# Set Theory in the UK @ Home

The next Set Theory in the UK workshop will take place online on Friday, 4 December 2020, from 9.30am-2pm.

How to participate: Information how to obtain a login will be available on the conference website soon. Please find this information in advance, on the day before the meeting.

09.30-09.55 Yair Hayut: Generics via ultrapowers
10.00-10.50 Arno Pauly: Luzin’s (N) and randomness reflection
11.00-11.50 Peter Holy: Ramsey-like operators
lunch break
13.30-13.55 Jiachen Yuan: Indestructibility of supercompactness and large cardinals

Titles and abstracts:

Yair Hayut (Hebrew University of Jerusalem): Generics via ultrapowers
Bukovský and Dehornoy observed (independently) that there is a generic for the Prikry forcing over the iterated ultrapower by the measure. I will show how one can use this fact in order to derive (without referring to the forcing) many interesting properties of the generic extension.

Arno Pauly (Swansea University): Luzin’s (N) and randomness reflection

Peter Holy (University of Udine): Ramsey-like operators
Starting from measurability upwards, larger large cardinals are usually characterized by the existence of certain elementary embeddings of the universe, or dually, the existence of certain ultrafilters. However, below measurability, we have a somewhat similar picture when we consider certain embeddings with set-sized domain, or ultrafilters for small collections of sets. I will present some new results, and also review some older ones, showing that not only large cardinals below measurability, but also several related concepts can be characterized in such a way, and I will also provide a sample application of these characterizations.

Jiachen Yuan (University of East Anglia): Indestructibility of supercompactness and large cardinals
It is well known that “there is a supercompact cardinal which is immune to any $\kappa-$directed closed set forcing” is relatively consistent with “there is a supercompact cardinal”. We also know that there is no analogue of such a theorem to any large cardinal stronger than extendible. In fact, provably in $ZFC$ such large cardinal properties will be destroyed by any $\kappa-$directed closed set forcing. For larger cardinals, according to a theorem of Usuba, they can not survive in any set-forcing extension which is not equivalent to a small forcing. However, it was not known if it is possible to have such a large cardinal notion with its supercompactness indestructible. It turns out that this is true for a lot of large cardinals by forcing from a ground model with the same strength.

See you at the meeting!
Andrew Brooke-Taylor, Asaf Karagila and Philipp Schlicht

# Online Activities 30 November – 6 December

Logic Seminar, Carnegie Mellon University
Time: Tuesday, December 1,  3:30 – 4:30pm Eastern Standard Time (21:30 – 22:30 CET)
Speaker: Nam Trang, University of North Texas
Title: Ideals and determinacy
Abstract: We present some ideas involved in the proof of the equiconsistency of AD_\reals + Theta is regular and the existence of a strong, pseudo-homogeneous ideal on P_{\omega_1}(\reals). Some variations of this hypothesis are also shown to be equiconsistent with AD_\reals + Theta is regular. This work is related to and partially answers a long-standing conjecture of Woodin regarding the equiconsistency of AD_\reals + Theta is regular and CH + the nonstationary ideal on \omega_1 is \omega_1-dense. We put this result in a broader context of the general program of understanding connections between canonical models of large cardinals, models of determinacy, and strong forcing axioms (e.g. PFA, MM).
Information: Zoom link https://cmu.zoom.us/j/621951121, meeting ID: 621 951 121

Hebrew University-Bar Ilan University Set Theory seminar
Time: Wednesday, December 2, 14:00-16:00 Israel Time (13:00-15:00 CET)
Speaker: Menachem Magidor, Hebrew University of Jerusalem
Title: Woodin’s extender algebra and its applications (part 2)
Abstract: I’ll continue to talk about Woodin’s extender algebra and absoluteness
Information: Contact Menachem Magidor, Asaf Rinot or Omer Ben-Neria ahead of time for the zoom link.

Münster research seminar on set theory
Time: Wednesday, December 2, 15:15-16:45 CET
Speaker: Tba
Title: Tba
Abstract: Tba
Information: Please check the seminar webpage to see if the seminar takes place. Contact rds@wwu.de ahead of time in order to participate.

Barcelona Set Theory Seminar
Time:
Wednesday, December 2, 16:00 CET
Title: What set theory could not be
Abstract: I am going to argue that set theory – playing its foundational
role – is not particularly concerned with sets. Despite appearances, I don’t
think this is a frivolous claim. Among other things, even if the conclusion
is doubted, it reveals an important split in attitudes researchers have
toward set theory.
Information: Online. If you wish to attend, please send an email to bagaria@ub.edu asking for the link.

Bristol Logic and Set Theory Seminar/Oxford Set Theory Seminar
Time:
Wednesday, December 2, 16:00-17:30 UK time (17:00-18:30 CET)
Speaker: Kameryn Williams, University of Hawai’i at Manoa
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, both positive and negative, about the sequence of inner mantles, answering some questions of Fuchs, Hamkins, and Reitz, results which are analogues of classic results about the sequence of iterated HODs. On the positive side: (Joint with Reitz) Every model of set theory is the eta-th inner mantle of a class forcing extension for any ordinal eta in the model. On the negative side: The sequence of inner mantles may fail to carry through at limit stages. Specifically, it is consistent that the omega-th inner mantle not be a definable class and it is consistent that it be a definable inner model of ¬AC.

Caltech Logic Seminar
Time: Wednesday, December 2, 12:00 – 1:00pm Pacific time (22:00 CET)
Speaker: Aristotelis Panagiotopoulos, University of Münster
Title: The definable content of homological invariants I: Ext⁡(−,−) and lim1(−)
Abstract: This is the first talk in a three-part series in which we illustrate how classical invariants of homological algebra and algebraic topology can be enriched with additional descriptive set-theoretic information.
In the first talk we will focus on the “definable enrichment” of the first derived functors of Hom(−,−) and lim(−). We will show that the resulting “definable Ext(B,F)” for pairs of countable abelian groups B,F; and the “definable lim1(A)” for towers A of Polish abelian groups substantially refine their purely algebraic counterparts. In the process, we will develop an Ulam stability framework for quotients of Polish groups G by Polishable subgroups H and we will provide several rigidity results in the case where the ambient Polish group G is abelian and non-archimedean. A special case of our rigidity results answers a question of Kanovei and Reeken regarding quotients of the p-adic groups.
This is joint work with Jeffrey Bergfalk and Martino Lupini.
Information: See the seminar webpage.

KGRC Research Seminar
Time:
Thursday, December 3, 15:00 CET
Speaker: Mirna Džamonja, CNRS & Panthéon Sorbonne, Paris and Czech Academy of Sciences, Prague
Title: On logics that make a bridge from the Discrete to the Continuous
Abstract: The talk starts with a surveys of some recent connections between logic and discrete mathematics. Then we discuss 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. We compare such logics and discuss some consequences of such comparisons, as well as some hopes for further results in this research project.
Information: Talk via zoom.

Set Theory in the UK @ Home
Time: Friday, December 4, 9:30am-2pm UK time (10:30-15:00 CET)
Schedule:
9:30-9:55 Yair Hayut: Generics via ultrapowers
10.00-10.50 Arno Pauly: Luzin’s (N) and randomness reflection
11.00-11.50 Peter Holy: Ramsey-like operators
lunch break
13.30-13.55 Jiachen Yuan:

CUNY Set Theory Seminar
Time: Friday, December 4, 3pm New York time (21:00 CET)
Speaker: Zach Norwood, University of Michigan
Title: Tba
Abstract: Tba
Information: The seminar will take place virtually. Please email Victoria Gitman (vgitman@nylogic.org) for the meeting id.

Toronto Set Theory Seminar
Time: Friday, December 4, 1.30-3pm Toronto time (19:30-21:00 CET)
Speaker: Thomas Gilton, University of Pittsburgh
Title: The Abraham-Rubin-Shelah Open Coloring Axiom with a Large Continuum
Abstract: Open Coloring Axioms may be viewed as consistent generalizations of Ramsey’s Theorem to ω1 in which topological restrictions are placed on the colorings. The first of these, denoted 𝖮𝖢𝖠ARS, appeared in the 1985 paper by Abraham, Rubin, and Shelah. There the authors showed that 𝖮𝖢𝖠ARS is consistent with 𝖹𝖥𝖢. To ensure that the posets which add the homogeneous sets satisfy the c.c.c., they construct a type of “diagonalization” object (for a continuous coloring χ) called a Preassignment of Colors, which guides the forcing to add the χ-homogeneous sets.
However, the only known constructions of effective preassignments require the 𝖢𝖧. Since a forcing iteration of ℵ1-sized posets all of whose proper initial segments satisfy the 𝖢𝖧 results in a model in which 2ℵ0 is at most ℵ2, this leads naturally to the question of whether 𝖮𝖢𝖠ARS is consistent, say, with 2ℵ0=ℵ3.
In joint work with Itay Neeman, we answer this question in the affirmative. In light of the 𝖢𝖧 obstacle, we only construct names for preassignments with respect to a small class  of 𝖢𝖧-preserving iterations. However, our preassignments are powerful enough to work even over models in which the 𝖢𝖧 fails.
Our final forcing is built by combining the members of  into a new type of forcing, called a Partition Product. A partition product is a type of restricted memory iteration with isomorphism and coherent-overlap conditions on the memories. In particular, each “memory” is isomorphic to a member of .
In this talk, we will describe in some detail the definition of a Partition Product. We will then discuss how to construct more general preassignments than those used by Abraham, Rubin, and Shelah, gesturing at the end towards the full construction which we use for our theorem.
Information: Email Ivan Ongay Valverde ahead of time for the zoom link.

