Abstracts
Talks by Invited Speakers
Shaun Allison
Title: A rank determining when a Polish automorphism group has less-than-maximal classification strength
Abstract:
The classification strength of a Polish group roughly measures how much it can "emulate" other orbit equivalence relations in the context of Borel reductions. We would say that a Polish group $G$ emulates an equivalence relation $E$ living on a Polish space if there is a Polish space $Y$ and a continuous action of $G$ on $Y$ such that $E$ Borel- reduces to the resulting orbit equivalence relation. In this talk, we focus on the class of Polish automorphism (non-Archimedean) Polish groups. At the top of the classification strength hierarchy for these groups is the class of "maximum strength" groups which can emulate any orbit equivalence relation induced by a Polish automorphism group.
In analogy with our work with Aristotelis Panagiotopoulos, where we developed a rank function inspired by work of Deissler and Malicki to detect when a given Polish group is cli (has a complete left-invariant metric), we develop here a rank function which fully captures the class of Polish automorphism groups with less-than-maximal classification strength, a class which strictly contains the cli Polish automorphism groups. This leads to a natural but very novel independence notion which weakens the notion of disjoint amalgamation in Fraïssé theory. We will start with an analysis of a structure due to Julia Knight whose automorphism group has low classification strength yet fails to be cli. Then we will show how this suggests a rank function, and we will show how it has the desired properties. Time permitting, we will gather many different equivalent conditions for a Polish group to have less-than-maximal classification strength, showing that this class is quite robust and interesting.
Omer Ben-Neria
Title: Advances in Iterated Forcing Methods
Abstract:
We shall describe advances in the theory of iterated forcing, including certain technical insights and applications.
Michal Doucha
Title: Global properties of transitive and minimal actions of groups on
the Cantor space
Abstract:
I will present joint work in progress with Julien Melleray and
Todor Tsankov on descriptive set theoretic properties of actions of
countable groups on the Cantor space. This includes eg the existence
of comeager and dense conjugacy classes in the space of transitive and
minimal actions for various countable groups, genericity of minimality
resp. transitivity in the space of all actions, and the description of
the closure of the set of transitive actions in the space of all
actions.
Sumun Iyer
Title: Generic actions of countable groups
Abstract:
Let $G$ be a countable discrete group. We consider a Polish space $\textrm{Act}(G)$ of all continuous $G$-actions on Cantor space. The orbit equivalence relation of a particular action of $G$ is a countable Borel equivalence relation. We show that if $G$ has finite asymptotic dimension, then the generic element of $\textrm{Act}(G)$ generates a hyperfinite equivalence relation. This is joint work with Forte Shinko.
Eyal Kaplan
Title: Failure of GCH on a measurable with the Ultrapower Axiom
Abstract:
The Ultrapower Axiom (UA) roughly states that any pair of ultrapowers can be compared by internal ultrapowers. The Axiom was extensively studied by Gabriel Goldberg, leading to a series of striking results.
Goldberg asked whether UA is consistent with a measurable cardinal that violates GCH. The main challenge is that UA is not easily preserved under forcing constructions, especially ones that achieve violation of GCH on a measurable from large cardinal assumptions. For example, such forcings might create normal measures which are incomparable in the Mitchell order – a property that negates UA.
In this talk, we show that the failure of GCH on the least measurable cardinal can indeed be forced while preserving UA, starting from the minimal canonical inner model carrying a $(\kappa,\kappa^{++})$-extender. We will present the forcing construction and provide a sketch of the main proof ideas. This is a joint work with Omer Ben-Neria.
Andreas Lietz
Title: On Mathias Characterizations for Generics for Variants Of Namba Forcing
Abstract:
Namba forcing is the first instance of a forcing which singularizes $\omega_2$ while preserving $\omega_1$. Several versions of Namba forcing exist which have shown to not be equivalent under CH by Magidor-Shelah as well as Jensen.
We generalize these theorems by removing the CH assumption and taking into account many more variations of Namba forcing. Further, we show that all “natural” variations of Namba forcing generate extensions which are minimal conditioned on $\cof(\omega_2^V)=\omega$ and moreover, we analyze exactly which and how many other sequences in such an extension are generic for a variation of Namba forcing. Further, we show that no Mathias-style characterization for variants of Namba forcing are possible except for Priky-style forcings. This answers a question of Gunter Fuchs.
Heike Mildenberger
Title: Namba Forcing and Singular Cardinals
Abstract:
Namba forcing was discovered independently by Bukovský and Namba to address
questions about singularizing $\aleph_2$ and the minimality of such extensions.
We present new results on singular Namba forcing. Our work pertains to minimality,
in some cases seemingly requiring a large cardinal assumption, and also to variations
of the Hamkins approximation property. Notably we deal with the challenges presented by the Laver versions of these forcings. The emphasis on this case is motivated in part by questions regarding the compactness of Jensen's square principle in the presence of good scales at $\aleph_\omega$. This is a joint work with Maxwell Levine.
Aristotelis Panagiotopoulos (University of Vienna)
Title: The class and dynamics of α-balanced Polish groups
Abstract:
A Polish group is TSI if it admits a two-side invariant metric. It is CLI if it admits complete and left-invariant metric. The class of CLI groups contains every TSI group but there are many CLI groups that fail to be TSI. In this talk we will introduce the class of α-balanced Polish groups where α ranges over all countable ordinals. We will show that these classes completely stratify the space between TSI and CLI. We will also introduce "generic $\alpha$-unbalancedness": a turbulence-like obstruction to classification by actions of $\alpha$-balanced Polish groups. Finally, for each $alpha$ we will provide an action of an $alpha$-balanced Polish group whose orbit equivalence relation is not classifiable by actions of any $\beta$-balanced Polish group with $\beta<\alpha$. This is a joint work with Shaun Allison.
