June 16

**Udine graduate course**

Time: Tuesday, June 16, 10:00-12:00 CEST

**Speaker:** Vincenzo Dimonte, University of Udine

**Title:** Generalized Descriptive Set Theory II, Lecture 4

**Abstract:** The objective of the course is to prove an analogue of Silver’s Theorem for the space $2^\lambda$, where $\lambda$ is an uncountable cardinal of cofinality $\omega$, using some large cardinal strength (the proof is still unpublished).

This result has been chosen as an example to show, more in general, how to generalize a deep classical theorem in this setting, which properties of singular cardinals are useful in that respect, and what are the main obstacles of the generalization. The proof will use some peculiarities of singular cardinal combinatorics and some large cardinal strength, and everything will be introduced in the first three lessons.

The course is self-contained (despite the name), the only prerequisite is to know basic set theory (the theory of forcing, the most basic descriptive set theory, maybe inaccessible cardinals).

The following is a tentative schedule:Tuesday 10.00-12.00 CEST, Friday 10.00-12.00 CEST, from 5 June 2020, for 5 lessons.

Lesson 1: Measurable cardinals

Lesson 2: Prikry forcing, diagonal Prikry forcing

Lesson 3: Strong Prikry condition, “double” diagonal Prikry forcing

Lesson 4: generalized G_0 dichotomy

Lesson 5: generalized Silver Theorem

**Information:** Via Microsoft Teams, to participate contact vincenzo.dimonte@uniud.it.

**Münster Set Theory Seminar**

**Time:** Tuesday, June 16, 4:15pm CEST

**Speaker:** Farmer Schlutzenberg

**Title:** Remarks on rank-into-rank embeddings part II

**Abstract:** Recall that Woodin’s large cardinal axiom I0 gives an ordinal λ and an elementary embedding j:L(V_{λ+1})→L(V_{λ+1}) with critical point <λ. Using methods due to Woodin, we show that if ZFC+I0 is consistent then so is ZF+DC(λ)+ there is an ordinal λ and an elementary j:V_{λ+2}→V_{λ+2}”. (A version with the added assumption that V_{λ+1}^sharp exists is due to the author, and Goldberg observed that the appeal to V_{λ+1}^sharp could actually be replaced by some further calculations of Woodin’s.)

Reference: https://arxiv.org/abs/2006.01077, “On the consistency of ZF with an elementary embedding from V_{λ+2} into V_{λ+2}”.

**Information:** contact rds@wwu.de ahead of time in order to participate.

June 17

**Bar-Ilan University and Hebrew University Set Theory Seminar**

**Time:** Wednesday, April 17, 11:00-13:00 Israel time (10:00-13:00 CEST)

**Speaker:** Mirna Dzamonja (University of East Anglia)

**Title:** Wide Aronszajn trees

**Abstract:** A wide Aronszajn tree is a tree is size and height omega_1 but with no uncountable branch. Such trees arise naturally in the study of model-theoretic notions on models of size aleph_1 as well as in generalised descriptive set theory. In their 1994 paper devoted to various aspects of such trees, Mekler and Väänänen studied the so called weak embeddings between such trees, which are simply defined as strict-order preserving functions. Their work raised the question if under MA there exists a universal wide Aronszajn tree under such embeddings. We present a negative solution to this question, obtained in a paper to appear, joint with Shelah. We also discuss various connected notions and the history of the problem.

**Information:** Contact Assaf Rinot for the zoom id.

**Bristol Logic and Set Theory Seminar** (recurring lecture series)

Time: Wednesday, June 17, 13:30-15:00 UK time (14:30-16:00 CEST)

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

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

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

**Information:** zoom via https://zoom.us/j/96803195711

**Paris-Lyon Séminaire de Logique**

Time: Wednesday, June 17, 16:00-17:15 CEST

**Speaker:** Michał Skrzypczak – Université de Varsovie

**Title:** Measure theory and Monadic Second-order logic over infinite trees

**Abstract: **Monadic Second-order (MSO) logic is a well-studied formalism featuring many decision procedures and effective transformations. It is the fundamental logic considered in automata theory, equivalent to various other ways of defining sets of objects. In this talk, I will speak about the expressive power of MSO over infinite binary trees (i.e. free structures of two successors) – the theory from the famous Rabin’s decidability result.