# Online Activities 23-29 November

Logic Seminar, Carnegie Mellon University
Time: Tuesday, November 24,  3:30 – 4:30pm Eastern Standard Time (21:30 – 22:30 CET)
Speaker: Shaun Allison
Title: An anticlassification result for TSI Polish groups
Abstract: We give a dynamical obstruction to classification by TSI Polish groups, and apply it to an equivalence relation of Clemens and Coskey. This yields the first known example of an equivalence relation that is classifiable by a CLI Polish group but not by TSI Polish groups. This work is joint with Aristotelis Panagiotopoulos.
Information: Zoom link https://cmu.zoom.us/j/621951121, meeting ID: 621 951 12

Helsinki Logic Seminar
Time: Wednesday, November 25, 12:15 Helsinki Time (11:15 CET)
Speaker: Carolin Antos-Kuby, University of Konstanz
Title: Two aspects of explanatoriness
Abstract: The phenomenon of explanation in mathematics is an interesting one: If there are different proofs for one theorem, all of them show that the theorem holds but often only some also show why the theorem holds, i.e. additionally also explain the theorem. Unlike in the natural sciences this phenomenon is not easily reducible to the phenomenon of causation. It is even unclear if there is only one form of explanatoriness or if it is a pluralistic notion. Here we give an example from recent descriptive set theory where we study two approaches to proving set-theoretic dichotomy theorems. We will see that both approaches provide explanations for the theorem, albeit in very different ways. We will use this to highlight two ways in which explanatoriness can be spelled out and distinguish between agent-dependent and agent-independent notions of explanatoriness.
Information: See the seminar webpage.

Hebrew University-Bar Ilan University Set Theory seminar
Time: Wednesday, November 25, 14:00-16:00 Israel Time (13:00-15:00 CET)
Speaker: Menachem Magidor, Hebrew University of Jerusalem
Title: Woodin’s extender algebra and its applications
Abstract: This talk will survey known results and will be the first of several talks which will not necessarily follow in the consecutive weeks.
Information: Contact Menachem Magidor, Asaf Rinot or Omer Ben-Neria ahead of time for the zoom link.

Barcelona Set Theory Seminar
Time:
Wednesday, November 25, 16:00 CET
Speaker: Andrew Brooke-Taylor, University of Leeds
Title: Categorifying Borel reducibility
Abstract: Borel reducibility is a framework that has been very successful in showing
that classification programmes in different areas of mathematics are not possible to
complete. However, a feature of many such classification programmes that is not
accounted for in the standard Borel reducibility framework is functoriality – a good
classification function is expected to respect all maps between objects, not just the
isomorphisms. I will present an extension of the Borel reducibility framework that
takes functoriality into account, and give some initial results showing that this is a
meaningful refinement of the standard framework.
Information: Online. If you wish to attend, please send an email to bagaria@ub.edu asking for the link.

KGRC Research Seminar
Time:
Thursday, November 26, 15:00 CET
Speaker: Damian Sobota, KGRC
Title: Convergence of Borel measures and filters on omega
Abstract: The celebrated Josefson–Nissenzweig theorem asserts, under certain interpretations, that for every infinite compact space K there exists a sequence of normalized signed Borel measures on K which converges to 0 with respect to every continuous real-valued function (i.e. the corresponding integrals converge to 0). We showed that in the case of products of two infinite compact spaces K and L one can construct such a sequence of measures with an additional property that every measure has finite support—let us call such a sequence “an fsJN-sequence” (i.e. a finitely supported Josefson–Nissenzweig sequence). We then studied the case when the spaces K and L are only pseudocompact and we proved in ZFC that if the product of K and L is pseudocompact, then it also admits an fsJN-sequence. On the other hand, we showed that under the Continuum Hypothesis, or Martin’s axiom, or even some weaker set-theoretic assumptions concerning weak P-points, there exists a pseudocompact space X such that its square is not pseudocompact and it does not admit any fsJN-sequences. During my talk I will discuss these as well as other results concerning the topic and obtained during a joint work with various combinations of J. Kakol, W. Marciszewski and L. Zdomskyy.
Information: Talk via zoom.

Münster research seminar on set theory
Time: Thursday, November 18, 16:15-17:45 CET
Speaker: Grigor Sargsyan, Gdansk
Title: Determinacy, forcing axioms and inner models (part 4)
Abstract: We will exposit some recent results of the speaker and others that connect determinacy axioms, forcing axioms and inner models. A culmination of this work is a recent proof that the most liberally backgrounded construction of a model build from an extender sequence cannot be shown to converge in ZFC alone. In this construction, which is a type of Kc construction, one uses extenders that are certified by a Mostowski collapse. This result challenges common perceptions of the role of the model Kc in the inner model program.
We will mention a specific consistency result showing that the failure of □ω3 and □(ω3) with 2ω=2ω1=ω2 and 2ω2=ω3 is weaker than a Woodin cardinal that is a limit of Woodin cardinals.
Many people have been involved in this project. The work is heavily based on the efforts of Steel, Jensen, Woodin, Schindler, Mitchell, Schimmerling, Trang, Larson, Neeman, Zeman, Schlutzenberg, the speaker and many others.
Information: contact rds@wwu.de ahead of time in order to participate.

Toronto Set Theory Seminar
Time: Friday, November 27, 1.30-3pm Toronto time (19:30-21:00 CET)
Speaker: Sandra Müller, University of Vienna
Title: The Large Cardinal Strength of Determinacy Axioms
Abstract: The study of inner models was initiated by Gödel’s analysis of the constructible universe L. Later, it became necessary to study canonical inner models with large cardinals, e.g. measurable cardinals, strong cardinals or Woodin cardinals, which were introduced by Jensen, Mitchell, Steel, and others. Around the same time, the study of infinite two-player games was driven forward by Martin’s proof of analytic determinacy from a measurable cardinal, Borel determinacy from ZFC, and Martin and Steel’s proof of levels of projective determinacy from Woodin cardinals with a measurable cardinal on top. First Woodin and later Neeman improved the result in the projective hierarchy by showing that in fact the existence of a countable iterable model, a mouse, with Woodin cardinals and a top measure suffices to prove determinacy in the projective hierarchy.
This opened up the possibility for an optimal result stating the equivalence between local determinacy hypotheses and the existence of mice in the projective hierarchy, just like the equivalence of analytic determinacy and the existence of x♯for every real x which was shown by Martin and Harrington in the 70’s. The existence of mice with Woodin cardinals and a top measure from levels of projective determinacy was shown by Woodin in the 90’s. Together with his earlier and Neeman’s results this estabilishes a tight connection between descriptive set theory in the projective hierarchy and inner model theory.
In this talk, we will outline some of the main results connecting determinacy hypotheses with the existence of mice with large cardinals and discuss a number of more recent results in this area.
Information: Email Ivan Ongay Valverde ahead of time for the zoom link.

# Online Activities 16-22 November

Helsinki Logic Seminar
Time: Wednesday, November 18, 12:15 Helsinki Time (11:15 CET)
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 a byproduct of the groundbreaking result of Schindler and Asperò showing that Woodin’s axiom (*) can be forced by a stationary set preserving forcing.
Information: See the seminar webpage.

Hebrew University-Bar Ilan University Set Theory seminar
Time: Wednesday, November 18, 14:00-16:00 Israel Time (13:00-15:00 CET)
Speaker: Vera Fischer, University of Vienna
Title: Independent families in the countable and the uncountable
Abstract: Independent families on $\omega$ are families of infinite sets of integers with the property that for any two finite subfamilies $A$ and $B$ the set $\bigcap A\backslash \bigcup B$ is infinite. Of particular interest are the sets of the possible  cardinalities of maximal independent families, which we refer to as the spectrum of independence. Even though we do have the tools to control the spectrum of independence at $\omega$ (at least to a large extent), there are many relevant questions regarding higher counterparts of independence in generalised Baire spaces, which remain widely open.
Information: Contact Menachem Magidor, Asaf Rinot or Omer Ben-Neria ahead of time for the zoom link.

