), is enough for the preservation results. GMP can be formulated also for larger cardinals. We will give an application of our result by showing that there is a model in which the negation of the weak Kurepa hypothesis holds at ℵω+1.

This talk will be given in mixed mode, in person as well as via Zoom.

If you want to attend in person, please be aware of the fact that you will be required to show proof of your COVID-19 “2.5G” status (vaccinated, recovered, PCR tested) upon entry of the buildings, or during sporadic random checks in the seminar rooms. During the lectures we will also pass around an attendance sheet to facilitate contact tracing. (According to the regulations, this form will be kept for 28 days and destroyed thereafter.)

**Information:**This talk will be given in mixed mode, in person as well as via Zoom.

**Konstanz Logik Kolloquium****Time:** Monday, 15 November, 15:15 CET)**Speaker:** Philip Welch, University of Bristol**Title:** Quasi-induction**Abstract:** Induction, whether appearing in the role of a proof by induction (“if 1 has property P , and (n has property P implies that n + 1 has property P , therefore all n have property P”) or as inductive definitions (e.g., that of the set of well formed formulae in a formal language) is a principle tool in mathematics. Whilst many inductive definitions involve only a passage through finite stages, and are often presented as definitions by recursion, many also involve transfinitely many stages beforecompletion. (Wegiveanexampleinvolving“infinitechess”.) Thetheory of such inductive definitions over general structures was magisterially laid down by Moschovakis in the 1970’s (in“Elementary Induction on Abstract Structures”).

We consider here a broader class of“quasi-inductive”processes that alter the rules of induction at limit stages of their production to a“liminf”rule rather than simple “union”. We give some examples that have arisen in the philosophical theory of truth, from computer science, and from set theory. Such q.i.-processes extend inductive ones and also result in a rich theory to which pleasingly many of the results from the Moschovakian theory can be extended.**Information:** See the seminar wegpage. Please contact Michele Serra in advance for the login.

8-14 November

**CUNY Set Theory Seminar****Time:** Friday, 12 November, 1pm New York time (19:00 CET)**Speaker:** Tom Benhamou, Tel Aviv University**Title:** Intermediate Prikry-type models, quotients, and the Galvin property II**Abstract:** We classify intermediate models of Magidor-Radin generic extensions. It turns out that similar to Gitik Kanovei and Koepke’s result, every such intermediate model is of the form V[C] where C is a subsequence of the generic club added by the forcing. The proof uses the Galvin property for normal filters to prove that quotients of some Prikry-type forcings are κ+-c.c. in the generic extension and therefore do not add fresh subsets to κ+. If time permits, we will also present results regarding intermediate models of the Tree-Prikry forcing.**Information:** The seminar will take place virtually. Please email Victoria Gitman (vgitman@nylogic.org) for the meeting id.

**Toronto Set Theory Seminar****Time:** Friday, 12 November, 10.30am-12.00pm Toronto time (16.30-18.00 CET)**Speaker:** Gianluca Paolini, University of Toronto**Title:** Torsion-Free Abelian Groups are Borel Complete**Abstract:** I will talk about my recent result joint with S. Shelah establishing that the Borel space of torsion-free Abelian groups with domain ω is Borel complete, i.e., the isomorphism relation on this Borel space is as complicated as possible, as an isomorphism relation. This solves a long-standing open problem in descriptive set theory, which dates back to the seminal paper on Borel reducibility of Friedman and Stanley from 1989. Time permitting, I will also talk about some recent results (also joint with S. Shelah) on the existence of uncountable Hopfian and co-Hopfian abelian groups and on anti-classification results for the countable co-Hopfian abelian and 2-nilpotent groups. In particular, we will see that the countable co-Hopfian groups are complete co-analytic in the Borel space of 2-nilpotent groups with domain ω, this solves an open question of Thomas, who posed the question for the space of all groups with domain ω.**Information:** Please see the semianr webpage.

**Cross-Alps Logic Seminar****Time:** Friday, 12 November, 16.00-17.00 CET **Speaker:** Sandra Müller, TU Wien**Title:** Large Cardinals and Determinacy**Abstract:** 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. I will survey some recent results in this flourishing area. This, in particular, includes results on connecting the determinacy of longer games to canonical inner models with many Woodin cardinals, a new lower bound for a combinatorial statement about infinite trees, as well as an application of determinacy answering a question in general topology.**Information:** Online on WebEx. Please see the seminar webpage.

**CMU Model Theory Seminar****Time:** Thursday, 11 November, 11:00-12:30 Pittsburgh time (17:00-18:30 CET)**Speaker:** Samson Leung, CMU**Title:** Axiomatizing AECs and applications, part 1**Abstract:** Let $K$ be an abstract elementary class and $\lambda=LS(K)$. $K$ can be axiomatized by a sentence in $L_{(2^\lambda)^+,\lambda^+}$, allowing game quantification. This extends Kueker’s result which assumes finite character and countable LS(K). It is also a parallel to Shelah-Villaveces result which demands a much higher complexity of junctions but without game quantification. Shelah’s presentation theorem gives $K=PC(T,\Gamma,L(K))$ where $T$ is a first-order theory in an expansion of $L(K)$ and $\Gamma$ is a set of $T$-types. We provide a better bound of $|\Gamma|$ in terms of $I_2(\lambda,K)$. We also give conditions under which the categoricity in two successive cardinals implies the existence of models in the next cardinal. This improves the result of Shelah and as a corollary we extend Shelah-Vasey’s result.**Information:** Please see the seminar webpage

**Caltech Logic Seminar****Time:** Wednesday, 10 November, 12:00 – 1:00pm Pacific time (21:00 CET)**Speaker:** Gianluca Paolini, University of Turin**Title:** Torsion-free abelian groups are Borel complete**Abstract:** I will talk about my recent result joint with S. Shelah establishing that the Borel space of torsion-free abelian groups with domain ωω is Borel complete, i.e., that the isomorphism relation on this Borel space is as complicated as possible, as an isomorphism relation. This solves a long-standing open problem in descriptive set theory, which dates back to the seminal paper on Borel reducibility of Friedman and Stanley from 1989. Time permitting, I will also talk about some recent results (also joint with S. Shelah) on the existence of uncountable Hopfian and co-Hopfian abelian groups and on anti-classification results for the countable co-Hopfian abelian and 22-nilpotent groups. In particular, we will see that the countable co-Hopfian groups are complete co-analytic in the Borel space of 22-nilpotent groups with domain ωω. This solves an open question of Thomas, who had posed the question for the space of all groups with domain ω. **Information:** Please see the seminar webpage.

**Barcelona Set Theory Seminar****Time:** Wednesday, 10 November, 16:00-17:30 CET**Speaker:** Yaroslav D. Sergeyev, University of Calabria**Title:** Some paradoxes of Infinity revisited**Abstract:** In this talk, some classical paradoxes of infinity such as Galileo’s paradox, Hilbert’s paradox of the Grand Hotel, Thomson’s lamp paradox, and the rectangle paradox of Torricelli are considered. In addition, three paradoxes regarding divergent series and a new paradox dealing with multiplication of elements of an infinite set are also described. It is shown that the surprising counting system of an Amazonian tribe, Piraha, working with only three numerals (one, two, many) can help us to change our perception of these paradoxes. A recently introduced methodology allowing one to work with finite, infinite, and infinitesimal numbers in a unique computational framework not only theoretically but also numerically is briefly described. This methodology is actively used nowadays in numerous applications in pure and applied mathematics and computer science as well as in teaching. It is shown in the talk that this methodology also allows one to consider the paradoxes listed above in a new constructive light. More information on this topic is available at https://www.theinfinitycomputer.com. **Information:** Online. If you wish to attend, please send an email to bagaria@ub.edu asking for the link.

**Leeds Models and Sets Seminar****Time:** Wednesday, 10 November, 13:45-15:00 UK time (14:45-16:00 CET)**Speaker: **Victoria Gitman, CUNY Graduate Center**Title:** Set theory without powerset**Abstract:** Many natural set-theoretic structures satisfy the basic axioms of set theory, but not the powerset axiom. These include the collections $H_{\kappa^+}$ of sets whose transitive closure has size at most $\kappa$, forcing extensions of models of ${\rm ZFC}$ by pretame (but not tame) forcing, and first-order models that are morally equivalent to models of the second-order Kelley-Morse set theory (with class choice). It turns out that a reasonable set theory in the absence of the powerset axiom is not simply ${\rm ZFC}$ with the powerset axiom removed. Without the powerset axiom, the Replacement scheme is not equivalent to the Collection scheme, and the various forms of the Axiom of Choice are not equivalent. In this talk, I will give an overview of the properties of a robust set theory without powerset, ${\rm ZFC}^-$, whose axioms are ${\rm ZFC}$ without the powerset axiom, with the Collection scheme instead of the Replacement scheme and the Well-Ordering Principle instead of the Axiom of Choice. While a great deal of standard set theory can be carried out in ${\rm ZFC}^-$, for instance, forcing works mostly as it does in ${\rm ZFC}$, there are several important properties that are known to fail and some which we still don’t know whether they hold. For example, the Intermediate Model Theorem fails for ${\rm ZFC}^-$, and so does ground model definability, and it is not known whether ${\rm HOD}$ is definable. I will also discuss a strengthening of ${\rm ZFC}^-$ obtained by adding the Dependent Choice Scheme, and some rather strange ${\rm ZFC}^-$-models.**Information:** Please see the seminar webpage.

**Hebrew University-Bar Ilan University Set Theory seminar****Time:** Wednesday, 10 November, 14:00-16:00 Israel Time (13:00-15:00 CET)**Speaker:** Yair Hayut**Title:** Stationary Reflection and the Successors of Singular cardinals – continued **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:** Please check on the seminar webpage to see if the seminar will take place. Contact Menachem Magidor, Asaf Rinot or Omer Ben-Neria ahead of time for the zoom link.

**KGRC Set Theory Seminar, ViennaTime:** Tuesday, 9 November , 15:00 CET

**Speaker:**Corey Switzer, KGRC

**Title:**The Special Tree Number

**Abstract:**A tree $T$ of height $\omega_1$ with no uncountable branch is {\em special} if there is a function $f:T \to \omega$ which is injective on chains. It’s well known that under $\mathrm{MA} + \neg \CH$ every tree of height $\omega_1$ with no uncountable branch of size less than the continuum is special, while in $\mathrm{ZFC}$ one can construct a non-special tree of height $\omega_1$ with no uncountable branch. At the same time there may be a Souslin tree while the continuum is as large as you like thus providing a model with a small non-special tree. These facts together suggest a new cardinal characteristic, the special tree number, denote $\mathfrak{st}$: the least size of a tree of height $\omega_1$ with no uncountable branch which is not special. By what was observed above, $\mathrm{MA} + \neg \CH$ implies that $\mathfrak{st} = 2^{\aleph_0}$ while it is consistent that $mathfrak{st} < 2^{\aleph_0}$ with the latter arbitrarily large. In this talk we will introduce the basic properties of $\mathfrak{st}$ and prove in particular that it is consistent on the one hand that $\mathfrak{st}$ is $\aleph_1$ while essentially all well-studied cardinal characteristics are arbitrarily large and on the other hand it is consistent that for any regular $\kappa$ we have $\mathfrak{a} = {\rm non}(\mathcal M) = \aleph_1 < \mathfrak{st} = {\rm cov}(\mathcal M) = 2^{\aleph_0} = \kappa$. In other words, $\mathfrak{st}$ is independent of the lefthand side of Cicho\'{n}’s diagram, $\mathfrak{p}$ and $\mathfrak{a}$. The latter model involves a careful analysis of reals added by the standard ccc forcing to specialize trees, which may be of independent interest. This is a relatively new investigation and there are many open questions I

hope to discuss as well, time permitting.

**Information:**This talk will be given in mixed mode, in person as well as via Zoom.

1-7 November

**CUNY Set Theory Seminar****Time:** Friday, 5 November, 1pm New York time (18:00 CET)**Speaker:** Tom Benhamou, Tel Aviv University**Title:** Intermediate Prikry-type models, quotients, and the Galvin property**Abstract:** We classify intermediate models of Magidor-Radin generic extensions. It turns out that similar to Gitik Kanovei and Koepke’s result, every such intermediate model is of the form V[C] where C is a subsequence of the generic club added by the forcing. The proof uses the Galvin property for normal filters to prove that quotients of some Prikry-type forcings are κ+-c.c. in the generic extension and therefore do not add fresh subsets to κ+. If time permits, we will also present results regarding intermediate models of the Tree-Prikry forcing.**Information:** The seminar will take place virtually. Please email Victoria Gitman (vgitman@nylogic.org) for the meeting id.

**Toronto Set Theory Seminar****Time:** Friday, 5 November, 12.30pm Toronto time (17:30 CET)**Speaker:** Philip Welch, University of Bristol**Title:** The universe constructed from a set (or class) of regular cardinals**Abstract:** The universe constructed from a set (or class) of regular cardinals**Abstract:** We continue some work on L[Card] (the universe constructed from

the predicate for the cardinals) to look at L[Reg] where Reg is the class of

uncountable regular cardinals. The latter is also a model of a rich combinatorial

structure being, as it turns out, a Magidor iteration of Prikry forcings (using

recent work of Ben-Neria). But it is limited in size, in fact is a rather ‘thin’

model. We show, letting O^s = O^sword be the least iterable structure with a

measure which concentrates on measurable cardinals:

Theorem (ZFC):

(a) Let S be a set, or proper class, of regular cardinals, then O^s is not an

element of L[S].

(b) This is best possible, in that no smaller mouse M can be substituted for O^s.

(c) L[S] is a Magidor generic extension of its core model and hence is a model

of: GCH, Square’s, Diamonds, Morasses etc., and has Ramsey cardinals, but

no measurable cardinals. **Information:** No webpage available. Email Ivan Ongay Valverde to receive the seminar announcements and for the zoom link.

**Turin-Udine logic seminar****Time:** Friday, 5 November, 16:00-18:00 CET**Speaker:** A. Zucker, University of California San Diego**Title:** Big Ramsey degrees in binary free amalgamation classes**Abstract:** In structural Ramsey theory, one considers a “small” structure A, a “medium” structure B, a “large” structure C and a number r, then considers the following combinatorial question: given a coloring of the copies of A inside C in r colors, can we find a copy of B inside C all of whose copies of A receive just one color? For example, when C is the rational linear order and A and B are finite linear orders, then this follows from the finite version of the classical Ramsey theorem. More generally, when C is the Fraisse limit of a free amalgamation class in a finite relational language, then for any finite A and B in the given class, this can be done by a celebrated theorem of Nesetril and Rodl. Things get much more interesting when both B and C are infinite. For example, when B and C are the rational linear order and A is the two-element linear order, a pathological coloring due to Sierpinski shows that this cannot be done. However, if we weaken our demands and only ask for a copy of B inside C whose copies of A receive “few” colors, rather than just one color, we can succeed. For the two-element linear order, we can get down to two colors. For the three-element order, 16 colors. This number of colors is called the big Ramsey degree of a finite structure in a Fraisse class. Recently, building on groundbreaking work of Dobrinen, I proved a generalization of the Nesetril-Rodl theorem to binary free amalgamation classes defined by a finite forbidden set of irreducible structures (for instance, the class of triangle-free graphs), showing that every structure in every such class has a finite big Ramsey degree. My work only bounded the big Ramsey degrees, and left open what the exact values were. In recent joint work with Balko, Chodounsky, Dobrinen, Hubicka, Konecny, and Vena, we characterize the exact big Ramsey degree of every structure in every binary free amalgamation class defined by a finite forbidden set.**Information:** Online on WebEx. Please see the seminar webpage.

**Ghent-Leeds Virtual Logic SeminarTime:** Wednesday, 3 November, 16:00 CET

**Speaker:**Mohammad Golshani, Institute for Research in Fundamental Sciences, IPM

**Title:**Ultraproducts and the continuum hypothesis

**Abstract:**In this talk, I will review some recent joint work with Shelah on the connection between ultraproducts and the continuum hypothesis.

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

**Barcelona Set Theory Seminar****Time:** Wednesday, 3 November, 16:00-17:30 CET**Speaker:** Sean Cox, Virginia Commonwealth University**Title:** Homological algebra, elementary submodels, and stationary logic**Abstract:** The talk will focus on the speaker’s use of set-theoretic elementary submodel

techniques to 1) show that Salce’s Problem (about completeness of cotorsion pairs) is

independent of ZFC, and 2) give a consistently positive solution to some precovering

problems in Gorenstein Homological Algebra.**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, 3 November, 14:00-16:00 Israel Time (13:00-15:00 CET)**Speaker:** Yair Hayut**Title:** Stationary reflection at successors of singular cardinals – continuation**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:** Please check on the seminar webpage if the seminar will take place. Contact Menachem Magidor, Asaf Rinot or Omer Ben-Neria ahead of time for the zoom link.

25-31 October

**CUNY Set Theory Seminar****Time:** Friday, 29 October, 2pm New York time (20:00 CET)**Speaker:** Kameryn Williams, Sam Houston State University**Title:** Potentialism about 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:** The seminar will take place virtually. Please email Victoria Gitman (vgitman@nylogic.org) for the meeting id.

**Toronto Set Theory Seminar****Time:** Friday, 29 October, 1.30-3pm Toronto time (19:30-21:00 CET)**Speaker:** Stefan Hoffelner, University of Münster **Title:** Forcing and the Separation, the Reduction and the Uniformization-property.**Abstract:** tba**Information:** No webpage available. Email Ivan Ongay Valverde to receive the seminar announcements and for the zoom link.

**Caltech Logic Seminar****Time:** Wednesday, 27 October, 12:00 – 1:00pm Pacific time (21:00 CET)**Speaker:** Linda Westrick, Penn State**Title:** Of the principles just slightly weaker than ATR, the most well-known are the theories of hyperarithmetic analysis (THA). By definition, such principles hold in HYP. Motivated by the question of whether the Borel dual Ramsey theorem is a THA, we consider several theorems involving Borel sets and ask whether they hold in HYP. To make sense of Borel sets without ATR, we formalize the theorems using completely determined Borel sets. We characterize the completely determined Borel subsets of HYP as precisely the sets of reals which are Δ11 in Lω1ck. Using this, we show that in HYP, Borel sets behave quite differently than in reality. For example, in HYP, the Borel dual Ramsey theorem fails, every nn-regular Borel acyclic graph has a Borel 22-coloring, and the prisoners have a Borel winning strategy in the infinite prisoner hat game. Thus the negations of these statements are not THA. Joint work with Henry Towsner and Rose Weisshaar.**Abstract:** Borel combinatorics fail in HYP**Information:** Please see the seminar webpage.

**Barcelona Set Theory Seminar****Time:** Wednesday, 27 October, 16:00-17:30 CET**Speaker:** Martina Iannella, Università degli Studi di Udine**Title:** Convex embeddability on linear/circular orders and connections to knot theory**Abstract:** Given two countable linear orders L and M, we say that L is convex

embeddable in M iff L is isomorphic to a convex set in M. We first show that, in contrast

to the usual embeddability between linear orders, convex embeddability is

combinatorially complicated. Then we study the complexity of the equivalence relation

induced by convex embeddability, proving that it “is not much more complicated” than

the isomorphism relation between linear orders. We use convex embeddability to look

at the complexity of the sub-arc relation among proper arcs, and we also consider an

analogue of convex embeddability on countable circular orders to obtain similar results

for knots. Finally, we give a generalization of convex embeddability on countable linear

orders and circular orders to provide a better overview of such embeddability relations.

This is joint work with Vadim Kulikov, Alberto Marcone, and Luca Motto Ros.**Information:** Online. If you wish to attend, please send an email to bagaria@ub.edu asking for the link.

**Leeds Models and Sets Seminar****Time:** Wednesday, 27 October, 16:00-17:00 UK time (17:00-18:00 CET) **note the time****Speaker: **Dilip Raghavan, National University of Singapore**Title:** Tba**Abstract:** Tba**Information:** Please see the seminar webpage.

**Hebrew University-Bar Ilan University Set Theory seminar****Time:** Wednesday, 27 October, 14:00-16:00 Israel Time (13:00-15:00 CET)**Speaker:** Yair Hayut**Title:** Stationary reflection at successors of singulars**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:** Please check on the seminar webpage if the seminar will take place. Contact Menachem Magidor, Asaf Rinot or Omer Ben-Neria ahead of time for the zoom link.

18-24 October

**KGRC Research Seminar, ViennaTime:** Thursday, 21 October , 15:00 – 15:45 CET

**Speaker:**Chris Lambie-Hanson, Czech Academy of Sciences

**Title:**Variations on a theorem of Silver

**Abstract:**Shortly after the advent of forcing in the 1960s, Easton proved that, modulo some trivial constraints concerning monotonicity and cofinality, the axioms of set theory place no restrictions on the behavior of exponentiation at regular cardinals. In a surprising turn of events, this turned out not to be the case for singular cardinals, and the last half-century has seen a procession of deep results uncovering nontrivial constraints on exponentiation at singular cardinals. One of the first of these results was Silver’s theorem, which in essence states that if λ is a singular cardinal of uncountable cofinality and there are “many” singular cardinals κ<λ such that 2κ=κ+, then it must also be the case that 2λ=λ+. In particular, if the Singular Cardinals Hypothesis fails, then it must fail first at a singular cardinal of countable cofinality. We will discuss this seminal theorem and a number of variations thereon, and we will end by sketching a proof of a version of Silver’s theorem pertaining to certain generalized cardinal characteristics.

**Information:**This talk will be given in mixed mode, in person as well as via Zoom.

**Caltech Logic Seminar****Time:** Wednesday, 20 October, 12:00 – 1:00pm Pacific time (21:00 CET)**Speaker:** Juan P. Aguilera, Ghent University, TU Wien**Title:** The axiom of real determinacy in admissible sets**Abstract:** The axiom of real determinacy (ADR) asserts the determinacy of every infinite two-player, perfect-information game with moves in the set of real numbers. By a theorem of Woodin, ZF+ADR is consistent if and only if ZFC is consistent together with the existence of a cardinal λλ which is a limit of Woodin cardinals and <λ<λ-strong cardinals.

In this talk, we explore the strength of ADR over the theory KP+“R exists”and observe that it is much weaker. Indeed, the theory R+“R exists” is weaker than ZFC+“there are ω2 Woodin cardinals”. This is a consequence of the following theorem: over ZFC, the existence of a transitive model of KP+ADR containing the set of all real numbers is equivalent to the determinacy of all open games of length ω3.**Information:** Please see the seminar webpage.

**Barcelona Set Theory Seminar****Time:** Wednesday, 20 October, 16:00-17:30 CET**Speaker:** Philip Welch, University of Bristol**Title:** The universe constructed from a set (or class) of regular cardinals**Abstract:** We continue some work on L[Card] (the universe constructed from

the predicate for the cardinals) to look at L[Reg] where Reg is the class of

uncountable regular cardinals. The latter is also a model of a rich combinatorial

structure being, as it turns out, a Magidor iteration of Prikry forcings (using

recent work of Ben-Neria). But it is limited in size, in fact is a rather ‘thin’

model. We show, letting O^s = O^sword be the least iterable structure with a

measure which concentrates on measurable cardinals:

Theorem (ZFC):

(a) Let S be a set, or proper class, of regular cardinals, then O^s is not an

element of L[S].

(b) This is best possible, in that no smaller mouse M can be substituted for O^s.

(c) L[S] is a Magidor generic extension of its core model and hence is a model

of: GCH, Square’s, Diamonds, Morasses etc., and has Ramsey cardinals, but

no measurable cardinals. **Information:** Online. If you wish to attend, please send an email to bagaria@ub.edu asking for the link.

**Leeds Models and Sets Seminar****Time:** Wednesday, 20 October, 13:45 UK time (14:45 CET)**Speaker:** Mirna Džamonja, CNRS – Université de Paris**Title:** On the universality problem for aleph_2-Aronszajn and wide aleph_2-Aronszajn trees.**Abstract:** We report on a joint work in progress with Rahman Mohammadpour in which we study the problem of the possible existence of a universal tree under weak embeddings in the classes of $\aleph_2$-Aronszajn and wide $\aleph_2$-Aronszajn trees. This problem is more complex than previously thought, in particular it seems not to be resolved under ShFA + CH using the technology of weakly Lipshitz trees. We show that under CH, for a given $\aleph_2$-Aronszajn tree T without a weak ascent path, there is an $\aleph_2$-cc countably closed forcing forcing which specialises T and adds an $\aleph_2$-Aronszajn tree which does not embed into T. One cannot however apply the ShFA to this forcing.

Further, we construct a model à la Laver-Shelah in which there are $\aleph_2$-Aronszajn trees, but none is universal. Work in progress is to obtain an analogue for universal wide $\aleph_2$-Aronszajn trees. We also comment on some negative ZFC results in the case that the embeddings are assumed to have a strong preservation property.**Information:** Please see the seminar webpage.

**Singapore Logic Seminar****Time:** Wednesday, 20 October, 16:00-17:00 Singapore time (10:00-11:00 CET)**Speaker:** Yang Yue, NUS**Title:** A recursive coloring without Delta-3 solutions for Hindman’s Theorem**Abstract:** This is a follow up of Liao Yuke’s talk last week. I will explain in detail his result which says that there exists a recursive coloring f: N -> {0,1} such that for all infinite subset X of N, if FS(X) is homogeneous for f, then X is not recursive in 0”. Here FS(X) is the set of all finite sums of distinct elements of X. Liao Yuke’s result improved Blass, Hirst and Simpson’s theorem in 1987 (from no 0′ recursive solutions to no 0” recursive ones).**Information:** See the seminar webpage.

**Logic Seminar, Carnegie Mellon University****Time:** Tuesday, 19 October, 3:30 – 4:30pm Eastern Standard Time (21:30 – 22:30 CET) **Speaker:** Andreas Blass, University of Michigan**Title:** Tukey Ordering and Forcing Preservation of Ultrafilters, part 1**Abstract: **I plan to describe two species of ultrafilters on the set of natural numbers and to speculate about a connection between them. For both species, the central question is whether ZFC can prove the existence of such ultrafilters.

One species is defined in terms of the Tukey ordering of directed sets, but it also admits a more combinatorial definition. The other species is defined in terms of preservation by forcing, but it also admits a combinatorial definition. The two combinatorial definitions, though different, have very similar “flavor”, and that leads to my speculations.

