Since March 2020, a number of set theory seminars worldwide have moved online, which provides an unprecedented opportunity for participation.

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

Besides our list, take a look at the seminar web pages linked below and the following for more seminar listings:

**Online seminars and talks in Logic** A list of online seminars in logic, collected by Miguel Moreno. **Logic Supergroup (UCONN)** An alliance of logicians in quarantine, comprised of logic groups across the world, hosting virtual talks.

Please email new announcements to us at philipp.schlicht@bristol.ac.uk, matteo.viale@unito.it and boban.velickovic@math.univ-paris-diderot.fr.

Week 21-27 September

**CUNY Set Theory Seminar****Time:** Friday, September 25, 11:00 New York time (17:00 CEST)**Speaker:** Ralf Schindler, University of Münster**Title:** Martin’s Maximum^++ implies the P_max axiom (*)**Abstract:** Forcing axioms spell out the dictum that if a statement can be forced, then it is already true. The P_max axiom (*) goes beyond that by claiming that if a statement is consistent, then it is already true. Here, the statement in question needs to come from a resticted class of statements, and ‘consistent’ needs to mean ‘consistent in a strong sense.’ It turns out that (*) is actually equivalent to a forcing axiom, and the proof is by showing that the (strong) consistency of certain theories gives rise to a corresponding notion of forcing producing a model of that theory. This is joint work with D. Asperó building upon earlier work of R. Jensen and (ultimately) Keisler’s ‘consistency properties’.**Information:** The seminar will take place virtually. Please email Victoria Gitman (vgitman@nylogic.org) for the meeting id.

**Caltech Logic Seminar****Time:** Wednesday, September 16, 12:00 – 1:00pm Pacific time (21:00 CEST)**Speaker:** Dana Bartošová, University of Florida**Title:** On phase spaces of universal minimal flows of groups with compact normal subgroups**Abstract:** For a topological group G, a G-flow is a continuous action of G on a compact Hausdorff space X; we call X the phase space of the G-flow. A G-flow on X is minimal if X has no closed non-trivial invariant subset. The universal minimal G-flow, M(G), has every minimal G-flow as a quotient and it is unique up to isomorphism. We show that whenever we have a short exact sequence 0→K→G→H→0 of topological groups with the image of K a compact normal subgroup of G, then the phase space of M(G) is homeomorphic to the product of the phase space of M(H) with K. For instance, if G is a Polish, non-Archimedean group, and the image of K is open in G, then H is a countable discrete group. The phase space of M(H) is homeomorphic to Gl(22ℵ0), the Stone space of the completion of the free Boolean algebra on 2ℵ0 generators by Balcar-Błaszczyk and Glasner-Tsankov-Weiss-Zucker. Therefore, the phase space of M(G) is homeomorphic to K×Gl(22ℵ0). When the sequence splits, that is, G≅H⋉K, then the homeomorphism witnesses an isomorphism of flows, recovering a result of Kechris and Sokić.**Information:** Online talk https://caltech.zoom.us/j/99296122790?pwd=bUN4RS94RVYrTEtGTGhqTHRJbm9nZz09

**Southern Illinois University Logic Seminar****Time:** Thursday, 10 September, 1pm US Central Daylight Time (20:00 CEST) **Speaker:** Arno Pauly, Swansea University**Title:** How computability-theoretic degree structures and topological spaces are related**Abstract:** We can generalize Turing reducibility to points in a large class of topological spaces. The point degree spectrum of a space is the collection of the degrees of its points. This is always a collection of Medvedev degrees, and it turns out that topological properties of the space are closely related to what degrees occur in it. For example, a Polish space has only Turing degrees iff it is countably dimensional. This connection can be used to bring topological techniques to bear on problems from computability theory and vice versa. The talk is based on joint work with Takayuki Kihara and Keng Meng Ng (https://arxiv.org/abs/1405.6866 and https://arxiv.org/abs/1904.04107).**Information:** The seminar will take place virtually via zoom.

Week 14-20 September

**CUNY Set Theory Seminar****Time:** Friday, September 18, 14:00 New York time (20:00 CEST)**Speaker:** Arthur Apter, CUNY**Title:** UA and the Number of Normal Measures over ℵω+1**Abstract:** The Ultrapower Axiom UA, introduced by Goldberg and Woodin, is known to have many striking consequences. In particular, Goldberg has shown that assuming UA, the Mitchell ordering of normal measures over a measurable cardinal is linear. I will discuss how this result may be used to construct choiceless models of ZF in which the number of normal measures at successors of singular cardinals can be precisely controlled.**Information:** The seminar will take place virtually. Please email Victoria Gitman (vgitman@nylogic.org) for the meeting id.

**CUNY Logic Seminar (MOPA)****Time:** Wednesday, September 16, 5pm New York time (23:00 CEST) – note the time**Speaker: **Sam Coskey, Boise State University**Title:** Classification of countable models of ZFC**Abstract:** In 2009 Roman Kossak and I showed that the classification of countable models of PA is Borel complete, which means it is as complex as possible. The proof is a straightforward application of Gaifman’s canonical I-models. In 2017 Sam Dworetzky, John Clemens, and I showed that the argument may also be used to show the classification of countable models of ZFC is Borel complete too. In this talk I’ll outline the original argument for models of PA, the adaptation for models of ZFC, and briefly consider several subclasses of countable models of ZFC. **Information:** The seminar will take place virtually. Please email Victoria Gitman (vgitman@nylogic.org) for the meeting id.

**Caltech Logic Seminar****Time:** Wednesday, September 16, 12:00 – 1:00pm Pacific time (21:00 CEST) **Speaker:** Steve Jackson, University of North Texas **Title:** Some complexity results in dynamics and number theory**Abstract:** The Ki-Linton theorem asserts that the set of base b normal numbers is a Π03-complete set. The base bb normal numbers can be viewed as the set of generic points for an associated dynamical system. This leads to the question of the complexity of the set of generic points for other numeration/dynamical systems, for example continued fractions, β-expansions, Lüroth expansions to name a few. We prove a general result which covers all of these cases, and involves a well-known property in dynamics, a form of the specification property. We then consider differences of these sets. Motivated by the descriptive set theory arguments, we are able to show that the set of continued fraction normal but not base b normal numbers is a complete D2(Π30) set. Previously, the best known result was that this set was non-empty (due to Vandehey), and this assumed the generalized Riemann hypothesis. The first part of the work is joint with Mance and Kwietniak, and the second part with Mance and Vandehey.

Information: Online talk https://caltech.zoom.us/j/95952118325?pwd=QzFPa3ZOeTJKWXJnSW5VbHhGOXJEZz09

**Logic Seminar, Carnegie Mellon University****Time:** Tuesday, September 15, 3:30 – 4:30pm Eastern Daylight Time (21:00 CEST) **Speaker:** William Chan (Carnegie Mellon University) **Title:** A Survey of Combinatorics and Cardinality under Determinacy **Abstract: **We will survey some recent work with Jackson and Trang concerning combinatorics under the axiom of determinacy. We will be especially concerned with ultrapowers of the first uncountable cardinal by the partition measures and related questions concerning club uniformization and continuity of functions around the first uncountable cardinal. We will show that the cardinals below the power set of the first and second uncountable cardinals have a very complicated and rich structure under determinacy axioms. We will summarize our knowledge of this structure under AD, AD+, and the axiom of real determinacy. **Information:** Zoom link https://cmu.zoom.us/j/621951121, meeting ID: 621 951 121

**Logic and Metaphysics Workshop**, CUNY **Time:** Monday, September 14th, 4.15-6.15 pm (22.15 CEST)**For zoom information, email Yale Weiss at: yweiss@gradcenter.cuny.edu. Speaker:** Chris Scambler (NYU)

**Title:**Cantor’s Theorem, Modalized

**Abstract:**I will present a modal axiom system for set theory that (I claim) reconciles mathematics after Cantor with the idea there is only one size of infinity. I’ll begin with some philosophical background on Cantor’s proof and its relation to Russell’s paradox. I’ll then show how techniques developed to treat Russell’s paradox in modal set theory can be generalized to produce set theories consistent with the idea that there’s only one size of infinity.

Week 7-13 September

**Southern Illinois University Logic Seminar****Time:** Thursday, 10 September, 1pm US Central Daylight Time (20:00 CEST) **Speaker:** Mirna Džamonja (IHPST, CNRS-Université Panthéon-Sorbonne Paris, France)**Title:** On logics that make a bridge from the Discrete to the Continuous**Abstract:** We study logics which model the passage between an infinite sequence of finite models to an uncountable limiting object, such as is the case in the context of graphons. Of particular interest is the connection between the countable and the uncountable object that one obtains as the union versus the combinatorial limit of the same sequence.**Information:** The seminar will take place virtually via zoom.

**CUNY Logic Seminar (MOPA)****Time:** Wednesday, September 9, 3pm New York time (21:00 CEST) – note the time**Speaker: **Saeideh Bahrami, Institute for Research in Fundamental Sciences, Tehran**Title:** Fixed Points of Initial Self-Embeddings of Models of Arithmetic**Abstract:** In 1973, Harvey Friedman proved his striking result on *initial self-embeddings* of countable nonstandard models of set theory and Peano arithmetic. In this talk, I will discuss my joint work with Ali Enayat focused on the fixed point set of initial self-embeddings of countable nonstandard models of arithmetic. Especially, I will survey the proof of some generalizations of well-known results on the fixed point set of automorphisms of countable recursively saturated models of PA, to results about the fixed point set of initial self-embeddings of countable nonstandard models of IΣ1.**Information:** The seminar will take place virtually. Please email Victoria Gitman (vgitman@nylogic.org) for the meeting id.

**Genoa Logic Seminar****Time:** Tuesday, September 8, 17:00-18:00 CEST**Speaker:** Gianluca Basso, Université de Lausanne**Title:** Topological dynamics beyond Polish groups**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 $\mathrm{Sym}(\kappa)$ is the space of linear orders on $\kappa$—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 show that the class of groups satisfying such properties is closed under some topological operations and use this to compute the universal minimal flows of some concrete groups, like $\mathrm{Homeo}(\omega_{1})$. This is joint work with Andy Zucker.**Information:** The seminar will take place 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 http://www.dima.unige.it/~camerlo/glhome.html

Note: Due to the current emergency situation, the web page might not be updated.