Münster research seminar on set theory
Time: Wednesday, November 18, 15:15-16:45 CET
Speaker: Grigor Sargsyan, Gdansk
Title: Determinacy, forcing axioms and inner models (part 3)
Abstract: We will exposit some recent results of the speaker and others that connect determinacy axioms, forcing axioms and inner models. A culmination of this work is a recent proof that the most liberally backgrounded construction of a model build from an extender sequence cannot be shown to converge in ZFC alone. In this construction, which is a type of Kc construction, one uses extenders that are certified by a Mostowski collapse. This result challenges common perceptions of the role of the model Kc in the inner model program.
We will mention a specific consistency result showing that the failure of □ω3 and □(ω3) with 2ω=2ω1=ω2 and 2ω2=ω3 is weaker than a Woodin cardinal that is a limit of Woodin cardinals.
Many people have been involved in this project. The work is heavily based on the efforts of Steel, Jensen, Woodin, Schindler, Mitchell, Schimmerling, Trang, Larson, Neeman, Zeman, Schlutzenberg, the speaker and many others.
Information: contact rds@wwu.de ahead of time in order to participate.

Paris-Lyon Séminaire de Logique
Time:
Wednesday, November 18, 16:00-17:00 CET
Speaker: Denis Osin
Title: A topological zero-one law and elementary equivalence of finitely generated groups
Abstract: The space of finitely generated marked groups, denoted by G, is a locally compact Polish space whose elements are groups with fixed finite generating sets; the topology on G is induced by the local convergence of the corresponding Caley graphs. We will discuss equivalent characterizations of closed subspaces S of G satisfying the following zero-one law: for any sentence sigma in the infinitary logic L_{\omega_1, \omega}, the set of all models of sigma in S is either meager or comeager. In particular, this zero-one law holds for certain natural spaces associated to hyperbolic groups and their generalizations. We will also discuss some open problems.

Barcelona Set Theory Seminar
Time:
Wednesday, November 18, 16:00 CET
Speaker: Monroe Eskew, University of Vienna
Title: Uncommon systems of embeddings
Abstract: We will survey some results from two papers about systems of
elementary embeddings that “narrowly avoid” Kunen’s inconsistency. The first
involves systems where the embedding is amenable to the target model. The
uncommon features are that the system can be densely ordered (even
isomorphic to the reals) and branch off into non-amalgamable models. The
second paper focuses on collections of distinct models that are pairwise
mutually embeddable. Some restrictions on the structure of the system are
imposed by requiring the models to satisfy V=HOD. This is joint work with Sy
Friedman, Yair Hayut, and Farmer Schlutzenberg.
Information: Online. If you wish to attend, please send an email to bagaria@ub.edu asking for the link.

Bristol Logic and Set Theory Seminar/Oxford Set Theory Seminar
Time:
Wednesday, November 18, 16:00-17:30 UK time (17:00-18:30 CET)
Speaker: Gabriel Goldberg, Harvard University
Title: Even ordinals and the Kunen inconsistency
Abstract: The Burali-Forti paradox suggests that the transfinite cardinals “go on forever,” surpassing any conceivable bound one might try to place on them. The traditional Zermelo-Frankel axioms for set theory fall into a hierarchy of axiomatic systems formulated by reasserting this intuition in increasingly elaborate ways: the large cardinal hierarchy. Or so the story goes. A serious problem for this already naive account of large cardinal set theory is the Kunen inconsistency theorem, which seems to impose an upper bound on the extent of the large cardinal hierarchy itself. If one drops the Axiom of Choice, Kunen’s proof breaks down and a new hierarchy of choiceless large cardinal axioms emerges. These axioms, if consistent, represent a challenge for those “maximalist” foundational stances that take for granted both large cardinal axioms and the Axiom of Choice. This talk concerns some recent advances in our understanding of the weakest of the choiceless large cardinal axioms and the prospect, as yet unrealized, of establishing their consistency and reconciling them with the Axiom of Choice.

Caltech Logic Seminar
Time: Wednesday, November 18, 12:00 – 1:00pm Pacific time (22:00 CET)
Speaker: Asger Törnquist, University of Copenhagen
Title: A new proof of Thoma’s theorem on type I groups
Abstract: In the theory of unitary group representations, the following theorem of Elmar Thoma from the early 1960s is fundamental: A countable discrete group is “type I” if and only if it has an abelian finite index subgroup. By way of a celebrated theorem of Glimm from the same period, a group being “type I” is equivalent to saying that the irreducible unitary representations of the group admits a smooth classification in the familiar sense of Borel reducibility, and in fact they are all finite-dimensional in this case. Glimm’s theorem, and later work by Hjorth, Farah and Thomas, implies that if a group is not type I, then it is quite hard to classify the irreducible unitary representations.
In this talk I will give an overview of the descriptive set-theoretic perspective on the classification of irreducible representations, and I will discuss a new proof of Thoma’s theorem due to F.E. Tonti and the speaker.
Information: See the seminar webpage.

KGRC Research Seminar
Time:
Thursday, November 19, 15:00 CET
Speaker: Gabriel Fernandes, Bar-Ilan University
Title: Local club condensation in extender models
Abstract: Local club condensation is a condensation principle defined by Friedman and Holy. It is a theorem due to Friedman and Holy that local club condensation holds in most of the extender models that are weakly iterable.
We prove that in any weakly iterable extender model with λ-indexing, given a cardinal κκ, the sequence ⟨Lα[E]∣α<κ++⟩ witnesses local club condensation on the interval (κ+,κ++) iff κ is not a subcompact cardinal in L[E].
We also prove that if κ is subcompact, then there is no sequence ⟨Mα∣α<κ++⟩∈L[E] with Mκ=(Hκ)L[E] and Mκ++=(Hκ++)L[E] which witnesses local club condensation in (κ+,κ++).
Using the equivalence between subcompact cardinals and ¬◻κ, due to Schimmerling and Zeman, it follows that ◻κ holds iff the sequence ⟨Lα[E]∣α<κ++⟩ witnesses local club condensation on the interval (κ+,κ++).
Information: Talk via zoom.

Southern Illinois University Logic Seminar
Time:
Thursday, November 19, 1pm CET (20:00 CET)
Speaker: Anush Tserunyan, Mc Gill University
Title: Tba
Abstract: Tba
Information: See the seminar webpage.

Toronto Set Theory Seminar
Time: Friday, November 20, 1.30pm Toronto time (19:30 CET)
Speaker: Andrea Medini, University of Vienna
Title: Topological applications of Wadge theory
Abstract: Wadge theory provides an exhaustive analysis of the
topological complexity of the subsets of a zero-dimensional Polish
space. Fons van Engelen pioneered its applications to topology by
obtaining a classification of the zero-dimensional homogeneous Borel
spaces (recall that a space $X$ is homogeneous if for all $x,y\in X$
there exists a homeomorphism $h:X\longrightarrow X$ such that $h(x)=y$).
As a corollary, he showed that all such spaces (apart from trivial
exceptions) are in fact strongly homogeneous (recall that a space $X$ is
strongly homogeneous if all non-empty clopen subspaces of $X$ are
homeomorphic to each other).
In a joint work with the other members of the “Wadge Brigade” (namely,
Raphaël Carroy and Sandra Müller), we showed that this last result
extends beyond the Borel realm if one assumes AD. We intend to sketch
the proof of this theorem, with a view towards a complete classification
of the zero-dimensional homogeneous spaces under AD.
Information: Email Ivan Ongay Valverde ahead of time for the zoom link.

CUNY Set Theory Seminar
Time: Friday, November 20, 3pm New York time (21:00 CET)
Speaker: Philipp Schlicht, University of Vienna
Title: The recognisable universe in the presence of measurable cardinals
Abstract: A set x of ordinals is called recognisable if it is defined, as a singleton, by a formula phi(y) with ordinal parameters that is evaluated in L[y]. The evaluation is always forcing absolute, in contrast to even Sigma_1-formulas with ordinal parameters evaluated in V. Furthermore, this notion is closely related to similar concepts in infinite computation and Hamkins’ and Leahy’s implicitly definable sets.
It is conjectured that the recognisable universe generated by all recognisable sets is forcing absolute, given sufficient large cardinals. Our goal is thus to determine the recognisable universe in the presence of large cardinals. The new main result, joint with Philip Welch, is a computation of the recognisable universe within the least inner model with infinitely many measurable cardinals.
Information: The seminar will take place virtually. Please email Victoria Gitman (vgitman@nylogic.org) for the meeting id.