I’ll present the original definitions as well as the combinatorial equivalents, and then I’ll discuss attempts to combine the key properties of the two species.**Information:** For connection details, please see the seminar webpage.

**KGRC Set Theory Seminar, ViennaTime:** Tuesday, 19 October , 15:00-16:30 CET

**Speaker:**Chris Lambie-Hanson, Czech Academy of Sciences

**Title:**Strongly unbounded subadditive colorings

**Abstract:**Given infinite cardinals κ and θ, functions of the form c:[κ]2→θ exhibiting certain unboundedness properties provide a strong counterexample to the generalization of Ramsey’s theorem to κ and have seen a wide variety of applications. In this talk, we will discuss the existence of such strongly unbounded colorings, focusing in particular on colorings with subadditivity properties. We will then present some applications to general topology. In particular, building on work of Chen-Mertens and Szeptycki, we will prove that the failure of the Singular Cardinals Hypothesis implies the existence of a Fréchet, α1-space whose Gδ-modification has large tightness. This is joint work with Assaf Rinot.

**Information:**This talk will be given in mixed mode, in person as well as via Zoom.

11-17 October

**CUNY Set Theory Seminar****Time:** Friday, 15 October, 2pm New York time (20:00 CET)**Speaker:** Yuxin Zhou, University of Florida**Title:** Color isosceles triangles countably in R2 and but not in R3**Abstract:** Let n>1 be a natural number, let Γn be the hypergraph of isosceles triangles in Rn. Under the axiom of choice, the existence of a countable coloring for Γn is true for every n. Without the axiom of choice, the coloring problems will be hard to answer. We often expect the case that the countable chromatic number of one hypergraph doesn’t imply the one for another. With an inaccessible cardinal, there is a model of ZF+DC in which Γ2 has countable chromatic number while Γ3 has uncountable chromatic number. This result is obtained by a balanced forcing over the symmetric Solovay model.**Information:** The seminar will take place virtually. Please email Victoria Gitman (vgitman@nylogic.org) for the meeting id.

**Caltech Logic Seminar****Time:** Wednesday, 13 October, 12:00 – 1:00pm Pacific time (21:00 CET)**Speaker:** Theodore Slaman, UC Berkeley**Title:** Algorithmic information theory, effective descriptive set theory and geometric measure theory**Abstract:** We will describe how the perspectives of Recursion Theory and Set Theory suggest lines of investigation into Geometric Measure Theory. We will discuss the extent of capacitability for Hausdorff dimension and the question of existence of sets of strong gauge dimension, which is a property generalizing that of strong measure zero.**Information:** Please see the seminar webpage.

**Hebrew University-Bar Ilan University Set Theory seminar****Time:** Wednesday, 13 October 14:00-16:00 Israel Time (13:00-15:00 CET)**Speaker:** Moshe Teutch**Title:** Structurability of Countable Borel Equivalence Relations**Abstract:** countable relational model M: The relations must define Borel subsets of a power of X, and the restriction to each E-class must give a model isomorphic to M. The structurability of E by various models can shed light on its properties as a CBER. One interesting question is, Which models structure every CBER? I’ll discuss an adaptation by Marks of a construction of Ackerman, Freer, and Patel which finds a large class of such models. I’ll also show some applications of their construction to this question. Time permitting I’ll discuss some related results and techniques from my thesis. **Information:** Please check on the seminar webpage if the seminar will take place. Contact Menachem Magidor, Asaf Rinot or Omer Ben-Neria ahead of time for the zoom link.

**KGRC Set Theory Seminar, ViennaTime:** Tuesday, 12 October , 15:00-16:30 CET

**Speaker:**Iian Smythe, University of Michigan

**Title:**Parametrizing the Ramsey theory of vector spaces

**Abstract:**In the late 90’s, Gowers proved a Ramsey-theoretic dichotomy for subspaces

of infinite-dimensional Banach spaces. The combinatorial essence of this

result was later extracted by Rosendal in the setting of discrete vector

spaces. Both dichotomies say, roughly, that given an analytic partition of

the set of infinite block sequences of vectors, there is an

infinite-dimensional subspace with a wealth of block sequences entirely

contained in, or disjoint from, one piece of the partition. We will

describe a new “parametrized” form of Rosendal’s dichotomy: Given an

analytic family of partitions indexed by the reals, there is a single

subspace which witnesses Rosendal’s dichotomy for uncountably many of the

partitions, simultaneously. An integral part of our proof is the

preservation of certain analogues of selective ultrafilters, by Sacks

forcing. We will also discuss applications to families of linear

transformations.

**Information:**Talk via zoom.

**Paris-Lyon Séminaire de Logique****Time:** Monday, 11 October,15:15 CET**Speaker:** Mirna Džamonja, IRIF (CNRS & Université de Paris)**Title:** Paracompactness of the box topology**Abstract:** The box topology on a product of topological spaces is given

by declaring as basic open sets any products of open sets of the

composants of the product.

It is a natural definition, but does not preserve many topological

properties, notably it miserably fails to preserve compactness. However,

a weakening of compactness, called paracompactness and introduced by

Jean Dieudonné in 1944 for its nice behaviour in analysis, is sometimes

preserved by box products. Investigating this for spaces obtained by the

product of countably many factors or aleph_1 many factors with either

full boxes or boxes of countable size, was a classical topic in

set-theoretic topology of the 1980s or so, with important works by Mary

Ellen Rudin, Kenneth Kunen, Eric van Douwen and others.

In our work in progress we are, rather, interested in an unexplored

territory of products with many coordinates. In particular, we consider

the following question:

Suppose that kappa is a cardinal such that for every lambda >= kappa,

the box product {}^{<\kappa} 2^\lambda is paracompact.

Is kappa a large cardinal ?

(the notation means that the topology on 2^lambda is generated by

boxes of size < kappa)

We present some partial results and the difficulties with the

consideration of the case kappa singular. This is somewhat connected

with the recent works on descriptive set theory of the space 2^kappa for

kappa singular.

This is joint work with David Buhagiar, University of Malta.**Information:** Join via the link on the seminar webpage

4-10 October

**CUNY Set Theory Seminar****Time:** Friday, 8 October, 2pm New York time (20:00 CET)**Speaker:** Brent Cody, Virginia Commonwealth University**Title:** Higher derived topologies**Abstract:** By beginning with the order topology on an ordinal δ, and iteratively declaring more and more derived sets to be open, Bagaria defined the derived topologies τξ on δ, where ξ is an ordinal. He showed that the non-isolated points in the space (δ,τξ) can be characterized using a strong form of iterated simultaneous stationary reflection called ξ-s-reflection, which is deeply connected to certain transfinite indescribability properties. However, Bagaria’s definitions break for ξ≥δ because, under his definitions, the δ-th derived topology τδ is discrete and no ordinal α can be α+1-s-stationary. We will discuss some new work in which we use certain diagonal versions of Bagaria’s definitions to extend his results. For example, we introduce the notions of diagonal Cantor derivative and use it to obtain a sequence of derived topologies on a regular δ that is strictly longer than that of Bagaria’s, under certain hypotheses.**Information:** The seminar will take place virtually. Please email Victoria Gitman (vgitman@nylogic.org) for the meeting id.

**CMU Model Theory Seminar****Time:** Thursday, 7 October, 11:00 Eastern Daylight Time (17:00 CET)**Speaker:** Samson Leung, CMU**Title:** Hanf number of the first stability cardinal in AECs, Part II**Abstract:** Adapting Shelah’s example, we show that the need for joint embedding property for two type-counting lemmas by Boney is independent of ZFC. Combining the example with coding of the power set, we deduce that $\beth_{2^{LS(K)}}$ is the lower bound to the Hanf number of order property length and of the first stability cardinal. Our example satisfies joint embedding property, no maximal models, $(<\aleph_0)$-tameness but not amalgamation property.**Information:** Please see the seminar webpage

**Caltech Logic Seminar****Time:** Wednesday, 6 October, 12:00 – 1:00pm Pacific time (21:00 CET)**Speaker:** Forte Shinko, Caltech**Title:** Realizations of equivalence relations and subshifts**Abstract:** Every continuous action of a countable group on a Polish space induces a Borel equivalence relation. We are interested in the problem of realizing (i.e. finding a Borel isomorphic copy of) these equivalence relations as continuous actions on compact spaces. We provide a number of positive results on this problem, and we investigate the connection to subshifts. Joint with Joshua Frisch, Alexander Kechris and Zoltán Vidnyánszky.**Information:** Please see the seminar webpage.

**Logic Seminar, Carnegie Mellon University****Time:** Tuesday, 4 October, 3:30 – 4:30pm Eastern Standard Time (21:30 – 22:30 CET) **Speaker:** Jindra Zapletal, University of Florida **Title:** Set theory of algebraic hypergraphs**Abstract: **I explain the main aim of the geometric set theory program: obtaining a careful calibration of Sigma two one sentences (typically, consequences of the axiom of choice) in choiceless set theory. As a specific class of such sentences, I consider the countable chromatic number of various (sigma-)algebraic hypergraphs on Euclidean spaces. A recent result deals with the graph G_n connecting points of rational distance in R^n: for every n>0, it is consistent with ZF+DC that the chromatic number of G_n is countable while that of G_{n+1} is not.**Information:** Zoom link https://cmu.zoom.us/j/621951121, meeting ID: 621 951 121

27 September-3 October

**NY Set Theory Seminar****Time:** Friday, 1 October, 11:30am New York time (17:30 CET)**Speaker:** Matteo Viale, University of Torino**Title:** Absolute model companionship, forcibility, and the continuum problem: Part II**Abstract:** See the seminar webpage. **Information:** The seminar will take place virtually. Please email Victoria Gitman (vgitman@nylogic.org) for the meeting id.

**CMU Model Theory Seminar****Time:** Thursday, 30 September, 11:00 Eastern Daylight Time (17:00 CET)**Speaker:** Samson Leung, CMU**Title:** Hanf number of the first stability cardinal in AECs, Part I**Abstract:** Adapting Shelah’s example, we show that the need for joint embedding property for two type-counting lemmas by Boney is independent of ZFC. Combining the example with coding of the power set, we deduce that $\beth_{2^{LS(K)}}$ is the lower bound to the Hanf number of order property length and of the first stability cardinal. Our example satisfies joint embedding property, no maximal models, $(<\aleph_0)$-tameness but not amalgamation property.**Information:** Please see the seminar webpage

**Caltech Logic Seminar****Time:** Wednesday, 29 September, 12:00 – 1:00pm Pacific time (21:00 CET)**Speaker:** Samuel Coskey, Boise State**Title:** Jumps in the Borel complexity hierarchy**Abstract:** There are several well-studied “jumps” on the class of Borel equivalence relations under Borel reducibility, namely, the Friedman-Stanley jump and the family of Louveau jumps. 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 recent developments (due to Shani and Allison) on Bernoulli jumps.**Information:** Please see the seminar webpage.

**Barcelona Set Theory Seminar****Time:** Wednesday, 29 September, 16:00-17:30 CET**Speaker:** Zhixing You, Chinese Academy of Sciences and Universitat de Barcelona**Title:** Hierarchies of delta-strongly but not full strongly compact cardinals**Abstract:** We show that if there is no measurable limit of supercompact cardinals, then (assuming the existence of a proper class of supercompact cardinals) for any class K of supercompact cardinals and any increasing sequence delta_k, k in K, of measurable cardinals, such that delta_k < k and any limit pointof these delta_k is not in K, we can force that for any k in K, k is exactly delta_k-strongly compact, and there are no strongly compact cardinals. This result gives an affirmative answer to a question of Bagaria and Magidor, and negative answers to some related questions of Boney-Unger and Brooke-Taylor. This is joint work with Jiachen Yuan.**Information:** Online. If you wish to attend, please send an email to bagaria@ub.edu asking for the link.

**Logic Seminar, Carnegie Mellon University****Time:** Tuesday, 28 September, 3:30 – 4:30pm Eastern Standard Time (21:30 – 22:30 CET) **Speaker:** Thomas Gilton, University of Pittsburgh**Title:** The tension between reflection/compactness and rigidity in combinatorial set theory**Abstract: **The aim of this talk is to provide background with which to motivate a recent joint result of the speaker with Omer Ben-Neria. This result concerns the tension between two classes of combinatorial principles in set theory, namely reflection/compactness principles on the one hand and incompactness/anti-reflection properties on the other. The rigidity implied by the latter class often suffices to prove the negation of principles in the former class, and as a result, a large research program in set theory is dedicated to investigating when principles from these two classes are jointly consistent. Our theorem – that the Special Aronszajn Tree Property is consistent with Club Stationary Reflection on $\omega_2$ – is such a result. We will discuss this tension historically before showing how, in our result, the tension shows up in our proof, especially in the radically different properties of our posets which we have to maintain throughout the course of our construction.**Information:** Zoom link https://cmu.zoom.us/j/621951121, meeting ID: 621 951 121

**Paris-Lyon Séminaire de Logique****Time:** Monday, 27 September,15:15 CET**Speaker:** Wieslaw Kubis, Czech Academy of Sciences**Title:** Generic evolutions**Abstract:** We shall present the concept of abstract evolution system” which, in particular, captures the main ideas of the theory of universal homogeneous structures (Fraisse limits). Evolution systems can also be viewed as a generalization of abstract rewriting systems. We shall present an analogue of Newman’s Lemma, saying that a locally confluent terminating system is confluent. Terminating evolution systems actually correspond to finite ultra-homogeneous structures.**Information:** Join via the link on the seminar webpage

20-26 September

**CUNY Set Theory Seminar****Time:** Friday, 24 September, 11:30am New York time (17:30 CET)**Speaker:** Matteo Viale, University of Torino**Title:** Absolute model companionship, forcibility, and the continuum problem**Abstract:** See the seminar webpage. **Information:** The seminar will take place virtually. Please email Victoria Gitman (vgitman@nylogic.org) for the meeting id.

**CMU Model Theory Seminar****Time:** Thursday, 23 September, 11:00 Eastern Daylight Time (17:00 CET)**Speaker:** Will Boney, Texas State University**Title:** Model Theoretic Characterizations of Large Cardinals, Part II**Abstract:** Large cardinals in set theory are typically characterized in a variety of ways. This talk focuses on characterizing large cardinals through model-theoretic compactness principles. Inspired by the work of Benda, we begin by showing that normality of ultrafilters corresponds to type omission. We will then move onto stronger logics and more exotic cardinals, including using Woodin cardinals to motivate an abstract definition of Henkin structures. Some of this work is joint with Dimopoulos, Gitman, and Magidor.**Information:** Please see the seminar webpage

**Caltech Logic Seminar****Time:** Wednesday, 22 September, 12:00 – 1:00pm Pacific time (21:00 CET)**Speaker:** Riley Thornton, UCLA**Title:** An algebraic approach to Borel CSPs**Abstract:** For a given finite relational structure RR, the associated constraint satisfaction problem (or CSP) is the problem of testing when a structure admits a homomorphism into RR. The algebraic approach studies a CSP by considering its algebra of so-called polymorphisms, operations (of all arities) which preserve the relations in RR. This approach has led to many classification results. For instance, CSPs solvable by polynomial time algorithms, linear relaxation, and constraint propagation have all been classified. In this talk I will present partial results towards similar classifications of Borel CSPs with various descriptive set theoretic properties: those problems which are Π11 in the codes, effectivizable, or equivalent to their classical version.**Information:** Please see the seminar webpage.

**Paris-Lyon Séminaire de Logique****Time:** Wednesday, 22 September, 16:00-17:00 CET**Speaker:** Gianluca Basso, Université Lyon I**Title:** Topological dynamics beyond Polish groups, and the ambitability question**Abstract:** When G is a Polish group, one way of knowing that it has nice dynamics is to show that M(G), the universal minimal flow of G, is metrizable. For non-Polish groups, this is not the relevant dividing line: the universal minimal flow of the symmetric group of a set of cardinality κ is the space of linear orders on κ–not a metrizable space, but still nice–, for example. In this talk, we present a set of equivalent properties of topological groups which characterize having nice dynamics. We then concentrate on an open question of Pachl and its consequences on the dynamics of topological groups. This is joint work with Andy Zucker.**Information:** Join via the link on the seminar webpage.

**Helsinki Logic Seminar****Time:** Wednesday, 22 September, 12:00 – 14:00 Helsinki time (11:00-13:00 CET)**Speaker:** Gabriel Goldberg, Berkeley**Title:** Large cardinals, maximality principles, and the multiverse**Abstract:** We discuss some mathematical results indicating a tension between large cardinal axioms and maximality principles such as forcing axioms and the Axiom of Choice. Topics will include the failure of the Ground Axiom in natural models, the optimality of Usuba’s theorem, the analogy between large cardinal axioms and determinacy principles, and the role of forcing in theory selection.**Information:** Please see the seminar webpage.

13-19 September

No talks

6-12 September

**CMU Model Theory Seminar****Time:** Thursday, 9 September, 11:00 Eastern Daylight Time (17:00 CET)**Speaker:** Will Boney, Texas State University**Title:** Model Theoretic Characterizations of Large Cardinals, Part I**Abstract:** Large cardinals in set theory are typically characterized in a variety of ways. This talk focuses on characterizing large cardinals through model-theoretic compactness principles. Inspired by the work of Benda, we begin by showing that normality of ultrafilters corresponds to type omission. We will then move onto stronger logics and more exotic cardinals, including using Woodin cardinals to motivate an abstract definition of Henkin structures. Some of this work is joint with Dimopoulos, Gitman, and Magidor.**Information:** Please see the seminar webpage

**Helsinki Logic Seminar****Time:** Wednesday, 8 September, 12:00 – 14:00 Helsinki time (11:00-13:00 CET)**Speaker:** Mark Kamsma **Title:** Independence Relations in Abstract Elementary Categories**Abstract:** In Shelah’s stability hierarchy we classify theories using combinatorial properties. Some important classes are: stable, simple and NSOP1 each being contained in the next. We can characterise these classes by the existence of a certain independence relation. For example, in vector spaces such an independence relation comes from linear independence. Part of this characterisation is canonicity of the independence relation: there can be at most one nice enough independence relation in a theory.

Lieberman, Rosický and Vasey proved canonicity of stable-like independence relations in accessible categories. Accessible categories are a very general framework. The category of models of some theory is an accessible category, every AEC (abstract elementary class) is an accessible category, but even then accessible categories are more general. Inspired by this we introduce the framework of AECats (abstract elementary categories) and prove canonicity for simple-like and NSOP1-like independence relations. This way we reconstruct part of the hierarchy that we have for first-order theories, but now in the very general category-theoretic setting.**Information:** Please see the seminar webpage.

**Caltech Logic Seminar****Time:** Wednesday, 8 September, 12:00 – 1:00pm Pacific time (21:00 CET)**Speaker:** Zoltán Vidnyánszky, Caltech**Title:** New examples of bounded degree acyclic graphs with large Borel chromatic number**Abstract:** Marks proved the existence of dd-regular acyclic Borel graphs with Borel chromatic number d+1. It can be shown that such statements cannot be proved using measures or Baire category, and, indeed, the Borel determinacy theorem had to be invoked.

We discuss a generalization of Marks’ method, which leads to an interesting new class of examples of 3-regular acyclic Borel graphs, which we call homomorphism graphs.

This yields new proofs of a number of known results. As a new application, we show that it is hard to decide whether the Borel chromatic number of a Borel graph is ≤3, even for acyclic 3-regular graphs (that is, such graphs form a Σ21-complete set).

Joint work with Jan Grebík.**Information:** Please see the seminar webpage.

30 August – 5 September

**Logic Seminar, Carnegie Mellon University****Time:** Tuesday, 31 August, 3:30 – 4:30pm Eastern Standard Time (21:30 – 22:30 CET) **Speaker:** Felix Weilacher, Carnegie Mellon University**Title:** Borel Edge Colorings for Finite Dimensional Groups**Abstract: **In Borel graph combinatorics, one often produces a structure (e.g. a coloring) by dividing a graph into subgraphs with finite connected components, then defining the structure on those components via some straightforward uniformization result. We first give an overview of some recent work formalizing these notions and applying them to various problems. We then present our own application to the problem of edge coloring. For Borel actions of certain groups, we find “degree plus one” Borel edge colorings, matching the classical bound of Vizing. Furthermore, for finitely generated abelian groups, we are able to exactly determine Borel edge chromatic numbers.**Information:** Zoom link https://cmu.zoom.us/j/621951121, meeting ID: 621 951 121

**Caltech Logic Seminar****Time:** Monday, 30 August, 12:00 – 1:00pm Pacific time (21:00 CET)**Speaker:** Gábor Kun, Alfréd Rényi Institute of Mathematics**Title:** Measurable perfect matchings**Abstract:** We go through the history of measurable perfect matchings from the Banach-Tarski paradox via circle squaring and report on recent progress. We construct a dd-regular treeing (for every d>2d>2) without a measurable perfect matching. We show that the Hall condition is essentially sufficient in the hyperfinite, one-ended, bipartite case. This allows us to characterize bipartite Cayley graphs with factor of i.i.d. perfect matchings extending the Lyons-Nazarov theorem. We apply these to Gardner’s conjecture for uniformly distributed sets, to balanced orientations, and to get new, simple proofs of the measurable circle squaring. We prove the analogous theorems in the context of rounding flows, too. Partially joint work with Matt Bowen and Marcin Sabok.**Information:** Check on the seminar webpage if the seminar will take place.

23-29 August

**Online Logic Seminar****Time:** Thursday, 26 August, 01:00pm US central time (20:00 CET)**Speaker:** Colin Jahel, Université Claude Bernard Lyon 1**Title:** Some progress on the unique ergodicity problem**Abstract:** In 2005, Kechris, Pestov and Todorcevic exhibited a correspondence between combinatorial properties of structures and dynamical properties of their automorphism groups. In 2012, Angel, Kechris and Lyons used this correspondence to show the unique ergodicity of all the minimal actions of some subgroups of S_{∞}. In this talk, I will give an overview of the aforementioned results and discuss recent work generalizing results of Angel, Kechris and Lyons in several directions.**Information:** See the seminar webpage.

16-22 August

**Singapore Logic Seminar****Time:** Wednesday, 18 August, 16:00-17:00 Singapore time (10:00-11:00 CET)**Speaker:** Yu Liang**Title:** Generalizing Besicovitch-Davis theorem**Abstract:** Besicovitch-Davis theorem says that the Hausdorff dimension of every analytic set can be approximated by its closed subset. But the Besicovitch-Davis theorem fails for co-analytic sets under the assumption V=L as observed by Slaman. We prove that the theorem holds for arbitrary sets under ZF+sTD. We also prove that the theorem holds for Σ^{1}_{2}-sets under Martin’s axiom.

This is joint work with Peng Yinhe and Wu Liuzhen. **Information:** See the seminar webpage.

9-15 August

**CUNY Set Theory Seminar****Time:** Friday, 13 August, 2pm New York time (20:00 CET)**Speaker:** Adrian Mathias, University of Freiburg**Title:** Linking descriptive set theory to symbolic dynamics: Part II**Abstract:** 1. I’ll begin by reviewing the work I did in 1993-6 on a problem raised by the dynamics group at the Universidad Autonomoa de Barcelona. They were interested in a phenomenon that resembles the Cantor-Bendixson sequence of derivatives, and hoped to prove that the sequence would always stop at a countable stage. Using ideas of Kunen and Martin I showed that it would always stop at or before stage omega_1.

2. In 2002/3, alerted by observations of David Fremlin, to the possibility that the barcelona conjecture was false, I succeeded in constructing an example with recursive initial data where the sequence stops exactly at stage omega_1.

My Réunion colleague Christian Delhommé simplified and extended my ideas.

I’ll outline the construction, as I think the underlying idea might have applications elsewhere in descriptive set theory.

3. I will outline more recent work using ideas of Blass and Fremlin to to study ‘uniform’ versions of the results of 1993-96.

I’ll end with listing some open problems which I hope will be found interesting.**Information:** The seminar will take place virtually. Please email Victoria Gitman (vgitman@nylogic.org) for the meeting id.

**Singapore Logic Seminar****Time:** Wednesday, 11 August, 16:00-17:00 Singapore time (10:00-11:00 CET)**Speaker:** Frank Stephan, National University of Singapore**Title:** A survey on the structures realised by positive equivalence relations**Abstract:** Let a positive equivalence relation to be an r.e. equivalence relation on the set of natural numbers with infinitely many equivalence relations. Khoussainov initiated with coauthors a deep study of the following question: Given a positive equivalence relation eta, which structures from a given set of structures does this equivalence relation realise? Here realisation means that functions in the structure are recursive and relations are r.e. with the equality itself given by the equivalence relation eta. In other words, the given r.e. structure divided by eta is the structure realised by eta. Now questions studied by Khoussainov and his coworkers included questions like “What is the partial ordering on positive equivalence relations eta,rho where eta is below rho iff every structure of the given type realised by eta is also realised by rho? Besides algebraic structures and orders, it has also been studied how the learnability notions behave with respect to uniformly r.e. one-one families realised by positive equivalence relations.**Information:** See the seminar webpage.

**Logic Seminar, Carnegie Mellon University****Time:** Tuesday, 10 August, 3:30 – 4:30pm Eastern Standard Time (21:30 – 22:30 CET) **Speaker:** Nathaniel Bannister, Carnegie Mellon University**Title:** Additivity of strong homology for locally compact separable metric spaces, part 6**Abstract: **This series of talks will cover the 2019 paper “On the additivity of strong homology for locally compact separable metric spaces” as well as recent work establishing a conceptual basis for the results therein. We will show that (relative to a weakly compact cardinal) it is consistent for strong homology to be additive and compactly supported on the class of locally compact separable metric spaces. In the process, we develop an equivalent algebraic statement and a sufficient cardinal-theoretic condition.