Alejandro Poveda
Title: Axiom $\mathcal{A}$ and supercompactness
Abstract:
In this presentation I would like to report on some recent results on the large cardinal hierarchy between the first supercompact cardinal and Vopenka's Principle. Specifically, we will explore the possible configurations among the classes of supercompact, $C^{(n)}$-supercompact and $C^{(n)}$-extendible cardinals. We will present several consistency results as well as a conjecture about how the large-cardinal hierarchy of $\mathrm{Ultimate-}L$ at these latitudes looks like. One of our main theorems is that (consistently) every supercompact cardinal is $C^{(1)}$-supercompact (ie, supercompact with strong limit targets). This configuration is consistent with (virtually) all large cardinals, yet at odds with the predictions made by the $\mathrm{Ultimate-}L$ program for the class of $C^{(1)}$-supercompacts. This new configuration is a consequence of a new axiom we introduce (named $\mathcal{A}$) which regards the mutual relationship between superstrong and tall cardinals with strong limit targets. Time permitting we shall also propose open problems and discuss possible strengthenings of axiom $\mathcal{A}$.
Grigor Sargsyan
Title: Recent advances in descriptive inner model theory
Abstract:
In recent years, there have been major advances in descriptive
inner model theory. The younger generation has advanced the subject both inward and
outward, proving some striking results of theoretical interest as well as building
bridges with other topics such as forcing axioms and ideal hypotheses.
In some cases, they have found ways of expressing older results without using the
technical language of inner model theory (eg Siskind's characterization of the Core
Model or Kasum's characterization of the minimal model of AD$_{\mathbb{R}}$ + $ \Theta$ is regular).
Such reformulations allow a wider use of these rudimentary descriptive inner model
theoretical structures. In this talk we will outline the contributions made to descriptive
inner model theory by Benjamin Siskind (Characterization of the core model, full
normalization), Takehiko Gappo (focusing over models of determinacy, building higher
models of determinacy), William Chan (AD combinatorics) , Douglas Blue (forcing over
models of determinacy), Obrad Kasum (combinatorial characterization of models of AD),
Derek Levinson (unreachability within the projective hierarchy), Lukas Koschat
(models of the Largest Suslin Axiom via purely large cardinal techniques) and
Jan Kruschewski (an answer to a question of Steel) in collaboration with Jackson,
Larson, Muller, Neeman, the speaker, Schlutzenberg, Steel and Trang.
John Steel
Title: The Comparison Lemma
Abstract:
What makes a canonical inner model $M$ canonical is the existence of
an iteration strategy for $M$. This implies that $M$
can be compared with other models of its kind, a key fact is known as The Comparison Lemma
for the models in question.
It turns out that there is a comparison lemma for iteration strategies too. As the models become
more complicated, so do their iteration strategies. In both cases, comparison establishes
a fine, well-ordered complexity hierarchy. We shall outline
what is known about it at present.
Šárka Stejskalová
Title: Kurepa trees
Abstract:
In the talk we will survey recent results related to Kurepa trees, non-saturated trees and almost Kurepa Suslin trees at $\omega_1$. In the first part, we will discuss the solution of the question of Jin and Shelah who asked whether one could add a Kurepa tree with a small (size $\omega_1$) ccc forcing over models with no Kurepa trees. In the second part of the talk we will focus on some natural generalizations of Jin and Shelah's question; for instance, we will consider models where the property that there are no Kurepa trees is a consequence of the Guessing model principle (GMP).
Asger Tornquist
Cancelled
Anush Tserunyan
Title: Quasi-treeable equivalence relations are treeable
Abstract:
A result from geometric group theory says that if a Cayley graph of a
finitely generated group is tree-like (more precisely, is quasi-isometric to a tree)
then the group is virtually free. We prove the analogue of this in the context of
countable Borel equivalence relations, thus answering a question of R. Tucker-Drob
from 2015. Our method exploits the Stone duality between certain families of cuts
in a graph and median graphs ($1$-skeleta of CAT(0) cube complexes) via cloning
ultrafilters on cuts. This is a joint work with R. Chen, A. Poulin, and R. Tao.
Ur Ya'ar
Title: Iterating the construction of inner models from extended logics
Abstract:
One way to generalize Gödel's constructible universe $L$ is to replace the notion of definability used at successor stage, by taking all subsets which are definable using a logic $\mathcal{L}^*$ extending first order logic. This will result in a model of ZF, denoted $C(\mathcal{L}^*)$. In many cases it will also be a model of AC.
Specific cases of this kind of construction were studied by Chang and by Myhill and Scott in the past, but recently Juliette Kennedy, Menachem Magidor and Jouko Väänänen have initiated a systematic study of these models, leading to many new results.
The topic of this talk will be iterating this construction. As in the case of $L$, we can formulate the axiom ``$V=C(\mathcal{L}^*)$'', but unlike $L$, $C(\mathcal{L^*}) $ might not satisfy it -- when we construct $C(\mathcal{L}^*)$ inside $C(\mathcal{L}^*)$ we might get a smaller model.
When this happens we can iterate the construction, and if the iteration does not stabilize, we continue transfinitely by taking intersections at limit stages.
We will present some results concerning these sequences of iterated constructible models, focusing mainly on the case of the logic obtained by adding the cofinality-$\omega$ quantifier, and the case of Stationary logic. We will show, as time permits, that in the first case the possibilities can be rather limited, while in the second case we can get almost everything we want.
Contributed Talks
Joan Bagaria
Title: Ultraexact cardinals
Authors: Juan P. Aguilera, Joan Bagaria, and Philipp Lücke
Abstract: We present some recent results on exact and ultraexact cardinals, large-cardinal notions at the fringe of consistency with ZFC. We answer a question of the second and third authors by showing that the consistency of ultraexact cardinals follows from the existence of an I0 embedding. In fact, in the presence of sharps, the existence of an ultraexact cardinal and the existence of an I0 embedding are equiconsistent.