# Online Activities 9-15 November

Helsinki Logic Seminar
Time: Wednesday, November 11, 12:15 Helsinki Time (11:15 CET)
Speaker: Jeffrey Schatz
Title: Axiom Selection after Large Cardinals: Maximize and the Question of CH
Abstract: There are two noted mathematical programs providing axioms extending the theory of Zermelo-Fraenkel set theory with large cardinals: the inner model program and the forcing axiom program. While these programs historically developed to serve different mathematical goals and ends, proponents of each have attempted to justify their preferred axiom candidate on the basis of its supposed maximization potential. Since ‘maximize’ proves central to the justification of ZFC+LCs itself, and shows up centrally in the current debate over how to best extend this theory, any attempt to resolve this debate will need to investigate the relationship between maximization notions and the candidates for a strong theory of sets. This talk will survey this project, discussing the history of ‘maximization’ considerations in set theory, introducing the main candidates for extending ZFC+LCs, and conclude by presenting recent results toward a resolution of these questions.
Information: See the seminar webpage.

Hebrew University-Bar Ilan University Set Theory seminar
Time: Wednesday, November 11, 14:00-16:00 Israel Time (13:00-15:00 CET)
Speaker: Yair Hayut, Hebrew University
Title: Higher Chang Conjecture
Abstract: In this talk, I will focus on a joint work with Eskew. The main result is the consistency of (\kappa^+, \kappa) –>> (\lambda^+, \lambda).
Information: Contact Menachem Magidor, Asaf Rinot or Omer Ben-Neria ahead of time for the zoom link.

Münster research seminar on set theory
Time: Wednesday, November 11, 15:15-16:45 CET
Speaker: Grigor Sargsyan, Gdansk
Title: Determinacy, forcing axioms and inner models
Abstract: Continuation: We will exposit some recent results of the speaker and others that connect determinacy axioms, forcing axioms and inner models. A culmination of this work is a recent proof that the most liberally backgrounded construction of a model build from an extender sequence cannot be shown to converge in ZFC alone. In this construction, which is a type of Kc construction, one uses extenders that are certified by a Mostowski collapse. This result challenges common perceptions of the role of the model Kc in the inner model program.
We will mention a specific consistency result showing that the failure of □ω3 and □(ω3) with 2ω=2ω1=ω2 and 2ω2=ω3 is weaker than a Woodin cardinal that is a limit of Woodin cardinals.
Many people have been involved in this project. The work is heavily based on the efforts of Steel, Jensen, Woodin, Schindler, Mitchell, Schimmerling, Trang, Larson, Neeman, Zeman, Schlutzenberg, the speaker and many others.
Information: contact rds@wwu.de ahead of time in order to participate.

Barcelona Set Theory Seminar
Time:
Wednesday, November 11, 16:00 CET
Speaker: Gabriel Goldberg, UC Berkeley
Title: On the uniqueness of elementary embeddings
Abstract: This talk will explore the connection between Woodin’s HOD Conjecture and certain uniqueness properties of elementary embeddings of models of set theory, which will lead to some consequences of the failure of the HOD Conjecture reminiscent of the consequences of choiceless large cardinal axioms.
Information: Online. If you wish to attend, please send an email to bagaria@ub.edu asking for the link.

MOPA (Models of Peano Arithmetic), CUNY
Time: Wednesday, November 11, 12pm New York time (18:00 CET)
Speaker: Joel David Hamkins, Oxford University
Title: Continuous models of arithmetic
Abstract: Ali Enayat had asked whether there is a model of Peano arithmetic (PA) that can be represented as ⟨Q,⊕,⊗⟩, where ⊕ and ⊗ are continuous functions on the rationals Q. We prove, affirmatively, that indeed every countable model of PA has such a continuous presentation on the rationals. More generally, we investigate the topological spaces that arise as such topological models of arithmetic. The reals R, the reals in any finite dimension Rn, the long line and the Cantor space do not, and neither does any Suslin line; many other spaces do; the status of the Baire space is open.
This is joint work with Ali Enayat, myself and Bartosz Wcisło.
Information: The seminar will take place virtually. Please email Victoria Gitman (vgitman@nylogic.org) for the meeting id.

Caltech Logic Seminar
Time: Wednesday, November 11, 12:00 – 1:00pm Pacific time (22:00 CET)
Speaker: Miroslav Zelený, Charles University
Title: On the Luzin-Novikov theorem
Abstract: We show that for every ordinal α∈[1,ω1)α∈[1,ω1), there is a closed set F⊂2ω×ωωF⊂2ω×ωω such that for every x∈2ωx∈2ω, the section {y∈ωω:(x,y)∈F}{y∈ωω:(x,y)∈F} is a two-point set and FF cannot be covered by countably many graphs B(n)⊂2ω×ωωB(n)⊂2ω×ωω of functions of the variable x∈2ωx∈2ω such that each B(n)B(n) is in the additive Borel class Σ0αΣα0. This rules out the possibility to have a quantitative version of the Luzin-Novikov theorem. The construction is a modification of the method of Harrington who invented it to show that there exists a countable Π01Π10 set in ωωωω containing a non-arithmetic singleton. By another application of the same method, we get closed sets excluding a quantitative version of the Saint Raymond theorem on Borel sets with σσ-compact sections. (Joint work with P. Holický)
Information: See the semianr webpage.

KGRC Research Seminar
Time:
Thursday, November 12, 15:00 CET
Speaker: Hossein Lamei Ramandi, University of Toronto, Ontario, Canada
Title: Can You Take Komjath’s Inaccessible Away?
Abstract: In this talk we aim to compare Kurepa trees and Aronszajn trees. Moreover, we talk about the affect of large cardinal assumptions on this comparison. Using the the method of walks on ordinals, we will show it is consistent with ZFC that there is a Kurepa tree and every Kurepa tree contains a Souslin subtree, if there is an inaccessible cardinal. This is stronger than Komjath’s theorem which asserts the same consistency from two inaccessible cardinals. We will briefly sketch the ideas to prove that our large cardinal assumption is optimal. If time permits, we talk about the comparison of Kurepa trees and Aronszajn trees in the presence of no large cardinal.
Information: Talk via zoom.

Ghent-Leeds Virtual Logic Seminar
Time:
Thursday, November 12, 15:00 CET
Speaker: Paul Levy, University of Birmingham
Title: Broad infinity and generation principles
Abstract: In set theory, Mahlo’s principle (also called “Ord is Mahlo”) states that the class of regular cardinals is stationary.  Although it has many intuitive consequences, such as the existence of Grothendieck universes, Mahlo’s principle itself is not so intuitive.
To resolve this situation, we give a new axiom scheme called Broad Infinity, resembling the axiom of Infinity.  It is equivalent over ZFC to Mahlo’s principle, but arguably is more intuitive.
We see that both Infinity and Broad Infinity give us principles for generating a set, a family or an ordinal.  We also track the use of Choice and Excluded Middle in proving these results.

CUNY Set Theory Seminar
Time: Friday, November 13, 3pm New York time (21:00 CET)
Speaker: Diana Montoya, University of Vienna
Title: Independence and uncountable cardinals
Abstract: The classical concept of independence, first introduced by Fichtenholz and Kantorovic has been of interest within the study of combinatorics of the subsets of the real line. In particular the study of the cardinal characteristic i defined as the minimum size of a maximal independent family of subsets of ω. In the first part of the talk, we will review the basic theory, as well as the most important results regarding the independence number. We will also point out our construction of a poset P forcing a maximal independent family of minimal size which turns out to be indestructible after forcing with a countable support iteration of Sacks forcing.
In the second part, we will talk about the generalization (or possible generalizations) of the concept of independence in the generalized Baire spaces, i.e. within the space κκ when κ is a regular uncountable cardinal and the new challenges this generalization entails. Moreover, for a specific version of generalized independence, we can have an analogous result to the one mentioned in the paragraph above.
This is joint work with Vera Fischer.
Information: The seminar will take place virtually. Please email Victoria Gitman (vgitman@nylogic.org) for the meeting id.

Toronto Set Theory Seminar
Time: Friday, November 13, 11am Toronto time (17:00 CET)
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: Email Ivan Ongay Valverde ahead of time for the zoom link.

Online Activities 2-8 November

University of Wisconsin Logic Seminar
Time:
Tuesday, November 3, 3:00pm CST (22:00 CET)
Speaker: Vera Fischer, University of Vienna
Title: Independent families in the countable and the uncountable
Abstract: Independent families on ω are families of infinite sets of integers with the property that for any two disjoint finite subfamilies A and B, the set ⋂ A \ ⋃ B is infinite. Of particular interest are the sets of the possible cardinalities of maximal independent families, which we refer to as the spectrum of independence. Even though we do have the tools to control the spectrum of independence at ω (at least to a large extent), there are many relevant questions regarding higher counterparts of independence in generalized Baire spaces, which remain wide open.
Information: Zoom Meeting ID: 970 9130 0913, Passcode: 926119

Helsinki Logic Seminar
Time: Wednesday, November 4, 12:15 Helsinki Time (11:15 CET)
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: Zoom, see the seminar webpage for login information.