This is joint work with Justin Moore, Jeffrey Bergfalk, and Stevo Todorcevic.**Information:** Zoom link https://cmu.zoom.us/j/621951121, meeting ID: 621 951 121

2-8 August

26 July – 1 August

**CUNY Set Theory Seminar****Time:** Friday, 30 July, 2pm New York time (20:00 CET)**Speaker:** Neil Barton, University of Konstanz**Title:** Countabilism and Maximality (or ‘Some Systems of Set Theory on which Every Set Is Countable’)**Abstract:** It is standard in set theory to assume that Cantor’s Theorem establishes that the continuum is an uncountable set. A challenge for this position comes from the observation that through forcing one can collapse any cardinal to the countable and that the continuum can be made arbitrarily large. In this paper, we present a different take on the relationship between Cantor’s Theorem and extensions of universes, arguing that they can be seen as showing that every set is countable and that the continuum is a proper class. We examine several theories based on maximality considerations in this framework (in particular countabilist analogues of reflection principles) and show how standard set theories (including ZFC with large cardinals added) can be incorporated. We conclude that the systems considered raise questions concerning the foundational purposes of set theory.**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 July, 1.30-3pm Toronto time (19:30-21:00 CET)**Speaker:** tba**Title:** tba**Abstract:** tba**Information:** No webpage available. Email Ivan Ongay Valverde to receive the seminar announcements and for the zoom link.

**Hebrew University-Bar Ilan University Set Theory seminar****Time:** Wednesday, 28 July 14:00-16:00 Israel Time (13:00-15:00 CET)**Speaker:** tba**Title:** tba**Abstract:** tba**Information:** Please check on the seminar webpage if the seminar will take place. Contact Menachem Magidor, Asaf Rinot or Omer Ben-Neria ahead of time for the zoom link.

**Logic Seminar, Carnegie Mellon University****Time:** Tuesday, 27 July, 3:30 – 4:30pm Eastern Standard Time (21:30 – 22:30 CET) **Speaker:** Nathaniel Bannister, Carnegie Mellon University**Title:** Additivity of strong homology for locally compact separable metric spaces, part 3**Abstract: **This series of talks will cover the 2019 paper “On the additivity of strong homology for locally compact separable metric spaces” as well as recent work establishing a conceptual basis for the results therein. We will show that (relative to a weakly compact cardinal) it is consistent for strong homology to be additive and compactly supported on the class of locally compact separable metric spaces. In the process, we develop an equivalent algebraic statement and a sufficient cardinal-theoretic condition.

This is joint work with Justin Moore, Jeffrey Bergfalk, and Stevo Todorcevic.**Information:** Zoom link https://cmu.zoom.us/j/621951121, meeting ID: 621 951 121

19-26 July

**Logic Colloquium 2021, Poznan****Time:** Monday, 19 July – Friday, 24 July

Free registration on the conference website https://lc2021.pl

**CUNY Set Theory Seminar****Time:** Friday, 23 July, 2pm New York time (20:00 CET)**Speaker:** Philip Welch, University of Bristol**Title:** The universe constructed from a set (or class) of regular cardinals**Abstract:** We continue some work on L[Card] (the universe constructed from the predicate for the cardinals) to look at L[Reg] where Reg is the class of uncountable regular cardinals. The latter is also a model of a rich combinatorial structure being, as it turns out, a Magidor iteration of prikry forcings (using recent work of Ben-Neria). But it is limited in size, in fact is a rather ‘thin’ model. We show, letting O^s = O^sword be the least iterable structure with a measure which concentrates on measurable cardinals:

Theorem (ZFC):

1. Let S be a set, or proper class, of regular cardinals, then O^s is not an element of L[S].

2. This is best possible, in that no smaller mouse M can be substituted for O^s.

3. L[S] is a model of: GCH, Square’s, Diamonds, Morasses etc and has Ramsey cardinals, but no measurable cardinals.**Information:** The seminar will take place virtually. Please email Victoria Gitman (vgitman@nylogic.org) for the meeting id.

**Toronto Set Theory Seminar****Time:** Friday, 23 July, 1.30-3pm Toronto time (19:30-21:00 CET)**Speaker:** Gianluca Paolini, University of Torino**Title:** Torsion-Free Abelian Groups are Borel Complete**Abstract:** We prove that the Borel space of torsion-free Abelian groups with domain *ω* is Borel complete, i.e., the isomorphism relation on this Borel space is as complicated as possible, as an isomorphism relation. This solves a long-standing open problem in descriptive set theory, which dates back to the seminal paper on Borel reducibility of Friedman and Stanley from 1989. **Information:** No webpage available. Email Ivan Ongay Valverde to receive the seminar announcements and for the zoom link.

**Münster research seminar on set theory****Time:** Wednesday, 21 July, 15:15-16:45 CET**Speaker:** tba**Title:** tba **Abstract:** tba **Information:** Please check the seminar webpage to see if the seminar takes place. Contact rds@wwu.de ahead of time in order to participate.

**Hebrew University-Bar Ilan University Set Theory seminar****Time:** Wednesday, 21 July 14:00-16:00 Israel Time (13:00-15:00 CET)**Speaker:** Dominik Adolf, Bar-Ilan University**Title:** Scales in inner models, part III**Abstract:** Tree like scales were introduced by Pereira in his PhD thesis. Unlike other properties like goodness, tree-likeness is known to be consistent with the existence of extremely strong large cardinals like I_0 -cardinals (Pereira) and supercompact cardinals (Cummings).

We will add to this by showing that tree-like scales also exist in all canonical (short extender) inner models. In fact, every product in such a model carries a continuous tree-like scale. If time permits we will also discuss when such scales exist in arbitrary universes by the use of core model theory. **Information:** Please check on the seminar webpage if the seminar will take place. Contact Menachem Magidor, Asaf Rinot or Omer Ben-Neria ahead of time for the zoom link.

**Logic Seminar, Carnegie Mellon University****Time:** Tuesday, 20 July, 3:30 – 4:30pm Eastern Standard Time (21:30 – 22:30 CET) **Speaker:** Nathaniel Bannister, Carnegie Mellon University**Title:** Additivity of strong homology for locally compact separable metric spaces, part 2**Abstract: **This series of talks will cover the 2019 paper “On the additivity of strong homology for locally compact separable metric spaces” as well as recent work establishing a conceptual basis for the results therein. We will show that (relative to a weakly compact cardinal) it is consistent for strong homology to be additive and compactly supported on the class of locally compact separable metric spaces. In the process, we develop an equivalent algebraic statement and a sufficient cardinal-theoretic condition.

This is joint work with Justin Moore, Jeffrey Bergfalk, and Stevo Todorcevic.**Information:** Zoom link https://cmu.zoom.us/j/621951121, meeting ID: 621 951 121

12-18 July

5-11 July

**CUNY Set Theory Seminar****Time:** Friday, 9 July, 2pm New York time (20:00 CET)**Speaker:** Paul Kindvall Gorbow, University of Gothenburg**Title:** The Copernican Multiverse of Sets: Part II**Abstract:** In these two talks, I will explain an untyped framework for the multiverse of set theory, developed in a joint paper with Graham Leigh. ZF is extended with semantically motivated axioms utilizing the new symbols Uni(U) and Mod(U, sigma), expressing that U is a universe and that sigma is true in the universe U, respectively. Here sigma ranges over the augmented language, leading to liar-style phenomena.

The framework is both compatible with a broad range of multiverse conceptions and suggests its own philosophically and semantically motivated multiverse principles. In particular, the framework is closely linked with a deductive rule of Necessitation expressing that the multiverse theory can only prove statements that it also proves to hold in all universes. We argue that this may be philosophically thought of as a Copernican principle, to the effect that the background theory of the multiverse does not hold a privileged position over the theories of its internal universes.

Our main mathematical result is a lemma encapsulating a technique for locally interpreting a wide variety of extensions of our basic framework in more familiar theories. This is applied to show, for a range of such semantically motivated extensions, that their consistency strength is at most slightly above that of the base theory ZF, and thus not seriously limiting to the diversity of the set-theoretic multiverse. I also plan to discuss connections with Hamkins’s multiverse theory, and the model of this constructed by Gitman and Hamkins. Throughout the talks I’m keen to discuss both philosophical and mathematical matters with the audience, concerning our Copernican approach to the multiverse of 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, 9 July, 1.30-3pm Toronto time (19:30-21:00 CET)**Speaker:** tba**Title:** tba**Abstract:** tba**Information:** No webpage available. Email Ivan Ongay Valverde to receive the seminar announcements and for the zoom link.

**Münster research seminar on set theory****Time:** Wednesday, 7 July, 15:15-16:45 CET**Speaker:** Omer Ben-Neria, Hebrew University, Jerusalem**Title:** tba **Abstract:** tba **Information:** Please check the seminar webpage to see if the seminar takes place. Contact rds@wwu.de ahead of time in order to participate.

**Hebrew University-Bar Ilan University Set Theory seminar****Time:** Wednesday, 7 July, 14:00-16:00 Israel Time (13:00-15:00 CET)**Speaker:** Dominik Adolf**Title:** Scales in inner models**Abstract:** Tree like scales were introduced by Pereira in his PhD thesis. Unlike other properties like goodness, tree-likeness is known to be consistent with the existence of extremely strong large cardinals like I_0 -cardinals (Pereira) and supercompact cardinals (Cummings).

We will add to this by showing that tree-like scales also exist in all canonical (short extender) inner models. In fact, every product in such a model carries a continuous tree-like scale. If time permits we will also discuss when such scales exist in arbitrary universes by the use of core model theory. **Information:** Please check on the seminar webpage if the seminar will take place. Contact Menachem Magidor, Asaf Rinot or Omer Ben-Neria ahead of time for the zoom link.

28 June – 4 July

**CUNY Set Theory Seminar****Time:** Friday, 2 July, 2pm New York time (20:00 CET)**Speaker:** Paul Kindvall Gorbow, University of Gothenburg**Title:** The Copernican Multiverse of Sets**Abstract:** In these two talks, I will explain an untyped framework for the multiverse of set theory, developed in a joint paper with Graham Leigh. ZF is extended with semantically motivated axioms utilizing the new symbols Uni(U) and Mod(U, sigma), expressing that U is a universe and that sigma is true in the universe U, respectively. Here sigma ranges over the augmented language, leading to liar-style phenomena.

The framework is both compatible with a broad range of multiverse conceptions and suggests its own philosophically and semantically motivated multiverse principles. In particular, the framework is closely linked with a deductive rule of Necessitation expressing that the multiverse theory can only prove statements that it also proves to hold in all universes. We argue that this may be philosophically thought of as a Copernican principle, to the effect that the background theory of the multiverse does not hold a privileged position over the theories of its internal universes.

Our main mathematical result is a lemma encapsulating a technique for locally interpreting a wide variety of extensions of our basic framework in more familiar theories. This is applied to show, for a range of such semantically motivated extensions, that their consistency strength is at most slightly above that of the base theory ZF, and thus not seriously limiting to the diversity of the set-theoretic multiverse. I also plan to discuss connections with Hamkins’s multiverse theory, and the model of this constructed by Gitman and Hamkins. Throughout the talks I’m keen to discuss both philosophical and mathematical matters with the audience, concerning our Copernican approach to the multiverse of sets.**Information:** The seminar will take place virtually. Please email Victoria Gitman (vgitman@nylogic.org) for the meeting id.

**Münster research seminar on set theory****Time:** Wednesday, 26 June, 15:15-16:45 CET**Speaker:** Diana Montoya, University of Vienna**Title:** Independence at uncountable cardinals **Abstract:** In this talk, we will discuss the concept of maximal independent

families for uncountable cardinals. First, we will mention a summary of

results regarding the existence of such families in the case of an

uncountable regular cardinal. Specifically, we will focus on joint work

with Vera Fischer regarding the existence of an indestructible maximal

independent family, which turns out to be indestructible after forcing with

generalized Sacks forcing.

In the second part, we will focus on the singular case and present two results obtained in joint work with Omer Ben-Neria. Finally, I will mention some open questions and future paths of research.**Information:** Please check the seminar webpage to see if the seminar takes place. Contact rds@wwu.de ahead of time in order to participate.

**Hebrew University-Bar Ilan University Set Theory seminar****Time:** Wednesday, 26 June 14:00-16:00 Israel Time (13:00-15:00 CET)**Speaker:** Tzoor Plotinkov**Title:** The Automorphism Tower of a Group, continued **Abstract:** We will talk about the operation of forming the automorphism tower over a certain group. Namely, looking at the automorphism group of a certain group, on the automorphism group of that group, and so forth, continuing transfinitely.

In the late 80’s Simon Thomas showed that for every centerless group , the automorphism tower of stabilizes in fewer than many steps.

The question of when the tower stabilizes has been studied by Thomas, Shelah, Just, Hamkins, Fuchs, Lucke and more, and turned out to have a lot of set theoretical content.

We will have two talks going over some of the proofs and techniques used in the subject. The first one will be more dedicated to outright ZFC results, and the second one will be more focused on consistency results. **Information:** Please check on the seminar webpage if the seminar will take place. Contact Menachem Magidor, Asaf Rinot or Omer Ben-Neria ahead of time for the zoom link.

21-27 June

**Toronto Set Theory Seminar****Time:** Friday, 25 June, 1.30-3pm Toronto time (19:30-21:00 CET)**Speaker:** tba**Title:** tba**Abstract:** tba**Information:** No webpage available. Email Ivan Ongay Valverde to receive the seminar announcements and for the zoom link.

**KGRC Research Seminar, Vienna****Time:** Thursday, 24 June , 15:00-16:30 CET**Speaker:** Johannes Schürz, TU Wien **Title:** Preserving levels of projective determinacy and regularity properties**Abstract:** Since \mathbf{\Pi}^1_1-determinacy is a desirable property on the reals,

the natural question arises as to how one can preserve it under forcing.

We will show using the technique of capturing that the statement ‘Every

real has a sharp’ is preserved under any countable support iteration of

‘simply’ definable forcing notions. By the famous results of L. Harrington

and D. Martin this shows that \mathbf{\Pi}^1_1-determinacy is preserved

under such iterations.

More generally, our theorem also shows that the statement ‘M_n^\sharp(x)

exists for every real x \in \omega^\omega’ is preserved. By the results of

I. Neeman and H. Woodin this generalizes our result to higher levels of

projective determinacy.

Without the existence of large cardinals the technique of capturing can

still be used to show preservation results for regularity properties such

as the \mathbf{\Delta}^1_2- or \mathbf{\Sigma}^1_2-Baire property.

This is a joint project with J. Schilhan and P. Schlicht.**Information:** Talk via zoom.

**Hebrew University-Bar Ilan University Set Theory seminar****Time:** Wednesday, 23 June, 14:00-16:00 Israel Time (13:00-15:00 CET)**Speaker:** Tzoor Plotinkov**Title:** The Automorphism Tower of a Group, continued **Abstract:** We will talk about the operation of forming the automorphism tower over a certain group. Namely, looking at the automorphism group of a certain group, on the automorphism group of that group, and so forth, continuing transfinitely.In the late 80’s Simon Thomas showed that for every centerless group , the automorphism tower of stabilizes in fewer than many steps.The question of when the tower stabilizes has been studied by Thomas, Shelah, Just, Hamkins, Fuchs, Lucke and more, and turned out to have a lot of set theoretical content. We will have two talks going over some of the proofs and techniques used in the subject. The first one will be more dedicated to outright ZFC results, and the second one will be more focused on consistency results. **Information:** Please check on the seminar webpage if the seminar will take place. Contact Menachem Magidor, Asaf Rinot or Omer Ben-Neria ahead of time for the zoom link.

**Caltech Logic Seminar****Time:** Monday, 21 June, 12:00 – 1:00pm Pacific time (21:00 CET)**Speaker:** Udayan Darji, University of Louisville**Title:** Local entropy and descriptive complexity**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 Π11Π11-rank. More importantly, she showed that the collection of CPE Z2Z2-SFT’s is a Π11Π11-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 Π11Π11-complete. This is joint work with García-Ramos.**Information:** Check on the seminar webpage if the seminar will take place.

14-20 June

**Toronto Set Theory Seminar****Time:** Friday, 18 June, 1.30-3pm Toronto time (19:30-21:00 CET)**Speaker:** Davdi Schrittesser**Title:** A taste of nonstandard analysis and statistical decision theory**Abstract:** Statistical decision theory takes inspiration from game theory to

provide a basic framework in which one can reason about optimality (or

lack thereof) of statistical methods, such as estimators and tests.

One (very weak) property of such methods is admissibility – roughly, a

method of estimation is admissible if there is no other which does

better under all circumstances (in a sense specified by the decision

theoretical framework).

Although a weak property, admissibility is notoriously hard to

characterize. Recently we have found a characterization of admissibility

(in a large class of statistical problems) in Bayesian terms, by using

prior probability distributions which can take on infinitesimal values.

(The talk will not presuppose any knowledge on statistics or nonstandard

analysis. Joint work with D. Roy and H. Duanmu.) **Information:** No webpage available. Email Ivan Ongay Valverde to receive the seminar announcements and for the zoom link.

**Turin-Udine logic seminar****Time:** Friday, 18 June, 16:30-18:30 CET**Speaker:** C. Brech, Universidade de São Paulo**Title:** tba **Abstract:** tba**Information:** Online on WebEx. Please see the seminar webpage.

**KGRC Research Seminar, ViennaTime:** Thursday, 17 June , 15:00-16:30 CET

**Speaker:**David Chodounský, Czech Academy of Sciences

**Title:**Big Ramsey degrees of 3-uniform hypergraphs are finite

**Abstract:**It is well known that the (universal countable) Rado graph has finite big Ramsey degrees. I.e., given a finite colouring of n-tuples of its vertices there is a copy of the Rado graph such that its n‑tuples have at most D(n)‑many colours. The proof of this fact uses a theorem of Milliken for trees, I will give sketch of the argument. I will moreover sketch an extension of the proof which works also for universal structures with higher arities, in particular 3‑uniform hypergraphs.

Joint work with M. Balko, J. Hubička, M. Konečný, and L. Vena, see https://arxiv.org/abs/2008.00268.

**Information:**Talk via zoom.

**Bristol Logic and Set Theory Seminar**/**Oxford Set Theory Seminar****Time:** Wednesday, 16 June, 16:30-18:00 UK time (17:30-19:00 CEST)

**Speaker:**Joan Bagaria, University of Barcelona

**Title:**Some recent results on Structural Reflection

**Abstract:**The general Structural Reflection (SR) principle asserts that for every definable, in the first-order language of set theory, possibly with parameters, class C of relational structures of the same type there exists an ordinal 𝛼α that

*reflects*C, i.e., for every 𝐴A in C there exists 𝐵B in ∩𝑉𝛼C∩Vα and an elementary embedding from 𝐵B into 𝐴A. In this form, SR is equivalent to Vopenka’s Principle (VP). In my talk I will present some different natural variants of SR which are equivalent to the existence some well-known large cardinals weaker than VP. I will also consider some forms of SR, reminiscent of Chang’s Conjecture, which imply the existence of large cardinal principles stronger than VP, at the level of rank-into-rank embeddings and beyond. The latter is a joint work with Philipp Lücke.

**Information:**For the Zoom access code, contact Samuel Adam-Day me@samadamday.com. Link: https://zoom.us/j/96803195711 (open 30 minutes before)

**Hebrew University-Bar Ilan University Set Theory seminar****Time:** Wednesday, 16 June, 14:00-16:00 Israel Time (13:00-15:00 CET)**Speaker:** Tzoor Plotinkov**Title:** The Automorphism Tower of a Group**Abstract:** We will talk about the operation of forming the automorphism tower over a certain group. Namely, looking at the automorphism group of a certain group, on the automorphism group of that group, and so forth, continuing transfinitely.In the late 80’s Simon Thomas showed that for every centerless group , the automorphism tower of stabilizes in fewer than many steps.The question of when the tower stabilizes has been studied by Thomas, Shelah, Just, Hamkins, Fuchs, Lucke and more, and turned out to have a lot of set theoretical content. We will have two talks going over some of the proofs and techniques used in the subject. The first one will be more dedicated to outright ZFC results, and the second one will be more focused on consistency results. **Information:** Please check on the seminar webpage if the seminar will take place. Contact Menachem Magidor, Asaf Rinot or Omer Ben-Neria ahead of time for the zoom link.

**Caltech Logic Seminar****Time:** Monday, 14 June, 12:00 – 1:00pm Pacific time (21:00 CET)**Speaker:** László Márton Tóth, EPFL**Title:** Schreier decorations of unimodular random graphs**Abstract:** It is a nice exercise in combinatorics to show that every 2d2d-regular finite graph arises as a Schreier graph of the free group FdFd. I will present generalizations of this fact to a measurable setting, as well as some examples showing the limitations. I will formulate these results using both the language of unimodular random networks and that of (p.m.p.) graphings, which are two sides of the same coin. Partially joint work with Ferenc Bencs and Aranka Hrušková.**Information:** Check on the seminar webpage if the seminar will take place.

7-13 June

**Toronto Set Theory Seminar****Time:** Friday, 11 June, 1.30-3pm Toronto time (19:30-21:00 CET)**Speaker:** tba**Title:** tba**Abstract:** tba**Information:** No webpage available. Email Ivan Ongay Valverde to receive the seminar announcements and for the zoom link.

**Turin-Udine logic seminar****Time:** Friday, 11 June, 16:30-18:30 CET**Speaker:** Victoria Gitman, CUNY Graduate Center **Title:** The old and the new of virtual large cardinals**Abstract:** The idea of defining a generic version of a large cardinal by asking that some form of the elementary embeddings characterizing the large cardinal exist in a forcing extension has a long history. A large cardinal (typically measurable or stronger) can give rise to several natural generic versions with vastly different properties. For a \emph{generic large cardinal}, a forcing extension should have an elementary embedding j:V→Mof the form characterizing the large cardinal where the target model M is an inner model of the forcing extension, not necessarily contained in V. The closure properties on Mmust correspondingly be taken with respect to the forcing extension. Very small cardinals such as ω1 can be generic large cardinals under this definition. Quite recently set theorists started studying a different version of generic-type large cardinals, called \emph{virtual large cardinals}. Large cardinals characterized by the existence of an elementary embedding j:V→M typically have equivalent characterizations in terms of the existence of set-sized embeddings of the form j:Vλ→M. For a virtual large cardinal, a forcing should have an elementary embedding j:Vλ→M of the form characterizing the large cardinal with M∈Vand all closure properties on M considered from V’s standpoint. Virtual large cardinals are actually large cardinals, they are completely ineffable and more, but usually bounded above by an ω-Erd\H os cardinal. Despite sitting much lower in the large cardinal hierarchy, they mimic the reflecting properties of their original counterparts. Several of these notions arose naturally out of equiconsistency results. In this talk, I will give an overview of the virtual large cardinal hierarchy including some surprising recent directions.**Information:** Please check on the semianr webpage if the seminar will take place. Online on WebEx. Please see the seminar webpage.

**Barcelona Set Theory Seminar****Time:** Wednesday, 9 June, 16:00-17:30 CET**Speaker:** Raffaella Cutolo, Università degli Studi di Napoli Federico II **Title:** N-Berkeley cardinals and the two futures of set theory**Abstract:** The talk will focus on Berkeley cardinals – the strongest known large

cardinal axioms – and their relativized version to inner models of ZFC, which

in fact play a decisive role in the current scenario of set theory. As we shall

see, by recent results of Woodin, there are just two, opposite possible futures

for set theory, and Berkeley cardinals are very involved in the question of

which of the two futures will prevail. In particular, the relativized version of

Berkeley cardinals turns out to be relevant with respect to that question, and

it is therefore worthy of attention.

We shall show the first example of the existence of a “N-Berkeley cardinal”

for an inner model N of ZFC, a result that is quite surprising as the involved

model N is a weak extender model, that is, N satisfies structural properties

making it very close to the set-theoretic universe V with respect to the large

cardinal axioms it is able to recognize; nevertheless, there exists (in V ) a

N-Berkeley cardinal, one that cannot exist in N, which models AC. We then

isolate a strong version of the notion of being N-Berkeley, and prove that

such strong version is inconsistent with the assumption that N is closed under

ω-sequences.

We finally illustrate the relevance of the results above with respect to the

crucial decision between the two futures of set theory.**Information:** Online. If you wish to attend, please send an email to bagaria@ub.edu asking for the link.

**Münster research seminar on set theory****Time:** Wednesday, 9 June, 15:15-16:45 CET**Speaker:** Gunter Fuchs, CUNY)**Title:** Fragments of (diagonal) strong reflection **Abstract:** Continuation of last week’s talk. **Information:** Please check the seminar webpage to see if the seminar takes place. Contact rds@wwu.de ahead of time in order to participate.

**Hebrew University-Bar Ilan University Set Theory seminar****Time:** Wednesday, 9 June, 14:00-16:00 Israel Time (13:00-15:00 CET)**Speaker:** tba**Title:** tba**Abstract:** tba**Information:** Please check on the seminar webpage if the seminar will take place. Contact Menachem Magidor, Asaf Rinot or Omer Ben-Neria ahead of time for the zoom link.

**Caltech Logic Seminar****Time:** Monday, 7 June, 12:00 – 1:00pm Pacific time (21:00 CET)**Speaker:** Julien Melleray, Université Lyon 1 **Title:** A new proof of a theorem of Giordano, Putnam, and Skau **Abstract:** A well-known result of Giordano-Putnam-Skau asserts that two minimal homeomorphisms of the Cantor space which have the same invariant Borel probability measures are orbit equivalent. I will present a new, rather elementary, proof of that fact, based on a strengthening of a 1979 theorem of Krieger concerning minimal actions of certain locally finite groups on the Cantor space. No familarity with topological dynamics will be assumed.

This is joint work with Simon Robert (Lyon).**Information:** Check on the seminar webpage if the seminar will take place.

31 May – 6 June

**CUNY Set Theory Seminar****Time:** Friday, 4 June, 2pm New York time (20:00 CET)**Speaker: **Gabriel Goldberg, University of Berkeley**Title:** The HOD conjecture and the structure of elementary embeddings: Part II**Abstract:** Woodin’s HOD conjecture asserts that in the context of very large cardinals, the inner model HOD closely approximates the universe of sets in the same way Gödel’s constructible universe does assuming 0# does not exist. The subject of these two talks is the relationship between Woodin’s conjecture and certain constraints on the structure of elementary embeddings of the universe of sets. For example, in the second talk, we will prove that any two elementary embeddings of the universe of sets into the same inner model agree on HOD, while if a local version of this theorem held, then the HOD conjecture would follow.**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 June, 1.30-3pm Toronto time (19:30-21:00 CET)**Speaker:** tba**Title:** tba**Abstract:** tba**Information:** No webpage available. Email Ivan Ongay Valverde to receive the seminar announcements and for the zoom link.