We prove that ultraexact cardinals are equivalent to a form of structural reflection, called "Square Root Reflection", thus showing that they are natural large-cardinal notions. However, the existence of an exact cardinal already implies that V is not HOD, and moreover, the existence of an ultraexact cardinal implies that the successor of the supremum of the critical sequence of the corresponding elementary embedding is omega-strongly measurable in HOD.
We also show that the consistency of the existence of a slight strenghtening of a Reinhardt cardinal, plus a supercompact cardinal greater than the supremum of its critical sequence, implies the consistency with ZFC of an ultraexact cardinal together with an extendible cardinal below that supremum, thus yielding the consistency with ZFC of large cardinal assumptions that imply a failure of Woodin's HOD Hypothesis.
Fernando Barrera
Title: The $\lambda$-PSP at $\lambda$-coanalytic sets
Abstract: I will show that if there is a strong limit cardinal $\lambda$ of countable cofinality such that all $\lambda$-coanalytic sets have the $\lambda$-PSP, then there is an inner model with $\lambda$-many measurable cardinals all below $\lambda^+$. The proof is quite an accessible example of the technique of establishing consistency strength lower bounds using core models and their covering properties. Time permitting, I will mention the obstacles that arise when trying to increase the given lower bound. This is joint work with Sandra Müller and Vincenzo Dimonte.
Mariam Beriashvili
Title: Luzin spaces and some point sets
Abstract:
There are known several notions of a Luzin topological space and there are some variations of its definition in use. We represent here two definitions:
1. A Luzin space (I) is an uncountable topological $T_1$ -space without isolated points in which every nowhere dense subset is at most countable.
2. A Luzin space (II) is an uncountable topological space X, such that there exists no nonzero sigma-fiFinite Borel measure on X vanishing at all singletons in X.
Let us remark, that these defInitions are not equivalent in general. It is consistent with ZFC theory that every metrizable Luzin space (I) is a Luzin space (II); the converse assertion is not true in general;
The classical example of a Luzin space (both (I) and (II)) is a Luzin set on the real line. More generally, an example of a Luzin set is the generic set of Cohen reals in the Cohen real model, which can we consider as a Luzin space.
In the talk we present some properties of Luzin spaces and establish their connections with Bernstein type sets.
Ben De Bondt
Title: $\mathsf{OCA}_{\mathrm T}$ and triviality of automorphisms of reduced products
Abstract:
Reduced products are structures obtained by modding out countable products by the Fréchet (or a more general) ideal. Shelah proved the consistency of the statement that the archetypical reduced product, namely the Boolean algebra $\mathcal{P}(\omega)/{\mathrm{Fin}}$, has only trivial automorphisms. This statement has since been shown to be a consequence of the Proper Forcing Axiom (Shelah--Steprans 1988) and the weaker assumption $\mathsf {OCA}_{\mathrm T} + \mathsf{MA} _{\aleph_1}$ (Velickovic 1993), where $\mathsf {OCA}_{\mathrm T}$ is Todorcevic’s Open Colouring Axiom. We will discuss some of the multiple directions that can be explored for extending these results and address the following question: for which other reduced products can it be proved from forcing axioms that all automorphisms are trivial? In particular, we show from $\mathsf {OCA}_{\mathrm T} + \mathsf{MA} _{\aleph_1}$ that various classes of reduced products, including reduced products of countable linear orders and the reduced power of the random graph, have only trivial automorphisms.
This is joint work with Ilijas Farah and Alessandro Vignati.
Ari Brodsky
Title: Proxy principles in combinatorial set theory.
Abstract:
The parameterized proxy principles were introduced by Brodsky and Rinot in a 2017 paper as new foundations for the construction of $\kappa$-Souslin trees in a uniform way that does not depend on the nature of the (regular uncountable) cardinal $\kappa$. Since their introduction, these principles have facilitated the construction of Souslin trees with complex combinations of features, and have enabled the discovery of completely new scenarios in which Souslin trees must exist. Furthermore, the proxy principles have found new applications beyond the construction of trees.
We shall open with a comprehensive exposition of the proxy principles. We motivate their very definition, emphasizing the utility of each of the parameters and the consequent flexibility that they provide. We then survey the findings surrounding them, presenting a rich spectrum of unrelated models and configurations in which the proxy principles are known to hold, and showcasing a gallery of Souslin trees constructed from the principles.
Miguel A. Cardona
Title: Controlling the covering number of the null ideal
Abstract:
Call a function $\pi\colon {}^{<\omega}2\to 2$ a predictor. We denote by $\Sigma_2$ the set of predictors $\pi\colon {}^{<\omega}2\to 2$. Given $\pi\in\Sigma_2$ and $f\in {}^{\omega}2$, say that $\pi$ predicts constantly $f$ denoted by $f\sqsubset^{\mathrm{pc}} \pi$ iff
$\exists k\in\omega\,\forall^\infty i\,\exists j\in[i,i+k)\colon f(j)=\pi(f{\upharpoonright}j)$.
We define the constant evasion number
\[\mathfrak{e}_2^{\mathrm{const}}:=\min\{|F| \colon\, F\subseteq {}^{\omega}2\textrm{\ and\ }\neg\exists \pi\in\Sigma_2\,\forall f\in F\colon f\sqsubset^{\mathrm{pc}}\pi\}\]
and constant prediction number
\[\mathfrak{v}_2^{\mathrm{const}}:=\min\{|S| \colon\, S\subseteq\Sigma_2\textrm{\ and\ }\forall x\in {}^{\omega}2\,\exists \pi\in S\colon f\sqsubset^{\mathrm{pc}}\pi\}.\]
These cardinals have been intensive researched for many people who have been studied for decades the relation between these cardinal invariants and other classical cardinal invariants of the continuum. In particular, it is known in ZFC that $\mathfrak{e}_2^\mathrm{const}\leq\mathrm{non}(\mathcal{E})$ and $\mathrm{cov}(\mathcal{E})\leq\mathfrak{v}_2^\mathrm{const}$ due to Kamo [KAM00], where $\mathrm{non}(\mathcal{E})$ and $\mathrm{cov}(\mathcal{E})$ denotes the uniformity number and covering number of the ideal $\mathcal{E}$ generated by the $F_\sigma$ measure zero subsets of the reals.