Hebrew University-Bar Ilan University Set Theory seminar
Time: Wednesday, November 4, 14:00-16:00 Israel Time (13:00-15:00 CET)
Speaker: Yair Hayut, Hebrew University
Title: Higher Chang Conjecture
Abstract: In this talk I will present some results regarding the consistency strength of Higher variants of Chang’s Conjecture. I will start with the classical result by Silver of Chang’s Conjecture from $\omega_1$-Erdos cardinal.  Then, I will give an upper bound for the consistency strength of  $(\aleph_{\omega+1}, \aleph_{\omega}) –>>(\aleph_1, \aleph_0)$and $(\aleph_4, \aleph_3) –>> (\aleph_2, \aleph_1)$ (joint with Eskew)from supercompactness assumptions. If time permits, I will describe the strategy for obtaining a global result:(\kappa^+,\kappa) –>> (\mu^+, \mu)for all regular $\kappa$, and $\mu < \kappa$, and talk about the barriers that we face when trying to extend this result.
Information: Contact Menachem Magidor, Asaf Rinot or Omer Ben-Neria ahead of time for the zoom link.

Paris-Lyon Séminaire de Logique
Time:
Wednesday, November 4, 16:00-17:00 CET
Speaker: Jeffrey Bergfalk, University of Vienna
Title: Set theory and strong homology: an overview
Abstract: Motivated by several recent advances, we will provide a research history of the main set-theoretic problems arising in the study of strong homology. We will presume no knowledge, in our audience, of the latter. The aforementioned advances close out a second major phase of research in this area, leaving just a few conspicuous last “first questions,” and our aim is to provide some context for engaging them. This research centers on multidimensional combinatorial phenomena generalizing the classical theme of \emph{nontrivial coherent families indexed by ωω}; its progress has involved an intriguing mix of classical (forcing axioms, iterations of large cardinal length) and novel (higher-dimensional Δ-systems, simplicial combinatorics) set-theoretic techniques.

Bristol Logic and Set Theory Seminar/Oxford Set Theory Seminar
Time:
Wednesday, November 4, 16:00-17:30 UK time (17:00-18:30 CET)
Speaker: Mirna Džamonja, CNRS & Panthéon Sorbonne, Paris and Czech Academy of Sciences, Prague
Title: On Wide Aronszajn Trees
Abstract: Aronszajn trees are a staple of set theory, but there are applications where the requirement of all levels being countable is of no importance. This is the case in set-theoretic model theory, where trees of height and size ω1 but with no uncountable branches play an important role by being clocks of Ehrenfeucht–Fraïssé games that measure similarity of model of size ℵ1. We call such trees wide Aronszajn. In this context one can also compare trees T and T’ by saying that T weakly embeds into T’ if there is a function f that map T into T’ while preserving the strict order <_T. This order translates into the comparison of winning strategies for the isomorphism player, where any winning strategy for T’ translates into a winning strategy for T’. Hence it is natural to ask if there is a largest such tree, or as we would say, a universal tree for the class of wood Aronszajn trees with weak embeddings. It was known that there is no such a tree under CH, but in 1994 Mekler and Väänanen conjectured that there would be under MA(ω1).
In our upcoming JSL paper with Saharon Shelah we prove that this is not the case: under MA(ω1) there is no universal wide Aronszajn tree.
The talk will discuss that paper. The paper is available on the arxiv and on line at JSL in the preproof version doi: 10.1017/jsl.2020.42.

CUNY Logic Seminar (MOPA)
Time: Wednesday, November 4, 3pm New York time (21:00 CET)
Speaker: Victoria Gitman, CUNY
Title: A model of second-order arithmetic satisfying AC but not DC: Part II
Abstract: One of the strongest second-order arithmetic systems is full second-order arithmetic Z2 which asserts that every second-order formula (with any number of set quantifiers) defines a set. We can augment Z2 with choice principles such as the choice scheme and the dependent choice scheme. The Σ1n-choice scheme asserts for every Σ1n-formula φ(n,X) that if for every n, there is a set Xwitnessing φ(n,X), then there is a single set Z whose n-th slice Zn is a witness for φ(n,X). The Σ1n-dependent choice scheme asserts that every Σ1n-relation φ(X,Y) without terminal nodes has an infinite branch: there is a set Z such that φ(Zn,Zn+1) holds for all n. The system Z2 proves the Σ12-choice scheme and the Σ12-dependent choice scheme. The independence of Π12-choice scheme from Z2 follows by taking a model of Z2 whose sets are the reals of the Feferman-Levy model of ZF in which every ℵLn is countable and ℵLω is the first uncountable cardinal.
We construct a model of ZF+ACω whose reals give a model of Z2 together with the full choice scheme in which Π12-dependent choice fails. This result was first proved by Kanovei in 1979 and published in Russian. It was rediscovered by Sy Friedman and myself with a slightly simplified proof.
Information: The seminar will take place virtually. Please email Victoria Gitman (vgitman@nylogic.org) for the meeting id.

Caltech Logic Seminar
Time: Wednesday, November 4, 12:00 – 1:00pm Pacific time (22:00 CET)
Speaker: Anush Tserunyan, McGill University
Title: A backward ergodic theorem and its forward implications
Abstract: In the classical pointwise ergodic theorem for a probability measure preserving (pmp) transformation TT, one takes averages of a given integrable function over the intervals {x,T(x),T2(x),…,Tn(x)}{x,T(x),T2(x),…,Tn(x)} in front of the point xx. We prove a “backward” ergodic theorem for a countable-to-one pmp TT, where the averages are taken over subtrees of the graph of TT that are rooted at xx and lie behind xx (in the direction of T−1T−1). Surprisingly, this theorem yields forward ergodic theorems for countable groups, in particular, for pmp actions of finitely generated groups, where the averages are taken along set-theoretic (but backward) trees on the generating set. This strengthens Bufetov’s theorem from 2000, which was the leading result in this vein. This is joint work with Jenna Zomback.

Ghent-Leeds Virtual Logic Seminar
Time:
Thursday, November 5, 15:00 CET
Speaker: Sandra Müller, University of Vienna
Title: Determinacy and inner models
Abstract: The study of inner models was initiated by Gödel’s analysis of the constructible universe L.  Later, it became necessary to study canonical inner models with large cardinals, e.g. measurable cardinals, strong cardinals or Woodin cardinals, which were introduced by Jensen, Mitchell, Steel, and others.  Around the same time, the study of infinite two-player games was driven forward by Martin’s proof of analytic determinacy from a measurable cardinal, Borel determinacy from ZFC, and Martin and Steel’s proof of levels of projective determinacy from Woodin cardinals with a measurable cardinal on top.  First Woodin and later Neeman improved the result in the projective hierarchy by showing that in fact the existence of a countable iterable model, a mouse, with Woodin cardinals and a top measure suffices to prove determinacy in the projective hierarchy.
This opened up the possibility for an optimal result stating the equivalence between local determinacy hypotheses and the existence of mice in the projective hierarchy, just like the equivalence of analytic determinacy and the existence of X^# for every real X which was shown by Martin and Harrington in the 70’s.  The existence of mice with Woodin cardinals and a top measure from levels of projective determinacy was shown by Woodin in the 90’s.  Together with his earlier and Neeman’s results this establishes a tight connection between descriptive set theory in the projective hierarchy and inner model theory.
In this talk, we will outline some of the main results connecting determinacy hypotheses with the existence of mice with large cardinals and discuss a number of more recent results in this area, some of which are joint work with Juan Aguilera.

KGRC Research Seminar
Time:
Thursday, November 5, 15:00 CET
Speaker: Omer Ben-Neria, Hebrew University
Title: On Continuous Tree-Like Scales and related properties of Internally Approachable structures
Abstract: In his PhD thesis, Luis Pereira isolated and developed several principles of singular cardinals that emerge from Shelah’s PCF theory; principles which involve properties of scales, such as the inexistence of continuous Tree-Like scales, and properties of internally approachable structures such as the Approachable Free Subset Property.
In the talk, we will discuss these principles and their relations, and present new results from a joint work with Dominik Adolf concerning their consistency and consistency strength.
Information: Talk via zoom.

Toronto Set Theory Seminar
Time: Friday, November 6, 1.30pm Toronto time (19:30 CET)
Speaker: Jonathan Schilhan, University of Vienna
Title: Definable maximal families of reals in forcing extensions
Abstract: Many types of combinatorial, algebraic or measure-theoretic
families of reals, such as mad families, Hamel bases or Vitali sets, can
be framed as maximal independent sets in analytic hypergraphs on Polish
spaces. Their existence is guaranteed by the Axiom of Choice, but
low-projective witnesses ($\mathbf{Delta}^1_2$) were only known to exist
in general in models of the form $L[a]$ for a real $a$. Our main result
is that, after a countable support iteration of Sacks forcing or for
example splitting forcing (a less known forcing adding splitting reals)
over L, every analytic hypergraph on a Polish space has a
$\mathbf{\Delta}^1_2$ maximal independent set. As a corollary, this
solves an open problem of Brendle, Fischer and Khomskii by providing a
model with a $\Pi^1_1$ mif (maximal independent family) while the
independence number $\mathfrak{i}$ is bigger than $\aleph_1$.
Information: Email Ivan Ongay Valverde ahead of time for the zoom link.