**Turin-Udine logic seminar****Time:** Friday, 4 June, 16:30-18:30 CET**Speaker:** M. Pinsker, Vienna University of Technology **Title:** tba **Abstract:** tba**Information:** Please check on the semianr webpage if the seminar will take place. Online on WebEx. Please see the seminar webpage.

**Bristol Logic and Set Theory Seminar**/**Oxford Set Theory Seminar****Time:** Wednesday, 2 June, 16:30-18:00 UK time (17:30-19:00 CEST)

**Speaker:**Christopher Turner, Bristol

**Title:**Forcing Axioms and Name Principles

**Abstract:**Forcing axioms are a well-known way of expressing the concept ”there are filters in V which are close to being generic”. \textit{Name principles} are another expression of this concept. A name principle says: ”Let $\sigma$ be any sufficiently nice name which is forced to have some property. Then there is a filter $g\in V$ such that $\sigma^g$ has that property.” Name principles have often been used on an ad-hoc basis in proofs, but have not been studied much as axioms in their own right. In this talk, I will present some of the connections between different name principles, and between name principles and forcing axioms. This is based on joint work with Philipp Schlicht.

**Information:**For the Zoom access code, contact Samuel Adam-Day me@samadamday.com. Link: https://zoom.us/j/96803195711 (open 30 minutes before)

**Barcelona Set Theory Seminar****Time:** Wednesday, 2 June, 16:00-17:30 CET**Speaker:** Michał Godziszewski**Title:** The Multiverse, Recursive Saturation and Well-Foundedness Mirage **Abstract:** Recursive saturation, introduced by J. Barwise and J. Schlipf is a robust notion which has proved to be important for the study of nonstandard models (in particular, it is ubiquitous in the model theory of axiomatic theories of truth, e.g. in the topic of satisfaction classes, where one can show that if M is a countable omega-nonstandard

model of ZFC, then M admits a satisfaction class iff M is recursively saturated). V. Gitman and J. Hamkins showed in A Natural Model of the Multiverse Axioms that the collection of countable, recursively saturated models of set theory satisfy the so-called Hamkins’s Multiverse Axioms. The property that forces all the models in the Multiverse to be recursively saturated is the so-called Well-Foundedness Mirage axiom which asserts that every universe is omega-nonstandard from the perspective of some larger universe, or to be more precise, that: if a model M is in the multiverse then there is a model N in the multiverse such that M is a set in N and N satisfies ‘M is omega-nonstandard.’ Inspection of the proof led to a question if the recursive saturation could be avoided in the Multiverse by weakening the Well-Foundedness Mirage axiom. Our main results answer this in the positive. We give two different versions of the Well-Foundedness Mirage axiom — what we call Weak Well-Foundedness Mirage (saying that if M is a model in the Multiverse then there is a model N in the Multiverse such that M is an element of N and N satisfies ‘M is nonstandard’ and Covering Well-Foundedness Mirage (saying that if M is a model in the Multiverse then there is a model N in the Multiverse with K in N such that K is an end-extension of M and N satisfies ‘K is omega-nonstandard.’ I will try to present constructions of two different Multiverses satisfying these two weakened axioms (with a promise concerning at least one of them, with the second one depending on what time permits). This is joint work with V. Gitman. T. Meadows and K. Williams. **Information:** Online. If you wish to attend, please send an email to bagaria@ub.edu asking for the link.

**Münster research seminar on set theory****Time:** Wednesday, 2 June, 15:15-16:45 CET**Speaker:** Gunter Fuchs, CUNY**Title:** Fragments of (diagonal) strong reflection.**Abstract:** I will talk about reflection principles that arose out of an attempt to find an analog of Todorcevic’s strong reflection principle SRP, which captures many of the major consequences of Martin’s Maximum, that works with forcing axioms for other forcing classes, in particular subcomplete forcing. Since SRP fails to encapsulate phenomena of diagonal reflection which follow from MM, I will propose a diagonal version of it that does have these consequences, as well as its fragments. The gist of these principles is that there is a natural strengthening of the concept of a projective stationary set, which I call “spread out”, which gives rise to the subcomplete fragments of these strong reflection principles. Part of this work is joint with Sean Cox.**Information:** Please check the seminar webpage to see if the seminar takes place. Contact rds@wwu.de ahead of time in order to participate.

**Hebrew University-Bar Ilan University Set Theory seminar****Time:** Wednesday, 2 June, 14:00-16:00 Israel Time (13:00-15:00 CET)**Speaker:** tba**Title:** tba**Abstract:** tba**Information:** Please check on the seminar webpage if the seminar will take place. Contact Menachem Magidor, Asaf Rinot or Omer Ben-Neria ahead of time for the zoom link.

24-30 May

**CUNY Set Theory Seminar****Time:** Friday, 28 May, 2pm New York time (20:00 CET)**Speaker: **Gabriel Goldberg, University of Berkeley**Title:** The HOD conjecture and the structure of elementary embeddings**Abstract:** Woodin’s HOD conjecture asserts that in the context of very large cardinals, the inner model HOD closely approximates the universe of sets in the same way Gödel’s constructible universe does assuming 0# does not exist. The subject of these two talks is the relationship between Woodin’s conjecture and certain constraints on the structure of elementary embeddings of the universe of sets. For example, in the second talk, we will prove that any two elementary embeddings of the universe of sets into the same inner model agree on HOD, while if a local version of this theorem held, then the HOD conjecture would follow.**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 May, 1.30-3pm Toronto time (19:30-21:00 CET)**Speaker:** tba**Title:** tba**Abstract:** tba**Information:** No webpage available. Email Ivan Ongay Valverde to receive the seminar announcements and for the zoom link.

**Turin-Udine logic seminar****Time:** Friday, 28 May, 16:30-18:30 CET**Speaker:** D. Bartosova, University of Florida **Title:** Short exact sequences and universal minimal flows**Abstract:** We will investigate an interplay between short exact sequences of topological groups and their universal minimal flows in case one of the factors is compact. We will discuss possible and impossible extensions of the results in a few directions. An indispensable ingredient in our technique is a description of the universal pointed flow of a given group in terms of filters on the group, which we will describe.**Information:** Please check on the semianr webpage if the seminar will take place. Online on WebEx. Please see the seminar webpage.

**KGRC Research Seminar, ViennaTime:** Thursday, 27 May, 15:00-16:30 CET

**Speaker:**Diana Carolina Montoya (KGRC)

**Title:**Independent families and singular cardinals

**Abstract:**In this talk, we will discuss the concept of independent families for uncountable cardinals. First, we will mention a summary of results regarding the existence of such families in the case of an uncountable regular cardinal. In the second part, we focus on the singular case and present two results of ours. This is joint work with Omer Ben-Neria.

**Information:**Talk via zoom.

**Bristol Logic and Set Theory Seminar**/**Oxford Set Theory Seminar****Time:** Wednesday, 26 May, 16:30-18:00 UK time (17:30-19:00 CEST)

**Speaker:**Sandra Müller, University of Vienna

**Title:**The strength of determinacy when all sets are universally Baire

**Abstract:**The large cardinal strength of the Axiom of Determinacy when enhanced with the hypothesis that all sets of reals are universally Baire is known to be much stronger than the Axiom of Determinacy itself. In fact, Sargsyan conjectured it to be as strong as the existence of a cardinal that is both a limit of Woodin cardinals and a limit of strong cardinals. Larson, Sargsyan and Wilson showed that this would be optimal via a generalization of Woodin’s derived model construction. We will discuss a new translation procedure for hybrid mice extending work of Steel, Zhu and Sargsyan and use this to prove Sargsyan’s conjecture.

**Information:**For the Zoom access code, contact Samuel Adam-Day me@samadamday.com. Link: https://zoom.us/j/96803195711 (open 30 minutes before)

**Barcelona Set Theory Seminar****Time:** Wednesday, 26 May, 16:00-17:30 CET**Speaker:** David Aspero, Norwich **Title:** Around (*)**Abstract:** Abstract: In this talk I will present work motivated by the derivation of the

Pmax axiom (*) from Martin’s Maximum++.**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, 26 May, 14:00-16:00 Israel Time (13:00-15:00 CET)**Speaker:** Menachem Magidor **Title:** When complicated relation on a Polish space can be Borel**Abstract:** tba**Information:** Please check on the seminar webpage if the seminar will take place. Contact Menachem Magidor, Asaf Rinot or Omer Ben-Neria ahead of time for the zoom link.

**Caltech Logic Seminar****Time:** Monday, 24 May, 12:00 – 1:00pm Pacific time (21:00 CET)**Speaker:** Spencer Unger, University of Toronto**Title:** Borel factor maps and embeddings between actions of Zd**Abstract:** We are interested in generalizations of some theorems of ergodic theory to the Borel context, particularly for natural spaces of tilings, colorings and Hamilton paths. The work combines dynamical properties of actions of ZdZd with the finite combinatorics of the ZdZd lattice. This is joint work with Nishant Chandgotia.**Information:** Check on the seminar webpage if the seminar will take place.

17-23 May

**CUNY Set Theory Seminar****Time:** Friday, 21 May, 1pm New York time (19:00 CET)**Speaker: **Omer Ben-Neria, Hebrew University**Title:** Mathias-type Criterion for the Magidor Iteration of Prikry forcings**Abstract:** In his seminal work on the identity crisis of strongly compact cardinals, Magidor introduced a special iteration of Prikry forcings for a set of measurable cardinals known as the Magidor iteration. The purpose of this talk is to present a Mathias-type criterion which characterizes when a sequence of omega-sequences is generic for the Magidor iteration. The result extends a theorem of Fuchs, who introduced a Mathias criterion for discrete products of Prikry forcings. We will present the new criterion, discuss several applications, and outline the main ideas of the proof.**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 May, 1.30-3pm Toronto time (19:30-21:00 CET)**Speaker:** tba**Title:** tba**Abstract:** tba**Information:** No webpage available. Email Ivan Ongay Valverde to receive the seminar announcements and for the zoom link.

**Turin-Udine logic seminar****Time:** Friday, 21 May, 16:30-18:30 CET**Speaker: **L. Westrick (Penn State University)**Title:** Borel combinatorics fail in HYP**Abstract:** Of the principles just slightly weaker than ATR, the most well-known are the theories of hyperarithmetic analysis (THA). By definition, such principles hold in HYP. Motivated by the question of whether the Borel Dual Ramsey Theorem is a THA, we consider several theorems involving Borel sets and ask whether they hold in HYP. To make sense of Borel sets without ATR, we formalize the theorems using completely determined Borel sets. We characterize the completely determined Borel subsets of HYP as precisely the sets of reals which are Delta^1_1 in L_{\omega_1^{ck}}. Using this, we show that in HYP, Borel sets behave quite differently than in reality. In HYP, the Borel dual Ramsey theorem fails, every n-regular Borel acyclic graph has a Borel 2-coloring, and the prisoners have a Borel winning strategy in the infinite prisoner hat game. Thus the negations of these statements are not THA. Joint work with Henry Towsner and Rose Weisshaar.**Information:** Please check on the semianr webpage if the seminar will take place. Online on WebEx. Please see the seminar webpage.

**KGRC Research Seminar, ViennaTime:** Thursday, 20 May, 15:00-16:30 CET

**Speaker:**Lev Bukovsky (Pavol Jozef Šafárik University in Košice, Slovakia)

**Title:**Extensions of inner models of ZFC

**Abstract:**I would like to present some results of members of Vopěnka’s seminary in 1960’s and 1970’s (B. Balcar, P. Vopěnka, P. Hájek and me), which were either not published or published in the language of semisets theory. Consequently, those results are not commonly known.

**Information:**Talk via zoom.

**Bristol Logic and Set Theory Seminar**/**Oxford Set Theory Seminar****Time:** Wednesday, 19 May, 16:30-18:00 UK time (17:30-19:00 CEST)

**Speaker:**Sam Adam-Day, Oxford

**Title:**The continuous gradability of the cut-point orders of R-trees

**Abstract:**An R-tree is a metric space tree in which every point can be branching. Favre and Jonsson posed the following problem in 2004: can the class of orders underlying R-trees be characterised by the fact that every branch is order-isomorphic to a real interval? In the first part of the talk, I answer this question in the negative: there is a branchwise-real tree order which is not continuously gradable. In the second part, I show that a branchwise-real tree order is continuously gradable if and only if every embedded well-stratified (i.e. set-theoretic) tree is R-gradable. This tighter link with set theory is put to work in the third part answering a number of refinements of the main question, yielding several independence results.

**Information:**For the Zoom access code, contact Samuel Adam-Day me@samadamday.com. Link: https://zoom.us/j/96803195711 (open 30 minutes before)

**Barcelona Set Theory Seminar****Time:** Wednesday, 19 May, 16:00-17:30 CET**Speaker:** Luca Incurvati**Title:** Iteration, dependence and structuralism**Abstract:** In the first part of the talk, I clarify what is at stake in the debate

between accounts of the iterative conception based on the notion of

metaphysical dependence and the minimalist account I have defended in

previous work. Special attention will be paid to the relationship between

this debate and the debate between actualist and potentialist accounts of

the cumulative hierarchy. In the second part of the talk, I use the

distinctions drawn in the first part of the talk to assess an objection leveled

by Mark Gasser against structuralist accounts of mathematics.**Information:** Online. If you wish to attend, please send an email to bagaria@ub.edu asking for the link.

**Münster research seminar on set theory****Time:** Wednesday, 19 May, 15:15-16:45 CET**Speaker:** Azul Fatalini**Title:** Forcing a Mazurkiewicz set.**Abstract:** A subset of the plane is called a Mazurkiewicz set iff its intersection with every line is exactly two points. There is a well-known construction of these sets in ZFC, using transfinite recursion of the length of the continuum. We will talk about the construction of a model of ZF+DC with no well-ordering of the reals that has a Mazurkiewicz set.**Information:** Please check the seminar webpage to see if the seminar takes place. Contact rds@wwu.de ahead of time in order to participate.

**Hebrew University-Bar Ilan University Set Theory seminar****Time:** Wednesday, 19 May, 14:00-16:00 Israel Time (13:00-15:00 CET)**Speaker:** Menachem Magidor**Title:** When complicated relation on a Polish space can be Borel**Abstract:** tba**Information:** Please check on the seminar webpage if the seminar will take place. Contact Menachem Magidor, Asaf Rinot or Omer Ben-Neria ahead of time for the zoom link.

**Caltech Logic Seminar****Time:** Monday, 17 May, 12:00 – 1:00pm Pacific time (21:00 CET)**Speaker:** Robin Tucker-Drob, Texas A&M**Title:** Orbit equivalence and wreath products**Abstract:** Let FF be a nonabelian free group. We show that, for any two nontrivial finite groups, the natural actions of the wreath product groups A≀FA≀F and B≀FB≀F, on AFAF and BFBFrespectively, are orbit equivalent. On the other hand, we show that these actions are not even stably orbit equivalent if FF is replaced with any ICC sofic group with property (T), and AA and BB have different cardinalities. This is joint work with Konrad Wrobel.**Information:** Check on the seminar webpage if the seminar will take place.

10-16 May

**CUNY Set Theory Seminar****Time:** Friday, 14 May, 2pm New York time (20:00 CET)**Speaker: **Corey Switzer, University of Vienna**Title:** Tight Maximal Eventually Different Families**Abstract:** Maximal almost disjoint (MAD) families and their relatives have been an important area of combinatorial and descriptive set theory since at least the 60s. In this talk I will discuss some relatives of MAD families, focussing on eventually different families of functions f:ω→ω and eventually different sets of permutations p∈S(ω). In the context of MAD families it has been fruitful to consider various strengthenings of the maximality condition to obtain several flavors of ‘strongly’ MAD families. One such strengthening that has proved useful in recent literature is that of *tightness*. Tight MAD families are Cohen indestructible and come with a properness preservation theorem making them nice to work with in iterated forcing contexts.

I will introduce a version of tightness for maximal eventually different families of functions f:ω→ω and maximal eventually different families of permutations p∈S(ω) respectively. These tight eventually different families share a lot of the nice, forcing theoretic properties of tight MAD families. Using them, I will construct explicit witnesses to ae=ap=ℵ1 in many known models of set theory where this equality was either not known or only known by less constructive means. Working over L we can moreover have the witnesses be Π11 which is optimal for objects of size ℵ1 in models where CH fails. These results simultaneously strengthen several known results on the existence of definable maximal sets of reals which are indestructible for various definable forcing notions. This is joint work with Vera Fischer.**Information:** The seminar will take place virtually. Please email Victoria Gitman (vgitman@nylogic.org) for the meeting id.

**Toronto Set Theory Seminar****Time:** Friday, 14 May, 1.30-3pm Toronto time (19:30-21:00 CET)**Speaker:** tba**Title:** tba**Abstract:** tba**Information:** No webpage available. Email Ivan Ongay Valverde to receive the seminar announcements and for the zoom link.

**Turin-Udine logic seminar****Time:** Friday, 14 May, 16:30-18:30 CET**Speaker:** R. Sklinos, Stevens Institute of Technology**Title:** Fields interpretable in the free group**Abstract:** After Sela and Kharlampovich-Myasnikov proved that nonabelian free groups share the same common theory, a model theoretic interest for the theory of the free group arose. Moreover, maybe surprisingly, Sela proved that this common theory is stable. Stability is the first dividing line in Shelah’s classification theory and it is equivalent to the existence of a nicely behaved independence relation – forking independence. This relation, in the theory of the free group, has been proved (Ould Houcine-Tent and Sklinos) to be as complicated as possible (n-ample for all n). This behavior of forking independence is usually witnessed by the existence of an infinite field. We prove that no infinite field is interpretable in the theory of the free group, giving the first example of a stable group which is ample but does not interpret an infinite field.**Information:** Please check on the semianr webpage if the seminar will take place. Online on WebEx. Please see the seminar webpage.

**Barcelona Set Theory Seminar****Time:** Wednesday, 12 May, 16:00-17:30 CET**Speaker:** Sakae Fuchino**Title:** Generically supercompact cardinals as reflection principles**Abstract:** Bernhard König proved in his 2004 paper that the assertion omega_2 is

generically supercompact by sigma-closed forcing” is equivalent to his “Strong

Game Reflection Principle”.

We consider a generalization of this result and discuss about the relationship of

generic supercompactness of omega_2 and the existence of a Laver-generically

supercompact cardinal. The talk is connected with the on-going research

programme of the speaker together with Hiroshi Sakai which was started with

[1] Sakaé Fuchino, André Ottenbreit Maschio Rodrigues, and Hiroshi

Sakai. Strong downward Löwenheim-Skolem theorems for stationary logics I,

Archive for Mathematical Logic, Vol.60, 1-2, (2021), 17–47.**Information:** Online. If you wish to attend, please send an email to bagaria@ub.edu asking for the link.

**Münster research seminar on set theory****Time:** Wednesday, 12 May, 15:15-16:45 CET**Speaker:** Jindrich Zapletal, University of Florida**Title:** Chromatic numbers of distance graphs on Euclidean spaces**Abstract:** Let Gn be the graph on n-dimensional Euclidean space connecting points of rational distance. I will show that it is consistent relative to an inaccessible cardinal that ZF+DC holds, chromatic number of G3 is countable, yet the chromatic number of G4 is uncountable. I will use the opportunity to explain the basic concepts,methods, and results of geometric set theory, as contained in a recent book with Paul Larson.

In the first lecture, I will provide a broad outline of geometric set theory. I will define balanced forcing, a class of partial orders which can be used to prove numerous independence results in ZF+DC, and prove its central theorems. In its usage and flexibility, balanced forcing is a parallel to proper forcing in the context of choiceless set theory. In the second lecture, I will discuss chromatic numbers of algebraic hypergraphs in general and the rational distance graphs in particular. Finally, I will construct a coloring poset which yields the consistency result mentioned in the first paragraph.**Information:** Please check the seminar webpage to see if the seminar takes place. Contact rds@wwu.de ahead of time in order to participate.

**Hebrew University-Bar Ilan University Set Theory seminar****Time:** Wednesday, 12 May, 14:00-16:00 Israel Time (13:00-15:00 CET)**Speaker:** Menachem Magidor**Title:** When complicated relation on a Polish space can be Borel**Abstract:** tba**Information:** Please check on the seminar webpage if the seminar will take place. Contact Menachem Magidor, Asaf Rinot or Omer Ben-Neria ahead of time for the zoom link.

**Caltech Logic Seminar****Time:** Monday, 10 May, 12:00 – 1:00pm Pacific time (21:00 CET)**Speaker:** Luca Motto Ros, University of Turin**Title:** Arcs, knots, and convex embeddability**Abstract:** Working in the framework of Borel reducibility, we analyze the complexity of the natural counterparts in terms of quasi-orders of the well-known relations of equivalence for arcs and knots. It turns out that this problem is related to the study of convex embeddability between countable linear orders (and of its analogue for circular orders), which is a topic of independent interest. This is work in progress, joint with Iannella, Kulikov and Marcone.**Information:** Check on the seminar webpage if the seminar will take place.

3-9 May

**CUNY Set Theory Seminar****Time:** Friday, 7 May, 2pm New York time (20:00 CET)**Speaker: **Benjamin Goodman, CUNY**Title:** tba**Abstract:** tba**Information:** The seminar will take place virtually. Please email Victoria Gitman (vgitman@nylogic.org) for the meeting id.

**Toronto Set Theory Seminar****Time:** Friday, 7 May, 1.30-3pm Toronto time (19:30-21:00 CET)**Speaker:** tba**Title:** tba**Abstract:** tba**Information:** No webpage available. Email Ivan Ongay Valverde to receive the seminar announcements and for the zoom link.

**Turin-Udine logic seminar****Time:** Friday, 7 May, 16:30-18:30 CET**Speaker:****Title:****Abstract:****Information:** Please check on the semianr webpage if the seminar will take place. Online on WebEx. Please see the seminar webpage.

**KGRC Research Seminar, ViennaTime:** Thursday, 6 May, 15:00-16:30 CET

**Speaker:**Matteo Viale, Università degli Studi di Torino, Italy

**Title:**tba

**Abstract:**tba

**Information:**Talk via zoom.

**Münster research seminar on set theory****Time:** Wednesday, 5 May, 15:15-16:45 CET**Speaker:** Jindrich Zapletal, University of Florida**Title:** Chromatic numbers of distance graphs on Euclidean spaces**Abstract:** Let Gn be the graph on n-dimensional Euclidean space connecting points of rational distance. I will show that it is consistent relative to an inaccessible cardinal that ZF+DC holds, chromatic number of G3 is countable, yet the chromatic number of G4 is uncountable. I will use the opportunity to explain the basic concepts,methods, and results of geometric set theory, as contained in a recent book with Paul Larson.

In the first lecture, I will provide a broad outline of geometric set theory. I will define balanced forcing, a class of partial orders which can be used to prove numerous independence results in ZF+DC, and prove its central theorems. In its usage and flexibility, balanced forcing is a parallel to proper forcing in the context of choiceless set theory. In the second lecture, I will discuss chromatic numbers of algebraic hypergraphs in general and the rational distance graphs in particular. Finally, I will construct a coloring poset which yields the consistency result mentioned in the first paragraph.**Information:** Please check the seminar webpage to see if the seminar takes place. Contact rds@wwu.de ahead of time in order to participate.

**Hebrew University-Bar Ilan University Set Theory seminar****Time:** Wednesday, 5 May, 14:00-16:00 Israel Time (13:00-15:00 CET)**Speaker:** tba**Title:** tba**Abstract:** tba**Information:** Please check on the seminar webpage if the seminar will take place. Contact Menachem Magidor, Asaf Rinot or Omer Ben-Neria ahead of time for the zoom link.

26 April – 2 May

**CUNY Set Theory Seminar****Time:** Friday, 30 April, 2pm New York time (20:00 CET)**Speaker:** Elliot Glazer, Harvard University**Title:** Paradoxes of perfectly small sets**Abstract:** We define a set of real numbers to be perfectly small if it has perfectly many disjoint translates. Such sets have a strong intuitive claim to being probabilistically negligible, yet no non-trivial measure assigns them all a value of 0. We will prove from a moderate amount of choice that any total extension of Lebesgue measure concentrates on a perfectly small set, suggesting that for any such measure, translation-invariance fails ‘as badly as possible.’ From the ideas of this proof, we will also derive analogues of well-known paradoxes of randomness, specifically Freiling’s symmetry paradox and the infinite prisoner hat puzzle, in terms of perfectly small sets. Finally, we discuss how these results constrain what a paradox-free set theory can look like and some related open questions.**Information:** The seminar will take place virtually. Please email Victoria Gitman (vgitman@nylogic.org) for the meeting id.

**Toronto Set Theory Seminar****Time:** Friday, 20 April, 1.30-3pm Toronto time (19:30-21:00 CET)**Speaker:** tba**Title:** tba**Abstract:** tba**Information:** No webpage available. Email Ingay Valverde to receive the seminar announcements and for the zoom link.

**Turin-Udine logic seminar****Time:** Friday, 30 April, 16:30-18:30 CET**Speaker:** S. Barbina, Open University**Title:** The theory of the universal-homogeneous Steiner triple system**Abstract:** A Steiner triple system is a set S together with a collection B of subsets of S of size 3 such that any two elements of S belong to exactly one element of B. It is well known that the class of finite Steiner triple systems has a Fraïssé limit, the countable homogeneous universal Steiner triple system M. In joint work with Enrique Casanovas, we have proved that the theory T of M has quantifier elimination, is not small, has TP2, NSOP1, eliminates hyperimaginaries and weakly eliminates imaginaries. In this talk I will review the construction of M, give an axiomatisation of T and prove some of its properties.**Information:** Please check on the semianr webpage if the seminar will take place. Online on WebEx. Please see the seminar webpage.

**KGRC Research Seminar, ViennaTime:** Thursday, 29 April, 15:00-16:30 CET

**Speaker:**Moreno Pierobon, Università di Pisa, Italy

**Title:**Fullness and mixing property for boolean valued models

**Abstract:**Besides being one of the classical approaches to forcing, boolean valued models provide a flexible tool to produce a variety of structures.

In this talk, we will investigate in details the fullness property and the mixing property for boolean valued models. The former is necessary to control the semantics when quotienting a boolean valued model by an ultrafilter. The latter implies the former and it is easier to check.