The goal of the talk is to survey recent results about measure properties of MSO-definable sets of infinite trees. First, I will argue that these sets are indeed measurable (which is not obvious, as there exist non-Borel sets definable in MSO). Then I will move to the question of our ability to compute the measure of the set defined by a given formula. Although the general question is still open (and seems to be demanding), I will speak about decidability results for fragments of MSO, focusing on the so-called weak-MSO.

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

**Oxford Set Theory Seminar**

Time: Wednesday, June 17, 16:00-17:30 UK time (17:00-18:30 CEST)

**Speaker:** Corey Bacal Switzer, City University of New York

**Title:** Some Set Theory of Kaufmann Models

**Abstract:** A Kaufmann model is an ω1-like, recursively saturated, rather classless model of PA. Such models were shown to exist by Kaufmann under the assumption that ♢ holds, and in ZFC by Shelah via an absoluteness argument involving strong logics. They are important in the theory of models of arithmetic notably because they show that many classic results about countable, recursively saturated models of arithmetic cannot be extended to uncountable models. They are also a particularly interesting example of set theoretic incompactness at ω1, similar to an Aronszajn tree.

In this talk we’ll look at several set theoretic issues relating to this class of models motivated by the seemingly naïve question of whether or not such models can be killed by forcing without collapsing ω1. Surprisingly the answer to this question turns out to be independent: under MAℵ1 no ω1-preserving forcing can destroy Kaufmann-ness whereas under ♢ there is a Kaufmann model M and a Souslin tree S so that forcing with S adds a satisfaction class to M (thus killing rather classlessness). The techniques involved in these proofs also yield another surprising side of Kaufmann models: it is independent of ZFC whether the class of Kaufmann models can be axiomatized in the logic Lω1,ω(Q) where Q is the quantifier “there exists uncountably many”. This is the logic used in Shelah’s aforementioned result, hence the interest in this level of expressive power.

**Information:** For the Zoom access code, contact Samuel Adam-Day: me@samadamday.com.

**CUNY Set Theory Seminar**

**Time:** Wednesday, June 17, 14:00 New York time (20:00 CEST)

**Speaker: **Mateusz Łełyk, University of Warsaw

**Title:** Partial Reflection over Uniform Disquotational Truth

**Abstract:** In the context of arithmetic, a reflection principle for a theory Th is a formal way of expressing that all theorems of Th are true. In the presence of a truth predicate for the language of Th this principle can be expressed as a single sentence (called the Global Reflection principle over Th) but most often is met in the form of a scheme consisting of all sentences of the form ∀x(ProvTh(ϕ(x˙))→ϕ(x)).

Obviously such a scheme is not provable in a consistent theory Th. Nevertheless, such soundness assertions are said to provide a natural and justified way of extending ones initial theory.

This perspective is nowadays very fruitfully exploited in the context of formal theories of truth. One of the most basic observations is that strong axioms for the notions of truth follow from formally weak types of axiomatizations modulo reflection principles. In such a way compositional axioms are consequences of the uniform disquotational scheme for for the truth predicate, which is ∀xT(ϕ(x˙))≡ϕ(x).

The above observation is also used in the recent approach to ordinal analysis of theories of predicative strength by Lev Beklemishev and Fedor Pakhomov. The assignment of ordinal notations to theories proceeds via partial reflection principles (for formulae of a fixed Σn-complexity) over (iterated) disquotational scheme. It becomes important to relate theories of this form to fragments of standard theories of truth, in particular the ones based on induction for restricted classes of formulae such as CT0 (the theory of compositional truth with Δ0-induction for the extended language. The theory was discussed at length in Bartek Wcisło’s talk). Beklemishev and Pakhomov leave the following open question: Is Σ1-reflection principle over the uniform disquotational scheme provable in CT0? The main goal of our talk is to present the proof of the affirmative answer to this question. The result significantly improves the known fact on the provability of Global Reflection over PA in

CT0. During the talk, we explain the theoretical context described above including the information on how the result fits into Beklemishev-Pakhomov project. In the meantime we give a different proof of their characterisation of

Δ_0-reflection over the disquotational scheme.Despite the proof-theoretical flavour of these results, our proofs rests on essentially model-theoretical techniques. The important ingredient is the Arithmetized Completeness Theorem.

**Information:** The seminar will take place virtually. Please email Victoria Gitman (vgitman@nylogic.org) for the meeting id.