Week 31 August – 6 September

**CUNY Set Theory Seminar****Time:** Friday, September 4, 14:00 New York time (20:00 CEST)**Speaker:** Mirna Džamonja, IHPST, CNRS-Université Panthéon-Sorbonne Paris**Title:** On logics that make a bridge from the Discrete to the Continuous**Abstract:** We study logics which model the passage between an infinite sequence of finite models to an uncountable limiting object, such as is the case in the context of graphons. Of particular interest is the connection between the countable and the uncountable object that one obtains as the union versus the combinatorial limit of the same sequence.**Information:** The seminar will take place virtually. Please email Victoria Gitman (vgitman@nylogic.org) for the meeting id.

**CUNY Logic Seminar (MOPA)****Time:** Wednesday, September 2, 14:00 New York time (20:00 CEST) – note the time**Speaker: **Petr Glivický, Universität Salzburg**Title:** The ω-iterated nonstandard extension of N and Ramsey combinatorics**Abstract:** In the theory of nonstandard methods (traditionally known as nonstandard analysis), each mathematical object (a set) x has a uniquely determined so called nonstandard extension ∗x. In general, ∗x⊋{∗y;y∈x} – that is, besides the original ‘standard’ elements ∗y for y∈x, the set ∗x contains some new ‘nonstandard’ elements.

For instance, some of the nonstandard elements of ∗R can be interpreted as infinitesimals (there is ε∈∗R such that 0<ε<1/n for all n∈N) allowing for nonstandard analysis to be developed in ∗R, while ∗N turns out to be an (at least ℵ1-saturated) nonstandard elementary extension of N (in the language of arithmetic).

While the whole nonstandard real analysis is most naturally developed in ∗R (with just a few advanced topics where using the second extension ∗∗R is convenient, though far from necessary), recent successful applications of nonstandard methods in combinatorics on N have utilized also higher order extensions (n)∗N=∗∗∗⋯∗N with the chain ∗∗∗⋯∗ of length n>2.

In this talk we are going to study the structure of the ω-iterated nonstandard extension ⋅N=⋃n∈ω(n)∗N of N and show how the obtained results shed new light on the complexities of Ramsey combinatorics on N and allow us to drastically simplify proofs of many advanced Ramsey type theorems such as Hindmann’s or Milliken’s and Taylor’s.**Information:** The seminar will take place virtually. Please email Victoria Gitman (vgitman@nylogic.org) for the meeting id.

**Paul Bernays Lectures 2020: Struggling with the size of infinity**, Lecture 3

**Time:**Tuesday, September 1, 16:30 CEST

**Speaker:**Prof. Saharon Shelah, Hebrew University Jerusalem

**Title:**Cardinal invariants of the continuum: are they all independent?

**Abstract:**Experience has shown that in almost all cases, if you define a bunch of cardinal invariants of the continuum, then modulo some easy inequalities, it follows by forcing (the method introduced by Cohen) that there are no more restrictions. Well, those independence results have been mostly for the case of the continuum being at most aleph2. But this seems to be just due to our lack of ability, as the problems are harder. However, this opinion ignores the positive side of having forcing, of us being able to prove independence results: clearing away the rubble of independence results, the cases where we fail may indicate that there are theorems there. We shall on the one hand deal with cases where this succeeds and on the other hand with cofinality arithmetic, and what was not covered in the first lecture.

**Information:**Due to the unusual circumstances of the COVID-19 pandemia, the Paul Bernays Lectures 2020 will take place as a webinar: Link for this webinar. All lectures are given in English and are self-contained. Lecture 1 is aimed at a general audience; lecture 2 and 3 address the scientific community.

** Paul Bernays Lectures 2020: Struggling with the size of infinity**, Lecture 2

**Time:**Tuesday, September 1, 14:15 CEST

**Speaker:**Prof. Saharon Shelah, Hebrew University Jerusalem

**Title:**How large is the continuum?

**Abstract:**Cantor discovered that in mathematics we can distinguish many infinities, called the aleph numbers. The works of Gödel and Cohen told us that we cannot decide what the value of the continuum is, that is, which of the aleph numbers is the answer to the question “How many real numbers are there?”. This still does not stop people from having opinions and arguments. One may like to assume extra axioms which will decide the question (usually as aleph1 or aleph2), and argue that they should and eventually will be adopted. We feel that assuming the continuum is small makes us have equalities which are incidental. So if we can define 10 natural cardinals which are uncountable but at most the continuum, and the continuum is smaller than aleph10, at least two of them will be equal, without any inherent reasons. Such numbers are called cardinal invariants of the continuum, and they arise naturally from various perspectives. We would like to show that they are independent, that is, that there are no non-trivial restrictions on their order. More specifically, we shall try to explain the Cichon diagram and what we cannot tell about it.

**Information:**Due to the unusual circumstances of the COVID-19 pandemia, the Paul Bernays Lectures 2020 will take place as a webinar: Link for this webinar. All lectures are given in English and are self-contained. Lecture 1 is aimed at a general audience; lecture 2 and 3 address the scientific community.

**Paul Bernays Lectures 2020: Struggling with the size of infinity**, Lecture 1**Time:** Monday, August 31, 17:00 CEST**Speaker:** Prof. Saharon Shelah, Hebrew University Jerusalem**Title:** Cardinal arithmetic: Cantor’s paradise**Abstract:** We will explain Hilbert’s first problem. Specifically, this asks what the value of the continuum is: Is the number of real numbers equal to aleph1 — the first infinite cardinal above aleph0=the number of natural numbers? Recall that Cantor introduces infinite numbers just as equivalence classes of sets under “there is a bijection”. The problem of the size of the continuum really means: ”What are the laws of cardinal arithmetic, i.e. the arithmetic of infinite numbers”. We will review its history, (including Gödel and Cohen), mention different approaches, explain what is undecidable, and mainly present some positive answers which we have nowadays. Those will be mainly about cofinality arithmetic, the so called pcf theory; but we will also mention cardinal invariants of the continuum.**Information:** Due to the unusual circumstances of the COVID-19 pandemia, the Paul Bernays Lectures 2020 will take place as a webinar: Link for this webinar. All lectures are given in English and are self-contained. Lecture 1 is aimed at a general audience; lecture 2 and 3 address the scientific community.

Week 24-30 August

**CUNY Set Theory Seminar****Time:** Friday, August 28, 14:00 New York time (20:00 CEST)**Speaker:** Miha Habic, Bard College at Simon’s Rock**Title: **Normal ultrapowers with many sets of ordinals**Abstract:** Any ultrapower M of the universe by a normal measure on a cardinal κ is quite far from V in the sense that it computes V_κ+2 incorrectly. If GCH holds, this amounts to saying that M is missing a subset of κ+. Steel asked whether, even in the absence of GCH, normal ultrapowers at κ must miss a subset of κ+. In the early 90s Cummings gave a negative answer, building a model with a normal measure on κ whose ultrapower captures the entire powerset of κ+. I will present some joint work with Radek Honzík in which we improved Cummings’ result to get this capturing property to hold at the least measurable cardinal.**Information:** Please email Victoria Gitman (vgitman@nylogic.org) for meeting id (this talk will have a different meeting ID!).

**CUNY Logic Seminar (MOPA)****Time:** Wednesday, August 26, 12:00 New York time (18:00 CEST) – note the time**Speaker: **Emil Jeřábek, Czech Academy of Sciences**Title:** Feasible reasoning with arithmetic operations**Abstract:** In bounded arithmetic, we study weak fragments of arithmetic that often correspond in a certain sense to computational complexity classes (e.g., polynomial time). Questions about provability in such theories can be thought of as a form of *feasible reasoning*: considering a natural object of interest from a complexity class C, can we prove its fundamental properties using only concepts from C?

Our objects of interest in this talk will be the elementary integer arithmetic operations +,−,×,/, whose complexity class is (uniform) TC0, a small subclass of P. The corresponding arithmetical theory is VTC0. Since we do not know yet if the theory can prove the totality of division and iterated multiplication ∏i<nXi which are in TC0 by an intricate result of Hesse, Allender, and Barrington, we will also consider an extension of the theory VTC0+IMUL.

Our main question is what can VTC0±IMUL prove about the elementary arithmetic operations. The answer is that more than one might expect: VTC0+IMUL proves induction for quantifier-free formulas in the basic language of arithmetic (IOpen), and even induction and minimization for Σb0 (sharply bounded) formulas in Buss’s language. This result is connected to the existence of TC0 constant-degree root-finding algorithms; the proof relies on a formalization of a form of the Lagrange inversion formula in VTC0+IMUL, and on model-theoretic abstract nonsense involving valued fields.

The remaining problem is if VTC0 proves IMUL. We will discuss issues with formalization of the Hesse–Allender–Barrington construction in VTC0, and some partial results (this is a work in progress).**Information:** The seminar will take place virtually. Please email Victoria Gitman (vgitman@nylogic.org) for the meeting id.

**Bar-Ilan-****J****erusalem Set Theory Seminar****Time:** Wednesday, August 26, 11:00am Israel Time (10:00 CEST)**Speaker:** Menachem Magidor, Hebrew University**Title:** Around weak diamond and uniformization**Abstract:** I’ll present some old results about weak diamond, uniformization and maybe some connections to Whitehead problem. In particular I’ll present Woodin’s elegant proof to the Devlin-Shelah equivalence of Weak diamond with 2^\aleph_0<2^\aleph_1. **Information:** Contact Menachem Magidor, Asaf Rinot or Omer Ben-Neria ahead of time for the zoom link.

**ALGOS 2020** – ALgebras, Graphs and Ordered Sets – August 26th to 28th (Online)**Time:** Wednesday, August 26, 9:30am – Friday, August 29, 19:30pm, CEST**Information:** Please see here for the program and advance registration.

Week 17-23 August

**CUNY Set Theory Seminar****Time:** Friday, August 21, 14:00 New York time (20:00 CEST)**Speaker:** Dan Hathaway University of Vermont**Title: **A relative of ZF+DC+‘ω1 is measurable’**Abstract:** Let Φ be the statement that for any function f:ω1×ω1→ω, there are functions g1,g2:ω1→ω such that for all (x,y)∈ω1×ω1, we have f(x,y)≤max {g1(x),g2(y)}. We will show that Φ follows from ZF+DC+‘ω1 is measurable’. On the other hand using core models, we will show that Φ+‘the club filter on ω1 is normal’ implies there are inner models with many measurable cardinals. We conjecture that Φ and ZF+DC+‘ω1 is measurable’ have the same consistency strength. The research is joint with Francois Dorais at the University of Vermont.**Information:** Please email Victoria Gitman (vgitman@nylogic.org) for meeting id (this talk will have a different meeting ID!).