CUNY Set Theory Seminar
Time: Friday, November 6, 3pm New York time (21:00 CET)
Speaker: Ernest Schimmerling, Carnegie Mellon University
Title: Covering at limit cardinals of K
Abstract: Theorem (Mitchell and Schimmerling, submitted for publication) Assume there is no transitive class model of ZFC with a Woodin cardinal. Let ν be a singular ordinal such that ν>ω_2 and cf(ν)<|ν|. Suppose ν is a regular cardinal in K. Then ν is a measurable cardinal in K. Moreover, if cf(ν)>ω, then oK(ν)≥cf(ν).
I will say something intuitive and wildly incomplete but not misleading about the meaning of the theorem, how it is proved, and the history of results behind it.
Information: The seminar will take place virtually. Please email Victoria Gitman (vgitman@nylogic.org) for the meeting id.

# Online Activities 26 October – 1 November

Helsinki Logic Seminar
Time: Wednesday, October 28, 12:15 Helsinki Time (11:15 CET)
Speaker: Yurii Khomskii, Amsterdam University College and Universität Hamburg
Title:  Bounded Symbiosis and Upwards Reflection
Abstract: In [1], Bagaria and Väänänen developed a framework for studying the large cardinal strength of Löwenheim-Skolem theorems of strong logics using  the notion of Symbiosis (originally introduced by Väänänen in [2]). Symbiosis provides a way of relating model theoretic properties of strong logics to definability in set theory. We continue the systematic investigation of Symbiosis and apply it to upwards Löwenheim-Skolem theorems and upwards reflection principles. To achieve this, the notion of Symbiosis is adapted to what we call “Bounded Symbiosis”.  As an application, we provide some upper and lower bounds for the large cardinal strength of upwards Löwenheim-Skolem principles of second order logic.
This is joint work with Lorenzo Galeotti and Jouko Väänänen.
[1] Joan Bagaria and Jouko Väänänen, “On the Symbiosis Between Model-Theoretic and Set-Theoretic Properties of Large Cardinals”, Journal of Symbolic Logic 81 (2) P. 584-604
[2] Jouko Väänänen, “Abstract logic and set theory. I. Definability.” In Logic Colloquium ’78 (Mons, 1978), volume 97 of Stud. Logic Foundations Math., pages 391–421. North-Holland, Amsterdam-New York, 1979.
Information: Zoom Meeting ID: 476 210 6037
Passcode: HLGrp

Bar-Ilan-Jerusalem Set Theory Seminar
Time: Wednesday, October 28, 14:00-16:00 Israel Time (13:00-15:00 CET)
Speaker: Omer Ben Neria, Hebrew University
Title: On Continuous Tree-Like Scales and related properties of Internally Approachable structures
Abstract: In his PhD thesis, Luis Pereira isolated and developed several principles of singular cardinals that emerge from Shelah’s PCF theory; principles which involve properties of scales, such as the inexistence of continuous Tree-Like scales, and properties of internally approachable structures such as the Approachable Free Subset Property.
In the talk, we will discuss these principles and their relations, and present new results from a joint work with Dominik Adolf concerning their consistency and consistency strength.
Information: Contact Menachem Magidor, Asaf Rinot or Omer Ben-Neria ahead of time for the zoom link.

Barcelona Set Theory Seminar
Time:
Wednesday, October 28, 16:00 CET
Speaker: Joel David Hamkins, University of Oxford
Title: A new proof of the Barwise extension theorem, and the universal finite sequence
Abstract: The Barwise extension theorem, asserting that every countable
model of ZF set theory admits an end-extension to a model of ZFC+V=L, is
both a technical culmination of the pioneering methods of Barwise in
admissible set theory and infinitary logic and also one of those rare
mathematical theorems that is saturated with philosophical significance.
In this talk, I shall describe a new proof of the theorem that omits any
need for infinitary logic and relies instead only on classical methods of
descriptive set theory. This proof leads directly to the universal finite
sequence, a Sigma_1-definable finite sequence, which can be extended
arbitrarily as desired in suitable end-extensions of the universe. The result
has strong consequences for the nature of set-theoretic potentialism. This
work is joint with Kameryn J. Williams.
Information: Online. If you wish to attend, please send an email to bagaria@ub.edu asking for the link.

Paris-Lyon Séminaire de Logique
Time:
Wednesday, October 28, 16:00-17:00 CET
Speaker: Marcin Sabok, Mc Gill University, Montreal
Title: Hyperfiniteness at Gromov boundaries
Abstract: I will discuss recent results establishing hyperfiniteness of equivalence relations induced by actions on Gromov boundaries of various hyperbolic spaces. This includes boundary actions of hyperbolic groups (joint work with T. Marquis) and actions of the mapping class group on boundaries of the arc graph and the curve graph (joint work with P. Przytycki)

CUNY Logic Seminar (MOPA)
Time: Wednesday, October 28, 3pm New York time (20:00 CET)
Speaker: Victoria Gitman, CUNY
Title: A model of second-order arithmetic satisfying AC but not DC
Abstract: One of the strongest second-order arithmetic systems is full second-order arithmetic Z2 which asserts that every second-order formula (with any number of set quantifiers) defines a set. We can augment Z2 with choice principles such as the choice scheme and the dependent choice scheme. The Σ1n-choice scheme asserts for every Σ1n-formula φ(n,X) that if for every n, there is a set Xwitnessing φ(n,X), then there is a single set Z whose n-th slice Zn is a witness for φ(n,X). The Σ1n-dependent choice scheme asserts that every Σ1n-relation φ(X,Y) without terminal nodes has an infinite branch: there is a set Z such that φ(Zn,Zn+1) holds for all n. The system Z2 proves the Σ12-choice scheme and the Σ12-dependent choice scheme. The independence of Π12-choice scheme from Z2 follows by taking a model of Z2 whose sets are the reals of the Feferman-Levy model of ZF in which every ℵLn is countable and ℵLω is the first uncountable cardinal.
We construct a model of ZF+ACω whose reals give a model of Z2 together with the full choice scheme in which Π12-dependent choice fails. This result was first proved by Kanovei in 1979 and published in Russian. It was rediscovered by Sy Friedman and myself with a slightly simplified proof.
Information: The seminar will take place virtually. Please email Victoria Gitman (vgitman@nylogic.org) for the meeting id.

Caltech Logic Seminar
Time: Wednesday, October 28, 12:00 – 1:00pm Pacific time (21:00 CET)
Speaker: Tyler Arant, UCLA
Title: From recursively presented metric spaces to recursive Polish spaces
Abstract: Beyond the Baire space, recursively presented metric spaces are structures which serve as a setting for effective descriptive set theory. Motivated by the classical distinction between a complete separable metric space and its corresponding Polish space topological structure, we will explore the notions and issues involved in moving from a recursively presented metric space to its effective Polish space structure. We will survey different approaches to these issues, in particular work by Moschovakis on recursive frames and work by Louveau on effective topology, and prove some original results which clarify some foundational problems in the area.
Information: Online talk https://caltech.zoom.us/j/89937794322?pwd=d2kzZEkrSWo0QW93RWdJMnVucE83Zz09

KGRC Research Seminar
Time:
Thursday, October 29, 15:00 CET
Speaker: Philipp Lücke, University of Barcelona, Spain
Title: Structural reflection and shrewd cardinals
Abstract: In my talk, I want to present work dealing with the interplay between extensions of the Downward Löwenheim–Skolem Theorem to strong logics, large cardinal axioms and set-theoretic reflection principles, focussing on the characterization of large cardinal notions through model- and set-theoretic reflection properties. The work of Bagaria and his collaborators shows that various important objects in the middle and upper reaches of the large cardinal hierarchy can be characterized through principles of structural reflection. I will discuss recent results dealing with possible characterizations of notions from the lower part of this hierarchy through the principle SR−SR−, introduced by Bagaria and Väänänen. These results show that the principle SR−SR− is closely connected to the notion of shrewd cardinals, introduced by Rathjen in a proof-theoretic context, and embedding characterizations of these cardinals that resembles Magidor’s classical characterization of supercompactness.
Information: Talk via zoom.