June 18

**Kurt Gödel Research Center Seminar** (organised by Ben Miller)

Time: Thursday, June 18, 16:00 CEST

**Speaker:** Anush Tserunyan, University of Illinois at Urbana-Champaign

**Title:** Hyperfinite subequivalence relations of treed equivalence relations

**Abstract:** A large part of measured group theory studies structural properties of countable groups that hold “on average”. This is made precise by studying the orbit equivalence relations induced by free Borel actions of these groups on a standard probability space. In this vein, the amenable groups correspond to hyperfinite equivalence relations, and the free groups to the treeable ones. In joint work with R. Tucker-Drob, we give a detailed analysis of the structure of hyperfinite subequivalence relations of a treed equivalence relation on a standard probability space, deriving the analogues of structural properties of amenable subgroups (copies of ℤZ) of a free group. Most importantly, just like every such subgroup is contained in a unique maximal one, we show that even in the non-pmp setting, every hyperfinite subequivalence relation is contained in a unique maximal one.

**Information:** Talk via zoom.

June 19

**Udine graduate course**

Time: Friday, June 19, 10:00-12:00 CEST

**Speaker:** Vincenzo Dimonte, University of Udine

**Title:** Generalized Descriptive Set Theory II, Lecture 5

**Abstract:** The objective of the course is to prove an analogue of Silver’s Theorem for the space $2^\lambda$, where $\lambda$ is an uncountable cardinal of cofinality $\omega$, using some large cardinal strength (the proof is still unpublished).

This result has been chosen as an example to show, more in general, how to generalize a deep classical theorem in this setting, which properties of singular cardinals are useful in that respect, and what are the main obstacles of the generalization. The proof will use some peculiarities of singular cardinal combinatorics and some large cardinal strength, and everything will be introduced in the first three lessons.

The course is self-contained (despite the name), the only prerequisite is to know basic set theory (the theory of forcing, the most basic descriptive set theory, maybe inaccessible cardinals).

The following is a tentative schedule:Tuesday 10.00-12.00 CEST, Friday 10.00-12.00 CEST, from 5 June 2020, for 5 lessons.

Lesson 1: Measurable cardinals

Lesson 2: Prikry forcing, diagonal Prikry forcing

Lesson 3: Strong Prikry condition, “double” diagonal Prikry forcing

Lesson 4: generalized G_0 dichotomy

Lesson 5: generalized Silver Theorem

**Information:** Via Microsoft Teams, to participate contact vincenzo.dimonte@uniud.it.

**CUNY Set Theory Seminar**

**Time:** Friday, June 19, 2pm New York time (8pm CEST)

**Speaker:** Boban Velickovic, University of Paris 7

**Title:** Strong guessing models

**Abstract:** The notion of a guessing model introduced by Viale and Weiss. The principle

GM(ω2,ω1) asserts that there are stationary many guessing models of size ℵ1 in Hθ, for all large enough regular θ. It follows from PFA and implies many of its structural consequences, however it does not settle the value of the continuum. In search of higher of forcing axioms it is therefore natural to look for extensions and higher versions of this principle. We formulate and prove the consistency of one such statement that we call SGM+(ω3,ω1).

It has a number of important structural consequences:

- the tree property at ℵ2 and ℵ3
- the failure of various weak square principles
- the Singular Cardinal Hypothesis
- Mitchell’s Principle: the approachability ideal agrees with the non stationary ideal on the set of cof(ω1) ordinals in ω2
- Souslin’s Hypothesis
- The negation of the weak Kurepa Hypothesis
- Abraham’s Principles: every forcing which adds a subset of

ω2 either adds a real or collapses some cardinals, etc.

The results are joint with my PhD students Rahman Mohammadpour.

**Information:** The seminar will take place virtually. Please email Victoria Gitman (vgitman@nylogic.org) for the meeting id.

**Toronto Set Theory Seminar**

**Time:** Friday, June 19, 1.30pm Toronto time (7.30pm CEST)

**Speaker:** David Schrittesser, University of Vienna

**Title:** Higher degrees of madness

**Abstract:** The notion of mad family can be generalized by replacing the finite ideal by an iterated Fubini product of the finite ideal. While these ideals are more complicated both combinatorially and in terms of Borel complexity, it turns out that the same assumptions of Ramsey theoretic regularity can rule out their existence. We sketch a proof of this and some related results. This talk is a sequel to my last talk at the Fields Institute Seminar.

**Information:** The seminar will take place virtually. ZOOM ID: https://yorku.zoom.us/j/96087161597