**CUNY Logic Seminar (MOPA)****Time:** Wednesday, August 19, 14:00 New York time (20:00 CEST) – note the time**Speaker: **Leszek Kołodziejczyk, University of Warsaw**Title:** Ramsey’s Theorem over RCA_0***Abstract:** The usual base theory used in reverse mathematics, RCA0, is the fragment of second-order arithmetic axiomatized by Δ01 comprehension and Σ01 induction. The weaker base theory RCA∗0 is obtained by replacing Σ01 induction with Δ01 induction (and adding the well-known axiom exp in order to ensure totality of the exponential function). In first-order terms, RCA0 is conservative over IΣ1 and RCA∗0 is conservative over BΣ1+exp.

Some of the most interesting open problems in reverse mathematics concern the first-order strength of statements from Ramsey Theory, in particular Ramsey’s Theorem for pairs and two colours. In this talk, I will discuss joint work with Kasia Kowalik, Tin Lok Wong, and Keita Yokoyama concerning the strength of Ramsey’s Theorem over RCA∗0.Given standard natural numbers n,k≥2, let RTnk stand for Ramsey’s Theorem for k-colourings of n-tuples. We first show that assuming the failure of Σ01 induction, RTnk is equivalent to its own relativization to an arbitrary Σ01-definable cut. Using this, we give a complete axiomatization of the first-order consequences of RCA∗0+RTnk for n≥3 (this turns out to be a rather peculiar fragment of PA) and obtain some nontrivial information about the first-order consequences of RT2k. Time permitting, we will also discuss the question whether our results have any relevance for the well-known open problem of characterizing the first-order consequences of RT22 over the traditional base theory RCA0.**Information:** The seminar will take place virtually. Please email Victoria Gitman (vgitman@nylogic.org) for the meeting id.

**Bar-Ilan-****J****erusalem Set Theory Seminar****Time:** Wednesday, August 19, 11:00am Israel Time (10:00 CEST)**Speaker:** Uri Abraham, Ben-Gurion University**Title:** Coding well ordering of the reals with ladders, part 5 **Abstract:** Results from the 2002 paper “Coding with Ladders a Well Ordering of the Reals” by Abraham and Shelah.**Information:** contact Menachem Magidor, Asaf Rinot or Omer Ben-Neria ahead of time for the zoom link.

Week 10-16 August

**CUNY Set Theory Seminar****Time:** Friday, August 14, 14:00 New York time (20:00 CEST)**Speaker:** Gunter Fuchs CUNY**Title: **Canonical fragments of the strong reflection principle**Abstract:** I have been working over the past few years on the project of trying to improve our understanding of the forcing axiom for subcomplete forcing. The most compelling feature of this axiom is its consistency with the continuum hypothesis. On the other hand, it captures many of the major consequences of Martin’s Maximum. It is a compelling feature of Martin’s Maximum that many of its consequences filter through Todorcevic’s Strong Reflection Principle SRP. SRP has some consequences that the subcomplete forcing axiom does not have, like the failure of CH and the saturation of the nonstationary ideal. It has been unclear until recently whether there is a version of SRP that relates to the subcomplete forcing axiom as the full SRP relates to Martin’s Maximum, but it turned out that there is: I will detail how to associate in a canonical way to an arbitrary forcing class its corresponding fragment of SRP in such a way that (1) the forcing axiom for the forcing class implies its fragment of SRP, (2) the stationary set preserving fragment of SRP is the full principle SRP, and (3) the subcomplete fragment of SRP implies the major consequences of the subcomplete forcing axiom. I will describe how this association works, describe some hitherto unknown effects of (the subcomplete fragment of) SRP on mutual stationarity, and say a little more about the extent of (3).**Information:** Please email Victoria Gitman (vgitman@nylogic.org) for meeting id (this talk will have a different meeting ID!).

**CUNY Logic Seminar (MOPA)****Time:** Wednesday, August 12, 19:00 New York time (1:00am July 30 CEST) – note the time**Speaker: **Athar Abdul-Quader Purchase College**Title:** CP-genericity and neutrality**Abstract:** In a paper with Kossak in 2018, we studied the notion of neutrality: a subset X of a model M of PA is called neutral if the definable closure relation in (M, X) coincides with that in M. This notion was suggested by Dolich. motivated by work by Chatzidakis-Pillay on generic expansions of theories. In this talk, we will look at a more direct translation of the Chatzidakis-Pillay notion of genericity, which we call ‘CP-genericity’, and discuss its relation to neutrality. The main result shows that for recursively saturated models, CP-generics are always neutral; previously we had known that not all neutral sets are CP-generic. **Information:** The seminar will take place virtually. Please email Victoria Gitman (vgitman@nylogic.org) for the meeting id.

**Bar-Ilan-****J****erusalem Set Theory Seminar****Time:** Wednesday, August 12, 11:00am Israel Time (10:00 CEST)**Speaker:** Uri Abraham, Ben-Gurion University**Title:** Coding well ordering of the reals with ladders, part 4 **Abstract:** Results from the 2002 paper “Coding with Ladders a Well Ordering of the Reals” by Abraham and Shelah.**Information:** contact Menachem Magidor, Asaf Rinot or Omer Ben-Neria ahead of time for the zoom link.

Week 3-9 August

**CUNY Set Theory Seminar****Time:** Friday, August 7, 14:00 New York time (20:00 CEST)**Speaker:** Brent Cody, Virginia Commonwealth University**Title: **Higher indescribability**Abstract:** To what extent can formulas from infinitary logics be used in set-theoretic reflection arguments? If κ is a measurable cardinal, any Lκ,κ sentence which is true in (κ,∈), must be true about some strictly smaller cardinal. Whereas, there are Lκ+,κ+ sentences of length κ which are true in (κ,∈)and which are not true about any smaller cardinal. However, if κ is a measurable cardinal and some Lκ+,κ+ sentence φ is true in (κ,∈), then there must be some strictly smaller cardinal α<κsuch that a canonically restricted version of φ holds about α. Building on work of Bagaria and Sharpe-Welch, we use canonical restriction of formulas to define notions of Π1ξ-indescribability of a cardinal κ for all ξ<κ+. In this context we show that such higher indescribability hypotheses are strictly weaker than measurability, we prove the existence of universal Π1ξ-formulas, study the associated normal ideals and notions of ξ-clubs and prove a hierarchy result. Time permitting we will discuss some applications.**Information:** Please email Victoria Gitman (vgitman@nylogic.org) for meeting id (this talk will have a different meeting ID!).

**Bar-Ilan-****J****erusalem Set Theory Seminar****Time:** Wednesday, August 5, 11:00am Israel Time (10:00 CEST) **Speaker:** Uri Abraham, Ben-Gurion University**Title:** Coding well ordering of the reals with ladders, part 3 **Abstract:** Results from the 2002 paper “Coding with Ladders a Well Ordering of the Reals” by Abraham and Shelah.**Information:** contact Menachem Magidor, Asaf Rinot or Omer Ben-Neria ahead of time for the zoom link.

Week 27 July – 2 August

**CUNY Set Theory Seminar****Time:** Friday, July 31, 12:00 New York time (18:00 CEST)**Speaker:** Corey Switzer, CUNY**Title:** Dissertation defense: Alternative Cichoń diagrams and forcing axioms compatible with CH**Abstract:** This dissertation surveys several topics in the general areas of iterated forcing, infinite combinatorics and set theory of the reals. There are two parts. In the first half I consider alternative versions of the Cichoń diagram. First I show that for a wide variety of reduction concepts there is a Cichoń diagram for effective cardinal characteristics relativized to that reduction. As an application I investigate in detail the Cichoń diagram for degrees of constructibility relative to a fixed inner model of ZFC. Then I study generalizations of cardinal characteristics to the space of functions from ωω to ωω. I prove that these cardinals can be organized into two diagrams analogous to the standard Cichoń diagram show several independence results and investigate their relation to cardinal invariants on omega. In the second half of the thesis I look at forcing axioms compatible with CH. First I consider Jensen’s subcomplete and subproper forcing. I generalize these notions to larger classes which are (apparently) much more nicely behaved structurally. I prove iteration and preservation theorems for both classes and use these to produce many new models of the subcomplete forcing axiom. Finally I deal with dee-complete forcing and its associated axiom DCFA. Extending a well-known result of Shelah, I show that if a tree of height ω1 with no branch can be embedded into an ω1 tree, possibly with uncountable branches, then it can be specialized without adding reals. As a consequence I show that DCFA implies there are no Kurepa trees, even if CH fails.**Information:** Note the different time! The seminar will take place virtually at 12pm US Eastern Standard Time. Please email Victoria Gitman (vgitman@nylogic.org) for meeting id (this talk will have a different meeting ID!).

**CUNY Logic Seminar (MOPA)****Time:** Wednesday, July 29, 14:00 New York time (20:00 CEST)**Speaker: **Kameryn Williams, University of Hawai‘i at Mānoa**Title:** End-extensions of models of set theory and the Σ1 universal finite sequence**Abstract:** Recall that if M⊆N are models of set theory then N end-extends M if N does not have new elements for sets in M. In this talk I will discuss a Σ1-definable finite sequence which is universal for end extensions in the following sense. Consider a computably axiomatizable extension ¯ZF of ZF. There is a Σ1-definable finite sequencea0,a1,…,anwith the following properties.

* ZF proves that the sequence is finite.

* In any transitive model of ¯ZF the sequence is empty.

* If M is a countable model of ¯ZF in which the sequence is s and t∈M is a finite sequence extending s then there is an end-extension N⊨¯ZF of M in which the sequence is exactly t.

* Indeed, for the previous statements it suffices that M⊨ZF and end-extends a submodel W⊨¯ZF of height at least (ωL1)M.

This universal finite sequence can be used to determine the modal validities of end-extensional set-theoretic potentialism, namely to be exactly the modal theory S4. The sequence can also be used to show that every countable model of set theory extends to a model satisfying the end-extensional maximality principle, asserting that any possibly necessary sentence is already true.

This talk is about joint work with Joel David Hamkins. The Σ1 universal finite sequence is a sister to the Σ2 universal finite sequence for rank-extensions of Hamkins and Woodin, and both are cousins of Woodin’s universal algorithm for arithmetic.**Information:** The seminar will take place virtually. Please email Victoria Gitman (vgitman@nylogic.org) for the meeting id.

