$ \newcommand{\setof}[2]{\{#1\,:\,#2\}} \newcommand{\ssetof}[1]{\{#1\}} $

In this talk, we will study the absoluteness of certain notions in real analysis. In particular, we will prove that the Riemann integral over $n$-rectangles in $\mathbb{R}^n$ is absolute for transitive models of ZFC. This is joint work with Carlos M. Parra-Londoño (Universidad Nacional de Colombia, Medellín).

The Rudin-Frolík ordering of ultrafilters over $\omega$ was introduced by Frolík in 1967, who used it to show that the topological space $\beta\omega\setminus\omega$ is not homogeneous. Two generalizations of the Rudin-Frolík ordering to ultrafilters on complete Boolean algebras have been proposed independently by Balcar and Dow in 1991 and by Murakami in 1999. In this talk, we compare the two notions and investigate to what extent they coincide on the class of c.c.c. Boolean algebras. This talk is based on joint work in progress with Jörg Brendle.

A cardinal invariant is a cardinal number lying between the least uncountable cardinal $\aleph_1$ and the cardinality of the continuum $\mathfrak c$. The study of cardinal invariants investigates how these cardinals can differ. In particular, some prediction principles admit various types (e.g., local type, constant types) and can produce cardinal characteristics such as $\textbf{add}(\mathcal N)$ and $\mathfrak b$, which serve as important examples of this framework. In this talk, we propose a local version of constant prediction, which we call the local constant type, and examine its connections with other cardinal invariants.

Gavalová and Mejía introduced generalizations of the measure zero ideal and closed measure zero ideal, denoted by $\mathcal{N} _J$ and $\mathcal{N} _J^\ast$, with ideals on $\omega$. To consider the notions of $\mathcal{N} _J$ and $\mathcal{N} _J\ast$ related to Borel sets such as idealzed forcings, we investigated the properties of the Borel supports of $\mathcal{N} _J$ and $\mathcal{N} _J\ast$. We prove the relations among the measure zero ideal $\mathcal{N}$, the $\sigma$-ideal $\mathcal{E}$ generated by closed measure zero sets, the meager ideal $\mathcal{M}, \mathcal{N} _J, \mathcal{N} _J\ast$ and their Borel supports.

For an ideal $\mathscr{I}$ on $\omega$, let $\mathcal{K}_\mathscr{I}$ denote the $\sigma$-ideal generated by sets of the form $\prod_{n<\omega}I_n$ where $I_n\in\mathscr{I}$ for each $n<\omega$. For example, when $\mathscr{I}=\mathrm{Fin}$ is the finite ideal, $\mathcal{K}_\mathscr{I}=\mathcal{K}$ is the $\sigma$-ideal generated by compact sets in $\omega^\omega$. The speaker will talk about several results concerning the cardinal invariants of $\mathcal{K}_\mathscr{I}$ for various ideals $\mathscr{I}$. This is a joint work with Aleksander Cieślak, Takehiko Gappo and Arturo Martínez-Celis.

For a class $\mathcal{P}$ of posets, the super-$C^{(\infty)}$-$\mathcal{P}$-Laver generic Large Cardinal Axioms for extendibility (super-$C^{(\infty)}$-$\mathcal{P}$-LgLCA for extendible, for short) is the axiom scheme asserting that $\kappa:=\kappa_{\mathfrak{refl}}\ (=\max\{2^{\aleph_0},\aleph_2\})$ satisfies: $\newcommand{\Elembed}[4]{#1:#2\stackrel{\prec\hspace{0.8ex}}{\rightarrow}_{#4}#3}$ $\newcommand{\utilde}[1]{\dot{#1}}$ $\newcommand{\forces}[2]{⫦_{\,#1\,}{“}\,#2\,\textrm{”}}$ $\newcommand{\cardof}[1]{|#1|}$

for any $n\in\mathbb{N}$, $\lambda_0>\kappa$, and $\mathbb{P}\in\mathcal{P}$, there are $\lambda\geq\lambda_0$ and a $\mathbb{P}$-name $\utilde{\mathbb{Q}}$ such that $V_\lambda\prec_{\Sigma_n}\mathsf{V}$, $\forces{\mathbb{P}}{\utilde{\mathbb{Q}}\in\mathcal{P}}$ and for any $(\mathsf{V},\mathbb{P}\ast\utilde{\mathbb{Q}})$-generic $\mathbb{H}$, there are $j$, $M\subseteq\mathsf{V}[\mathbb{H}]$ such that ${V_{j(\lambda)}}^{\mathsf{V}[\mathbb{H}]}\prec_{\Sigma_n}\mathsf{V}[\mathbb{H}]$, $\Elembed{j}{\mathsf{V}}{M}{\kappa}$, $j(\kappa)>\lambda$, ($M$ is transitive), $\mathbb{P},\mathbb{P}\ast\utilde{\mathbb{Q}}, \mathbb{H}, {V_{j(\lambda)}}^{\mathsf{V}[\mathbb{H}]}\in M$, and $\cardof{RO(\mathbb{P}\ast\utilde{\mathbb{Q}})}\leq j(\kappa)$.