We will show that not every model is full, and the mixing property in not equivalent to fullness. Moreover, we will improve the classical Łoś Theorem for boolean valued models.

In the end, we will give a simple characterization of the mixing property using étalé spaces. This last result is an easy corollary of a more general study we made on the categorical equivalence between boolean valued models and presheaves.

This is a joint work with Matteo Viale.

**Information:**Talk via zoom.

**Barcelona Set Theory Seminar****Time:** Wednesday, 28 April, 16:00-17:30 CET**Speaker:** Yair Hayut**Title:** omega-strongly measurable cardinals**Abstract:** In his profound work towards the identification of the ultimate-L (the ultimate canonical inner model), Woodin isolated a key ingredient: the w-

strongly measurable cardinals. Those cardinals are regular in V and measurable in HOD for a very simple reason – the intersection of their club filter

with HOD splits into a small collection of isolated normal measures. Woodin’s

HOD-dichotomy implies that if one can prove that there are class many regular

cardinals which are not strongly measurable, and there exists an extendible

cardinal, then some covering theorem holds between HOD and V.

In this talk I will present a recent joint result with Omer Ben-Neria, proving the

consistency of the existence of class many strongly measurable cardinals

(indeed, all the successors of regular cardinals), from a rather mild large

cardinal hypothesis, in the realm of o(κ) = κ.

I will focus on the details of the proof for the first two cardinals À1 and À2.**Information:** Online. If you wish to attend, please send an email to bagaria@ub.edu asking for the link.

**Münster research seminar on set theory****Time:** Wednesday, 28 April, 15:15-16:45 CET**Speaker:** Farmer Schlutzenberg **Title:** Local mantles of L[x]**Abstract:** Recall that for a cardinal κ, a <κ-ground is an inner model W of ZFC such that V is a set-generic extension of W, as witnessed by a forcing of size <κ, and the κ-mantle is the intersection of all <κ-grounds. We will start with a brief overview

of some known facts on the κ-mantle. Following this, assuming sufficient large cardinals, we will analyze the κ-mantle M of L[x], where x is a real of sufficiently high complexity, and κ is a limit cardinal of uncountable cofinality in L[x]. We will show in particular that M models ZFC + GCH + “There is a Woodin cardinal”. We will also discuss a variant, joint with John Steel, for the κ-cc mantle, where κ is regular uncountable in L[x] and κ≤ the least Mahlo of L[x]. The proof relies on Woodin’s analysis of HODL[x,G] and Schindler’s generation of grounds, and is motivated by work of Fuchs, Sargsyan, Schindler and the author on Varsovian models and the mantle.**Information:** Please check the seminar webpage to see if the seminar takes place. Contact rds@wwu.de ahead of time in order to participate.

**Hebrew University-Bar Ilan University Set Theory seminar****Time:** Wednesday, 28 April, 14:00-16:00 Israel Time (13:00-15:00 CET)**Speaker:** Menachem Magidor**Title:** When is a nice complicated equivalence relation Borel some where**Abstract:** The original problem is due to Kanovei, Sabok and Zapletal: Given an analytic equivalence relation , which is not Borel. Can we find a non trivial Borel set , such that the restriction of the relation to it is Borel.

‘Non trivial” here means positive with respect to some sigma-complete ideal on the Borel algebra It turns out that in order to avoid simple counter example we have to make some assumptions about the equivalence relation and about the ideal.There are some results due (independently) to Chan and Drucker, about this problem assuming some large cardinals.

We shall survey some of these results and discuss the issue of trying to generalize these results to larger family of equivalence relations (e.g. Universally Baire)-These are joint results with W. Chan. **Information:** Please check on the seminar webpage if the seminar will take place. Contact Menachem Magidor, Asaf Rinot or Omer Ben-Neria ahead of time for the zoom link.

**Caltech Logic Seminar****Time:** Monday, 26 April, 12:00 – 1:00pm Pacific time (21:00 CET)**Speaker:** Noé de Rancourt, University of Vienna**Title:** A dichotomy for countable unions of smooth Borel equivalence relations**Abstract:** I will present a dichotomy for equivalence relations on Polish spaces that can be expressed as countable unions of smooth Borel subequivalence relations. It can be seen as an extension of Kechris-Louveau’s dichotomy for hypersmooth Borel equivalence relations. A generalization of our dichotomy, for equivalence relations that can be expressed as countable unions of Borel equivalence relations belonging to certain fixed classes, will also be presented. This is a joint work with Benjamin Miller.**Information:** Check on the seminar webpage if the seminar will take place.

19-25 April

**CUNY Set Theory Seminar****Time:** Friday, 23 April, 2pm New York time (20:00 CET)**Speaker:** Andres Villaveces, CUNY**Title:** Two logics, and their connections with large cardinals / Questions for BDGM: Part II**Abstract:** In the past couple of years I have been involved (joint work with Väänänen and independently with Shelah) with some logics in the vicinity of Shelah’s L1κ (a logic from 2012 that has Interpolation and a very weak notion of compactness, namely Strong Undefinability of Well-Orderings, and in some cases has a Lindström-type theorem for those two properties). Our work with Väänänen weakens the logic but keeps several properties. Our work with Shelah explores the connection with definability of AECs.

These logics seem to have additional interesting properties under the further assumption of strong compactness of a cardinal, and this brings them close to recent work of Boney, Dimopoulos, Gitman and Magidor [BDGM].

During the first lecture, I plan to describe two games and a syntax of two logics: Shelah’s L1κ and my own logic (joint work with Väänänen) L1,cκ. I will stress some of the properties of these logics, without any use of large cardinal assumptions. During the second lecture, I plan to enter rather uncharted territory. I will describe some constructions done by Shelah (mostly) under the assumption of strong compactness, but I also plan to bring these logics to a territory closer to the work of [BDGM]. This second lecture will have more conjectures, ideas, and (hopefully interesting) discussions with some of the authors of that paper.**Information:** The seminar will take place virtually. Please email Victoria Gitman (vgitman@nylogic.org) for the meeting id.

**Toronto Set Theory Seminar****Time:** Friday, 23 April, 1.30-3pm Toronto time (19:30-21:00 CET)**Speaker:** tba**Title:** tba**Abstract:** tba**Information:** No webpage available. Email Ivan Ongay Valverde to receive the seminar announcements and for the zoom link.

**Turin-Udine logic seminar****Time:** Friday, 23 April, 16:30-18:30 CET**Speaker:** F. Loregian, Tallinn University of Technology**Title:** Functorial Semantics for Partial Theories**Abstract:** We provide a Lawvere-style definition for partial theories, extending the classical notion of equational theory by allowing partially defined operations. As in the classical case, our definition is syntactic: we use an appropriate class of string diagrams as terms. This allows for equational reasoning about the class of models defined by a partial theory. We demonstrate the expressivity of such equational theories by considering a number of examples, including partial combinatory algebras and cartesian closed categories. Moreover, despite the increase in expressivity of the syntax we retain a well-behaved notion of semantics: we show that our categories of models are precisely locally finitely presentable categories, and that free models exist.**Information:** Online on WebEx. Please see the seminar webpage.

**KGRC Research Seminar, ViennaTime:** Thursday, 22 April, 15:00-16:30 CET

**Speaker:**Osvaldo Guzmán, Universidad Nacional Autónoma de México

**Title:**MAD families and strategically bounding forcings

**Abstract:**The notion of strategically bounding forcings is a natural game-theoretic strengthening of the bounding property for partial orders. In this talk, we will study the basic properties of strategically bounding forcings and talk about indestructibility of MAD families. The motivation for this work is the problem of Roitman.

**Information:**Talk via zoom.

**Barcelona Set Theory Seminar****Time:** Wednesday, 21 April, 16:00-17:30 CET**Speaker:** Sam Roberts**Title:** Reinhardt’s potentialism**Abstract:** Reflection principles have been of interest to philosophers and mathematicians because they promise to be well-motivated additions to the standard axioms of set theory that nonetheless settle many of the questions left open by those axioms. Although William Reinhardt’s work on reflection principles has been immensely influential, some of his central ideas have remained unclear. The purpose of my talk will be to rectify this. I will start by outlining and formalising his primary contribution to the literature on reflection principles, which is a version of potentialism. I will then show that it is remarkably strong and discuss a number of criticisms.**Information:** Online. If you wish to attend, please send an email to bagaria@ub.edu asking for the link.

**Münster research seminar on set theory****Time:** Wednesday, 21 April, 15:15-16:45 CET**Speaker:** tba**Title:** tba**Abstract:** tba**Information:** Please check the seminar webpage to see if the seminar takes place. Contact rds@wwu.de ahead of time in order to participate.

**Hebrew University-Bar Ilan University Set Theory seminar****Time:** Wednesday, 21 April, 14:00-16:00 Israel Time (13:00-15:00 CET)**Speaker:** Omer Ben Neria**Title:** Strong Prikry property for Magidor Iterations**Abstract:** In his celebrated work on the identity crisis of strongly compact cardinals, Magidor introduced a special iteration of Prikry forcings for a set of measurable cardinals, known as the Magidor iteration.The purpose of this talk is to state and prove a version of the strong Prikry Lemma for such iterations, extending a result of Fuchs for the case where the set of measurables is discrete. We will also describe several applications regarding the genericity of sequences of critical points in iterated ultrapowers.**Information:** Please check on the seminar webpage if the seminar will take place. Contact Menachem Magidor, Asaf Rinot or Omer Ben-Neria ahead of time for the zoom link.

**Caltech Logic Seminar****Time:** Monday, 19 April, 12:00 – 1:00pm Pacific time (21:00 CET)**Speaker:** Francois Le Maitre, University of Paris**Title:** A characterization of high transitivity for groups acting on trees**Abstract:** A countable group is highly transitive if it admits an embedding in the permutation group of the integers with dense image. I will present a joint work with Pierre Fima, Soyoung Moon and Yves Stalder where we show that a large class of groups acting on trees are highly transitive, which yields a characterization of high transitivity for groups admitting a minimal faithful action of general type on a tree thanks to the work of Le Boudec and Matte Bon. Our proof is new even for the free group on two generators and I will give a detailed overview in this very particular case, showing that the generic transitive action of the free group on two generators is highly transitive.**Information:** Check on the seminar webpage if the seminar will take place.

12-18 April

**CUNY Set Theory Seminar****Time:** Friday, 16 April, 2pm New York time (20:00 CET)**Speaker:** Andres Villaveces, CUNY**Title:** tba**Abstract:** tba**Information:** The seminar will take place virtually. Please email Victoria Gitman (vgitman@nylogic.org) for the meeting id.

**Toronto Set Theory Seminar****Time:** Friday, 16 April, 1.30-3pm Toronto time (19:30-21:00 CET)**Speaker:** Micheal Hrusak**Title:** Ultrafiters, MAD families and the Katetov order**Abstract:** We shall survey recent results concerning classification of MAD

families and ultrafilters using the Katetov order, concentrating on

open problems.**Information:** No webpage available. Email Ivan Ongay Valverde to receive the seminar announcements and for the zoom link.

**Turin-Udine logic seminar****Time:** Friday, 16 April, 16:30-18:30 CET**Speaker:** Alejandro Poveda, Hebrew University of Jerusalem**Title:** Forcing iterations around singulars cardinals and an application to stationary reflection**Abstract:** In this talk we will give an overview of the theory of \Sigma-Prikry forcings and their iterations, recently introduced in a series of papers. We will begin motivating the class of \Sigma-Prikry forcings and showing that this class is broad enough to encompass many Prikry-type posets that center on countable cofinalities. Afterwards, we will present a viable iteration scheme for this family and discuss an application of the framework to the investigation of stationary reflection at the level of successors of singular cardinals. This is joint work with A. Rinot and D. Sinapova.**Information:** Online on WebEx. Please see the seminar webpage.

**Münster research seminar on set theory****Time:** Friday, 16 April, 10:00-12:00 CET (Unusual time!)**Speaker:** Liang Yu (Nanjing University).**Title:** A basis theorem for Π11-sets.**Abstract:** It was claimed by Harrington, but never published, that every non-thin Π11-set ranges over an upper cone of hyperarithmetic degrees. We shall give a proof via a full approximation argument.**Information:** Please check the seminar webpage to see if the seminar takes place. Contact rds@wwu.de ahead of time in order to participate.

**KGRC Research Seminar, ViennaTime:** Thursday, 15 April, 15:00-16:30 CET

**Speaker:**Andreas Blass, University of Michigan

**Title:**Choice, Groups, and Topoi

**Abstract:**Work of Tarski, Mostowski, Gauntt, and Truss provides finite, group-theoretic criteria for ZF-provability of implications between weak choice axioms of the form “every family of n-element sets has a choice function” or “every countable family of n-element sets has a choice function.” From a sufficiently broad, category-theoretic viewpoint, these implications and the equivalent group-theoretic criteria look like exactly the same statements but interpreted in different categories, namely certain particular sorts of topoi. The main result is that this equivalence applies not only to these particular sorts of topoi but to all topoi. I plan to describe the ingredients of this work — choice principles, group properties, and topoi — and, if time permits, give a hint about the ideas in the proofs.

**Information:**Talk via zoom.

**Ghent-Leeds Virtual Logic SeminarTime:** Thursday, 8 April, 2pm UK time (15:00 CET)

**Speaker:**Adrian R. D. Mathias, LIM, Université de la Réunion

**Title:**The eternal question: Where should definitions go? Part 2: The logophilia of economists

**Abstract:**This enquiry was prompted by a discussion at the BLC meeting in Edinburgh in 2016 and has led me to pursue two themes:

– the fear of logic evinced by many mathematicians

– the strong interest in logic evinced by many economist.

I shall give many examples of the first in the first talk, and of the second in the second The ideas of Husserl illuminate the relationship of economics to mathematics and logic. The Campanella principle (1590?) is that definitions should be at the beginning for teaching but at the end for research. The Humboldt principle (1810) is that in universities, teaching and research should be done by the same people. There is a hint of contradiction here, but the aim of teaching future researchers is to build confidence as well as to transmit learning.

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

**Barcelona Set Theory Seminar****Time:** Wednesday, 14 April, 16:00-17:30 CET**Speaker:** Erin Carmody, Fordham University**Title:** The relationships between measurable and strongly compact cardinals**Abstract:** This talk is about the ongoing investigation of the relationships

between measurable and strongly compact cardinals. I will present some of

the history of the theorems in this theme, including Magidor’s identity crisis,

and give new results. The theorems presented are in particular about the

relationships between strongly compact cardinals and measurable cardinals

of different Mitchell orders. One of the main theorems is that there is a

universe where k1 and k2 are the first and second strongly compact cardinals,

respectively, and where k1 is least with Mitchell order 1, and k2 is the least

with Mitchell order 2. Another main theorem is that there is a universe where

k1 and k2 are the first and second strongly compact cardinals,

respectively, with k1 the least measurable cardinal such that o(k1) = 2 and k2

the least measurable cardinal above k1. This is a joint work in progress with

Victoria Gitman and Arthur Apter.**Information:** Online. If you wish to attend, please send an email to bagaria@ub.edu asking for the link.

**Paris-Lyon Séminaire de Logique****Time:** Wednesday, 24 March, 16:00-17:00 CET**Speaker:** Joel Hamkins, University of Oxford**Title:** Determinacy for proper class games**Abstract:** The principle of open determinacy for class games — two-player games of perfect information with plays of length ω, where the moves are chosen from a possibly proper class, such as games on the ordinals — is not provable in Zermelo-Fraenkel set theory ZFC or Gödel-Bernays set theory GBC, if these theories are consistent, because provably in ZFC there is a definable open proper class game with no definable winning strategy. In fact, the principle of open determinacy and even merely clopen determinacy for class games implies Con(ZFC) and iterated instances Con(Con(ZFC)) and more, because it implies that there is a satisfaction class for first-order truth, and indeed a transfinite tower of truth predicates for iterated truth-about-truth, relative to any class parameter. This is perhaps explained, in light of the Tarskian recursive definition of truth, by the more general fact that the principle of clopen determinacy is exactly equivalent over GBC to the principle of elementary transfinite recursion ETR over well-founded class relations. Meanwhile, the principle of open determinacy for class games is strictly stronger, although it is provable in the stronger theory GBC+ Pi^1_1-comprehension, a proper fragment of Kelley-Morse set theory KM.

http://jdh.hamkins.org/determinacy-for-proper-class-games-seminaire-de-logique-lyon-paris-april-2021/**Information:** Join via the link on the seminar webpage.

**Caltech Logic Seminar****Time:** Monday, 12 April, 12:00 – 1:00pm Pacific time (21:00 CET)**Speaker:** Riccardo Camerlo, University of Genoa**Title:** The Wadge hierarchy on Zariski topologies**Abstract:** In this talk I will discuss the structure of the relation of continuous reducibility on affine varieties. If time permits, I will also present some results on polynomial reducibility. The results are joint work with C. Massaza.**Information:** See the seminar webpage.

5-11 April

**CUNY Set Theory Seminar****Time:** Friday, 9 April, 2pm New York time (20:00 CET)**Speaker:** Sandra Müller, University of Vienna**Title:** The exact consistency strength of ‘AD + all sets are universally Baire’**Abstract:** The large cardinal strength of the Axiom of Determinacy when enhanced with the hypothesis that all sets of reals are universally Baire is known to be much stronger than the Axiom of Determinacy itself. In fact, Sargsyan conjectured it to be as strong as the existence of a cardinal that is both a limit of Woodin cardinals and a limit of strong cardinals. Larson, Sargsyan and Wilson showed that this would be optimal via a generalization of Woodin’s derived model construction. We will discuss a new translation procedure for hybrid mice extending work of Steel, Zhu and Sargsyan and use this to prove Sargsyan’s conjecture.**Information:** Please check on the seminar webpage if the seminar will take place. The seminar will take place virtually. Please email Victoria Gitman (vgitman@nylogic.org) for the meeting id.

**Toronto Set Theory Seminar****Time:** Friday, 9 April, 10:30am to 12:00pm Toronto time (16:30-18:00 CET)**Speaker:** Joerg Brendle**Title:** Combinatorics of ultrafilters on complete Boolean algebras **Abstract:** The combinatorial structure of ultrafilters on the natural numbers has been investigated intensively for many decades, and a lot is known about the order structure of such ultrafilters (under either the Tukey or the Rudin-Keisler ordering), about special classes of ultrafilters (like P-points),or about cardinal invariants related to ultrafilters (like the ultrafilter number). Yet, very little has beendone so far concerning combinatorial aspects of ultrafilters on general Boolean algebras, and thepurpose of this talk will be to present some basic results in this direction.

Focus will be put on the Tukey ordering, on (non)existence of non-Tukey-maximal ultrafilters, on ultrafilter numbers, and on an analogue of the Rudin-Keisler ordering in the context of complete Boolean algebras. We will in particular deal with Cohen and random algebras. This is joint work with Francesco Parente.**Information:** No webpage available. Email Ivan Ongay Valverde to receive the seminar announcements and for the zoom link.

**Turin-Udine logic seminar****Time:** Friday, 9 April, 16:30-18:30 CET**Speaker:** A. Berarducci, University of Pisa**Title:** Asymptotic analysis of Skolem’s exponential functions**Abstract:** Skolem (1956) studied the germs at infinity of the smallest class of real valued functions on the positive real line containing the constant 1, the identity function x, and such that whenever f and g are in the set, f+g, fg and f^g are also in the set. This set of germs is well ordered and Skolem conjectured that its order type is epsilon-zero. Van den Dries and Levitz (1984) computed the order type of the fragment below 2^(2^x). They did so by studying the possible limits at infinity of the quotient f(x)/g(x) of two functions in the fragment: if g is kept fixed and f varies, the possible limits form a discrete set of real numbers of order type omega. Using the surreal numbers, we extend the latter result to the whole class of Skolem functions and we discuss some additional progress towards the conjecture of Skolem. This is joint work with Marcello Mamino ( http://arxiv.org/abs/1911.07576 , to appear in the JSL). **Information:** Online on WebEx. Please see the seminar webpage.

**Ghent-Leeds Virtual Logic SeminarTime:** Thursday, 8 April, 2pm UK time (15:00 CET)

**Speaker:**Adrian R. D. Mathias, LIM, Université de la Réunion

**Title:**The eternal question: Where should definitions go? Part 1: The logophobia of mathematicians

**Abstract:**This enquiry was prompted by a discussion at the BLC meeting in Edinburgh in 2016 and has led me to pursue two themes:

– the fear of logic evinced by many mathematicians

– the strong interest in logic evinced by many economist.

I shall give many examples of the first in the first talk, and of the second in the second The ideas of Husserl illuminate the relationship of economics to mathematics and logic. The Campanella principle (1590?) is that definitions should be at the beginning for teaching but at the end for research. The Humboldt principle (1810) is that in universities, teaching and research should be done by the same people. There is a hint of contradiction here, but the aim of teaching future researchers is to build confidence as well as to transmit learning.

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

**Barcelona Set Theory Seminar****Time:** Wednesday, 7 April, 16:00-17:30 CET**Speaker:** Farmer Schlutzenberg, University of Münster**Title:** Some results on restricted mantles**Abstract:** Recall that a ground of the set theoretic universe V is a class W

modelling ZFC, such that V is a set-generic extension of W. The mantle M is

the intersection of all grounds. By restricting the size or kind of forcings

permitted, one obtains variants of the mantle, intermediate between M and V.

We will discuss some work on such restricted mantles M’, with some general

results on partial choice principles in M’, and an analysis of certain M’

assuming that V=L[x] for x a real of high complexity; we will also discuss a

restriction on the nature of possible grounds of mice L[E].**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, 7 April, 14:00-16:00 Israel Time (13:00-15:00 CET)**Speaker:** Omer Ben Neria**Title:** Strong Prikry Property for Magidor Iterations**Abstract:** In his celebrated work on the identity crisis of strongly compact cardinals, Magidor introduced a special iteration of Prikry forcings for a set of measurable cardinals, known as the Magidor iteration.

The purpose of this talk is to state and prove a version of the strong Prikry Lemma for such iterations, extending a result of Fuchs for the case where the set of measurables is discrete. We will also describe several applications regarding the genericity of sequences of critical points in iterated ultrapowers.**Information:** Please check on the seminar webpage if the seminar will take place. Contact Menachem Magidor, Asaf Rinot or Omer Ben-Neria ahead of time for the zoom link.

**Caltech Logic Seminar****Time:** Monday, 5 April, 12:00 – 1:00pm Pacific time (21:00 CET)**Speaker:** Justin Moore, Cornell University**Title:** Subgroups of PLoIPLoI which do not embed into Thompson’s group **Abstract:** The group PLoIPLoI of piecewise linear orientation preserving homeomorphisms of the unit interval, equipped with composition, has a rich array of finitely generated subgroups. A basic question one can ask is when one of these groups embeds into another. One group which seems to play a particularly important role in this quasi-order is Richard Thompson’s group FF. For instance it is conjectured that every finitely generated subgroup of PLoIPLoI either contains a copy of FF or else embeds into FF. I will describe a general dynamical criterion for when a subgroup of PLoIPLoI does not embed into FF which covers all known examples. This is joint work with James Hyde. **Information:** See the seminar webpage.

29 March-4 April

**CUNY Set Theory Seminar****Time:** Friday, 2 April, 2pm New York time (20:00 CET)**Speaker:** Monroe Eskew, University of Vienna**Title:** The approximation property and generic embeddings**Abstract:** The approximation property was introduced by Hamkins for his Gap Forcing Theorem, and it has turned out to be a very useful notion, appearing for example in the partial equiconsistency result of Viale and Weiss on PFA, and in the proof of Woodin’s HOD Dichotomy Theorem. In the context of generic embeddings, there can be a useful interplay between elementarity and approximation. We discuss some recent work in this direction: (1) tensions between saturated ideals on ω2 and the tree property (with Sean Cox), (2) fragility of the strong independence spectra (with Vera Fischer), and (3) mutual inconsistency of Foreman‘s minimal generic hugeness axioms. **Information:** The seminar will take place virtually. Please email Victoria Gitman (vgitman@nylogic.org) for the meeting id.

**Toronto Set Theory Seminar****Time:** Friday, 2 April, 1.30-3pm Toronto time (19:30-21:00 CET)**Speaker:** tba**Title:** tba**Abstract:** tba**Information:** No webpage available. Email Ivan Ongay Valverde to receive the seminar announcements and for the zoom link.

**Hebrew University-Bar Ilan University Set Theory seminar****Time:** Wednesday, 31 March, 14:00-16:00 Israel Time (13:00-15:00 CET)**Speaker:** tba **Title:** tba **Abstract:** tba **Information:** Please check on the seminar webpage if the seminar will take place. Contact Menachem Magidor, Asaf Rinot or Omer Ben-Neria ahead of time for the zoom link.

**Münster research seminar on set theory****Time:** Wednesday, 31 March, 15:15-16:45 CET**Speaker:** tba**Title:** tba**Abstract:** tba**Information:** Please check the seminar webpage to see if the seminar takes place. Contact rds@wwu.de ahead of time in order to participate.

**Caltech Logic Seminar****Time:** Monday, 29 March, 12:00 – 1:00pm Pacific time (21:00 CET)**Speaker:** Natasha Dobrinen, University of Denver**Title:** Fraïssé classes with simply characterized big Ramsey degrees**Abstract:** Analogues of the infinite Ramsey Theorem to infinite structures have been studied since the 1930s, when Sierpiński gave a coloring of pairs of rationals into two colors such that, in any subset of the rationals forming a dense linear order, both colors persist. In the 1970s Galvin showed that two is the optimum number for pairs of rationals, while Erdős, Hajnal and Pósa extended Sierpiński’s result to colorings of edges in the Rado graph. The next several decades saw a steady advance of results for other structures, a pinnacle of which was the 2006 work of Laflamme, Sauer, and Vuksanović, characterizing the exact number of colors for unavoidable colorings of finite graphs inside the Rado graph, and for similar Fraïssé structures with finitely many binary relations, including the generic tournament. This exact number is called the “big Ramsey degree”, a term coined by Kechris, Pestov, and Todorčević.