CUNY Set Theory Seminar
Time: Friday, October 30, 1pm New York time (18:00 CET)
Speaker: Benedikt Löwe, University of Hamburg
Title: Analysis in higher analogues of the reals
Abstract: The real numbers are up to isomorphism the only completely ordered field with a countable dense subset. We consider non-Archimedean ordered fields whose smallest dense subset has cardinality kappa and investigate whether anything resembling ordinary analysis works on these fields.
In particular, we look at generalisations of the intermediate value theorem and the Bolzano-Weierstrass theorem, and realise that there is some mathematical tension between these theorems: the intermediate value theorem requires some saturation whereas Bolzano-Weierstrass fails if the field is saturated. We consider weakenings of Bolzano-Weierstrass compatible with saturation and realise that these are equivalent to the weak compactness of kappa.
This is joint work with Merlin Carl, Lorenzo Galeotti, and Aymane Hanafi.
Information: The seminar will take place virtually. Please email Victoria Gitman (vgitman@nylogic.org) for the meeting id.

Toronto Set Theory Seminar
Time: Friday, October 30, 1.30pm Toronto time (18:30pm CET)
Speaker: Spencer Unger
Title: Reflection properties at successors of singulars
Abstract: We survey some recent advances in techniques for getting reflection properties at successors of singulars with particular attention to the tree property and stationary reflection
Information: Email Ivan Ongay Valverde ahead of time for the zoom link.

# Online Activities 19-25 October

Logic Seminar, Carnegie Mellon University
Time: Tuesday, October 20,  3:30 – 4:30pm Eastern Daylight Time (21:00 CEST)
Speaker: Garrett Ervin, Carnegie Mellon University
Title: Maximally splitting pruned trees in locally finite graphs
Abstract: Let G be an infinite but locally finite connected graph, and let x be a vertex in G. We prove that G contains a pruned tree T rooted at x that splits as early and as often as possible. While this tree isn’t completely canonical, its levels are.
The main point of the talk will be that the phrases “as early” and “as often” here are actually meaningful. Toward seeing this, we’ll show that in many graphs there is actually a maximal set of minimal boundary containing a given vertex x. We call this set the diamond of x, and denote it D(x).
Diamonds can be used to prove many of the classical duality results in graph theory, including Hall’s matching theorem, König’s lemma, and Menger’s theorem. We’ll use diamonds to establish the existence of the tree T.
Information: Zoom link https://cmu.zoom.us/j/621951121, meeting ID: 621 951 121

Barcelona Set Theory Seminar
Time:
Wednesday, October 21, 16:00 CEST
Speaker: Mirna Dzamonja, Université Panthéon-Sorbonne, Paris and Czech Academy of Science
Title: On wide Aronszajn trees
Abstract: Aronszajn trees are a staple of set theory, but there are applications where the requirement of all levels being countable is of no importance. This is the case in set-theoretic model theory, where trees of height and size ω1 but with no uncountable branches play an important role by being clocks of Ehrenfeucht-Fraïssé games that measure similarity of models of size א1. We call such trees wide Aronszajn. In this context one can also compare trees T and T’ by saying that T weakly embeds into T’ if there is a function f that maps T into T’ while preserving the strict order <T. This order translates into the comparison of winning strategies for the isomorphism player, where any winning strategy for T’ translates into a winning strategy for T’. Hence it is natural to ask if there is a largest such tree, or as we would say, a universal tree for the class of wide Aronszajn trees with weak embeddings. It was known that there is no such a tree under CH, but in 1994 Mekler and Väänanen conjectured that there would be under MA(ω1). In our upcoming JSL paper with Saharon Shelah we prove that this is not the case: under MA(ω1) there is no universal wide Aronszajn tree. The talk will discuss that paper, which is available on the arxiv and online at JSL in the preproof version DOI: 10.1017/jsl.2020.42.
Information: Online. If you wish to attend, please send an email to bagaria@ub.edu asking for the link.

Bristol Logic and Set Theory Seminar/Oxford Set Theory Seminar
Time:
Wednesday, October 21, 16:00-17:30 UK time (17:00-18:30 CEST)
Speaker: Andreas Blass, University of Michigan
Title: tba
Abstract: tba

Paris-Lyon Séminaire de Logique
Time:
Wednesday, October 21, 17:00-18:00 CEST
Speaker: Dima Sinapova, University of Illinois at Chicago
Title: Iteration, reflection, and singular cardinals
Abstract: There is an inherent tension between stationary reflection and the failure of the singular cardinal hypothesis (SCH). The former is a compactness type principle that follows from large cardinals. Compactness is the phenomenon where if a certain property holds for every smaller substructure of an object, then it holds for the entire object. In contrast, failure of SCH is an instance of incompactness. Two classical results of Magidor are: (1) from large cardinals it is consistent to have reflection at ℵω+1, and (2) from large cardinals it is consistent to have the failure of SCH at ℵω. As these principles are at odds with each other, the natural question is whether we can have both. We show the answer is yes. We describe a Prikry style iteration, and use it to force stationary reflection in the presence of not SCH. Then we obtain this situation at ℵω by interleaving collapses. This is joint work with Alejandro Poveda and Assaf Rinot.

Caltech Logic Seminar
Time: Wednesday, October 21, 12:00 – 1:00pm Pacific time (21:00 CEST)
Speaker: Ronnie Chen, UIUC
Title: Borel and analytic sets in locales
Abstract: A locale is, informally, a topological space without an underlying set of points, with only an abstract lattice of “open sets”. Various results in the literature suggest that locale theory behaves in many ways like a generalization of descriptive set theory with countability restrictions removed. This talk will introduce locale theory from a descriptive set-theoretic point of view, and survey some known and new results which are common to both contexts. In particular, we will introduce the “∞∞-Borel hierarchy” of a locale, and sketch the existence of “σσ-analytic, non-∞∞-Borel sets”.
Information: Online talk https://caltech.zoom.us/j/84553965424?pwd=NHRyclZlZ1cydjZBNWkvTlF5QVFmdz09

Bar-Ilan-Jerusalem Set Theory Seminar
Time: Thursday, October 22, 10:00am Israel Time (10:00 CEST)
Speaker: Gabriel Fernandes, Bar-Ilan University
Title: Local club condensation in extender models
Abstract: Local club condensation is an abstraction of the condensation properties of the constructible hierarchy.
We will prove that for extender models that are countably iterable, given a cardinal kappa, the J_alpha^{E} hierarchy witnesses local club condensation in the interval
From the above and the equivalence between subcompact cardinals and square, due to Schimmerling and Zeman, it follows that in such extender models \square_kappa holds iff the J_alpha^{E} hierarchy witnesses that local club condensation holds in the interval (kappa^+,kappa^++).
Information: Contact Menachem Magidor, Asaf Rinot or Omer Ben-Neria ahead of time for the zoom link.

Kurt Gödel Research Center Seminar (organised by Ben Miller)
Time:
Thursday, October 22, 15:00 CEST
Speaker: Philipp Schlicht, KGRC
Title: Tree forcings, sharps and absoluteness
Abstract: In joint results with Fabiana Castiblanco from 2018, we showed that several classical tree forcings preserve sharps for reals and levels of projective determinacy, and studied their impact on definable equivalence relations (in particular, the question whether they add equivalence classes to thin projective equivalence relations). I will discuss these results and natural open problems on tree forcings and absoluteness that arise from them.
Information: Talk via zoom.

CUNY Set Theory Seminar
Time: Friday, October 23, 15:00 New York time (21:00 CEST)
Speaker: Gabriel Goldberg, University of Berkeley
Title: Ultrapowers and the approximation property
Abstract: Countably complete ultrafilters are the combinatorial manifestation of strong large cardinal axioms, but many of their basic properties are undecidable no matter the large cardinal axioms one is willing to adopt. The Ultrapower Axiom (UA) is a set theoretic principle that permits the development of a much clearer picture of countably complete ultrafilters and, consequently, the large cardinals from which they derive. It is not known whether UA is (relatively) consistent with very large cardinals, but assuming there is a canonical inner model with a supercompact cardinal, the answer should be yes: this inner model should satisfy UA and yet inherit all large cardinals present in the universe of sets. The predicted resemblance between the large cardinal structure of this model and that of the universe itself is so extreme as to suggest that certain consequences of UA must in fact be provable outright from large cardinal axioms. While the inner model theory of supercompact cardinals remains a major open problem, this talk will describe a technique that already permits a number of consequences of UA to be replicated from large cardinals alone. Still, the technique rests on the existence of inner models that absorb large cardinals, but instead of building canonical inner models, one takes ultrapowers.
Information: The seminar will take place virtually. Please email Victoria Gitman (vgitman@nylogic.org) for the meeting id.