**Bar-Ilan-****J****erusalem Set Theory Seminar** – cancelled**Time:** Wednesday, July 29, 11:00am Israel Time (10:00 CEST) – cancelled**Speaker:** Uri Abraham, Ben-Gurion University**Title:** Coding well ordering of the reals with ladders, part 3 **Abstract:** Results from the 2002 paper “Coding with Ladders a Well Ordering of the Reals” by Abraham and Shelah.**Information:** contact Menachem Magidor, Asaf Rinot or Omer Ben-Neria ahead of time for the zoom link.

Week 20-26 July

July 24

**CUNY Set Theory Seminar****Time:** Friday, July 24, 14:00 New York time (20:00 CEST)**Speaker:** Andrew Brooke-Taylor, University of Leeds**Title:** Measurable cardinals and limits in the category of sets**Abstract:** An old result of Isbell characterises measurable cardinals in terms of certain canonical limits in the category of sets. After introducing this characterisation, I will talk about recent work with Adamek, Campion, Positselski and Rosicky teasing out the importance of the canonicity for this and related results. The language will be category-theoretic but the proofs will be quite hands-on combinatorial constructions with sets.**Information:** The seminar will take place virtually. Please email Victoria Gitman (vgitman@nylogic.org) for the meeting id.

July 23

**Southern Illinois University Logic Seminar****Time:** Thursday, 23 July, 1pm US Central Daylight Time (20:00 CEST) **Speaker:** Dana Bartošová, University of Florida **Title:** Dynamics of finite products of groups and of group extensions **Abstract:** tba **Information:** The seminar will take place virtually via zoom.

July 22

**CUNY Logic Seminar (MOPA)****Time:** Wednesday, July 22, 14:00 New York time (20:00 CEST)**Speaker: **Tin Lok Wong, National University of Singapore**Title:** Properties preserved in cofinal extensions**Abstract:** Cofinal extensions generally preserve many more properties of a model of arithmetic than their sisters, end extensions. Exactly how much must or can they preserve? The answer is intimately related to how much arithmetic the model can do. I will survey what is known and what is not known about this question, and report on some recent work on this line.**Information:** The seminar will take place virtually. Please email Victoria Gitman (vgitman@nylogic.org) for the meeting id.

**Bar-I****l****a****n****–****J****erusalem Set Theory Seminar****Time:** Wednesday, July 22, 11:00am (Israel Time)**Speaker:** Uri Abraham, Ben-Gurion University**Title:** Coding well ordering of the reals with ladders, part 2 **Abstract:** Results from the 2002 paper “Coding with Ladders a Well Ordering of the Reals” by Abraham and Shelah.**Information:** contact Menachem Magidor, Asaf Rinot or Omer Ben-Neria ahead of time for the zoom link.

Week 13-19 July

July 17

**CUNY Set Theory Seminar****Time:** Friday, July 17, 14:00 New York time (20:00 CEST)**Speaker:** Kaethe Minden, Bard College at Simon’s Rock**Title:** Maximality and Resurrection**Abstract:** The maximality principle (MP) is the assertion that any sentence which can be forced in such a way that after any further forcing the sentence remains true, must already be true. In modal terms, MPstates that forceably necessary sentences are true. The resurrection axiom (RA) asserts that the ground model is as existentially closed in its forcing extensions as possible. In particular, RArelative to Hc states that for every forcing Q there is a further forcing R such that HVc≺HV[G][H]c, for G∗H⊆Q∗˙R generic.

It is reasonable to ask whether MP and RA can consistently both hold. I showed that indeed they can, and that RA+MP is equiconsistent with a strongly uplifting fully reflecting cardinal, which is a combination of the large cardinals used to force the principles separately. In this talk I give a sketch of the equiconsistency result.**Information:** The seminar will take place virtually. Please email Victoria Gitman (vgitman@nylogic.org) for the meeting id.

Week 6-12 July

July 10

**CUNY Set Theory Seminar****Time:** Friday, July 10, 14:00 New York time (20:00 CEST)**Speaker:** Peter Holy, University of Udine**Title:** Uniform large cardinal characterizations and ideals up to measurability**Abstract:** Many prominent large cardinal notions up to measurability can be characterized by the existence of certain ultrafilters for small models of set theory. Most prominently, this includes weakly compact, ineffable, Ramsey and completely ineffable cardinals, but there are many more, and our characterization schemes also give rise to many new natural large cardinal concepts. Moreover, these characterizations allow for the uniform definition of ideals associated to these large cardinals, which agree with the ideals from the set-theoretic literature (for example, the weakly compact, the ineffable, the Ramsey or the completely ineffable ideal) whenever such had been previously established. For many large cardinal notions, we can show that their ordering with respect to direct implication, but also with respect to consistency strength corresponds in a very canonical way to certain relations between their corresponding large cardinal ideals. This is all material from a fairly extensive joint paper with Philipp Luecke, and I will try to provide an overview as well as present some particular results from this paper.**Information:** The seminar will take place virtually. Please email Victoria Gitman (vgitman@nylogic.org) for the meeting id.

July 8

**CUNY Logic Seminar (MOPA)****Time:** Wednesday, July 8, 14:00 New York time (20:00 CEST)**Speaker: **Corey Switzer, CUNY**Title:** Axiomatizing Kaufmann models in strong logics**Abstract:** A Kaufmann model is an ω1-like, recursively saturated, rather classless model of PA. Such models were constructed by Kaufmann under the ♢ assumption and then shown to exist in ZFC by Shelah using an absoluteness argument involving the logic Lω1,ω(Q) where Q is the quantifier ‘there exists uncountably many…’. It remains an intriguing, if vague, open problem whether one can construct a Kaufmann model in ZFC ‘by hand’ i.e. without appealing to some form of absoluteness or other very non-constructive methods. In this talk I consider the related problem of axiomatizing Kaufmann models in Lω1,ω(Q) and show that this is independent of ZFC. Along the way we’ll see that it is also independent of ZFC whether there is an ω1-preserving forcing notion adding a truth predicate to a Kaufmann model.**Information:** The seminar will take place virtually. Please email Victoria Gitman (vgitman@nylogic.org) for the meeting id.

**Bristol Logic and Set Theory Seminar** (recurring lecture series – last lecture!)**Time:** Wednesday, July 8, 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 XI

**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 (open 30 minutes before).

July 7

**Münster Set Theory Seminar****Time:** Tuesday, June 23, 4:15pm CEST**Speaker:** Ralf Schindler, University of Münster**Title:** MM and (**∗)++****Abstract:** The axiom (∗)++ is a strengthening of (∗) which was also introduced by Woodin. (∗)++ says that the set of sets of reals, or equivalently, Hc+, is contained in a Pmax extension of a determinacy model. Woodin showed that (∗)++ is false in all the known models of Martin’s Maximum. We will give a proof of this result. It remains open if MM++ refutes (∗)++.**Information:** contact rds@wwu.de ahead of time in order to participate.

Week 29 June-5 July

July 3

**CUNY Set Theory Seminar****Time:** Friday, July 3, 14:00 New York time (20:00 CEST)**Speaker:** Vera Fischer, University of Vienna**Title:** More ZFC inequalities between cardinal invariants**Abstract:** We will discuss some recent ZFC results concerning the generalized Baire spaces, and more specifically the generalized bounding number, relatives of the generalized almost disjointness number, as well as generalized reaping and domination.**Information:** The seminar will take place virtually. Please email Victoria Gitman (vgitman@nylogic.org) for the meeting id.

July 1

**CUNY Logic Seminar (MOPA)****Time:** Wednesday, July 1, 14:00 New York time (20:00 CEST)**Speaker: **Zachiri McKenzie**Title:** Initial self-embeddings of models of set theory: Part II**Abstract:** In the 1973 paper ‘Countable models of set theory’, H. Friedman’s investigation of embeddings between countable models of subsystems of ZF yields the following two striking results:

1. Every countable nonstandard model of PA is isomorphic to a proper initial segment of itself.

2. Every countable nonstandard model of a sufficiently strong subsystem of ZF is isomorphic to a proper initial segment that is a union of ranks of the original model.

Note that, in contrast to PA, in the context of set theory there are three alternative notions of ‘initial segment’: transitive subclass, transitive subclass that is closed under subsets and rank-initial segment. Paul Gorbow, in his Ph.D. thesis, systematically studies versions of H. Friedman’s self-embedding that yield isomorphisms between a countable nonstandard model of set theory and a rank-initial segment of itself. In these two talks I will discuss recent joint work with Ali Enayat that investigates models of set theory that are isomorphic to a transitive subclass of itself. We call the maps witnessing these isomorphisms ‘initial self-embeddings’. I will outline a proof of a refinement of H. Friedman’s Theorem that guarantees the existence of initial self-embeddings for certain subsystems of ZF without the powerset axiom. I will then discuss several examples including a nonstandard model of ZFC minus the powerset axiom that admits no initial self-embedding, and models that separate the three different notions of self-embedding for models of set theory. Finally, I will discuss two interesting applications of our version of H. Freidman’s Theorem. The first of these is a refinement of a result due to Quinsey that guarantees the existence of partially elementary proper transitive subclasses of non-standard models of ZF minus the powerset axiom. The second result shows that every countable model of ZF with a nonstandard natural number is isomorphic to a transitive subclass of the hereditarily countable sets of its own constructible universe.**Information:** The seminar will take place virtually. Please email Victoria Gitman (vgitman@nylogic.org) for the meeting id.