On the other hand, regarding the consistency Brendle [Bre03] proved $\mathrm{CON}(\mathfrak{e}_2^\mathrm{const}>\mathfrak{b})$ where $\mathfrak{b}$ is the bounding number; Kamo [KAM00, KAM01] prove $\mathrm{CON}(\mathfrak{d}<\mathfrak{v}_2^\mathrm{const})$ and $\mathfrak{v}_2^\mathrm{const}>\mathrm{cof}(\mathcal{N})$ where $\mathfrak{d}$ and $\mathrm{cof}(\mathcal{N})$ denotes the dominating number and the cofinality of $\mathcal{N}$ the null ideal, respectively; and Brendle and Shelah [BS03] proved $\mathrm{CON}(\mathfrak{e}_2^\mathrm{const}<\mathrm{add}(\mathcal{N}))$ where $\mathrm{add}(\mathcal{N})$ is the additivity of $\mathcal{N}$ the null ideal.
In addition, concerning the consistency of the constant evasion number and the constant prediction number, the following summarises the current open questions.
Question (KAM00, KAM01). Are each one of the following statements consistent with ZFC?
- $\mathfrak{e}_2^{\textrm{const}}>\mathrm{cov}(\mathcal{N})$.
- $\mathfrak{v}_2^{\textrm{const}}<\mathrm{non}(\mathcal{N})$.
This talk aims to give a positive answer to (1). This work is joint with Saharon Shelah.
References
[Bre03] Jörg Brendle. Evasion and prediction. III. Constant prediction and dominating reals.
J. Math. Soc. Japan, 55(1):101–115, 2003.
[BS03] Jörg Brendle and Saharon Shelah. Evasion and prediction. IV. Strong forms of constant
prediction. Arch. Math. Logic, 42(4):349–360, 2003.
[Kam00] Shizuo Kamo. Cardinal invariants associated with predictors. In Logic Colloquium ’98
(Prague), volume 13 of Lect. Notes Log., pages 280–295. Assoc. Symbol. Logic, Urbana,
IL, 2000.
[Kam01] Shizuo Kamo. Cardinal invariants associated with predictors. II. J. Math. Soc. Japan,
53(1):35–57, 2001.
Valentin Haberl
Title: Menger and consonant spaces in the Sacks model
Abstract:
By a space we mean a metrizable separable zero-dimensional space.
A space $X$ is Menger if for any sequence $\mathcal{U}_0, \mathcal{U}_1,\dots$ of open covers of $X$, there are finite families $\mathcal{F}_0\subseteq \mathcal{U}_0, \mathcal{F}_1\subseteq\mathcal{U}_1,\dots$ such that the family $\bigcup_{n\in\omega}\mathcal{F}_n$ covers $X$. If, moreover, the $\mathcal{F}_n$'s can be chosen in such a way that for every $x \in X$, $x \in \bigcup \mathcal{F}_n$ holds for all but finitely many $n$, $X$ is said to be Hurewicz.
We call a space totally imperfect if it contains no copy of $2^\omega$.
We shall discuss our result that there are no totally imperfect Menger sets of reals of size $\mathfrak{c}$ in the Sacks model. Therefore, the Menger property behaves in the Sacks model as an instance of the Perfect Set Property, sets are either small or contain a perfect set. For models, which satisfy that $\mathfrak{d} = \mathfrak{c}$, there is always a totally imperfect Menger set of size continuum. (There are also models with small dominating number, where such sets exist.)
Thus, combined with our result the existence of totally imperfect Menger sets of reals of size $\mathfrak{c}$ is independent from ZFC.
Consonant spaces were introduced by Dolecki, Greco and Lechicki in 1995 and for the case $X \subseteq 2^\omega$ charaterized by Jordan in 2020 using a topological game on the complement $2^\omega \setminus X$. By considering a grouped version of the Menger game and using a similar approach like for the Menger space result, we conclude that every consonant and every Hurewicz subspace of $2^\omega$, as well as their complements, can be written as the union of $\omega_1$-many compact sets in the Sacks model. In particular, there are only continuum many consonant spaces and Hurewicz spaces in this model.
This is joint work with Piotr Szewczak (Cardinal Stefan Wyszy\'nski University in Warsaw) and Lyubomyr Zdomskyy (TU Vienna).
Cecelia Higgins
Title: Measurable Brooks's Theorem for Directed Graphs
Abstract: A greedy algorithm argument demonstrates that any undirected graph of degree bounded by $d$ has chromatic number at most $d + 1$. This upper bound is sharp; the obvious obstructions to a smaller upper bound are odd cycles and complete graphs. In 1941, Brooks proved that these obstructions are the only ones: Any undirected graph of degree bounded by $d$ not containing odd cycles or complete graphs admits a proper $d$-coloring. Work of Marks shows that Brooks's theorem for undirected graphs fails in the Borel setting. However, in 2016, Conley, Marks, and Tucker-Drob proved a measurable version of Brooks's theorem, demonstrating a divide between definable and descriptive combinatorics.
In the 1980s, combinatorialists introduced a coloring notion for directed graphs known as dicoloring. Work of Mohar and Harutyunyan and Mohar resulted in a Brooks-like characterization of the directed graphs of maximum degree bounded by $d$ that admit $d$-dicolorings. In this talk, we present and sketch a proof of a measurable version of this theorem. We explain also why the theorem has no Borel analogue.