# Online activities 12-18 October 2020

Logic Seminar, Carnegie Mellon University
Time: Tuesday, October 13,  3:30 – 4:30pm Eastern Daylight Time (21:00 CEST)
Speaker: Farmer Schlutzenberg, University of Muenster
Title: Definability of elementary embeddings beyond the axiom of choice
Abstract: Large cardinal axioms play a central role in set theory and our understanding of relative consistency strength. Such axioms are typically exhibited by elementary (truth-preserving) embeddings of the form $j:V\to M$, where $V$ is the universe of all sets and $M$ a sub-universe. Demanding greater resemblance between $V$ and $M$ tends to lead to stronger large cardinal notions. Kunen famously showed that the strongest possible resemblance — $V=M$ — is inconsistent with the assumption that $V$ models ZFC (including Choice). Suzuki showed later that just assuming $V$ models ZF, there can be no elementary $j:V\to V$ which is definable from parameters. Now one can ask whether (or to what extent) Suzuki’s result still holds if we weaken ZF. In this talk we will discuss various results — and counterexamples — along these lines. Some of the results I’ll mention were first due to Gabe Goldberg. (References: arXiv 2006.10574, 2006.01103)
Information: Zoom link https://cmu.zoom.us/j/621951121, meeting ID: 621 951 121

Helsinki Logic Seminar
Time: Wednesday, October 14,  12:15 Helsinki Time (11:15 CEST)
Speaker: Miguel Moreno, University of Vienna
Title:  Filter Reflection and Generalised Descriptive Set Theory
Abstract: Filter reflection is an abstract version of stationary reflection motivated from many results in generalised descriptive set theory. In this talk we will define filter reflection and different avatars of it. We will focus on its consequences in generalised descriptive set theory. We will also discuss how to force filter reflection and how to force the failure of filter reflection.
This is a joint work with Gabriel Fernandes and Assaf Rinot.
Information: Zoom meeting ID: 684 3587 2772, Passcode: 110592

Paris-Lyon Séminaire de Logique
Time:
Wednesday, October 14, 16:00-17:15 CEST
Speaker: Tomás Ibarlucía, University of Paris
Title: Automorphism groups acting on Hilbert spaces without almost invariant vectors
Abstract: We will discuss, first, how to construct automorphisms of countable/separable saturated models (and, more interestingly, pairs of automorphisms) that act “very freely” on the structure, in a sense given by stability theory. Then we will see how to use this to show that automorphism groups of aleph_0-categorical metric structures have Kazhdan’s Property (T), which roughly means that their unitary actions on Hilbert spaces do not have almost invariant vectors in non-trivial ways.

Illinois Logic Group Logic Seminar
Time: Wednesday, October 14, 1pm US Central Daylight Time (20:00 CEST)
Speaker: Jenna Zomback, UIUC Math
Title: A backward ergodic theorem and its forward implications
Abstract: A pointwise ergodic theorem for the action of a transformation T on a probability space equates the global property of ergodicity of the transformation to its pointwise combinatorics. Our main result is a backward (in the direction of T−1) ergodic theorem for countable-to-one probability measure preserving (pmp) transformations T. We discuss various examples of such transformations, including the shift map on Markov chains, which yields a new (forward) pointwise ergodic theorem for pmp actions of finitely generated countable groups, as well as one for the (non-pmp) actions of free groups on their boundary. This is joint work with Anush Tserunyan.
Information: The seminar will take place remotely.

Caltech Logic Seminar
Time: Wednesday, October 14, 12:00 – 1:00pm Pacific time (21:00 CEST)
Speaker: Jindřich Zapletal, University of Florida
Title: Geometric Set Theory
Abstract: The talk will be an outline of the book we published with Paul Larson recently. In particular, I will show how amalgamation problems in algebra naturally appear in consistency results for the choiceless set theory ZF+DCZF+DC, and how they can be stratified from a set-theoretic point of view.
Information: Online talk https://caltech.zoom.us/j/88977992358?pwd=UjJjdFhZZEVuMjFmZ0VLZFd5dGhqQT09

Kurt Gödel Research Center Seminar (organised by Ben Miller)
Time:
Thursday, October 15, 15:00 CEST
Speaker: Ziemowit Kostana, University of Warsaw
Title: Fraïssé theory, and forcing absoluteness of rigidity for linear orders
Abstract: During the talk I would like to introduce the theory of Cohen-like first-order structures. These are countable or uncountable structures which are “generic” much in the same sense as the Cohen reals. They can be added to the universe of set theory using finite or, say, countable conditions and exhibit different properties. I will focus on the construction of a rigid linear order, whose rigidity is absolute for ccc extensions.
Information: Talk via zoom.

CUNY Set Theory Seminar
Time: Friday, October 16, 15:00 New York time (21:00 CEST)
Speaker: Richard Matthews, University of Leeds
Title: Taking Reinhardt’s Power Away
Abstract: Many large cardinals can be defined through elementary embeddings from the set-theoretic universe to some inner model, with the guiding principle being that the closer the inner model is to the universe the stronger the resulting theory. Under ZFC, the Kunen Inconsistency places a hard limit on how close this can be. One is then naturally led to the question of what theory is necessary to derive this inconsistency with the primary focus having historically been embeddings in ZF without Choice.
In this talk we take a different approach to weakening the required theory, which is to study elementary embeddings from the universe into itself in ZFC without Power Set. We shall see that I1, one of the largest large cardinal axioms not known to be inconsistent with ZFC, gives an upper bound to the naive version of this question. However, under reasonable assumptions, we can reobtain this inconsistency in our weaker theory.
Information: The seminar will take place virtually. Please email Victoria Gitman (vgitman@nylogic.org) for the meeting id.

# Online activities 5-11 October 2020

Logic Seminar, Carnegie Mellon University
Time: Tuesday, October 6,  3:30 – 4:30pm Eastern Daylight Time (21:00 CEST)
Speaker: Dima Sinapova, University of Illinois at Chicago
Title: Iteration, reflection, and Prikry forcing
Abstract: There is an inherent tension between stationary reflection and the failure of the singular cardinal hypothesis (SCH). The former is a compactness type principle that follows from large cardinals. Compactness is the phenomenon where if a certain property holds for every smaller substructure of an object, then it holds for the entire object.
In contrast, failure of SCH is an instance of incompactness. It is 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 $\aleph_\omega$, combining two classical results of Magidor. This is joint work with Alejandro Poveda and Assaf Rinot.
Organizers’ note: This is a more general talk to be immediately followed by a more technical talk.
Information: Zoom link https://cmu.zoom.us/j/621951121, meeting ID: 621 951 121

Helsinki Logic Seminar
Time: Wednesday, October 7,  12:15 Helsinki Time (11:15 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 a byproduct of the groundbreaking result of Schindler and Asperò showing that Woodin’s axiom (*) can be forced by a stationary set preserving forcing.

Barcelona Set Theory Seminar
Time: Wednesday, October 7, 16:00 CEST
Speaker: Philipp Lücke, IMUB, Universitat de Barcelona
Title: Structural reflection, shrewd cardinals and the size of the continuum
Abstract:
Information: Online talk https://eu.bbcollab.com/guest/2bd01807690a4ae39a8f102e19911a31

Caltech Logic Seminar
Time: Wednesday, October 7, 12:00 – 1:00pm Pacific time (21:00 CEST)
Speaker: Riley Thornton, UCLA
Title: Factor of i.i.d. processes and Cayley diagrams
Abstract: A Cayley diagram for a Cayley graph G=Cay(Γ,E) is an edge labelling of G with generators from E so that a path is labelled with a relation in Γ if and only if it is a cycle. I will show how Aut(G)-f.i.i.d. Cayley diagrams can be used to lift Γ-f.i.i.d. solutions of local combinatorial problems to Aut⁡(G)-f.i.i.d. solutions. And, I will investigate which graphs admit Aut⁡(G)-f.i.i.d. Cayley diagrams, answering a question of Weilacher on approximate chromatic numbers in the process.
Information: Online talk https://caltech.zoom.us/j/99296122790?pwd=bUN4RS94RVYrTEtGTGhqTHRJbm9nZz09

Kurt Gödel Research Center Seminar (organised by Ben Miller)
Time:
Thursday, October 8, 15:00 CEST
Speaker: Colin Jahel, Claude Bernard University Lyon 1
Title: Actions of automorphism groups of Fraïssé limits on the space of linear orderings
Abstract: In 2005, Kechris, Pestov and Todorčević exhibited a correspondence between combinatorial properties of structures and dynamical properties of their automorphism groups. In 2012, Angel, Kechris and Lyons used this correspondence to show the unique ergodicity of all the actions of some groups. In this talk, I will give an overview of the aforementioned results and discuss recent work generalizing results of Angel, Kechris and Lyons.
Information: Talk via zoom.

CUNY Set Theory Seminar
Time: Friday, October 9, 11:00 New York time (17:00 CEST)
Speaker: Heike Mildenberger, Albert-Ludwigs-Universität Freiburg
Title: Forcing with variants of Miller trees
Abstract: Guzmán and Kalajdzievski introduced a variant of Miller forcing P(F) that diagonalises a given filter F over ω and has Axiom A. We investigate the effect of P(F) for particularly chosen Canjar filters F. This is joint work with Christian Bräuninger.
Information: The seminar will take place virtually. Please email Victoria Gitman (vgitman@nylogic.org) for the meeting id.