**Bristol Logic and Set Theory Seminar** (recurring lecture series)**Time:** Wednesday, July 1, 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 X

**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 (open 30 minutes before).

Week 22-28 June

June 26

**CUNY Set Theory Seminar****Time:** Friday, June 26, 14:00 New York time (20:00 CEST)**Speaker:** Joel David Hamkins, Oxford University**Title:** **Categorical cardinals****Abstract:** Zermelo famously characterized the models of second-order Zermelo-Fraenkel set theory ZFC2 in his 1930 quasi-categoricity result asserting that the models of ZFC2 are precisely those isomorphic to a rank-initial segment Vκ of the cumulative set-theoretic universe V cut off at an inaccessible cardinal κ. I shall discuss the extent to which Zermelo’s quasi-categoricity analysis can rise fully to the level of categoricity, in light of the observation that many of the Vκ universes are categorically characterized by their sentences or theories. For example, if κ is the smallest inaccessible cardinal, then up to isomorphism Vκ is the unique model of ZFC2 plus the sentence ‘there are no inaccessible cardinals.’ This cardinal κ is therefore an instance of what we call a first-order *sententially categorical* cardinal. Similarly, many of the other inaccessible universes satisfy categorical extensions of ZFC2 by a sentence or theory, either in first or second order. I shall thus introduce and investigate the categorical cardinals, a new kind of large cardinal. This is joint work with Robin Solberg (Oxford). **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 26, 1.30pm Toronto time (7.30pm CEST)**Speaker:** Will Brian, UNC Charlotte**Title:** Limited-information strategies in Banach-Mazur games.**Abstract:** The Banach-Mazur game is an infinite-length game played on a topological space X, in which two players take turns choosing members of an infinite decreasing sequence of open sets, the first player trying to ensure that the intersection of this sequence is empty, and the second that it is not. A *limited-information strategy* for one of the players is a game plan that, on any given move, depends on only a small part of the game’s history. In this talk we will discuss Telgársky’s conjecture, which asserts roughly that there must be topological spaces where winning strategies for the Banach-Mazur game cannot be too limited, but must rely on large parts of the game’s history in a significant way. Recently, it was shown that this conjecture fails in models of set theory satisfying GCH + square. In such models it is always possible for one player to code all information concerning a game’s history into a small piece of it. We will discuss these so-called coding strategies, why assuming GCH + square makes them work so well, and what can go wrong in other models of set theory.**Information:** The seminar will take place virtually. Follow the link below: https://yorku.zoom.us/j/96087161597

June 25

**Kurt Gödel Research Center Seminar** (organised by Ben Miller)**Time:** Thursday, June 25, 16:00 CEST

**Speaker:**Victoria Gitman, CUNY

**Title:**Class forcing in its rightful setting

**Abstract:**The use of class forcing in set theoretic constructions goes back to the proof Easton’s Theorem that GCH can fail at all regular cardinals. Class forcing extensions are ubiquitous in modern set theory, particularly in the emerging field of set-theoretic geology. Yet, besides the pioneering work by Friedman and Stanley concerning pretame and tame class forcing, the general theory of class forcing has not really been developed until recently. A revival of interest in second-order set theory has set the stage for understanding the properties of class forcing in its natural setting. Class forcing makes a fundamental use of class objects, which in the first-order setting can only be studied in the meta-theory. Not surprisingly it has turned out that properties of class forcing notions are fundamentally determined by which other classes exist around them. In this talk, I will survey recent results (of myself, Antos, Friedman, Hamkins, Holy, Krapf, Schlicht, Williams and others) regarding the general theory of class forcing, the effects of the second-order set theoretic background on the behavior of class forcing notions and the numerous ways in which familiar properties of set forcing can fail for class forcing even in strong second-order set theories.

**Information:**Talk via zoom.

June 24

**CUNY Logic Seminar (MOPA)****Time:** Wednesday, June 24, 14:00 New York time (20:00 CEST)**Speaker: **Bartosz Wcisło, Polish Academy of Sciences**Title:** Tarski boundary III**Abstract:** Truth theories investigate the notion of truth using axiomatic methods. To a fixed base theory (typically Peano Arithmetic PA) we add a unary predicate T(x) with the intended interpretation ‘x is a (code of a) true sentence’. Then we analyse how adding various possible sets of axioms for that predicate affects its behaviour. One of the aspects which we are trying to understand is which truth-theoretic principles make the added truth predicate ‘strong’ in that the resulting theory is not conservative over the base theory. Ali Enayat proposed to call this demarcating line between conservative and non-conservative truth theories ‘the Tarski boundary’. Research on Tarski boundary revealed that natural truth theoretic principles extending compositional axioms tend to be either conservative over PA or exactly equivalent to the principle of global reflection over A. It says that sentences provable in PA are true in the sense of the predicate T. This in turn is equivalent to Δ_0-induction for the compositional truth predicate which turns out to be a surprisingly robust theory. The equivalences between nonconservative truth theories are typically proved by relatively direct ad hoc arguments. However, certain patterns seem common to these proofs. The first one is construction of various arithmetical partial truth predicates which provably in a given theory have better properties than the original truth predicate. The second one is deriving induction for these truth predicates from internal induction, a principle which says that for any arithmetical formula, the set of those elements for which that formula is satisfied under the truth predicate satisfies the usual induction axioms. As an example of this phenomenon, we will present two proofs. First, we will show that global reflection principle is equivalent to local induction. Global reflection expresses that any sentence provable in PA is true. Local induction says that any predicate obtained by restricting truth predicate to sentences of a fixed syntactic complexity satisfies full induction. This is an observation due to Mateusz Łełyk and the author of this presentation. The second example is a result by Ali Enayat who showed that CT_0, a theory compositional truth with Δ_0-induction, is arithmetically equivalent to the theory of compositional truth together with internal induction and disjunctive correctness.This talk is intended as a continuation of ‘Tarski boundary II’ presentation at the same seminar. However, we will try to avoid excessive assumptions on familiarity with the previous part.**Information:** The seminar will take place virtually. Please email Victoria Gitman (vgitman@nylogic.org) for the meeting id.

**Bristol Logic and Set Theory SeminarTime:** Wednesday, June 24, 15:00-16:30 UK time (16:00-17:30 CEST)

**Speaker:**Alessandro Andretta, University of Torino

**Generalised Iteration Trees**

**Title:****A theorem of Gaifman states that any internal linear iteration whose length belongs to the model it is applied to has a well-founded direct limit.We have isolated a notion of “generalized iteration trees” for which a similar result is possible, at least if the length of the tree is $\omega$. These iterations are more general than the objects introduced by Martin and Steel over three decades ago in that the extender $E_n$ used to construct $M_{n+1}$ need not to belong to the last model $M_n$. In other words $E_n \in M_{d(n+1)}$, with $d(n+1) \leq n$. We isolate a simple property of the function $d$ characterizing continuous ill-foundedness of generalized iteration trees.Any generalized iteration trees satisfying this property is not continuously ill-founded. Conversely, any tree order with a $d$ function failing such property can be realized as a continuously ill-founded iteration tree on V.**

**Abstract:**This is joint work with John Steel.

**Information:**Via zoom: https://zoom.us/j/97281665521 (open 15 minutes before)

**Paris-Lyon Séminaire de LogiqueTime:** Wednesday, June 24, 16:00-17:15 CEST

**Speaker:**Ludovic Patey, University of Lyon

**Title:**The computability-theoretic aspects of Milliken’s tree theorem and

applications

**Abstract:**Milliken’s tree theorem states that for every countable, finitely

branching tree T with no leaves, and every finite coloring f of the

strong subtrees of height n, there is an infinite strong subtree over

which the strong subtrees of height n are monochromatic. This theorem

has several applications, among which Devlin’s theorem about finite

coloring of the rationals, and a theorem about the Rado graph. In this

talk, we give a survey of the computability-theoretic aspects of these

statements seen as mathematical problems, in terms of instances and

solutions. Our main motivation is reverse mathematics. This is a joint

work with Paul-Elliot Anglès d’Auriac, Peter Cholak and Damir Dzhafarov.

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

**Bristol Logic and Set Theory Seminar** (recurring lecture series)**Time:** Wednesday, June 24, 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 IX

**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 (open 30 minutes before).

**Bar-Ilan University and Hebrew University Set Theory Seminar****Time:** Wednesday, June 24, 11:00-13:00 Israel time (10:00-13:00 CEST)**Speaker:** Istvan Juhasz**Title:** On the free subset number of a topological space and their G_\delta modification**Abstract:** pdf available on seminar website.**Information:** Contact Assaf Rinot for the zoom id.

June 23

**Münster Set Theory Seminar****Time:** Tuesday, June 23, 4:15pm CEST**Speaker:** Farmer Schlutzenberg**Title:** Remarks on rank-into-rank embeddings part III**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.

Week 15-21 June

June 19

**CUNY Set Theory Seminar****Time:** Friday, June 19, 14:00 New York time (20:00 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

**Udine graduate courseTime:** 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.

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 17

**New York Logic Seminar (MOPA)****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.

**Oxford Set Theory SeminarTime:** 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.

**Paris-Lyon Séminaire de LogiqueTime:** 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.

**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 (open 30 minutes before).

**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.

June 16

**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.

**Udine graduate courseTime:** 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.

Week 8-14 June

June 12

**Udine graduate courseTime:** Friday, June 12, 10:00-12:00 CEST

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

**Title:**Generalized Descriptive Set Theory II, Lecture 1

**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.

June 10

**CUNY Set Theory Seminar****Time:** Wednesday, June 10, 7pm New York time (June 11, 1am CEST)**Speaker: **Zachiri McKenzie**Title:** Initial self-embeddings of models of set theory: Part II**Abstract:** In the 1973 paper ‘Countable models of set theory’, H. Friedman’s investigation of embeddings between countable models of subsystems of ZF yields the following two striking results:

1. Every countable nonstandard model of PA is isomorphic to a proper initial segment of itself.

2. Every countable nonstandard model of a sufficiently strong subsystem of ZF is isomorphic to a proper initial segment that is a union of ranks of the original model.

Note that, in contrast to PA, in the context of set theory there are three alternative notions of ‘initial segment’: transitive subclass, transitive subclass that is closed under subsets and rank-initial segment. Paul Gorbow, in his Ph.D. thesis, systematically studies versions of H. Friedman’s self-embedding that yield isomorphisms between a countable nonstandard model of set theory and a rank-initial segment of itself. In these two talks I will discuss recent joint work with Ali Enayat that investigates models of set theory that are isomorphic to a transitive subclass of itself. We call the maps witnessing these isomorphisms ‘initial self-embeddings’. I will outline a proof of a refinement of H. Friedman’s Theorem that guarantees the existence of initial self-embeddings for certain subsystems of ZF without the powerset axiom. I will then discuss several examples including a nonstandard model of ZFC minus the powerset axiom that admits no initial self-embedding, and models that separate the three different notions of self-embedding for models of set theory. Finally, I will discuss two interesting applications of our version of H. Freidman’s Theorem. The first of these is a refinement of a result due to Quinsey that guarantees the existence of partially elementary proper transitive subclasses of non-standard models of ZF minus the powerset axiom. The second result shows that every countable model of ZF with a nonstandard natural number is isomorphic to a transitive subclass of the hereditarily countable sets of its own constructible universe.**Information:** The seminar will take place virtually. Please email Victoria Gitman (vgitman@nylogic.org) for the meeting id.