Radek Honzik
Title: Higher cardinal invariants and compactness principles
Abstract:
We will discuss several results related to compactness principles, such as stationary reflection, the tree property and the negation of the weak Kurepa hypothesis, and their connection to cardinal invariants in the higher Baire spaces $\kappa^\kappa$ for an uncountable $\kappa$. We will focus on the ultrafilter number $u_\kappa$, tower number $t_\kappa$ and also on the invariants of the meager ideal. $\kappa$ can be singular or regular (in fact inaccessible); the compactness principles we will discuss live on $\kappa^{++}$ (or on $\kappa^+$ for the negation of the weak Kurepa hypothesis). Our method is based on isolating and proving indestructibility results for several compactness principles and applying them to obtain desired patterns of cardinal invariants.
Martina Iannella
Title: The complexity of (piecewise) convex embeddability between countable scattered linear orders
Abstract: Given a nonempty class $\mathcal{L}$ of countable linear orders, we say that the countable linear order $L$ is $\mathcal{L}$-convex embeddable into the countable linear order $L'$ if it is possible to partition $L$ into convex sets, indexed by some element of $\mathcal{L}$, which are isomorphic to convex subsets of $L'$ ordered in the same way. This notion generalizes convex embeddability and embeddability, which arise from the special cases $\mathcal{L}=\{1\}$ and $\mathcal{L}$ equal to the whole class of countable linear orders.
The present talk focuses on the behaviour of these relations on the set SCAT of countable scattered linear orders. We first look at the complexity of the embeddings witnessing that $L$ is $\mathcal{L}$-convex embeddable into $L'$, for some $L$ and $L'$ in SCAT, and show that, for some classes $\mathcal{L}$, there exists an embedding which is hyperarithmetic in $(L,L')$. We then analyse the complexity (in terms of Wadge reducibility) of these relations, extending the study started in [1].
The above results stem from a joint work with Juan Aguilera.
[1] M. Iannella, A. Marcone, L. Motto Ros, and V. Weinstein. Piecewise convex embeddability on linear orders. 2023. arXiv:2312.01198
Hannes Jakob
Title: Cascading variants of internal approachability
Abstract:
The variants of internal approachability were introduced by Foreman and
Todorcevic in 2005. By a folklore result, under $2^{\mu}=\mu^+$, the
failure of the approachability property at $\mu$ is equivalent to the
assertion that there are stationarily many substructures of $H(\mu^+)$
with size $\mu$ which are internally unbounded but not internally
approachable. We will show that this result relies on the assumption
$2^{\mu}=\mu^+$ by showing that whenever $\mu$ is regular and
$\kappa>\mu$ is $\kappa^+$-supercompact, there is a forcing extension
in which $\mu$ remains regular, $\operatorname{AP}_{\mu}$ holds and
there are stationarily many $N\in[H(\mu^+)]^{\mu}$ which are internally
stationary but not internally club.
This contrasts a result of Krueger: If $2^{\mu}=\mu^+$, the existence
of a disjoint stationary sequence on $\mu^+$ is equivalent to the
existence of stationarily many $N\in[H(\mu^+)]^{\mu}$ which are
internally unbounded but not internally club. Since the existence of a
disjoint stationary sequence on $\mu^+$ implies the failure of
$\AP_{\mu}$, our result shows that this theorem also depends on the
cardinal arithmetic assumption.
Additionally, we will prove that there can be stationarily many
substructures $M$ of $H(\Theta)$ such that the intersections $M\cap
H(\nu)$ are internally approachable in different ways for different
$\nu$, answering a question of Foreman.
Menachem Kojman
Title: Rudin's Dawker space and the SCH.
Abstract:
M. E. Rudin's Dawker space was constructed in ZFC in 1969 with techniques that in retrospect were seen to be pcf theory techiques,
inside the product of all $\aleph_n$-s. Whether replacing the sequence of the first $\omega$ cardinals by an arbitrary sequence of regular cardinals in
Rudin's construction always yields a Dowker space is a natural question. While for many instances the answer is positive, currently it is known to follow
for all sequences in ZFC and a mild form of Shelah's Strong Hypothesis.
Miloš Kurilić
Title: Vaught's Conjecture for More Classes of Partial Orders
Abstract:
Vaught's conjecture [10], stated by Robert Vaught in 1959, is the statement that the number I(T,ω) of non-isomorphic countable models of a complete countable first-order theory T is either at most countable or continuum. The conjecture is equivalent to its restriction to (complete theories of) partial orders (Arnold Miller, see [9]) and it was confirmed for linear orders with countably many unary predicates by Matatyahu Rubin [7], for (model-theoretic) trees by John Steel [9], for arborescent structures by James Schmerl [8], for Boolean algebras by Paul Iverson [2]. Roughly, if C is a class of structures for which the conjecture was confirmed, our intention is to confirm it for the structures belonging to a closure [C] of C obtained in some reasonable way. So, in [4] a sharp version of Vaught's conjecture, VC#: I(T,ω) is 0,1 or c, was confirmed for the structures definable by quantifier-free formulas in linear orders colored in finitely many convex colors (FMD structures, detected and investigated by Maurice Pouzet and Nicolas Thiéry [6]) and, in particular, for monomorphic structures [3] (explored by Roland Fraïssé [1]). In [5] the minimal closure of a class C under finite products and finite disjoint unions was denoted by [C], and VC# was confirmed for [C], when C is the class of linear orders or the class of rooted FMD trees. Here we confirm VC for the partial orders from the closure [C], where C is the class of all rooted trees.
This research was supported by the Science Fund of the Republic of Serbia, Program IDEAS, Grant No. 7750027: SMART.