In this talk, we will provide a brief overview of the area of big Ramsey degrees infinite structures. Then we will present recent joint work with Coulson and Patel, showing that free amalgamation classes, in which any forbidden substructures are 3-irreducible, have big Ramsey degrees which are simply characterized. These results extend to certain strong amalgamation classes as well, extending the results of Laflamme, Sauer, and Vuksanović. This is in contrast to the more complex characterization of big Ramsey degrees for binary relational free amalgamation classes with forbidden 2-irreducible substructures, obtained in joint work of the speaker with Balko, Chodounský, Hubička, Konečný, Vena, and Zucker. The work with Coulson and Patel develops coding trees of quantifier-free 1-types and uses forcing to do an unbounded search for monochromatic finite objects. Furthermore, we work with skew subtrees with branching degree two which still code the Fraïssé limit. This allows for more ease when working with relations of arity greater than two, and also allows us to give the first proof of exact big Ramsey degrees bypassing the standard method of “envelopes”. It also sets the stage for current work of the speaker on infinite-dimensional Ramsey theory, in the vein of Galvin-Příkrý, for Fraïssé limits of free amalgamation classes in which any forbidden substructures are 3-irreducible.**Information:** See the seminar webpage.

22-28 March

**CUNY Set Theory Seminar****Time:** Friday, 26 March, 11am New York time (16:00 CET)**Speaker:** Carolin Antos, University of Konstanz**Title:** The ‘algebraic’ vs. ‘non-algebraic’ distinction: New impulses for the universe/multiverse debate?**Abstract:** The distinction between ‘algebraic’ and ‘non-algebraic fields in mathematics, coined by Shapiro (1997), plays an important role in discussions about the status of set theory and connects back to the so-called universe/multiverse debate in the philosophy of set theory. In this talk we will see, that this distinction is not as clear cut as is usually assume when using it in the debate. In particular, we will see that in more recent formulations of this distinction, multiversism seems to split into a a strong and a weaker form. This can be translated to a meta-level, when considering the background theory in which set-theoretic multiversism can take place. This offers a more fine-grained picture of multiversism and allows us to mitigate a standard universist objection based on the conception of a multiversist background theory.**Information:** The seminar will take place virtually. Please email Victoria Gitman (vgitman@nylogic.org) for the meeting id.

**Toronto Set Theory Seminar****Time:** Friday, 26 March, 10.30-12pm Toronto time (15:30-17:00 CET)**Speaker:** Sakaé Fuchino**Title:** Laver-generically large cardinal and the Continuum Problem**Abstract:** Let us call a class $\calP$ of posets iterable, if, for any $\poP\in\calP$ and $\calP$-name

$\utpoQ$\vspace{-0.5\smallskipamount} \st\ $\forces{\poP}{\utpoQ\in\calP}$, we have

$\poP\ast\utpoQ\in\calP$.

For an iterable class $\mathcal{P}$ of posets, a cardinal $\mu$ is called {\it Laver-generically

supercompact for $\mathcal{P}$}, if, for any $\mathbb{P}\in\mathcal{P}$ and $\lambda\in\On$,

there is a $\poP$-name $\utpoQ$\vspace{-0.5\smallskipamount} with $\forces{\poP}{\utpoQ\in\calP}$ \st, letting

$\poQ=\poP\ast\utpoQ$,

there are $j$, $M\subseteq\uniV[\genH]$ for $(\uniV, \mathbb{Q})$-generic

$\genH$ such that

1) $\elembed{j}{V}{M}$,\smallskip

2) $crit(j)=\mu$, $j(\mu)>\lambda$,\smallskip

3) $\cardof{\poQ}\leq j(\mu)$,\smallskip

4) $\poP$, $\genH\in M$ and \smallskip

5) $j\imageof\lambda\in M$.\\\\

The notion of Laver-generically superhugeness is obtained when \assert{5} is replaced by

5′) $j\imageof j(\mu)\in M$.

The notion of Laver-generically large cardinal for $\calP$ given here is stronger than the one introduced in \cite{II} and is called there the {\it strongly} and {\it tightly}

Laver-generically large cardinal (the strongness corresponds the usage of two-step

iteration in the definition instead of just $\poP\circleq\poQ$, and the tightness the

condition \assert{3}).

In my talk, I will give a proof of the following:\quad

For many natural iterable class of proper posets $\mathcal{P}$, a

Laver-generically supercompact cardinal $\mu$ for $\poP$ is either $\aleph_2$ or very large (if it exists), and the continuum is either $\aleph_1$ or $\aleph_2$, or $\geq\mu$ in case of very large $\mu$, where it depends on $P$ which scenario we have.

If time allows, I will also sketch a proof of the following theorem:\quad

If $\mathcal{P}$ is the class of c.c.c.\ posets (or some other iterable class $\calP$ of posets preserving all cardinalities but adding some real), and if $\mu$ is Laver-generically superhuge for $\mathcal{P}$, then $\mu=2^{\aleph_0}$.

At the moment, it is open if the same theorem holds for a Laver-generically supercompact.**Information:** No webpage available. Email Ivan Ongay Valverde to receive the seminar announcements and for the zoom link.

**Turin-Udine logic seminar** **Time:** Friday, 26 March, 16:30-18:30 CET **Speaker:** Vincenzo Dimonte, University of Udine**Title:** tba**Abstract:** tba**Information:** Online on WebEx. Please see the seminar webpage.

**KGRC Research Seminar, ViennaTime:** Thursday, 25 March, 15:00-16:30 CET

**Speaker:**Iván Ongay-Valverde, University of Wisconsin–Madison, USA

**Title:**Splitting Localization and prediction numbers

**Abstract:**In 1993, Newelski and Roslanowski studied some cardinal characteristics related to the unsymmetric game (I, as Geschke, called them the localization numbers). While doing this, they found the n-localization property. When a forcing has this property, you can ensure that all new reals are ‘tame’ somehow (for example, you do not add Cohen or Random reals).

In a different line of study, Andreas Blass worked with some cardinal characteristics related to the idea of guessing correctly a real number given certain amount of information (he called them evasion and prediction numbers). In 2010, it was an open question whether some possible variations of these numbers were known cardinal characteristics or not.

Impressively, these two notions are related.

In this talk, we will show that the k global adaptive prediction numbers are not any other cardinal characteristic. In particular, they are not the localization numbers. To do this, we will use techniques analogue to Newelski and Roslanowski and we will show that the n-localization can be weakened to get their result.

**Information:**Talk via zoom.

**Barcelona Set Theory Seminar** **Time:** Wednesday, 24 March, 16:00-17:30 CET**Speaker:** Peter Koellner, Harvard University**Title:** Minimal models and $\beta$-categoricity**Abstract:** Let us say that a theory T in the language of set theory is β-consistent at

α if there is a transitive model of T of height α, and let us say that it is

β-categorical at α iff there is at most one transitive model of T of height α.

The sentence V = L has the feature that ZFC + V = L is β-categorical at

α, for every α. If we assume in addition that ZFC + V = L is β-consistent at

α, then the uniquely determined model is Lα, and the minimal such model,

Lα0, is model of determined by the β-categorical theory ZFC + V = L + M,

where M is the statement “There does not exist a transitive model of ZFC.”

It is natural to ask whether V = L is the only sentence that can be β-

categorical at a countable ordinal α; that is, whether there can be a sentence φ such that for some countable α, ZFC + φ is β-categorical and β-consistent

at α, where the unique transitive model of height α is not Lα.

In the early 1970s Harvey Friedman proved a partial result in this direc-

tion. For a given ordinal α, let n(α) be the next admissible ordinal above α, and, for the purposes of this discussion, let us say that an ordinal α is

minimal iff a bounded subset of α appears in Ln(α) r Lα. [Note that α0 is

minimal (indeed a new subset of ω appears as soon as possible, namely, in

a Σ1-definable manner over Lα0+1) and an ordinal α is non-minimal iff Ln(α)

satisfies that α is a cardinal.] Friedman showed that for all countable α which

are non-minimal, V = L is the only sentence that is β-categorical at α. The

question of whether this is also true for α which are minimal has remained

open.

In this talk I will describe some joint work with Hugh Woodin that bears

on this question. In general, when approaching a “lightface” question (such as the one under consideration) it is easier to first address the “boldface”

analogue of the question by shifting from the context of L to the context

of L[x], where x is a real. In this new setting everything is relativized to

the real x: For an ordinal α, we let nx(α) be the first x-admissible ordinal

above α, and we say that α is x-minimal iff a bounded subset of α appears

in Lnx(α)[x] r Lα[x].

Theorem. Assume that there is an inner model with a Woodin cardinal and

that for all X, X# exists. There is a sentence φ in the language of set theory

with two additional constants, ̊c and ̊d, such that for a Turing cone of x,

interpreting ̊c by x, for all countable α,

(1) if Lα[x] |= ZFC, then there is an interpretation of ̊d by something in

Lα[x] such that there is a β-model of ZFC + φ of height α, and any

such model is not equal to Lα[x], and

(2) if, in addition, α is x-minimal, then there is a unique β-model of ZFC+φ

of height α, and this model is not equal to Lα[x].

The sentence φ asserts the existence of an object which is external to

Lα[x] and which, in the case where α is minimal, is canonical. The object

is a branch b through a certain tree in Lα[x], and the construction uses

techniques from the HOD analysis of models of determinacy.

In this talk I will sketch the proof, describe some additional features of

the singleton, and say a few words about why the lightface version looks

difficult.**Information:** Online. If you wish to attend, please send an email to bagaria@ub.edu asking for the link.

**Paris-Lyon Séminaire de Logique****Time:** Wednesday, 24 March, 16:00-17:00 CET**Speaker:** Laura Fontanella, Paris**Title:** Realizability and the Axiom of Choice**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 general technique that generalizes the method of Forcing. However realizing the full Axiom of Choice is quite problematic. After a brief presentation of Krivine’s techniques, we will discuss the major obstacles for realizing the Axiom of Choice and I will present my recent joint work with Guillaume Geoffroy that led to realize weak versions of the Axiom of Choice for arbitrarily large cardinals.**Information:** Join via the link on the seminar webpage.

**Münster research seminar on set theory****Time:** Wednesday, 24 March, 15:15-16:45 CET**Speaker:** tba**Title:** tba**Abstract:** tba**Information:** Please check the seminar webpage to see if the seminar takes place. Contact rds@wwu.de ahead of time in order to participate.

**Hebrew University-Bar Ilan University Set Theory seminar****Time:** Wednesday, 24 March, 14:00-16:00 Israel Time (13:00-15:00 CET)**Speaker:** Omer Ben-Neria**Title:** The strong Prikry Property for Magidor iterations**Abstract:** In his celebrated work on the identity crisis of strongly compact cardinals, Magidor introduced a special iteration of Prikry forcings for a set of measurable cardinals, known as the Magidor iteration.

The purpose of this talk is to state and prove a version of the strong Prikry Lemma for such iterations, extending a result of Fuchs for the case where the set of measurables is discrete. We will also describe several applications regarding the genericity of sequences of critical points in iterated ultrapowers.**Information:** Please check on the seminar webpage if the seminar will take place. Contact Menachem Magidor, Asaf Rinot or Omer Ben-Neria ahead of time for the zoom link.

15-21 March

**CUNY Set Theory Seminar****Time:** Friday, 19 March, 2pm New York time (19:00 CET)**Speaker:** Paul Blain Levy, University of Birmingham**Title:** Broad Infinity and Generation Principles**Abstract:** Broad Infinity is a new and arguably intuitive axiom scheme in set theory. It states that ‘broad numbers’, which are three-dimensional trees whose growth is controlled, form a set. If the Axiom of Choice is assumed, then Broad Infinity is equivalent to the Ord-is-Mahlo scheme: every closed unbounded class of ordinals contains a regular ordinal.

Whereas the axiom of Infinity leads to generation principles for sets and families and ordinals, Broad Infinity leads to more advanced versions of these principles. The talk explains these principles and how they are related under various prior assumptions: the Axiom of Choice, the Law of Excluded Middle, and weaker assumptions.**Information:** The seminar will take place virtually. Please email Victoria Gitman (vgitman@nylogic.org) for the meeting id.

**Toronto Set Theory Seminar****Time:** Friday, 19 March, 1.30-3pm Toronto time (18:30-20:00 CET)**Speaker:** Anush Tserunyan**Title:** Ergodic theorems along trees**Abstract:** In the classical pointwise ergodic theorem for a probability measure preserving (pmp) transformation $T$, one takes averages of a given integrable function over the intervals $\{x, T(x), T^2(x), \hdots, T^n(x)\}$ in the forward orbit of the point $x$. In joint work with Jenna Zomback, we prove a “backward” ergodic theorem for a countable-to-one pmp $T$, where the averages are taken over subtrees of the graph of $T$ that are rooted at $x$ and lie behind $x$ (in the direction of $T^{-1}$). Surprisingly, this theorem yields (forward) ergodic theorems for countable groups, in particular, one for pmp actions of free groups of finite rank where the averages are taken along subtrees of the standard Cayley graph rooted at the identity. For free group actions, this strengthens the best known result in this vein due to Bufetov (2000).**Information:** No webpage available. Email Ivan Ongay Valverde to receive the seminar announcements and for the zoom link.

**Turin-Udine logic seminar****Time:** Friday, 19 March, 16:30-18:30 CET**Speaker:** G. Paolini, Turin**Title:** Torsion-Free Abelian Groups are Borel Complete**Abstract:** We prove that the Borel space of torsion-free Abelian groups with domain \omega is Borel complete, i.e., the isomorphism relation on this Borel space is as complicated as possible, as an isomorphism relation. This solves a long-standing open problem in descriptive set theory, which dates back to the seminal paper on Borel reducibility of Friedman and Stanley from 1989.**Information:** Online on WebEx. Please see the seminar webpage.

**KGRC Research Seminar, ViennaTime:** Thursday, 18 March, 15:00-16:30 CET

**Speaker:**Jaroslav Šupina, Pavol Jozef Šafárik University in Košice, Slovakia

**Title:**Partition forcing

**Abstract:**A. Miller introduced in 1980 a forcing notion we refer to as a partition forcing. Although it is a variant of Sacks’ perfect set forcing, it is closely related to Miller’s rational perfect set forcing.

The talk is devoted to our application of partition forcing in a proof of consistency of u=i<aT. Here, i is the minimal cardinality of a maximal independent family, u a minimal size of an ultrafilter base, and aT is the minimal cardinality of a maximal family of pairwise almost disjoint subtrees of 2<ω.

This is a joint work with Vera Fischer.

**Information:**Talk via zoom.

**Ghent-Leeds Virtual Logic SeminarTime:** Thursday, 18 March, 2pm UK time (15:00 CET)

**Speaker:**Aleksandra Kwiatkowska, University of Münster and University of Wrocław

**Title:**Simplicity of the automorphism groups of countable structures

**Abstract:**The program of understanding the normal subgroup structure of groups that arise as automorphism groups of countable structures dates back at least to the ’50s, when Higman described all proper normal subgroups of the automorphism group of rationals (Q,<). In recent several years Tent-Ziegler, following the work of Macpherson-Tent, proved simplicity for many automorphism groups of countable graphs and metric spaces. In the talk, we prove simplicity for the automorphism groups of order and tournament expansions of ultrahomogeneous structures like the bounded Urysohn metric space and the random graph. In particular, we show that the automorphism group of the linearly ordered random graph is a simple group. This is joint work with Filippo Calderoni and Katrin Tent.

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

**Münster research seminar on set theory****Time:** Wednesday, 17 March, 15:15-16:45 CET**Speaker:** tba**Title:** tba**Abstract:** tba**Information:** Please check the seminar webpage to see if the seminar takes place. Contact rds@wwu.de ahead of time in order to participate.

**Barcelona Set Theory Seminar****Time:** Wednesday, 17 March, 16:00-17:30 CET**Speaker:** Wojciech Woloszyn, University of Oxford**Title:** Modal graph theory as a foundation of mathematics**Abstract:** One can consider the class of all graphs as a Kripke model of possible worlds, where a graph extends or accesses a larger graph just in case it is an induced subgraph thereof. In this way, we can introduce modal operators of possibility and necessity. A statement is possible at a graph if it is true in some extension of that graph, and it is necessary if it is true at all such extensions. We can thus enlarge the first-order language of graphs by closing it under modal operators, Boolean connectives, and quantification. The resulting modal language of graph theory turns out to be rather fruitful—it can express finiteness, countability, size continuum, size א1, א2, אω, first ב fixed-point, first ב-hyper-fixed-point, and so on. Perhaps most remarkably, modal graph theory can interpret set-theoretic truth in Vθ for quite a long way into the

cumulative hierarchy. Does it run out of steam or can it interpret truth in the full set-theoretic universe V, and serve as a foundation of mathematics? **Information:** Online. If you wish to attend, please send an email to bagaria@ub.edu asking for the link.

**Caltech Logic Seminar****Time:** Monday, 15 March, 12:00 – 1:00pm Pacific time (20:00 CET)**Speaker:** Vladimir Kanovei, IITP, Moscow**Title:** An unpublished theorem of Solovay on OD partitions of reals into two non-OD parts, revisited**Abstract:** A definable pair of disjoint non-OD sets of reals (hence, indiscernible sets) exists in the Sacks and E0-large generic extensions of the constructible universe LL. More specifically, if aa is a real either Sacks generic or E0 generic over L, then it is true in L[a] that: there is a Π21 equivalence relation Q on the set U, of all nonconstructible reals, with exactly two equivalence classes, and both those classes are non-OD sets. This is joint work with Ali Enayat.**Information:** See the seminar webpage.

8-14 March

**CUNY Set Theory Seminar****Time:** Friday, 12 March, 2pm New York time (20:00 CET)**Speaker:** Hossein Lamei Ramandi, Cornell University**Title:** Galvin’s question on non-σ-well ordered linear orders**Abstract:** Assume C is the class of all linear orders L such that L is not a countable union of well ordered sets, and every uncountable subset of L contains a copy of ω1. We show it is consistent that C has minimal elements. This answers an old question due to Galvin. **Information:** The seminar will take place virtually. Please email Victoria Gitman (vgitman@nylogic.org) for the meeting id.

**Toronto Set Theory Seminar****Time:** Friday, 12 March, 1.30-3pm Toronto time (19:30-21:00 CET)**Speaker:** Menachem Kojman**Title:** Strong colorings over partitions**Abstract:** Strong colorings over partitions were introduced last year by Chen-Mertens, Kojman and Steprans.

In the talk I will present the subject and continue to present the next step of the theory, which was developed in a recent joint work by Kojman, Rinot and Steprans.

The advances include stretching arguments which use Walks on Ordinals. I will present this new technique. **Information:** No webpage available. Email Ivan Ongay Valverde to receive the seminar announcements and for the zoom link.

**Turin-Udine logic seminar****Time:** Friday, 12 March, 16:30-18:30 CET**Speaker:** Clinton Conley, Carnegie Mellon University**Title:** Dividing the sphere by rotations**Abstract:** We say that a subset A of the sphere r-divides it if r-many rotations of A perfectly tile the sphere’s surface. Such divisions were first exhibited by Robinson (47) and developed by Mycielski (55). We discuss a colorful approach to finding these divisions which are Lebesgue measurable or possess the property of Baire. This includes joint work with J. Grebik, A. Marks, O. Pikhurko, and S. Unger.**Information:** Online on WebEx. Please see the seminar webpage.

**KGRC Research Seminar, ViennaTime:** Thursday, 11 March, 15:00-16:30 CET

**Speaker:**Sandra Müller, TU Wien and Universität Wien

**Title:**The exact consistency strength of “AD+ + all sets are universally Baire”

**Abstract:**The large cardinal strength of the Axiom of Determinacy when enhanced with the hypothesis that all sets of reals are universally Baire is known to be much stronger than the Axiom of Determinacy itself. In fact, Sargsyan conjectured it to be as strong as the existence of a cardinal that is both a limit of Woodin cardinals and a limit of strong cardinals. Larson, Sargsyan and Wilson showed in 2014 that this would be optimal via a generalization of Woodin’s derived model construction. We will discuss a new translation procedure for hybrid mice extending work of Steel, Zhu and Sargsyan and use this to prove Sargsyan’s conjecture.

**Information:**Talk via zoom.

**Bristol Logic and Set Theory Seminar**/**Oxford Set Theory Seminar****Time:** Wednesday, 10 March, 16:00-17:30 UK time (17:00-18:30 CEST)

**Speaker:**Peter Koellner, Harvard University

**Title:**Minimal Models and β-Categoricity

**Abstract:**Let us say that a theory T in the language of set theory is β-consistent at α if there is a transitive model of T of height α, and let us say that it is β-categorical at α iff there is at most one transitive model of T of height α. Let us also assume, for ease of formulation, that there are arbitrarily large α such that ZFC is β-consistent at α.

The sentence V = L has the feature that ZFC + V = L is β-categorical at α, for every α. If we assume in addition that ZFC + V = L is β-consistent at α, then the uniquely determined model is Lα, and the minimal such model, Lα0 , is model of determined by the β-categorical theory ZFC + V = L + M, where M is the statement “There does not exist a transitive model of ZFC.”

It is natural to ask whether V=L is the only sentence that can be β- categorical at α; that is, whether, there can be a sentence φ such that ZFC+φ is β-categorical at α, β-consistent at α, and where the unique model is not Lα. In the early 1970s Harvey Friedman proved a partial result in this direction. For a given ordinal α, let n(α) be the next admissible ordinal above α, and, for the purposes of this discussion, let us say that an ordinal α is minimal iff a bounded subset of α appears in Ln(α) Lα. [Note that α0 is minimal (indeed a new subset of ω appears as soon as possible, namely, in a Σ_1-definable manner over L(α0+1) and an ordinal α is non-minimal iff Ln(α) satisfies that α is a cardinal.] Friedman showed that for all α which are non-minimal, V = L is the only sentence that is β-categorical at α. The question of whether this is also true for α which are minimal has remained open.

In this talk I will describe some joint work with Hugh Woodin that bears on this question. In general, when approaching a “lightface” question (such

1

as the one under consideration) it is easier to first address the “boldface” analogue of the question by shifting from the context of L to the context of L[x], where x is a real. In this new setting everything is relativized to the real x: For an ordinal α, we let nx(α) be the first x-admissible ordinal above α, and we say that α is x-minimal iff a bounded subset of α appears in Lnx(α)[x] \ Lα[x].

Theorem. Assume M1# exists and is fully iterable. There is a sentence φ in the language of set theory with two additional constants, ̊c and ̊d, such that for a Turing cone of x, interpreting ̊c by x, for all α

(1) if Lα[x] |= ZFC then there is an interpretation of ̊d by something in Lα[x] such that there is a β-model of ZFC+φ of height α and not equal to Lα[x], and

(2) if, in addition, α is x-minimal, then there is a unique β-model of ZFC+φ of height α and not equal to Lα[x].

The sentence φ asserts the existence of an object which is external to Lα[x] and which, in the case where α is minimal, is canonical. The object is a branch b through a certain tree in Lα[x], and the construction uses techniques from the HOD analysis of models of determinacy.

In this talk I will sketch the proof, describe some additional features of the singleton, and say a few words about why the lightface version looks difficult.

**Information:**For the Zoom access code, contact Samuel Adam-Day me@samadamday.com. Link: https://zoom.us/j/96803195711 (open 30 minutes before)

**Barcelona Set Theory Seminar****Time:** Wednesday, 10 March, 16:00-17:30 CET**Speaker:** Carolin Antos, University of Konstanz**Title:** The “algebraic” vs. “non-algebraic” distinction: New impulses for the universe/multiverse debate?**Abstract:** The distinction between “algebraic” and “non-algebraic” fields in mathematics, coined by Shapiro (1997), plays an important role in discussions about the status of set theory and connects back to the so-called universe/multiverse debate in the philosophy of set theory. In this talk we will see that this distinction is not as clear cut as is usually assumed when using it in the debate. In particular, we will see that in more recent formulations of this distinction, multiversism seems to split into a strong and a weaker form. This can be translated to a meta-level, when considering the background theory in which set-theoretic multiversism can take place. This offers a more fine-grained picture of multiversism and allows us to mitigate a standard universist objection based on the conception of a multiversist background theory.**Information:** Online. If you wish to attend, please send an email to bagaria@ub.edu asking for the link.

**Paris-Lyon Séminaire de Logique****Time:** Wednesday, 10 March, 16:00-17:00 CET**Speaker:** Zoltán Vidnyánszky**Title:** Bases for Borel graphs of large chromatic number**Abstract:** The first part of my talk will be an introduction to the field of Borel combinatorics. I will survey some of the most important results and discuss the connections to other fields. In the second part, I will talk about the structure of the collection of graphs with large Borel chromatic number, and whether it is possible to simply characterize them.**Information:** Join via the link on the seminar webpage.

**Münster research seminar on set theory****Time:** Wednesday, 10 March, 15:15-16:45 CET**Speaker:** tba**Title:** tba**Abstract:** tba**Information:** Please check the seminar webpage to see if the seminar takes place. Contact rds@wwu.de ahead of time in order to participate.

**Hebrew University-Bar Ilan University Set Theory seminar****Time:** Wednesday, 10 March, 14:00-16:00 Israel Time (13:00-15:00 CET)**Speaker:** Yair Hayut **Title:** Stationary reflection and Prikry forcing – Part II**Abstract:** In 1982, Magidor proved the consistency of stationary reflection at \aleph_{\omega+1}, relative to an \omega-sequence of supercompact cardinals. Square based heuristics indicated that a much weaker large cardinal hypothesis is the correct strength. In a sequence of results of various authors, Magidor’s result was gradually improved to stationary reflection at all sets except one “bad” stationary set at \aleph_{\omega+1}, starting with a large cardinal property weaker than \kappa^+-supercompactness. In a joint work with Unger, we managed to obtain the consistency of (full) stationary reflection, from what seems to be close to the optimal hypothesis.In this talk I will present the main ideas behind the proof (which is the interplay between Prikry type forcings and iterated ultrapowers). This method shares some features with the Sigma-Prikry framework, where the main difference is its non-iterative nature.In a joint work with Ben-Neria, we tackled the problem of combining the failure of SCH with stationary reflection, starting with a similar large cardinal hypothesis. In order to do that, we used a similar analysis of the extender based Prikry forcing. **Information:** Contact Menachem Magidor, Asaf Rinot or Omer Ben-Neria ahead of time for the zoom link.