**Bristol Logic and Set Theory SeminarTime:** Wednesday, June 10, 16:15-17:45 UK time (17:15-18:45 CEST)

**Speaker:**Peter Holy, University of Udine

**Ideal and Tree Forcing Topologies**

**Title:****While the usual topology on the kappa-reals is based on the bounded ideal, in the sense that the basic open sets are generated by bounded partial functions from kappa to 2, we consider generalized topologies based on arbitrary ideals on kappa, in particular on the nonstationary ideal on regular and uncountable cardinals kappa. We will illustrate the connections of these topologies with certain tree forcing topologies, and in particular the connection of the nonstationary topology with the topology generated by kappa-Silver forcing. We will show how properties of this forcing notion carry over to properties of the nonstationary topology, and we will also generalize results on kappa-Silver forcing of Friedman, Khomskii and Kulikov. This is joint work with Marlene Koelbing, Philipp Schlicht and Wolfgang Wohofsky.**

**Abstract:****Information:**Please login to the zoom meeting https://zoom.us/j/95916684321 a few minutes before the talk.

**Paris-Lyon Séminaire de LogiqueTime:** Wednesday, June 10, 16:00-17:15 CEST

**Speaker:**Assaf Rinot

**Title:**Transformations of the transfinite plane

**Abstract:**We study the existence of transformations of the transfinite

plane that allow to reduce Ramsey-theoretic statements concerning

uncountable Abelian groups into classic partition relations for

uncountable cardinals. This is joint work with Jing Zhang.

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

**Bristol Logic and Set Theory SeminarTime:** Wednesday, June 10, 14:00-15:30 UK time (15:00-16:30 CEST)

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

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

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

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

June 9

**Münster Set Theory Seminar****Time:** Tuesday, June 9, 4:15pm CEST**Speaker:** Farmer Schlutzenberg**Title:** Remarks on rank-into-rank embeddings**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.

**Udine graduate courseTime:** Tuesday, June 9, 10:00-12:00 CEST

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

**Title:**Generalized Descriptive Set Theory II, Lecture 1

**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.

Week 1-7 June

June 5

**Udine graduate courseTime:** Friday, June 5, 10:00-12:00 CEST

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

**Title:**Generalized Descriptive Set Theory II, Lecture 1

**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.

June 4

**Kurt Gödel Research Center Seminar** (organised by Ben Miller)**Time:** Thursday, June 4, 16:00 CEST

**Speaker:**Stefan Hoffelner, Universität Münster

**Title:**Forcing the Sigma-1-3 separation property

**Abstract:**The separation property, introduced in the 1920s, is a classical notion in descriptive set theory. It is well-known due to Moschovakis, that Delta-1-2-determinacy implies the Sigma-1-3 separation property; yet Detla-1-2-determinacy implies an inner model with a Woodin cardinal. The question whether the Sigma-1-3-separation property is consistent relative to jsut ZFC remained open however since Mathias “Surrealist Landscape”-paper. We show that one can force it over L.

**Information:**Talk via zoom.

June 3

**Bristol Logic and Set Theory SeminarTime:** Wednesday, June 3, 14:00-15:30 UK time (15:00-16:30 CEST)

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

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

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

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

Week 25-31 May

May 29

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

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

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

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

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

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

**Toronto Set Theory Seminar****Time:** Friday, May 29, 1.30pm Toronto time (7.30pm CEST)**Speaker:** Michael Hrusak (UNAM)**Title:** Invariant Ideal Axiom**Abstract:** We introduce the Invariant Ideal Axiom and discuss its impact on the structure of countable topological groups. (joint work with Alexander Shibakov)**Information:** The seminar will take place virtually. ZOOM ID: https://yorku.zoom.us/j/96087161597

May 28

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

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

**Title:**Preserving splitting families

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

**Information:**Talk via zoom.

May 27

**Oxford Set Theory SeminarTime:** Wednesday, May 27, 16:00-17:30 UK time (17:00-18:30 CEST)

**Speaker:**Ali Enayat, Gothenberg

**Title:**Leibnizian and anti-Leibnizian motifs in set theory

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

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

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

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

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

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

**Bristol Logic and Set Theory SeminarTime:** Wednesday, May 27, 14:00-15:30 UK time (15:00-16:30 CEST)

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

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

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

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

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

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

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

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

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

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

May 26

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

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

Week 18-24 May

May 22

**CUNY Set Theory Seminar****Time:** Friday, May 22, 2pm New York time (8pm CEST)**Speaker:** Ali Enayat, University of Gothenburg**Title:** Recursively saturated models of set theory and their close relatives: Part II**Abstract:** A model M of set theory is said to be ‘condensable’ if there is an ‘ordinal’ α of M such that the rank initial segment of M determined by α is both isomorphic to M, and an elementary submodel of M for infinitary formulae appearing in the well-founded part of M. Clearly if M is condensable, then M is ill-founded. The work of Barwise and Schlipf in the mid 1970s showed that countable recursively saturated models of ZF are condensable.

In this two-part talk, we present a number of new results related to condensable models, including the following two theorems.

Theorem 1. Assuming that there is a well-founded model of ZFC plus ‘there is an inaccessible cardinal’, there is a condensable model M of ZFC which has the property that every definable element of M is in the well-founded part of M (in particular, M is ω-standard, and therefore not recursively saturated).

Theorem 2. The following are equivalent for an ill-founded model M of ZF of any cardinality:

(a) M is expandable to Gödel-Bernays class theory plus Δ11-Comprehension.

(b) There is a cofinal subset of ‘ordinals’ α of M such that the rank initial segment of M determined by α is an elementary submodel of M for infinitary formulae appearing in the well-founded part of M.

Moreover, if M is a countable ill-founded model of ZFC, then conditions (a) and (b) above are equivalent to:

(c) M is expandable to Gödel-Bernays class theory plus Δ11-Comprehension + Σ12-Choice.**Information:** The seminar will take place virtually. Please email Victoria Gitman (vgitman@nylogic.org) for the meeting id.

May 20

**Oxford Set Theory SeminarTime:** Wednesday, May 20, 16:00-17:30 UK time (17:00-18:30 CEST)

**Speaker:**Joel David Hamkins, University of Oxford

**Title:**Bi-interpretation of weak set theories

**Abstract:**Set theory exhibits a truly robust mutual interpretability phenomenon: in any model of one set theory we can define models of diverse other set theories and vice versa. In any model of ZFC, we can define models of ZFC + GCH and also of ZFC + ¬CH and so on in hundreds of cases. And yet, it turns out, in no instance do these mutual interpretations rise to the level of bi-interpretation. Ali Enayat proved that distinct theories extending ZF are never bi-interpretable, and models of ZF are bi-interpretable only when they are isomorphic. So there is no nontrivial bi-interpretation phenomenon in set theory at the level of ZF or above. Nevertheless, for natural weaker set theories, we prove, including ZFC- without power set and Zermelo set theory Z, there are nontrivial instances of bi-interpretation. Specifically, there are well-founded models of ZFC- that are bi-interpretable, but not isomorphic—even ⟨Hω1,∈⟩ and ⟨Hω2,∈⟩ can be bi-interpretable—and there are distinct bi-interpretable theories extending ZFC-. Similarly, using a construction of Mathias, we prove that every model of ZF is bi-interpretable with a model of Zermelo set theory in which the replacement axiom fails. This is joint work with Alfredo Roque Freire.

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

**Paris-Lyon Séminaire de LogiqueTime:** Wednesday, May 6, 16:00-17:15 CEST

**Speaker:**Michale Hrusak, University of Morelia

**Title:**Strong measure zero in Polish groups

**Abstract:**We study the extent to which the Galvin-Mycielski-Solovay characterization of strong measure zero subsets of the real line extends to arbitrary Polish group. In particular, we show that an abelian Polish group satisfies the GMS characterization if and only if it is locally compact. We shall also consider the non-abelian case, and discuss the use and existence of invariant submeasures on Polish groups. (Joint with W. Wohofsky, J. Zapletal and/or O. Zindulka.)

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

**Bristol Logic and Set Theory SeminarTime:** Wednesday, May 20, 14:00-15:30 UK time (15:00-16:30 CEST)

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

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

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

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

Week 11-17 May

May 15

**CUNY Set Theory Seminar****Time:** Friday, May 15, 2pm New York time (8pm CEST)**Speaker:** Ali Enayat, University of Gothenburg**Title:** Recursively saturated models of set theory and their close relatives: Part I**Abstract:** A model M of set theory is said to be ‘condensable’ if there is an ‘ordinal’ α of M such that the rank initial segment of M determined by α is both isomorphic to M, and an elementary submodel of M for infinitary formulae appearing in the well-founded part of M. Clearly if M is condensable, then M is ill-founded. The work of Barwise and Schlipf in the mid 1970s showed that countable recursively saturated models of ZF are condensable.

In this two-part talk, we present a number of new results related to condensable models, including the following two theorems.

Theorem 1. Assuming that there is a well-founded model of ZFC plus ‘there is an inaccessible cardinal’, there is a condensable model M of ZFC which has the property that every definable element of M is in the well-founded part of M (in particular, M is ω-standard, and therefore not recursively saturated).

Theorem 2. The following are equivalent for an ill-founded model M of ZF of any cardinality:

(a) M is expandable to Gödel-Bernays class theory plus Δ11-Comprehension.

(b) There is a cofinal subset of ‘ordinals’ α of M such that the rank initial segment of M determined by α is an elementary submodel of M for infinitary formulae appearing in the well-founded part of M.

Moreover, if M is a countable ill-founded model of ZFC, then conditions (a) and (b) above are equivalent to:

(c) M is expandable to Gödel-Bernays class theory plus Δ11-Comprehension + Σ12-Choice.**Information:** The seminar will take place virtually. Please email Victoria Gitman (vgitman@nylogic.org) for the meeting id.

May 14

**Kurt Gödel Research Center Seminar** (organised by Ben Miller)**Time:** Thursday, May 14, 16:00 CEST

**Speaker:**Andrew Brooke-Taylor, University of Leeds

**Title:**Product of CW complexes

**Abstract:**CW spaces are often presented as the “spaces of choice” in algebraic topology courses, being relatively nice spaces built up by successively gluing on Euclidean balls of increasing dimension. However, the product of CW complexes need not be a CW complex, as shown by Dowker soon after CW complexes were introduced. Work in the 1980s characterised when the product is a CW complex under the assumption of CH, or just

*b*=ℵ1. In this talk I will give and prove a complete characterisation of when the product of CW complexes is a CW complex, valid under ZFC. The characterisation however involves

*b*; the proof is point-set-topological (I won’t assume any knowledge of algebraic topology) and uses Hechler conditions.