[1] R. Fraïssé, Theory of relations, Revised edition, With an appendix by Norbert Sauer, Studies in Logic and the Foundations of Mathematics, 145. North-Holland, Amsterdam, (2000)
[2] P. Iverson, The number of countable isomorphism types of complete extensions of the theory of Boolean algebras, Colloq. Math. 62,2 (1991) 181-187.
[3] M. S. Kurilić, Vaught's conjecture for monomorphic theories, Ann. Pure Appl. Logic 170,8 (2019) 910-920.
[4] M. S. Kurilić, Vaught's conjecture for theories admitting finite monomorphic decompositions, Fund. Math. 256,2 (2022) 131--169.
[5] M. S. Kurilić, Sharp Vaught's conjecture for some classes of partial orders, Ann. Pure Appl. Logic 175,4 (2024) Paper No. 103411, 18 pp.
[6] M. Pouzet, N. M. Thiéry, Some relational structures with polynomial growth and their associated algebras I: Quasi-polynomiality of the profile, El. J. Comb. 20 (2013) Paper 1, 35 pp.
[7] M. Rubin, Theories of linear order, Israel J. Math. 17 (1974) 392-443.
[8] J. H. Schmerl, Arborescent structures. II. Interpretability in the theory of trees, Trans. Amer. Math. Soc. 266,2 (1981) 629-643.
[9] J. R. Steel, On Vaught's conjecture. Cabal Seminar 76/77 (Proc. Caltech-UCLA Logic Sem., 1976/77), pp. 193-208, Lecture Notes in Math., 689, Springer, Berlin, 1978.
[10] R. L. Vaught, Denumerable models of complete theories, 1961 Infinitistic Methods (Proc. Sympos. Foundations of Math., Warsaw, 1959) pp. 303-21Pergamon, Oxford; PWN, Warsaw.
Boriša Kuzeljević
Title: Rudin-Keisler lower bounds for families of P-point ultrafilters
Abstract:
It is a result of Blass that each countable collection of P-points that has a Rudin-Keisler upper bound which is a P-point, also has an RK lower bound which is necessarily a P-point. In the talk we will present how to extend this result to larger families, but with a cost. Namely, we show that if MA$_{\kappa}$ holds, then each collection of at most $\kappa$ many P-points that has a P$_{\mathfrak c}$-point as an RK upper bound, also has a lower bound which is necessarily a P$_{\mathfrak c}$-point. This is joint work with Dilip Raghavan and Jonathan Verner.
Chris Lambie-Hanson
Title: Higher-dimensional coherence
Abstract:
We present some recent progress in the emerging study of higher-dimensional analogues of coherent Aronszajn trees arising from considerations in homological algebra. We will begin by focusing on the special case of two-dimensional coherence at $\aleph_2$ before discussing some more general results. This is joint work with Jeffrey Bergfalk and Jing Zhang.
Sven Manthe
Title: The Borel monadic theory of order is decidable
Abstract:
The monadic second-order theory S1S of (ℕ,<) is decidable (it essentially describes ω-automata). Undecidability of the monadic theory of (ℝ,<) was proven by Shelah. Previously, Rabin proved decidability if the monadic quantifier is restricted to Fσ-sets.
We discuss decidability for Borel sets, or even σ-combinations of analytic sets. Moreover, the Boolean combinations of Fσ-sets form an elementary substructure. Under determinacy hypotheses, the proof extends to larger classes of sets.
Pedro Marun
Title: Labelled sets
Abstract:
A theorem of Dilworth asserts that, if a poset $P$ has no antichains whose
size is larger than $m$, where $m$ is a natural number, then $P$ can be written as a
union of $m$ many chains. If $m$ is instead an infinite cardinal, then the analogous
statement is false, counterexamples were constructed by Perles. In recent work,
Abraham and Pouzet gave a basis for the class of such counterexamples, and asked
if it could be somewhat simplified. Labelled sets arise in connection with these
counterexamples. We show that, when the underlying sets are $\aleph_1$-dense, then
any two labelled sets embed into each other.
Jayde Massmann
Title: "Stability, interpretability and nonprojectibility"
Abstract:
In this talk I discuss my theorem that $\Sigma_2$-nonprojectility can be considered analogous to subtlety; formally, that a notion which many would agree encapsulates the strength of subtlety is in fact equivalent to $\Sigma_2$-nonprojectibility. I therefore focus on two polar opposites in the ordinals: countable ordinals and large cardinals. We consider analogies in the structure of these.
Kranakis, Richter, Aczel, Klev, Rathjen & Kaufmann have in the past discussed countable analogues to large cardinals and the difficulty of formalizing such a general notion. As part of my case I share some previously known results on the structure of stable ordinals and an equivalence of subtle cardinals (a combinatorial principle) with a significant strengthening of indescribability."
Julia Millhouse
Title:
Projective witnesses of optimal complexity
Abstract:
The spectrum of maximal almost disjoint families-- the set of cardinals having
a mad family of the size-- has been studied since
Hechler in the 1970's, around which time the work of Mathias initiated the study
of the definability of such families. Recently these two interests have been
combined, addressing questions such as how to obtain witnesses to an
element of the mad spectrum which are of optimal complexity in the projective
hierarchy. In this talk I will discuss how to obtain a model in which the
continuum is size $\aleph_2$, and there are tight mad families of size
$\aleph_1$ and $\aleph_2$, each having the best possible complexity for their
respective size. I will also discuss what is needed to have the analogous result
for models with larger continuum. This is joint work with Vera Fischer.
