Announcements are updated continuously on the website. For a list of talks in the coming weeks, see https://ests.wordpress.com/online-activities-2022.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

led to realize uncountable versions of the Axiom of Dependent Choice.**Information:** Online. If you wish to attend, please send an email to bagaria@ub.edu asking for the link.

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

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

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

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

**Abstract:**A

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

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

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

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

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