**Information:**Talk via zoom.

May 13

**CUNY Set Theory Seminar****Time:** Wednesday, May 13, 7pm New York time (1am May 14 CEST)**Speaker:** **Laurence Kirby**, CUNY**Title:** **Bounded finite set theory****Abstract:** There is a well-known close logical connection between PA and finite set theory. Is there a set theory that corresponds in an analogous way to bounded arithmetic IΔ0? I propose a candidate for such a theory, called IΔ0S, and consider the questions: what set-theoretic axioms can it prove? And given a model M of IΔ0 is there a model of IΔ0S whose ordinals are isomorphic to M? The answer is yes if M is a model of Exp; to obtain the answer we use a new way of coding sets by numbers.**Information:** The seminar will take place virtually. Please email Victoria Gitman (vgitman@nylogic.org) for the meeting id.

**Bristol Logic and Set Theory SeminarTime:** Wednesday, May 13, 14:00-15:30 UK time (15:00-16:30 CEST)

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

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

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

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

**Jerusalem Set Theory Seminar****Time:** Wednesday, May 13, 11:00am (Israel Time)**Speaker:** Alejandro Poveda (Universitat de Barcelona)**Title:** Sigma-Prikry forcings and their iterations (Part II)**Abstract:**In the previous talk, we introduced the notion of \Sigma-Prikry forcing and showed that many classical Prikry-type forcing which center on countable cofinalities fall into this framework.

The aim of this talk is to present our iteration scheme for \Sigma-Prikry forcings.

In case time permits, we will also show how to use this general iteration theorem to derive as a corollary the following strengthening of Sharon’s theorem: starting with \omega-many supercompact cardinals one can force a generic extension where Refl(<\omega,\kappa^+) holds and the SCH_\kappa fails, for \kappa being a strong limit cardinal with cofinality \omega**Information:** contact omer.bn@mail.huji.ac.il ahead of time in order to participate.

**Paris-Lyon Séminaire de LogiqueTime:** Wednesday, May 13, 16:00-17:15 CEST

**Speaker:**Caroline Terry (University of Chicago)

**Title:**Speeds of hereditary properties and mutual algebricity

(joint work with Chris Laskowski.)

**Abstract:**A hereditary graph property is a class of finite graphs closed under isomorphism and induced subgraphs. Given a hereditary graph property H, the speed of H is the function which sends an integer n to the number of distinct elements in H with underlying set {1,…,n}. Not just any function can occur as the speed of hereditary graph property. Specifically, there are discrete “jumps” in the possible speeds. Study of these jumps began with work of Scheinerman and Zito in the 90’s, and culminated in a series of papers from the 2000’s by Balogh, Bollob\'{a}s, and Weinreich, in which essentially all possible speeds of a hereditary graph property were characterized. In contrast to this, many aspects of this problem in the hypergraph setting remained unknown. In this talk we present new hypergraph analogues of many of the jumps from the graph setting, specifically those involving the polynomial, exponential, and factorial speeds. The jumps in the factorial range turned out to have surprising connections to the model theoretic notion of mutual algebricity, which we also discuss.

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

May 12

**Cornell Logic Seminar****Time:** Tuesday, May 12, 2:55pm New York time (20:55pm CEST)**Speaker:** Konstantin Slutsky, University of Paris 7**Title:** Smooth Orbit equivalence relation of free Borel R^d-actions**Abstract:** Smooth Orbit Equivalence (SOE) is an orbit equivalence relation between free ℝ*d*-flows that acts by diffeomorphisms between orbits. This idea originated in ergodic theory of ℝ-flows under the name of time-change equivalence, where it is closely connected with the concept of Kakutani equivalence of induced transformations. When viewed from the ergodic theoretical viewpoint, SOE has a rich structure in dimension one, but, as discovered by Rudolph, all ergodic measure-preserving ℝ*d*-flows, d > 1, are SOE. Miller and Rosendal initiated the study of this concept from the point of view of descriptive set theory, where phase spaces of flows aren’t endowed with any measures. This significantly enlarges the class of potential orbit equivalences, and they proved that all nontrivial free Borel ℝ-flows are SOE. They posed a question of whether the same remains to be true in dimension d>1. In this talk, we answer their question in the affirmative, and show that all nontrivial Borel ℝ*d*-flows are SOE.**Information:** contact solecki@cornell.edu ahead of time to participate.

**Münster Set Theory Seminar****Time:** Tuesday, May 12, 4:15pm CEST**Speaker:** Farmer Schlutzenberg, University of Muenster**Title:** $j:V_\delta\to V_\delta$ in $L(V_\delta)$**Abstract:** Assuming $\mathrm{ZF}+V=L(V_\delta)$ where $\delta$ is a

limit ordinal of uncountable cofinality, we show there is no

non-trivial $\Sigma_1$-elementary $j:V_\delta\to V_\delta$. Reference:

Section 8 of “Reinhardt cardinals and non-definability”, arXiv

2002.01215.**Information:** Please contact Ralf Schindler (rds@wwu.de ) ahead of time to participate.

Week 4-10 May

May 8

**CUNY Set Theory Seminar****Time:** Friday, May 8, 2pm New York time (8pm CEST)**Speaker:** Sandra Müller, Universität Wien**Title:** How to obtain lower bounds in set theory**Abstract:** Computing the large cardinal strength of a given statement is one of the key research directions in set theory. Fruitful tools to tackle such questions are given by inner model theory. The study of inner models was initiated by Gödel’s analysis of the constructible universe L. Later, it was extended to canonical inner models with large cardinals, e.g. measurable cardinals, strong cardinals or Woodin cardinals, which were introduced by Jensen, Mitchell, Steel, and others.

We will outline two recent applications where inner model theory is used to obtain lower bounds in large cardinal strength for statements that do not involve inner models. The first result, in part joint with J. Aguilera, is an analysis of the strength of determinacy for certain infinite two player games of fixed countable length, and the second result, joint with Y. Hayut, involves combinatorics of infinite trees and the perfect subtree property for weakly compact cardinals κ. Finally, we will comment on obstacles, questions, and conjectures for lifting these results higher up in the large cardinal hierarchy.**Information:** The seminar will take place virtually. Please email Victoria Gitman (vgitman@nylogic.org) for the meeting id.

**Toronto Set Theory Seminar****Time:** Friday, May 8, 1:30-3:00pm EDT (19:30-21:00 CEST)**Speaker:** Dima Sinapova, University of Chicago**Title:** Iteration, reflection, and Prikry forcing**Abstract:** There is an inherent tension between stationary reflection and the failure of SCH. The former is a compactness type principle that follows from large cardinals. The latter is an instance of incompactness, and usually obtained using Prikry forcing. We describe a Prikry style iteration, and use it to force stationary reflection in the presence of not SCH. Then we discuss the situation at smaller cardinals. This is joint work with Alejandro Poveda and Assaf Rinot.**Information:** The talk will take place via zoom: https://yorku.zoom.us/j/96087161597.

May 7

**Kurt Gödel Research Center Seminar** (organised by Ben Miller)**Time:** Thursday, May 7, 16:00 CEST

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

**Title:**Tba

**Abstract:**Tba

**Information:**Talk via zoom.

May 6

**Oxford Set Theory SeminarTime:** Wednesday, May 6, 16:00-17:30 UK time (17:00-18:30 CEST)

**Speaker:**Victoria Gitman, City University of New York

**Title:**Elementary embeddings and smaller large cardinals

**Abstract:**A common theme in the definitions of larger large cardinals is the existence of elementary embeddings from the universe into an inner model. In contrast, smaller large cardinals, such as weakly compact and Ramsey cardinals, are usually characterized by their combinatorial properties such as existence of large homogeneous sets for colorings. It turns out that many familiar smaller large cardinals have elegant elementary embedding characterizations. The embeddings here are correspondingly ‘small’; they are between transitive set models of set theory, usually the size of the large cardinal in question. The study of these elementary embeddings has led us to isolate certain important properties via which we have defined robust hierarchies of large cardinals below a measurable cardinal. In this talk, I will introduce these types of elementary embeddings and discuss the large cardinal hierarchies that have come out of the analysis of their properties. The more recent results in this area are a joint work with Philipp Schlicht.

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

**Paris-Lyon Séminaire de LogiqueTime:** Wednesday, May 6, 16:00-17:15 CEST

**Speaker:**Ilijas Farah, York University (Toronto)

**Title:**Between ultrapowers and reduced products

**Abstract:**Ultrapowers and reduced powers are two popular tools for studying countable (and separable metric) structures. Once an ultrafilter on N is fixed, these constructions are functors into the category of countably saturated structures of the language of the original structure. The question of the exact relation between these two functors has been raised only recently by Schafhauser and Tikuisis, in the context of Elliott’s classification programme. Is there an ultrafilter on N such that the quotient map from the reduced product associated with the Fréchet filter onto the ultrapower has the right inverse? The answer to this question involves both model theory and set theory. Although these results were motivated by the study of C*-algebras, all of the results and proofs will be given in the context of classical (discrete) model theory.

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

**Bristol Logic and Set Theory SeminarTime:** Wednesday, May 6, 14:00-15:30 UK time (15:00-16:30 CEST)

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

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

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

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

May 5

**Cornell Logic Seminar****Time:** Tuesday, May 5, 2:55pm New York time (20:55pm CEST)**Speaker:** Andrew Zucker, University of Lyon**Title:** Topological dynamics beyond Polish groups**Abstract:** When *G* is a Polish group, one way of knowing that *G* has “nice” dynamics is to show that *M*(*G*), the universal minimal flow of *G*, is metrizable. However, works of Bartosova, Gheysens, and Krupinski–Pillay investigate groups beyond the Polish realm, such as *Sym*(*κ*), *Homeo*(*ω*1), and automorphism groups of uncountable, *ω*-homogeneous structures. For example, Bartosova shows that the universal minimal flow of *Sym*(*κ*) is the space of linear orders on *κ*–not a metrizable space, but still “nice.” In this talk, we seek to put these results into a general framework which encompasses all topological groups.