Miguel Moreno
Title: The Borel reducibility Main Gap
Abstract
One of the biggest motivations in Generalized Descriptive Set Theory has been the connection with model theory, in particular the program of identifying counterparts of classification theory in the setting of Borel reducibility. The most important has been the division line classifiable vs non-classifiable, i.e. identify Shelah's Main Gap in the Borel reducibility hierarchy. This was studied by Friedman, Hyttinen and Weinstein (ne Kulikov) in their book "Generalized descriptive set theory and classification theory". Their work led them to conjecture the following:
If T is a classifiable theory and T' is a non-classifiable theory, then the isomorphism relation of T is Borel reducible to the isomorphism relation of T’.
In this talk we will sketch the proof of this conjecture, discuss the objects introduced, and provide a detailed overview of the gaps between classifiable shallow theories, classifiable non-shallow theories, and non-classifiable theories.
Francesco Parente
Title: On the ultrafilter number of Boolean algebras
Abstract:
I shall discuss some recent results concerning two cardinal characteristics of measure and category, namely the ultrafilter numbers of random and Cohen forcing algebras, respectively. Our main goal is to completely determine their relations with the cardinals in Cichoń's diagram. This is achieved by means of technical tools which may be of independent interest, such as an evaluation of the reaping number of reduced powers of Boolean algebras, as well as a fine analysis of the refinement relation between maximal antichains.
Joint work with Jörg Brendle and Michael Hrušák.
Elena Pozzan
Title: A Stone duality for the class of compact Hausdorff spaces
Abstract:
It is well known that we can characterize T0-topological spaces in terms of preorders
describing a base for the space. In particular, any T0-topological space can be represented
as the space whose points are the neighborhood filters of one of its basis for the open sets.
Conversely, we show that any dense family of filters on a preorder defines a topological
space whose characteristics are strictly connected to the ones of the preorder. Therefore,
we show how the separation properties of the topological space can be described in terms
of the algebraic properties of the corresponding preorder and family of filters.
Furthermore, we outline the algebraic conditions on a selected base of the topological space
ensuring that the space is compact and T1. This allows us to establish a duality between the
category of compact T1 spaces with continuous closed maps and an appropriate category
of lattices. Moreover, we could specialize this duality to the category of compact Hausdorff
spaces with continuous maps.
Weakening these results, we will also present two contravariant adjunctions between these
categories of topological spaces and some category of elementary lattices that are first-
order describable.
These characterizations allow us to give a description of the Stone–Cech compactification ˇ
of a topological space in terms of lattices.
This is joint work with Matteo Viale.
References
[1] William H. Cornish. Normal lattices. Journal of the Australian Mathematical Society, 14:200 – 215,
1972.
[2] Orrin Frink. Compactifications and semi-normal spaces. American Journal of Mathematics,
86(3):602–607, 1964.
[3] Matteo Viale. Notes on forcing, 2017. http://www.logicatorino.altervista.org/matteo_viale/
DispenseTI2014.pdf.
Juan Santiago
Title: Infinitary logics, concistency properties and forcing.
Abstract:
The relation between forcing, Boolean valued models and consistency properties was implicit from the beginning of forcing in the works of Mansfield, Keisler, Solovay, Scott and Makkai. However, no concrete application was given. The aim of this talk is to argue that consistency properties and infinitary logics provide a natural setting for building forcing notions.
First, we will discuss the failure of compactness for infinitary logics and how one may recover it. Once the tools are ready, we will present general logic results such as completeness, interpolation, omitting types and a variant of compactness.
Second, we will analyse the relation between consistency properties and forcing. Motivated by the fact that every generic filter can be seen as the model built by a consistency property, we will discuss for what formulae there is a consistency property forcing a model in a nice way. The results presented here build on the proof of MM$^{++}$ implies (*) by Asperò and Schindler together with the AS condition recently isolated by Kasum and Velickovic. We will conclude presenting some results about SSP forcings.
This work is joint with Matteo Viale.
Lukas Schembecker
Title: Forcing and combinatorics of (non) Van Douwen families
Abstract:
A family of eventually different reals is called Van Douwen if every
infinite partial function agrees infinitely often with one of its
members.
This is a combinatorial strengthening of the notion of a maximal
eventually different family and Raghavan(2010) proved that there always
is a Van Douwen family of size continuum.
I will present some recent results related to these Van Douwen families;
in particular, I will show that the set of all possible sizes of Van
Douwen families is closed under singular limits.
One of the main open questions in this context is the consistency of a
maximal eventually different family smaller than any Van Douwen family
(i.e. the consistency of $a_e < a_v$).
Towards this end (while still open), I will also present some results
regarding forcing indestructiblity of non Van Douwen families.
Benjamin Siskind
Title: Order-preserving Martin’s Conjecture and Inner Model Theory
Abstract:
Martin’s Conjecture is a proposed classification of Turing-invariant functions under the Axiom of Determinacy. Whether the classification holds for the ostensibly smaller class of order-preserving functions is open, but more tractable. In this talk, we’ll explain an approach to proving Martin’s Conjecture for order-preserving functions via inner model theory. This is joint work with Patrick Lutz.
Boris Šobot
Title: Ultrafilters basic for divisibility
Abstract:
A divisibility relation on ultrafilters can be defined as follows:
${\cal F}\hspace{1mm}\widetilde{\mid}\hspace{1mm}{\cal G}$ if and only
if every set in $\cal F$ upward closed for divisibility also belongs to
$\cal G$. Having describing the first $\omega$ levels of this quasiorder
in a previous paper, we now identify the basic divisors of an
ultrafilter; they are, in a sense, powers of prime ultrafilters. (An
ultrafilter is prime if it contains the set of prime numbers.) The goal
is then not only to find a possible factorization of a given ultrafilter
$\cal F$ into basic ultrafilters (since such a factorization is not
possible for most $\cal F$), but to determine its pattern: the
collection of its basic divisors as well as the multiplicity of each of
them. All such patterns have a certain closure property in an
appropriate topology. We isolate the family of sets belonging to every
ultrafilter with a given pattern. Finally, we show that every pattern
with the closure property is realized by an ultrafilter.