**Singapore Logic Seminar****Time:** Wednesday, 10 March, 17:00-18:00 Singapore time (10:00-11:00 CET)**Speaker:** Zekun Jia**Title:** Two Ramsey-theoretic statements in models where AC fails**Abstract:** There are a lot of theorems in Ramsey theory whose proof explicitly or implicitly uses the Axiom of Choice. This talk will focus on Ramsey’s Theorem and Open Ramsey Theorem in three models of set theory where the Axiom of Choice fails (the basic Cohen model, the basic Fraenkel model, and the ordered Mostowski model), as well as some consistency and independence results that follow. Also, the usual proof of Open Ramsey Theorem on omega given by Galvin and Prikry assumes the Axiom of Dependent Choice, and this talk will sketch an improvement on that proof to make it purely constructive. This talk is about a project advised by Zach Norwood. **Information:** See the seminar webpage.

**Panglobal Algebra and Logic Seminar, Boulder****Time:** Tuesday, 9 March, 1pm Colorado time (21:00 CET)**Speaker:** Alexandra Pasi, Baylor University**Title:** Forcing ℵ1-Free Groups to Be Free**Abstract:** ℵ1-free groups, abelian groups whose countable subgroups are free, are objects of both algebraic and set-theoretic interest. Illustrating this, we note that ℵ1-free groups, and in particular the question of when ℵ1-free groups are free, were central to the resolution of the Whitehead problem as undecidable. In elucidating the relationship between ℵ1-freeness and freeness, we prove the following result: an abelian group G is ℵ1-free in a countable transitive model of ZFC (and thus by absoluteness, in every transitive model of ZFC) if and only if it is free in some generic model extension. We would like to answer the more specific question of when an ℵ1-free group can be forced to be free while preserving the cardinality of the group. For groups of size ℵ1, we establish a necessary and sufficient condition for when such forcings are possible. We also identify a number of existing and novel forcings which force such ℵ1-free groups of size ℵ1 to become free with cardinal preservation. These forcings lay the groundwork for a larger project which uses forcing to explore various algebraic properties of ℵ1-free groups and develops new set-theoretical tools for working with them.**Information:** Zoom Meeting ID:https://cuboulder.zoom.us/j/96896272260, Passcode: PALS2021

**Caltech Logic Seminar****Time:** Monday, 8 March, 12:00 – 1:00pm Pacific time (21:00 CET)**Speaker:** Maciej Malicki, IMPAN**Title:** Infinitary continuous logic and descriptive set theory**Abstract:** There are deep connections between model theory of the infinitary logic Lω1ωLω1ω and descriptive set theory: Scott analysis, the López-Escobar theorem or the Suzuki theorem are well known examples of this phenomenon. In this talk, I would like to present results of an ongoing research devoted to generalizing these connections to the setting of continuous infinitary logic and Polish metric structures. In particular, I will discuss a continuous counterpart of a theorem of Hjorth and Kechris characterizing essential countability of the isomorphism relation on a given Borel class of countable structures. As an application, I will give a short model-theoretic proof of a result of Kechris saying that orbit equivalence relations induced by continuous actions of locally compact Polish groups are essentially countable. This is joint work with Andreas Hallbäck and Todor Tsankov.**Information:** See the seminar webpage.

1-7 March

**CUNY Set Theory Seminar****Time:** Friday, 5 March, 2pm New York time (20:00 CET)**Speaker:** Hiroshi Sakai, Kobe University**Title:** Generalized stationary reflection and cardinal arithmetic**Abstract:** The stationary reflection principle, which is often called the Weak Reflection Principle, is known to have many interesting consequences. As for cardinal arithmetic, it implies that λω=λ for all regular cardinal λ≥ω2. In this talk, we will discuss higher analogues of this stationary reflection principle and their consequences on cardinal 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, 5 March, 1.30-3pm Toronto time (19:30-21:00 CET)**Speaker:** Alan Dow**Title:** On the cardinality of separable pseudoradial spaces**Abstract:** A point is in the radial closure of a set A if there is a well-ordered sequence from A converging to the point. A set is radially closed if all points in the radial closure are in the set. A space is radial if the radial closure of a set equals its closure and is pseudoradial if every radially closed set is closed.

One can observe that the notions of Frechet-Urysohn and sequential are the related notions when restricted to the usual countable sequences. Motivatedby some work and questions by Santi Spadaro, Istvan Juhasz asked about the implicit question raised by the title. We discuss our progress on the problem in joint work with Istvan Juhasz.**Information:** Email Ivan Ongay Valverde ahead of time for the zoom link.

**Turin-Udine logic seminar****Time:** Friday, 5 March, 16:30-18:30 CET**Speaker:** Noe de Rancourt, University of Vienna**Title:** A dichotomy for countable unions of smooth Borel equivalence relations**Abstract:** I will present a dichotomy for equivalence relations on Polish spaces that can be expressed as countable unions of smooth Borel equivalence relations. It can be seen as an extension of Kechris-Louveau’s dichotomy for hypersmooth Borel equivalence relations. A generalization of our dichotomy, for equivalence relations that can be expressed as countable unions of Borel equivalence relations belonging to certain fixed classes, will also be presented. This is a joint work with Benjamin Miller.**Information:** Online on WebEx. Please see the seminar webpage.

**KGRC Research Seminar, ViennaTime:** Thursday, 4 March, 15:00-16:30 CET

**Speaker:**Allen Gehret, University of Vienna

**Title:**Asymptotic differential algebra and logarithmic transseries

**Abstract:**In this talk I will give a brief introduction to the subject ‘Asymptotic Differential Algebra’ and an overview of the logarithmic transseries programme. The intuition originates in freshman calculus (specifically: limits, l’hopital’s rule, convergence/divergence of integrals and series, asymptotic expansions). The mathematical concepts primarily involve various flavors of fields (equipped with a derivation and/or a valuation and/or an ordering). The logical content will be minimal: first-order languages, model completeness, quantifier elimination.

**Information:**Talk via zoom.

**MOPA (Models of Peano Arithmetic), CUNY****Time:** Wednesday, 3 March, 7pm New York time (01:00 CET on 4 March)**Speaker: **Ali Enayat, University of Gothenburg**Title:** PA with a class of indiscernibles**Abstract:** This talk focuses on the theory PAI (I for Indiscernibles), a theory formulated in the language of PA augmented with a unary predicate I(x). Models of PAI are of the form (M,I) where (1) M is a model of PA, (2) I is a proper class of M, i.e., I is unbounded in M and (M,I) satisfies PA*, and (3) I forms a class of indiscernibles over M. The formalizability of the Infinite Ramsey Theorem in PA makes it clear that PAI is a conservative extension of PA. As we will see, nonstandard models of PA (of any cardinality) that have an expansion to a model of PAI are precisely those nonstandard models PA that can carry an inductive partial satisfaction class. The formulation and investigation of PAI was inspired by my work on the set theoretical sibling ZFI of PAI, whose behavior I will also compare and contrast with that of PAI. **Information:** The seminar will take place virtually. Please email Victoria Gitman (vgitman@nylogic.org) for the meeting id.

**Hebrew University-Bar Ilan University Set Theory seminar****Time:** Wednesday, 3 March, 14:00-16:00 Israel Time (13:00-15:00 CET)**Speaker:** Yair Hayut **Title:** Stationary reflection and Prikry forcing**Abstract:** In 1982, Magidor proved the consistency of stationary reflection at \aleph_{\omega+1}, relative to an \omega-sequence of supercompact cardinals. Square based heuristics indicated that a much weaker large cardinal hypothesis is the correct strength. In a sequence of results of various authors, Magidor’s result was gradually improved to stationary reflection at all sets except one “bad” stationary set at \aleph_{\omega+1}, starting with a large cardinal property weaker than \kappa^+-supercompactness. In a joint work with Unger, we managed to obtain the consistency of (full) stationary reflection, from what seems to be close to the optimal hypothesis.In this talk I will present the main ideas behind the proof (which is the interplay between Prikry type forcings and iterated ultrapowers). This method shares some features with the Sigma-Prikry framework, where the main difference is its non-iterative nature.In a joint work with Ben-Neria, we tackled the problem of combining the failure of SCH with stationary reflection, starting with a similar large cardinal hypothesis. In order to do that, we used a similar analysis of the extender based Prikry forcing. **Information:** Contact Menachem Magidor, Asaf Rinot or Omer Ben-Neria ahead of time for the zoom link.

**Singapore Logic Seminar****Time:** Wednesday, 3 March, 17:00-18:00 Singapore time (10:00-11:00 CET)**Speaker:** Desmond Lau**Title:** On the unification of two “maximal” axioms**Abstract:** Martin’s Maximum^{++} and Woodin’s axiom (*) are two statements independent of, but consistent with, ZFC. I will present the common reasons they are appealing as set-theoretic axioms, before comparing the sense in which they are “maximal”. I will also run through an exposition of the recent work by Aspero and Schindler, which shows Martin’s Maximum^{++} implies (*), effectively “unifying” the statements.**Information:** See the seminar webpage.

**Caltech Logic Seminar****Time:** Monday, 1 March, 12:00 – 1:00pm Pacific time (21:00 CET)**Speaker:** Filippo Calderoni, University of Illinois at Chicago**Title:** Anticlassification results for groups acting freely on the line**Abstract:** In this talk we shall discuss some anticlassification results for orderable groups. First, we introduce the space of Archimedean orderings Ar(G)Ar(G) for a given countable orderable group GG. We prove that the equivalence relation induced by the natural action of GL2(Q)GL2(Q) on Ar(Q2)Ar(Q2) is not concretely classifiable. Then we shall discuss the complexity of the isomorphism relation for countable ordered Archimedean groups. In particular, we show that its potential class is not Π03Π30. This topological constraint prevents classifying ordered Archimedean groups using countable subsets of reals. Our proofs combine classical results on Archimedean groups, the theory of Borel equivalence relations, and analyzing definable sets in the basic Cohen model and other models of Zermelo-Fraenkel set theory without choice. This is joint work with Dave Marker, Luca Motto Ros, and Assaf Shani.**Information:** See the seminar webpage.

22-28 February

**CUNY Set Theory Seminar****Time:** Friday, 26 February, 2pm New York time (20:00 CET)**Speaker:** Farmer Schlutzenberg, University of Münster**Title:** (Non)uniqueness and (un)definability of embeddings beyond choice**Abstract:** Work in ZF and let j:Vα→Vα be an elementary, or partially elementary, embedding. One can examine the degree of uniqueness, definability or constructibility of j. For example, is there β<αsuch that j is the unique (partially) elementary extension of j↾Vβ? Is j definable from parameters over Vα? We will discuss some results along these lines, illustrating that answers can depend heavily on circumstances. Some of the work is due independently and earlier to Gabriel Goldberg.**Information:** The seminar will take place virtually. Please email Victoria Gitman (vgitman@nylogic.org) for the meeting id.

**Turin-Udine logic seminar****Time:** Friday, 26 February, 16:30-18:30 CET**Speaker:** S. Barbina, Open University**Title:** The theory of the universal-homogeneous Steiner triple system**Abstract:** A Steiner triple system is a set S together with a collection B of subsets of S of size 3 such that any two elements of S belong to exactly one element of B. It is well known that the class of finite Steiner triple systems has a Fraïssé limit, the countable homogeneous universal Steiner triple system M. In joint work with Enrique Casanovas, we have proved that the theory T of M has quantifier elimination, is not small, has TP2, NSOP1, eliminates hyperimaginaries and weakly eliminates imaginaries. In this talk I will review the construction of M, give an axiomatisation of T and prove some of its properties.**Information:** Online on WebEx. Please see the seminar webpage.

**Bristol Logic and Set Theory Seminar**/**Oxford Set Theory Seminar****Time:** Wednesday, 24 February, 16:00-17:30 UK time (17:00-18:30 CEST)

**Speaker:**Andrew Marks, UCLA

**Title:**The decomposability conjecture

**Abstract:**We characterize which Borel functions are decomposable into

a countable union of functions which are piecewise continuous on

\Pi^0_n domains, assuming projective determinacy. One ingredient of

our proof is a new characterization of what Borel sets are \Sigma^0_n

complete. Another important ingredient is a theorem of Harrington that

there is no projective sequence of length \omega_1 of distinct Borel

sets of bounded rank, assuming projective determinacy. This is joint

work with Adam Day.

**Information:**For the Zoom access code, contact Samuel Adam-Day me@samadamday.com. Link: https://zoom.us/j/96803195711 (open 30 minutes before)

**Barcelona Set Theory Seminar****Time:** Wednesday, 24 February, 16:00-17:30 CET**Speaker:** Brent Cody, Virgina Commonwealth University**Title:** Higher indescribability and ideal operators**Abstract:** Higher indescribability refers to a type of reflection property

possessed by certain large cardinals in which attributes that reflect are

expressible by infinitary formulas whose lengths are strictly larger than

the cardinal under consideration. We will discuss some new results, joint

with Peter Holy, involving such indescribability properties and their

relationship with the hierarchies associated to ineffability and

Ramseyness. For example, we show that between any two degrees in

Baumgartner’s original ineffability hierarchy there lies another hierarchy

obtained naturally by using the ineffability operator and higher

indescribability hypotheses. We will also discuss some open questions

regarding indescribability and the Ramsey hierarchy.**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, 24 February, 14:00-16:00 Israel Time (13:00-15:00 CET)**Speaker:** Assaf Rinot, Bar-Ilan University**Title:** All colorings are strong – but some colorings are stronger than others**Abstract:** Strong colorings are everywhere – they can be obtained from

analysis of basis problems, transfinite diagonalizations, oscillations,

or walks on ordinals. They give rise to interesting topological spaces

and partial orders.

In this talk, I’ll be looking at all aspects mentioned above, reporting

on findings from my joint projects with Kojman, Lambie-Hanson, Inamdar,

Steprans and Zhang. **Information:** Please check on the seminar webpage if the seminar will take place.

Contact Menachem Magidor, Asaf Rinot or Omer Ben-Neria ahead of time for the zoom link.

**Caltech Logic Seminar****Time:** Monday, 22 February, 12:00 – 1:00pm Pacific time (21:00 CET)**Speaker:** Oleg Pikhurko, University of Warwick**Title:** Circle squaring with Jordan measurable pieces**Abstract:** We discuss our result with András Máthé and Jonathan Noel that a disk in the plane can be partitioned into finitely many Jordan measurable pieces that can be moved by translations to form a partition of a square.**Information:** See the seminar webpage.

15-21 February

**CUNY Set Theory Seminar****Time:** Friday, 19 February, 2pm New York time (20:00 CET)**Speaker:** Philipp Lücke, University of Barcelona**Title:** Magidor-style embedding characterizations of large cardinals**Abstract:** Motivated by a classical theorem of Magidor, I will present results providing characterizations of important objects from the lower end of the large cardinal hierarchy through the existence of elementary embeddings between set-sized models that map their critical point to the large cardinal in question. Focusing on the characterization of *shrewd cardinals*, introduced by Rathjen in a proof-theoretic context, I will show how these results can be used in the study of the combinatorics of strong chain conditions and the investigation of *principles of structural reflection* formulated by Bagaria.**Information:** The seminar will take place virtually. Please email Victoria Gitman (vgitman@nylogic.org) for the meeting id.

**Toronto Set Theory Seminar****Time:** Friday, 19 February, 1.30-3pm Toronto time (19:30-21:00 CET)**Speaker:** Assaf Rinot, Bar-Ilan University**Title:** All colorings are strong – but some colorings are stronger than

the others**Abstract:** Strong colorings are everywhere – they can be obtained from

analysis of basis problems, transfinite diagonalizations, oscillations,

or walks on ordinals. They give rise to interesting topological spaces

and partial orders.

In this talk, I’ll be looking at all aspects mentioned above, reporting

on findings from my joint projects with Kojman, Lambie-Hanson, Inamdar,

Steprans and Zhang.**Information:** Email Ivan Ongay Valverde ahead of time for the zoom link.

**Turin-Udine logic seminar****Time:** Friday, 19 February, 16:30-18:30 CET**Speaker:** Paul Shafer, University of Leeds**Title:** An inside-outside Ramsey theorem in the Weihrauch degrees**Abstract:** Recall Ramsey’s theorem for pairs and two colors, which, in terms of graphs, may be phrased as follows: For every countably infinite graph G, there is an infinite set of vertices H such that either every pair of distinct vertices from H is adjacent or no pair of distinct vertices from H is adjacent. The conclusion of Ramsey’s theorem gives complete information about how the vertices in H relate to each other, but it gives no information about how the vertices outside H relate to the vertices inside H. Rival and Sands (1980) proved the following theorem, which weakens the conclusion of Ramsey’s theorem with respect to pairs of vertices in H, but does add information about how the vertices outside H relate to the vertices inside H: For every countably infinite graph G, there is an infinite set of vertices H such that each vertex of G is either adjacent to no vertices of H, to exactly one vertex of H, or to infinitely many vertices of H. We give an exact characterization of the computational strength of the Rival-Sands theorem by showing that it is strongly Weihrauch equivalent to the double-jump of weak König’s lemma (which is the problem of producing infinite paths through infinite trees that are given by Delta^0_3 approximations). In terms of Ramsey’s theorem, this means that solving one instance of the Rival-Sands theorem is equivalent to simultaneously solving countably many instances of Ramsey’s theorem for pairs and two colors in parallel. This work is joint with Marta Fiori Carones and Giovanni Soldà.**Information:** Online on WebEx. Please see the seminar webpage.

**Caltech Logic Seminar****Time:** Wednesday, 17 February, 12:00 – 1:00pm Pacific time (21:00 CET)**Speaker:** Matthew Foreman, UC Irvine**Title:** Anti-classification and independence results in smooth dynamical systems**Abstract:** We discuss two equivalence relations for diffeomorphisms of finite dimensional smooth manifolds. The first is *measure isomorphism* between ergodic Lebesgue measure preserving diffeomorphisms of the 2-torus. The second is topological conjugacy for diffeomorphisms of smooth manifolds. In both cases we show that the equivalence relation is unclassifiable. For topological conjugacy, for dimension ≥2 we can embed E0, while for dimension ≥5 we can embed graph isomorphism.

We finish by showing that the equivalence relations are Π10-hard: for measure preserving diffeomorphisms of the torus we exhibit a primitive recursive association of Π01Π10-statements with diffeomorphisms of the torus so that for each ϕ, ϕ is true if and only if Tϕ≅T−1ϕ.So, as Erdős used to say, *Independence rears its ugly head*.

These results are joint work with B. Weiss and A. Gorodetski.**Information:** See the seminar webpage.

**Barcelona Set Theory Seminar****Time:** Wednesday, 17 February, 16:00-17:30 CET**Speaker:** Hiroshi Sakai, Kobe University**Title:** **Abstract:** tba**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, 17 February, 14:00-16:00 Israel Time (13:00-15:00 CET)**Speaker:** Mark Poor**Title:** The spectra of cardinalities of branches of Kurepa trees, Part 2**Abstract:** Kurepa trees may look like. It turns out that assuming GCH below the second uncountable cardinal (and possibly the existence of an inaccessible) we can force every set of cardinals (satisfying some obvious necessary conditions) to be the Kurepa spectrum. This is a joint work with S. Shelah.**Information:** Please check on the seminar webpage if the seminar will take place.

Contact Menachem Magidor, Asaf Rinot or Omer Ben-Neria ahead of time for the zoom link.

**Singapore Logic Seminar****Time:** Wednesday, 17 February, 17:00-18:00 Singapore time (10:00-11:00 CET)**Speaker:** Xiao Ming**Title:** Borel Order Dimensions**Abstract:** Order dimension is a classical combinatorial object and has been widely studied by set theorists, combinatorists and computer scientists since its introduction by Dushnik and Miller in 1941. We focus on the partial orderings that are definable as a Borel subsets in a Polish space and analyze the order dimension that can be realized by Borel definable orders and show that there are some interesting behaviors that can be quite different from the classical order dimension using arbitrary realization.

This is a joint work with Dilip Raghavan. **Information:** See the seminar webpage.

8-14 February

**CUNY Set Theory Seminar****Time:** Friday, 12 February, 2pm New York time (20:00 CET)**Speaker:** Bea Adam-Day, University of Leeds **Title:** Indestructibility (or otherwise) of subcompactness and C(n)-supercompactness**Abstract:** Indestructibility results of large cardinals have been an area of interest since Laver’s 1978 proof that the supercompactness of κ may be made indestructible by any <κ-directed closed forcing. I will present a continuation of this work, showing that α-subcompact cardinals may be made suitably indestructible, but that for C(n)-supercompact cardinals this is largely not possible. **Information:** The seminar will take place virtually. Please email Victoria Gitman (vgitman@nylogic.org) for the meeting id.

**Turin-Udine logic seminar****Time:** Friday, 12 February, 16:30-18:30 CET**Speaker: **Alexandra Kwiatkowska, University of Münster**Title:** tba**Abstract:** tba**Information:** Online on WebEx. Please see the seminar webpage.

**Paris-Lyon Séminaire de Logique****Time:** Wednesday, 10 February, 16:00-17:00 CET**Speaker:** Rahman Mohammadpour, Université de Paris**Title:** Specializing trees of height \omega_2 with finite approximation**Abstract:** It is well-known that under CH one can attempt to specialize trees of height ω_2 without cofinal branches using a naive forcing with countable approximations. However, one has to require more (the nonexistence of ascending paths) than the lack of cofinal branches to make sure that the naive attempt does not fail. I will discuss these possible obstacles to specialize trees of height ω_2 , and then use models as side conditions to construct a forcing notion with finite conditions, which under PFA specializes a given tree of height ω_2 without cofinal branches. If time permits, I will mention generalizations of this result to taller trees.**Information:** Join via the link on the seminar webpage 10 minutes before the talk.

**Barcelona Set Theory Seminar****Time:** Wednesday, 10 February, 16:00-17:30 CET**Speaker:** Matteo Viale, University of Torino**Title:** The model-companionship spectrum of set theory, generic absoluteness, and the continuum problem**Abstract:** We show that for Pi2-properties of second or third order arithmetic as formalized in appropriate natural signatures the apparently weaker notion of forcibility overlaps with the standard notion of consistency (assuming large cardinal axioms). Among such Pi2-properties we mention: the negation of the continuum hypothesis, Souslin’s Hypothesis, the negation of Whitehead’s conjecture on free groups, the non-existence of outer automorphisms for the Calkin algebra, etc. In particular this gives an a posteriori explanation of the success forcing (and forcing axioms) met in producing models of such properties. Our main results relate generic absoluteness theorems for second order arithmetic, Woodin’s axiom (*) and forcing axioms to Robinson’s notion of model companionship (as applied to set theory).**Information:** Online. If you wish to attend, please send an email to bagaria@ub.edu asking for the link.

**Hebrew University-Bar Ilan University Set Theory seminar****Time:** Wednesday, 10 February, 14:00-16:00 Israel Time (13:00-15:00 CET)**Speaker:** Mark Poor**Title:** The spectra of cardinalities of branches of Kurepa trees**Abstract:** Kurepa trees may look like. It turns out that assuming GCH below the second uncountable cardinal (and possibly the existence of an inaccessible) we can force every set of cardinals (satisfying some obvious necessary conditions) to be the Kurepa spectrum. This is a joint work with S. Shelah. **Information:** Please check on the seminar webpage if the seminar will take place.

Contact Menachem Magidor, Asaf Rinot or Omer Ben-Neria ahead of time for the zoom link.

**Caltech Logic Seminar****Time:** Monday, 8 February, 12:00 – 1:00pm Pacific time (21:00 CET)**Speaker:** Márton Elekes, Alfréd Rényi Institute of Mathematics**Title:** Games, their values, and Baire class 1 functions**Abstract:** We consider interesting descriptive set-theoretic problems emerging from theoretical economics. First, we investigate a certain two-player game coming from gambling theory. Then, as a by-product, we obtain a novel game that characterises the Baire class 1 functions. We mention how games can be used to define new natural ranks on the Baire class 1 functions. Finally, we determine the exact complexity of the so-called value of the above game, which turns out to be a less well-known class, namely analytic-inductive.**Information:** See the seminar webpage.

1-7 February

**CUNY Set Theory Seminar****Time:** Friday, 5 February, 2pm New York time (20:00 CET)**Speaker:** Andreas Blass, University of Michigan**Title:** Choice from Finite Sets: A Topos View**Abstract:** Tarski proved (but didn’t publish) the theorem that choice from pairs implies choice from four-element sets. Mostowski (1937) began a systematic study of such implications between choice axioms for families of finite sets. Gauntt (1970) completed that study (but didn’t publish the results), obtaining equivalent characterizations in terms of fixed points of permutation groups. Truss (1973) extended Gauntt’s results (and published this work).

It turns out that these finite choice axioms and their group-theoretic characterizations are instances of the same topos-theoretic statements, interpreted in two very different classes of topoi. My main result is an extension of that observation to the class of all topoi.

Most of my talk will be explaining the background: finite choice axioms, permutation groups, and a little bit about topoi – just enough to make sense of the main result. If time permits, I’ll describe some of the ingredients of the proof.**Information:** The seminar will take place virtually. Please email Victoria Gitman (vgitman@nylogic.org) for the meeting id.

**Toronto Set Theory Seminar****Time:** Friday, 5 February, 1.30-3pm Toronto time (19:30-21:00 CET)**Speaker:** Andrés Villaveces, Universidad Nacional de Colombia**Title:** Axiomatizations of abstract elementary classes and natural logics for model theory: the role of partition relations**Abstract:** Two seemingly unrelated questions (the quest for natural logics of abstract elementary classes on the one hand, and the quest for logics adequate to model theory on the other hand) converge around the same combinatorial core: partition relations for scattered order types (due to Kómjath and Shelah). I will present recent results concerning the first question (and axiomatizing a.e.c.’s – joint work with Shelah) and the second question (joint work with Väänänen).**Information:** Email Ivan Ongay Valverde ahead of time for the zoom link.

**Turin-Udine logic seminar****Time:** Friday, 5 February, 16:30-18:30 CET**Speaker:** M. Viale, University of Turin **Title:** Tameness for set theory**Abstract:** We show that (assuming large cardinals) set theory is a tractable (and we dare to say tame) first order theory when formalized in a first order signature with natural predicate symbols for the basic definable concepts of second and third order arithmetic, and appealing to the model-theoretic notions of model completeness and model companionship. Specifically we develop a general framework linking generic absoluteness results to model companionship and show that (with the required care in details) a -property formalized in an appropriate language for second or third order number theory is forcible from some T extending ZFC + large cardinals if and only if it is consistent with the universal fragment of T if and only if it is realized in the model companion of T. Part (but not all) of our results are a byproduct of the groundbreaking result of Schindler and Asperò showing that Woodin’s axiom (*) can be forced by a stationary set preserving forcing.**Information:** Online on WebEx. Please see the seminar webpage.