This is joint work with Gianluca Basso

**Münster Set Theory Seminar****Time:** Tuesday, May 5, 4:15pm CEST**Speaker:** Andreas Lietz, University of Muenster**Title:** How to force (*) from less than a supercompact**Abstract:** Asperò-Schindler have shown that Woodin’s axiom (*) is a consequence of MM^{++} and the latter is known to be forceable from a supercompact cardinal. (*) however has consistency strength of \omega-many Woodin cardinals, so it should be possible to force it from a much weaker assumption. We present a construction that does so from strictly less than a \kappa^{+++}-supercompact cardinal \kappa (+GCH). The strategy will be to iterate the forcing from the proof of MM^{++}\Rightarrow(\ast). Two main difficulties arise: Whenever we want to use that forcing we will have to make sure that it is semiproper and that NS_{\omega_1} is saturated. We hope that the large cardinal assumption can be lowered to around the region of an inaccessible limit of Woodin cardinals. This is joint work with Ralf Schindler.**Information:** The seminar will be held remotely via zoom. Please contact rds@wwu.de ahead of time in order to participate.

Week 27 April-3 May

May 1

**CUNY Set Theory Seminar****Time:** Friday, May 1, 2pm New York time (8pm CEST)**Speaker:** Joan Bagaria, Universitat de Barcelona**Title:** From Strong to Woodin cardinals: A level-by-level analysis of the Weak Vopenka Principle**Abstract:** In May 2019 Trevor Wilson proved that the Weak Vopenka Principle (WVP), which asserts that the opposite of the category of Ordinals cannot be fully embedded into the category of Graphs, is equivalent to the class of ordinals being Woodin. In particular this implies that WVP is not equivalent to Vopenka’s Principle, thus solving an important long-standing open question in category theory. I will report on a joint ensuing work with Trevor Wilson in which we analyse the strength of WVP for definable classes of full subcategories of Graphs, obtaining exact level-by-level characterisations in terms of a natural hierarchy of strong 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, May 1, 1:30-3:00pm EDT (19:30-21:00 CEST)**Speaker:** Paul Szeptycki**Title:** Strong convergence properties and an example from a square-sequence**Abstract:** We present an example of a space constructed from square(kappa), answering some questions of Arhangel’skii. Coauthors Bill Chen and Cesar Corral-Rojas. **Information:** The talk will take place via zoom: https://yorku.zoom.us/j/925557716.

April 30

**Kurt Gödel Research Center Seminar** (organised by Ben Miller)**Time:** Thursday, April 23, 16:00 CEST

**Speaker:**Sandra Müller, KGRC

**Title:**How to obtain lower bounds in set theory

**Abstract:**Computing the large cardinal strength of a given statement is one of the key research directions in set theory. Fruitful tools to tackle such questions are given by inner model theory. The study of inner models was initiated by Gödel’s analysis of the constructible universe LL. Later, it was extended to canonical inner models with large cardinals, e.g. measurable cardinals, strong cardinals or Woodin cardinals, which were introduced by Jensen, Mitchell, Steel, and others.

We will outline two recent applications where inner model theory is used to obtain lower bounds in large cardinal strength for statements that do not involve inner models. The first result, in part joint with J. Aguilera, is an analysis of the strength of determinacy for certain infinite two player games of fixed countable length, and the second result, joint with Y. Hayut, involves combinatorics of infinite trees and the perfect subtree property for weakly compact cardinals κκ.

**Information:**Talk via zoom.

April 29

**Paris-Lyon Séminaire de LogiqueTime:** Wednesday, April 29, 16:00-17:15 CEST

**Speaker:**Christian Rosendal – University of Illinois at Chicago

**Title:**Continuity of universally measurable homomorphisms

**Abstract:**We show that a universally measurable homomorphism between Polish groups is automatically continuous. Using our general analysis of continuity of group homomorphisms, this result is used to calibrate the strength of the existence of a discontinuous homomorphism between Polish groups. In particular, it is shown that, modulo ZF+DC, the existence of a discontinuous homomorphism between Polish groups implies that the Hamming graph on {0, 1}N has finite chromatic number. This solves a classical problem originating in JPR Christensen’s work on Haar null sets.

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

April 28

**Münster Set Theory Seminar****Time:** Tuesday, April 28, 4:15pm CEST**Speaker:** Matteo Viale, University of Torino**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 Pi_2-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 conditional to the proof of Schindler and Asperò that Woodin’s axiom (*) can be forced by a stationary set preserving forcing.**Information:** The seminar will be held remotely via zoom. Please contact rds@wwu.de ahead of time in order to participate.

Week 20-26 April

April 24

**CUNY Set Theory Seminar****Time:** Friday, April 24, 2pm New York time (8pm CEST)**Speaker:** Arthur Apter, CUNY**Title:** Indestructibility and the First Two Strongly Compact Cardinals**Abstract:** Starting from a model of ZFC with two supercompact cardinals, I will discuss how to force and construct a model in which the first two strongly compact cardinals κ1 and κ2 are also the first two measurable cardinals. In this model, κ1’s strong compactness is indestructible under arbitrary κ1-directed closed forcing, and κ2’s strong compactness is indestructible under Add(κ2,λ) for any ordinal λ. This answers a generalized version of a question of Sargsyan.**Information:** The seminar will take place virtually. Please email Victoria Gitman (vgitman@nylogic.org) for the meeting id.

**Toronto Set Theory Seminar****Time:** Friday, April 24, 1:30-3:00pm EDT (19:30-21:00 CEST)**Speaker:** Todd Eisworth, Ohio University **Title:** Representability and pseudopowers.**Abstract:** We will prove some basic facts about Shelah’s pseudopower function, and derive some new (?) ZFC results in cardinal arithmetic using basic topological ideas. This talk is designed to be an introduction to this part of pcf theory.**Information:** The talk will take place via zoom: https://yorku.zoom.us/j/925557716.

April 23

**Kurt Gödel Research Center Seminar** (organised by Ben Miller)**Time:** Thursday, April 23, 16:00 CEST

**Speaker:**Noé de Rancourt, KGRC

**Title:**Weakly Ramsey ultrafilters

**Abstract:**Weakly Ramsey ultrafilters are ultrafilters on ωω satisfying a weak local version of Ramsey’s theorem; they naturally generalize Ramsey ultrafilters. It is well known that an ultrafilter on ωω is Ramsey if and only if it is minimal in the Rudin-Keisler ordering; in joint work with Jonathan Verner, we proved that similarly, weakly Ramsey ultrafilters are low in this ordering: there are no infinite chains below them. This generalizes a result of Laflamme’s. In this talk, I will outline a proof of this result, and the construction of a counterexample to the converse of this fact, namely a non-weakly Ramsey ultrafilter having exactly one Rudin-Keisler predecessor. This construction is partly based on finite combinatorics.

**Information:**Talk via zoom.

April 22

**New York Logic Seminar** (MOPA)**Time:** Wednesday, April 22, 7pm New York time (Thursday, April 23, 1am CEST)**Speaker:** Corey Switzer, CUNY**Title:** **Hanf Numbers of Arithmetics****Abstract:** Recall that given a complete theory T and a type p(x) the *Hanf number for* p(x) is the least cardinal κ so that any model of T of size κ realizes p(x) (if such a κ exists and ∞ otherwise). The *Hanf number for* T, denoted H(T), is the supremum of the successors of the Hanf numbers for all possible types p(x) whose Hanf numbers are <∞. We have seen so far in the seminar that for any complete, consistent T in a countable language H(T)≤ℶω1 (a result due to Morley). In this talk I will present the following theorems: (1) The Hanf number for true arithmetic is ℶω (Abrahamson-Harrington-Knight) but (2) the Hanf number for False Arithmetic is ℶω1 (Abrahamson-Harrington)**Information:** The seminar will take place virtually. Please email Victoria Gitman for the meeting id.

**Bar-Ilan University and Hebrew University Set Theory Seminar** **Time:** Wednesday, April 22, 11am IST (10am CEST) **Speaker:** Jing Zhang**Title:** Transformations of the transfinite plane**Abstract:** We discuss the existence of certain transformation functions turning pairs of ordinals into triples (or pairs) of ordinals, that allows reductions of complicated Ramsey theoretic problems into simpler ones. We will focus on the existence of various kinds of strong colorings. The basic technique is Todorcevic’s walks on ordinals. Joint work with Assaf Rinot.**Information:** The zoom meeting ID is 243-676-331 and no password.

April 21

**Münster Set Theory Seminar** **Time:** Tuesday, April 21, 4:15pm CEST**Speaker:** Farmer Schlutzenberg, University of Münster**Title:** Non-definability of embeddings $j:V_\lambda\to V_\lambda$**Abstract:** Assume $\ZF$. We show that there is no limit ordinal $\lambda$ and $\Sigma_1$-elementary $j:V_\lambda\to V_\lambda$ which is definable from parameters over $V_\lambda$.**Information:** The seminar will be held remotely via zoom. Please contact rds@wwu.de ahead of time in order to participate.

Week 13-19 April

April 17

**CUNY Set Theory Seminar****Time:** Friday, April 17, 2pm New York time (8pm CEST)**Speaker:** **Corey Switzer**, CUNY**Title:** Specializing Wide Trees Without Adding Reals**Abstract:** An important advancement in iterated forcing was Jensen’s proof that CH does not imply ♢ by iteratively specializing Aronszajn trees with countable levels without adding reals thus producing a model of CH plus ‘all Aronszajn trees are special’. This proof was improved by Shelah who developed a general method around the notion of dee-complete forcing. This class (under certain circumstances) can be iterated with countable support and does not add reals. However, neither Jensen’s nor Shelah’s posets will specialize trees of uncountable width and it remains unclear when one can iteratively specialize wider trees. Indeed a very intriguing example, due to Todorčević, shows that there is always a wide Aronszajn tree which cannot be specialized without adding reals. By contrast the ccc forcing for specializing Aronszajn trees makes no distinction between trees of different widths (but may add many reals). In this talk we will provide a general criteria a wide trees Aronszajn tree can have that implies the existence of a dee-complete poset specializing it. Time permitting we will discuss applications of this forcing to forcing axioms compatible with CH and some open questions related to set theory of the reals.**Information:** The seminar will take place virtually. Please email Victoria Gitman (vgitman@nylogic.org) for the meeting id.

**Toronto Set Theory Seminar****Time:** Friday, April 17, 1:30-3:00pm EDT (19:30-21:00 CEST) **Speaker:** Matteo Viale, University of Torino**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 conditional to the proof of Schindler and Asperò that Woodin’s axiom (*) can be forced by a stationary set preserving forcing.**Information:** The talk will take place via zoom: https://yorku.zoom.us/j/925557716.