Sebastiano Thei
Title: $\Sigma$-Prikry forcings and regularity properties for singular cardinals
Abstract:
It is well-known that every set of reals in $L(\mathbb{R})$ enjoys many regularity properties. Among those, we have the Perfect Set Property (PSP) and the Baire property (BP). It was realized after Solovay's theorem that AD is not necessary to obtain this configuration. Indeed, it is enough to collapse an inaccessible to $\omega_1$ and look at $L(\mathbb{R})$ of the generic extension. More recently, Cramer and Woodin showed that under $I_0$ every subset of $^\omega\lambda$ in $L(V_{\lambda+1})$ has the $\lambda$-PSP; a suitable analogue of the classical PSP for strong limit singular cardinals of countable cofinality. This is one of the pieces of evidence that (in spite the parallelism is not perfect) $L(V_{\lambda+1})$ behaves under $I_0$ very much like as $L(\mathbb{R})$ does under AD.
Our main results are as follows. First, we obtain a substantial improvement of Cramer and Woodin's result by proving the consistency of the $\lambda$-PSP for subsets in $L(V_{\lambda+1})$, starting with an inaccessible $\theta$ and a $<\theta$-supercompact cardinal. By strengthening our assumption (yet still below the realm of $I_0$) we prove the consistency of both the $\lambda$-PSP and the $\Vec{\mathcal{U}}$-BP for a rich class of subsets. This is an application of a general theory underpinned by the $\Sigma$-Prikry framework. This is joint work with Vincenzo Dimonte and Alejandro Poveda.
Andrés F. Uribe-Zapata
Title: Singular values in Cichoń's diagram: a general theory of iterated forcing using finitely additive measures
Abstract:
Saharon Shelah in [She00] introduced a finite-support iteration using finitely
additive measures to prove that, consistently, the covering of the null ideal may
have countable cofinality. In [KST19], Jakob Kellner, Saharon Shelah, and Anda
R. Tănasie achieved some new results and applications using such iterations.
In this talk, based on the works mentioned above, we present a general
theory of iterated forcing using finitely additive measures, which was developed
in the speaker's master's thesis [UZ23]. For this purpose, we introduce two
new notions: on the one hand, we define a new linkedness property, which
we call “$\mu$-FAM- linkedness ” and, on the other hand, we generalize the idea of
intersection number to forcing notions [UZ24], which justifies the limit steps of
our iteration theory. Finally, we use this theory to get a new constellation of
Cichoń’s diagram: a separation of the left side where the covering of the null
ideal is singular, even with countable cofinality.
This is a joint work with Miguel Cardona (Pavol Jozef Šafárik University,
Slovakia) and Diego A. Mejía (Shizuoka University, Japan).
References:
[KST19] Jakob Kellner, Saharon Shelah, and Anda R. Tănasie. Another ordering of
the ten cardinal characteristics in Cichoń's diagram. Comment. Math. Univ.
Carolin., 60(1):61–95, 2019.
[She00] Saharon Shelah. Covering the null ideal may have countable cofinality.
Find. Math., 166(1-2):109–136, 2000.
[UZ23] Andres Uribe-Zapata. Iterated forcing with finitely additive measures: applications of probability to forcing theory. Master's thesis, Universidad Nacional de Colombia, sede Medellín, 2023. https://shorturl.at/sHY59.
[UZ24] Andrés F. Uribe-Zapata. The intersection number for forcing notions. Kyoto Daigaku Sūrikaiseki Kenkyūsho Kōkyūroku, 2024. To appear, arXiv:2401.14552.
Tristan van der Vlugt
Title:
The horizontal direction
Abstract:
The Cichoń diagram consists of 10 cardinal numbers between $\aleph_1$ and the cardinality of the continuum, defined in terms of the meagre ideal, the lebesgue null ideal, and the domination order on the Baire space. It is well-known that for any pair of these cardinals there is a forcing extension in which they have different values.
In recent years, there has been significant progress in generalizing cardinal characteristics of the continuum to the context of the higher Baire space, that is, the space of functions $\kappa\to\kappa$, where $\kappa$ is uncountable, often a large cardinal. Some forcing notions (specifically Cohen, Hechler and Sacks forcing) can be generalized to higher context to show consistency results analogous to those from the classical Cichoń diagram. However, contrary to the classical case, there are multiple pairs of higher cardinal characteristics for which the consistency of different values is unknown. These problematic cases tend to be pairs of cardinal characteristics that are placed in the same column in the Cichoń diagram. Hence, this calls for forcing notions that separate the diagram in a horizontal direction.
In this talk we will discuss generalizations of some forcing notions that classically are used for separation in the horizontal direction, and the difficulties that arise in attempts to reproduce this for the higher Cichoń diagram. Specifically we will take a look at higher analogues of random, Laver, Mathias, Miller and eventually different forcing.
Bartosz Wcislo
Title: Dependent choice for reals and the Axiom of Determinacy
Abstract:
One of the most prominent alternatives to the Axiom of Choice is the
Axiom of Determinacy (AD). Even though AD is inconsistent with the AC,
all the known models of AD satisfy an important fragment of AC, so called
Dependent Choice for real numbers (DC$_{\mathbb{R}}$) which states that every tree of
reals which has no leaves, has an infinite branch. It is a major open problem
whether AD in fact implies DC$_{\mathbb{R}}$.
In our talk, we will present an argument obtained in a joint work with
Sandra Müller which provides a partial result towards that problem. By
modifying a construction described by Friedman, Gitman, and Kanovei, we
can show that determinacy for $\Pi^1_1$ sets does not imply DC$_{\mathbb{R}}$ for trees which have complexity $\Pi^1_3 $. We will also discuss possible extensions of this theorem to higher complexity classes.