**Ghent-Leeds Virtual Logic SeminarTime:** Thursday, 4 February, 2pm UK time (15:00 CET)

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

**Title:**Ordinal oddities

**Abstract:**Ordinals are a generalisation of natural numbers to allow us to arrange a possibly infinite set of objects in some order and this concept has become an incredibly useful tool in mathematics. However, in intuitionistic theories, the concept of an ordinal becomes more vague and many of the useful characteristics for ordinals may no longer hold. In this talk we shall explore some of the many strange properties that ordinals can have when we are no longer assuming the law of excluded middle. In particular, we shall see that it is possible for there to be an ordinal which is not in the constructible universe, answering a question of Lubarsky.

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

**Bristol Logic and Set Theory Seminar**/**Oxford Set Theory Seminar****Time:** Wednesday, 3 February, 16:00-17:30 UK time (17:00-18:30 CEST)

**Speaker:**Spencer Unger, University of Toronto

**Title:**Stationary reflection at successors of singular cardinals

**Abstract:**We survey some recent progress in understanding stationary reflection at successors of singular cardinals and its influence on cardinal arithmetic:

1) In joint work with Yair Hayut, we reduced the consistency strength of stationary reflection at aleph_{omega+1} to an assumption weaker than kappa is kappa+ supercompact.

2) In joint work with Yair Hayut and Omer Ben-Neria, we prove that from large cardinals it is consistent that there is a singular cardinal nu of uncountable cofinality where the singular cardinal hypothesis fails at nu and every collection of fewer than cf(nu) stationary subsets of nu+ reflects at a common point.

The statement in the second theorem was not previously known to be consistent. These results make use of analysis of Prikry generic objects over iterated ultrapowers.

**Information:**For the Zoom access code, contact Samuel Adam-Day me@samadamday.com. Link: https://zoom.us/j/96803195711 (open 30 minutes before)

**Barcelona Set Theory Seminar****Time:** Wednesday, 3 February, 16:00-17:30 CET**Speaker:** Adrian Mathias, Professeur émérite, LIM, Université de la Réunion**Title:** Power-admissible sets and ill-founded omega-models of weak subsystems of ZFC**Abstract:** In 1971 Harvey Friedman introduced what he called power-admissible sets, which are transitive models of a system I call KPP. Nearly thirty years later, in 2000, I proved results about them which

were added to my paper “The strength of Mac Lane set theory” at a late

stage. Their proofs used forcing over ill-founded w-models. Nineteen years after that I proved a further theorem about set forcing over power-admissible sets, again using forcing over a carefully chosen ill-founded omega-model. In this talk I shall review these results and the methods of proof.**Information:** Online. If you wish to attend, please send an email to bagaria@ub.edu asking for the link.

**Caltech Logic Seminar****Time:** Monday, 1 February, 12:00 – 1:00pm Pacific time (21:00 CET)**Speaker:** Su Gao, University of North Texas**Title:** Omnigenous groups**Abstract:** Generalizing a Urysohn-like extension property for Hall’s countable universal locally finite group, we define a concept of omnigenous groups and prove some results about such groups. One of the main results is that any countable omnigenous locally finite group can be embedded as a dense subgroup of the isometry group of the Urysohn space for all ΔΔ-metric spaces, for any countable distance value set ΔΔ. This implies a conjecture of Vershik from 2008. I will also talk about the current progress on the converse problem, namely to characterize all countable (locally finite) dense subgroups of the isometry groups of Urysohn spaces. This is joint work with Mahmood Etedadialiabadi, Francois Le Maître, and Julien Melleray. **Information:** See the seminar webpage.

**Genova logic seminar****Time:** Monday, 1 February, 14.00-15.30 CET**Speaker:** Mirna Džamonja, Institut d’Histoire et de Philosophie des Sciences et

des Techniques, Paris, and Institute of Mathematics Czech Academy,

Prague**Title:** Formalising Ordinal Partition Relations Using Isabelle/HOL**Abstract:** Joint work with with Angeliki Koutsoukou-Argyraki and Lawrence C. Paulson, FRS, Cambridge. This talk is about an application in set theory of what is sometimes called ‘automated theorem proving’ by mathematicians. This actually refers to several different things, including what computer scientists call formalisation. After briefly discussing general aspects of formalisation, we shall give an overview of a formalisation project in the proof assistant Isabelle/HOL of a number of results in ordinal partition relations : theorems by Erdős–Milner, Specker, Larson and Nash-Williams, leading to Jean Larson’s proof of the unpublished result by E.C. Milner asserting that for all m∈ℕ, ωω→(ωω,m).

Ordinal partition relations are notoriously hard to study by classical methods and have the uncanny feature to be mostly interesting for countable ordinals, where modern set theory seems to be quite silent.

Our approach has been to see if formalising might bring us closer to resolving some of the many unsolved problems in the area. The talk will focus on the process of formalisation, the difficulties and the hopes of the process. In particular, no new proof has yet been obtained by proof assistants. We are hoping that some of the counter-example finding methods in ordinal partitions we developed in this formalisation might allow us to make some modest progress in that direction. The actual formalisation behind our paper was done by Paulson and is available on the Archive of Formal Proofs. This project is also a demonstration of working with Zermelo–Fraenkel set theory in higher-order logic, as developed in this context by Paulson.**Information:** The seminar will be held on Microsoft Teams, at the page of the Genoa logic group. The access code is: fpedcxn. Alternatively, you can write to camerlo@dima.unige.it to have an access link. Further information on the activities of the Genoa logic group can be found at logic.dima.unige.it

25-31 January

**CUNY Set Theory Seminar****Time:** Friday, 29 January, 2pm New York time (20:00 CET)**Speaker:** Erin Carmody, Fordham University**Title:** The relationships between measurable and strongly compact cardinals, part 2**Abstract:** This talk is about the ongoing investigation of the relationships between measurable and strongly compact cardinals. I will present some of the history of the theorems in this theme, including Magidor’s identity crisis, and give new results. The theorems presented are in particular about the relationships between strongly compact cardinals and measurable cardinals of different Mitchell orders. One of the main theorems is that there is a universe where κ1 and κ2 are the first and second strongly compact cardinals, respectively, and where κ1 is least with Mitchell order 1, and κ2is the least with Mitchell order 2. Another main theorem is that there is a universe where κ1 and κ2are the first and second strongly compact cardinals, respectively, with κ1 the least measurable cardinal such that o(κ1)=2 and κ2 the least measurable cardinal above κ1. This is a joint work in progress with Victoria Gitman and Arthur Apter.**Information:** The seminar will take place virtually. Please email Victoria Gitman (vgitman@nylogic.org) for the meeting id.

**Toronto Set Theory Seminar****Time:** Friday, 29 January, 1:30-3pm Toronto time (19:30-21:00 CET)**Speaker:** Corey Switzer, University of Vienna**Title:** Higher Dimensional Cardinal Characteristics for Sets of Real Valued Functions**Abstract:** Cardinal characteristics on the generalized Baire and Cantor spaces $\kappa^\kappa$ and $2^\kappa$ have recently generated significant interest. In this talk I will introduce a different generalization of cardinal characteristics, namely to the space of functions $f:\omega^\omega \to \omega^\omega$. Given an ideal $I$ on Baire space and a relation $R$ let us define $f R_I g$ for $f$ and $g$ functions from $\omega^\omega$ to $\omega^\omega$ if and only if $f(x) R g(x)$ for an $I$-measure one set of $x \in \omega^\omega$. By letting $I$ vary over the null ideal, the meager ideal and the bounded ideal; and $R$ vary over the relations $\leq^*$, $\neq^*$ and $\in^*$ we get 18 new cardinal characteristics by considering the bounding and dominating numbers for these relations. These new cardinals form a diagram of provable implications similar to the Cichoń diagram. They also interact in several surprising ways with the cardinal characteristics on $\omega$. For instance, they can be arbitrarily large in models of CH, yet they can be $\aleph_1$ in models where the continuum is arbitrarily large. They are bigger in the Sacks model than the Cohen model. I will introduce these cardinals, show some of the provable implications and discuss what is known about consistent inequalities, focusing on the $\mathfrak{b}$-numbers in well-known models such as the Cohen and Random model. This is joint work with Jörg Brendle.**Information:** Email Ivan Ongay Valverde ahead of time for the zoom link.

**Turin-Udine logic seminar****Time:** Friday, 29 January, 16:30-18:30 CET **Speaker:** Vera Fischer, University of Vienna **Title:** The spectrum of independence**Abstract:** Families of infinite sets of natural numbers are said to be independent if for very two disjoint non-empty subfamilies the intersection of the members of the first subfamily with the complements of the members of the second family is infinite. Maximal independent families are independent families which are maximal under inclusion. In this talk, we will consider the set of cardinalities of maximal independent families, referred to as the spectrum of independence, and show that this set can be quite arbitrary. This is a joint work with Saharon Shelah.**Information:** Online on WebEx. Please see the seminar webpage.

**KGRC Research Seminar, ViennaTime:** Thursday, 28 January, 15:00-16:30 CET

**Speaker:**Marlene Koelbing and Wolfgang Wohofsky, University of Vienna

**Title:**Distributivity spectrum of forcing notions

**Abstract:**tba

**Information:**Talk via zoom.

**Barcelona Set Theory Seminar****Time:** Wednesday, 27 January, 16:00-17:30 CET**Speaker:** Richard Matthews, University of Leeds**Title:** Taking Reinhardt’s Power Away**Abstract:** **Information:** Online. If you wish to attend, please send an email to bagaria@ub.edu asking for the link.

**Paris-Lyon Séminaire de Logique****Time:** Wednesday, 27 January, 14:00-15:00 CET**Speaker:** Matthew de Brecht, Kyoto University**Title:** Quasi-Polish spaces as spaces of ideals, with applications to computable topology**Abstract:** We give a brief introduction to quasi-Polish spaces, which are a class of well-behaved countably based T0-spaces that generalize both Polish spaces and ω-continuous domains. We then present more recent results on a characterization of quasi-Polish spaces as spaces of ideals of a transitive relation on a countable set, and investigate some applications of this characterization to computable topology.**Information:** Join via the link on the seminar webpage 10 minutes before the talk.

**Hebrew University-Bar Ilan University Set Theory seminar****Time:** Wednesday, 27 January, 14:00-16:00 Israel Time (13:00-15:00 CET)**Speaker:** Philipp Lücke, University of Barcelona**Title:** Fresh subset of measurable ultrapowers**Abstract:** In my talk, I want to present recent results studying the closureand non-closure properties of measurable ultrapowers with respect to Hamkin’snotion of freshness. These results show that the extent of these propertieshighly depends on the combinatorics of the underlying model of set theory.While a result of Sakai shows that it is possible to obtain ultrapowers withmaximal closure properties by forcing over a model containing a strongly com-pact cardinal, it turns out that measurable ultrapowers of canonical innermodels possess the minimal amount of closure properties. The proof of thisresult heavily makes use of the existence of various square sequences in thesemodels. This is joint work with Sandra Muller (Vienna). **Information:** Contact Menachem Magidor, Asaf Rinot or Omer Ben-Neria ahead of time for the zoom link.

**Caltech Logic Seminar****Time:** Monday, 25 January, 12:00 – 1:00pm Pacific time (21:00 CET)**Speaker:** Vassilios Gregoriades, National Technical University of Athens**Title:** A method for showing Π30-completeness**Abstract:** We present a method for showing that a given Π03Π30 subset of a Polish space is in fact Π03-complete. This is motivated by some questions from V. Nestoridis about the sequential spaces ℓpℓp and more generally about families of FF-spaces (Xi)i∈(I,⪯) that form ⊆⊆-chains, where ⪯ is a linear ordering. The intersection ∩p>aℓp∩p>aℓp is known to be a Π03 subset of ℓq for all a,q with 0≤a<q<∞ (Nestoridis). We show that it is in fact a Π03-complete set. It turns out the proof can be generalized to the context of Polish spaces with no additional structure like linearity. This gives a method for showing Π03-completeness and in fact there are strong indications that it also gives a characterization of the latter property.**Information:** See the seminar webpage.

18-24 January

**Toronto Set Theory Seminar****Time:** Friday, 21 January, 1:30-3pm Toronto time (19:30-21:00 CET)**Speaker:** Marcos Mazari Armida, Carnegie Mellon University**Title:** Universal models in classes of abelian groups and modules**Abstract:** The search for universal models began in the early twentieth century when Hausdorff showed that there is a universal linear order of cardinality $\aleph_{n+1}$ if $2^{\aleph_n}= \aleph_{n + 1}$, i.e., a linear order $U$ of cardinality $\aleph_{n+1}$ such that every linear order of cardinality $\aleph_{n+1}$ embeds in $U$. In this talk, we will study universal models in several classes of abelian groups and modules with respect to pure embeddings. In particular, we will present a complete solution below $\aleph_\omega$, with the exception of $\aleph_0$ and $\aleph_1$, to Problem 5.1 in page 181 of \emph{Abelian Groups} by L\'{a}szl\'{o} Fuchs, which asks to find the cardinals $\lambda$ such that there is a universal abelian p-group for purity of cardinality $\lambda$. The solution presented will use both model-theoretic and set-theoretic ideas.**Information:** Email Ivan Ongay Valverde ahead of time for the zoom link.

**CUNY Set Theory Seminar****Time:** Friday, 22 January, 2pm New York time (20:00 CET)**Speaker:** Erin Carmody, Fordham University**Title:** The relationships between measurable and strongly compact cardinals**Abstract:** This talk is about the ongoing investigation of the relationships between measurable and strongly compact cardinals. I will present some of the history of the theorems in this theme, including Magidor’s identity crisis, and give new results. The theorems presented are in particular about the relationships between strongly compact cardinals and measurable cardinals of different Mitchell orders. One of the main theorems is that there is a universe where κ1 and κ2 are the first and second strongly compact cardinals, respectively, and where κ1 is least with Mitchell order 1, and κ2is the least with Mitchell order 2. Another main theorem is that there is a universe where κ1 and κ2are the first and second strongly compact cardinals, respectively, with κ1 the least measurable cardinal such that o(κ1)=2 and κ2 the least measurable cardinal above κ1. This is a joint work in progress with Victoria Gitman and Arthur Apter.**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 January, 11:00am-12:30pm Toronto time (17:00-18:30 CET)**Speaker:** Dima Sinapova, UIC, University of Illinois at Chicago**Title:** Iteration, reflection, and singular cardinals**Abstract:** Two classical results of Magidor are: (1) from large cardinals it is consistent to have reflection at $\aleph_{\omega+1}$ and (2) from large cardinals it is consistent to have the failure of SCH at $\aleph_{\omega}$.

These principles are at odds with each other. The former is a compactness type principle. (Compactness is the phenomenon where if a certain property holds for every smaller substructure of an object, then it holds for the entire object.) In contrast, failure of SCH is an instance of incompactness. The natural question is whether we can have both of these simultaneously. We show the answer is yes.

We describe a Prikry style iteration, and use it to force stationary reflection in the presence of not SCH. Then we obtain this situation at $\aleph_{\omega}$ . This is joint work with Alejandro Poveda and Assaf Rinot.**Information:** Email Ivan Ongay Valverde ahead of time for the zoom link.

**KGRC Research Seminar, ViennaTime:** Thursday, 21 January, 15:00-16:30 CET

**Speaker:**Juris Steprans, York University, Toronto, Canada

**Title:**Strong colourings over partitions

**Abstract:**The celebrated result of Todorcevic that ℵ1↛[ℵ1]2ℵ1 provides a well known example of a strong colouring. A mapping c:[ω1]2→κ is a strong colouring over a partition p:[ω1]2→ω if for every uncountable X⊆ω1 there is n∈ω such that the range of c on [X]2∩p−1{n} is all of κ. I will discuss some recent work with A. Rinot and M. Kojman on negative results concerning strong colourings over partitions and their relation to classical results in this area.

**Information:**Talk via zoom.

**Caltech Logic Seminar****Time:** Wednesday, 20 January, 12:00 – 1:00pm Pacific time (21:00 CET)**Speaker:** Todor Tsankov, Université Lyon 1**Title:** Universal minimal flows of homeomorphism groups of high-dimensional manifolds**Abstract:** The first interesting case of a non-trivial, metrizable universal minimal flow (UMF) of a Polish group was computed by Pestov who proved that the UMF of the homeomorphism group of the circle is the circle itself. This naturally led to the question whether a similar result is true for homeomorphism groups of other manifolds (or more general topological spaces). A few years later, Uspenskij proved that the action of a group on its UMF is never 3-transitive, thus giving a negative answer to the question for a vast collection of topological spaces. Still, the question of metrizability of their UMFs remained open and he asked specifically whether the UMF of the homeomorphism group of the Hilbert cube is metrizable. We give a negative answer to this question for the Hilbert cube and all closed manifolds of dimension at least 2, thus showing that metrizability of the UMF of a homeomorphism group is essentially a one-dimensional phenomenon. This is joint work with Yonatan Gutman and Andy Zucker.**Information:** See the seminar webpage.

**Bristol Logic and Set Theory Seminar**/**Oxford Set Theory Seminar****Time:** Wednesday, 20 January, 16:00-17:30 UK time (17:00-18:30 CEST)

**Speaker:**Dima Sinapova, University of Illinois at Chicago

**Title:**Iteration, reflection, and singular cardinals

**Abstract:**Two classical results of Magidor are:

(1) from large cardinals it is consistent to have reflection at $\aleph_{\omega+1}$, and

(2) from large cardinals it is consistent to have the failure of SCH at $\aleph_\omega$.These principles are at odds with each other. The former is a compactness type principle. (Compactness is the phenomenon where if a certain property holds for every smaller substructure of an object, then it holds for the entire object.) In contrast, failure of SCH is an instance of incompactness. The natural question is whether we can have both of these simultaneously. We show the answer is yes.

We describe a Prikry style iteration, and use it to force stationary reflection in the presence of not SCH. Then we obtain this situation at $\aleph_\omega$. This is joint work with Alejandro Poveda and Assaf Rinot.

**Information:**For the Zoom access code, contact Samuel Adam-Day me@samadamday.com. Link: https://zoom.us/j/96803195711 (open 30 minutes before)

**Barcelona Set Theory Seminar****Time:** Wednesday, 20 January, 16:00-17:30 CET**Speaker:** Vera Fischer, University of Vienna**Title:** Independent families in the countable and the uncountable**Abstract:** Independent families on w are families of infinite sets of integers with the property that for any two finite subfamilies A and B the set Ç A\È B is infinite. Of particular interest are the sets of the possible cardinalities of maximal independent families, which we refer to as the spectrum of independence. Even though we do have the tools to control the spectrum of independence at w (at least to a large extent), there are many relevant questions regarding higher counterparts of independence in generalised Baire spaces still remaining open.**Information:** Online. If you wish to attend, please send an email to bagaria@ub.edu asking for the link.

**Paris-Lyon Séminaire de LogiqueTime:** Wednesday, 20 January, 16:00-17:00 CEST

**Speaker:**Gianluca Basso, University of Lyon

**Title:**Compact metrizable structures via projective Fraïssé theory

**Abstract:**The goal of projective Fraïssé theory is to approximate compact metrizable spaces via classes of ﬁnite structures and glean topological or dynamical properties of a space by relating them to combinatorial features of the associated class of structures. We will discuss general results, using the framework of compact metrizable structures, as well as applications to the study a class of one-dimensional compact metrizable spaces, that of smooth fences, and to a particular smooth fence with remarkable properties, which we call the Fraïssé fence.

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

11-17 January

**CUNY Set Theory Seminar****Time:** Friday,15 January, 2pm New York time (20:00 CET)**Speaker:** Trevor Wilson, Miami University**Title:** tba **Abstract:** tba **Information:** The seminar will take place virtually. Please email Victoria Gitman (vgitman@nylogic.org) for the meeting id.

**Turin-Udine logic seminar****Time:** Friday, 15 January, 16:30-18:30 CET**Speaker:** A. Freund, TU Darmstadt**Title:** Ackermann, Goodstein, and infinite sets**Abstract:** In this talk, I show how Goodstein’s classical theorem can be turned into a statement that entails the existence of complex infinite sets, or in other words: into an object of reverse mathematics. This more abstract approach allows for very uniform results of high explanatory power. Specifically, I present versions of Goodstein’s theorem that are equivalent to arithmetical comprehension and arithmetical transfinite recursion. To approach the latter, we will study a functorial extension of the Ackermann function to all ordinals. The talk is based on a joint paper with J. Aguilera, M. Rathjen and A. Weiermann.**Information:** Online on WebEx. Please see the seminar webpage.

**KGRC Research Seminar, ViennaTime:** Thursday, 14 January, 15:00-16:30 CET

**Speaker:**Jeffrey Bergfalk, University of Vienna

**Title:**Infinitary combinatorics and strong homology

**Abstract:**Motivated by several recent advances, we will provide a research history of the main set-theoretic problems arising in the study of strong homology. As such, this talk will overlap with one on the same theme given in Paris-Lyon Logic Seminar last fall. We will presume no awareness in our audience either of strong homology or of that talk, but will aim in this one to provide, along with the necessary background, some sketch of the main ideas behind several recent arguments. This is an area in which simplicial principles and infinitary combinatorics come together. Its questions, at heart, have tended to be questions about higher-dimensional variants of classical set-theoretic concerns (like nontrivial coherence, Δ systems, etc.); these questions, in turn, increasingly appear to be of some interest in their own right.

**Information:**Talk via zoom.

**Set theory workshop Sao Paulo, for the World Logic DayTime:** Thursday, 14 January, 9:00-18:35 Brazil time (13:00-22:35 CET)

**Speakers:**Christina Brech (São Paulo), Vera Fischer (Vienna), Yurii Khomskii (Hamburg , Victor dos Santos Ronchim (São Paulo), Dorottya Sziráki (Budapest), Artur Hideyuki Tomita (São Paulo)

**Information:**Please register on the conference webpage ahead of time.

**Barcelona Set Theory Seminar****Time:** Wednesday, 13 January, 16:00-17:30 CET**Speaker:** Trevor Wilson, Miami University**Title:** tba**Abstract:** tba**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, 13 January, 14:00-16:00 Israel Time (13:00-15:00 CET)**Speaker:** Juris Steprans, York University, Toronto **Title:** Universal functions, strong colourings and ideas from PID**Abstract:** A construction of Shelah will be reformulated using the PID to provide alternative models of the failure of CH and the existence of a universal colouring of cardinality aleph_1. The impact of the range of the colourings will be examined. An application to the theory of strong colourings over partitions will **Information:** Contact Menachem Magidor, Asaf Rinot or Omer Ben-Neria ahead of time for the zoom link.

**Caltech Logic Seminar****Time:** Monday, 11 January, 12:00 – 1:00pm Pacific time (21:00 CET)**Speaker:** Zoltán Vidnyánszky, Caltech**Title:** A new regularity property of the Haar null ideal **Abstract:** Christensen’s Haar null ideal is a well-behaved generalization of Haar null sets to groups, which admit no Haar measure. We show that in the group ZωZω, every Haar positive (that is, non-Haar null) analytic set contains a Haar positive closed set. Using this result, we determine the exact Wadge class of the family of Haar null closed subsets of ZωZω. **Information:** See the seminar webpage.

**Genova logic seminar****Time:** Monday, 11 January, 15.00-16.30 CET **Speaker:** Filippo Calderoni, University of Illinois at Chicago**Title:** Categorifying Borel reducibility**Abstract:** The theory of Borel classification is a central research area in modern descriptive set theory. It provides a logical treatment to the process of classification and has been used effectively in different areas of mathematics as a tool to detect structural obstructions against classification theorems. The idea of making Borel reducibility functorial goes back to the start of the area, being raised already in one of the initial papers by Friedman and Stanley. In this talk we will discuss yet another attempt to formalize Borel reducibility in a categorical framework. This is joint work in progress with Andrew Brooke-Taylor. **Information:** The seminar will be held on Microsoft Teams, at the page of the Genoa logic group. The access code is: fpedcxn. Alternatively, you can write to camerlo@dima.unige.it to have an access link. Further information on the activities of the Genoa logic group can be

found at logic.dima.unige.it

4-10 January

**CUNY Set Theory Seminar****Time:** Friday, 8 January, 2pm New York time (20:00 CET)**Speaker:** Thilo Weinert, University of Vienna **Title:** A miscellany of observations regarding cardinal characteristics of the continuum**Abstract:** We are going to talk about some inequalities between cardinal characteristics of the continuum. In particular we are going to relate cardinal characteristics pertaining to the convergenve of series, recently isolated by Blass, Brendle, Brian and Hamkins, other characteristcs concerning equitable splitting defined comparatatively recently by Brendle, Halbeisen, Klausner, Lischka and Shelah and some characteristics defined less recently by Miller, Blass, Laflamme and Minami. All proofs in question are elementary.**Information:** The seminar will take place virtually. Please email Victoria Gitman (vgitman@nylogic.org) for the meeting id.

**Turin-Udine logic seminar****Time:** Friday, 8 January, 16:30-18:30 CET**Speaker:** F. Calderoni, University of Illinois at Chicago**Title:** The Borel structure on the space of left-orderings**Abstract:** In this talk we shall present some results on left-orderable groups and their interplay with descriptive set theory. We shall discuss how Borel classification can be used to analyze the space of left-orderings of a given countable group modulo the conjugacy action. In particular, we shall see that if G is not locally indicable then the conjugacy relation on LO(G) is not smooth. Also, if G is a nonabelian free group, then the conjugacy relation on LO(G) is a universal countable Borel equivalence relation. Our results address a question of Deroin-Navas-Rivas and show that in many cases LO(G) modulo the conjugacy action is nonstandard. This is joint work with A. Clay.**Information:** Online on WebEx. Please see the seminar webpage.

**Turin logic seminar****Time:** Friday, 8 January, 09.30-10.30 CET**Speaker:** A. Conversano, Massey, New Zealand**Title:** Model theory and groups**Abstract:** tba**Information:** Online. Please see the seminar webpage.