Super-$C^{(\infty)}$-LgLCAs for extendible imply practically all known (consistent) instances of Reflection Principles, Resurrection Principles, Maximality Principles, and Absoluteness Theorems. In particular,

Theorem 1. Super-$C^{(\infty)}$-$\mathcal{P}$-LgLCA for extendible implies $\mathsf{MP}(\mathcal{P},\mathcal{H}(\kappa_{\mathfrak{refl}}))$.

Note that $\mathsf{MP}(\mathcal{P},\mathcal{H}(\kappa_{\mathfrak{refl}}))$ is also an axiom scheme.
Super-$C^{(\infty)}$-LgLCAs for extendible have relatively low consistency strength:

Theorem 2. For a transfinitely iterable class $\mathcal{P}$ of posets, an almost huge cardinal implies the existence of a model of $\mathsf{ZFC}$ satisfying super-$C^{(\infty)}$-$\mathcal{P}$-LgLCA for extendible.

Theorem 3. (1) For the class of $\mathcal{P}$ all stationary preserving posets, $\mathcal{P}$-LgLCA for supercompact implies $\mathsf{MM}^{++}$ but $\mathsf{MM}^{++}$ does not imply $\mathcal{P}$-LgLCA for supercompact.
(2) For transfinitely iterable $\Sigma_2$-definable class $\mathcal{P}$ of posets, $\mathcal{P}$-LgLCA for supercompact does not imply $\mathcal{P}$-LgLCA for extendible.

In my talk, I shall give a proof of Theorem 2. I will also sketch proof(s) of Theorem 1 and/or 3 if there is still enough time left.

[1] Sakaé Fuchino, Extendible cardinals, and Laver-generic large cardinal axioms for extendibility, extended version of a note to appear in RIMS Kôkyûroku.

In the talk we will present a dichotomy statement concerning a class of transitive lists which at the level of $ \aleph_1$ is a consequence of Martin's Axiom and in fact follows both from $ \mathscr{K}'_2$ and from Martin's Axiom for Y-c.c. posets. At the level of $ \aleph_2$, the consistency of the dichotomy with CH holds assuming the existence of a weakly compact cardinal. We show that the dichotomy at the level of $ \lambda^+$ has an impact on the structure of natural transitive objects. For example, we prove that it implies: every $ \lambda^+$-Aronszajn tree is special, every $ \lambda^+$-tower in $ (\mathcal P(\lambda),\subseteq^*)$ is Hausdorff, the nonexistence of $\lambda^+$-Souslin lower semi-lattices, the nonexistence of certain strongly unbounded colorings and the nonexistence of Todorcevic $(\lambda^+,\lambda^+)$-gaps in $\mathcal P(\lambda)$. Joint work with Borisa Kuzeljevic and Stevo Todorcevic. p

This talk is purely expository and educational.

We examine the Tarski's undefinability of truth theorem in the set theoretic framework, and show how we can deduce from it, that the statement $V_{\underline{\kappa}}\prec\mathsf{V}$ is not formalizable as a formula with a free variable $\underline{\kappa}$ in the (first-order) language of ${\sf ZFC}$.
We shall also check that (in spite of the undefinability of truth theorem) $V_{\underline{\kappa}}\prec_{\Sigma_n}\mathsf{V}$ is formalizable as a $\Pi_n$-formula and $j:\mathsf{V}\stackrel{\prec\hspace{0.8ex}}{\rightarrow}_{\underline{\kappa}}M$ can be formalizable in ${\sf ZFC}$ (for definable $M$ and $j$) or in ${\sf NBGC}$ (even with the second order variables $M$ and $j$).

The well-known Isomorphism Theorem for measures states that any Borel probability measure space (whose points have measure zero) is isomorphic with $[0,1]$ (with the Lebersgue measure on the Borel $\sigma$-algebra). A similar result in the context of category is also known: for any perfect Polish space, there is a Borel isomorphism with $[0,1]$ respecting the Borel sets.

For a Borel measure $\mu$ on a topological space $X$, we denote by $\mathcal{E}(\mu)$ the ideal generated by the $F_\sigma$ measure zero sets. We prove the following Isomorphism theorem: if $\mu$ is a probability measure such that points have measure zero, then there is a Borel isomoprhism $f\colon X\to [0,1]$ respecting measure and the ideal $\mathcal{E}$. In the case that the open non-empty sets have positive measure, we obtain in addition that $f$ respects the meager ideal. This theorem is also valid when $\mu$ is a $\sigma$-finite measure such that points have measure zero and $\mu(X)=\infty$, but replacing $[0,1]$ by $\mathbb{R}$.

This is part of a joint work in progress with Andrés Uribe-Zapata. We make use of the theory of probability trees developed with the same coauthor (https://arxiv.org/abs/2501.07023).

Recently, many cardinal invariants have been defined using the notion of arithmetic density, particularly to create variants of the splitting or reaping number (see [1]), or to study the minimal number of permutations required to alter the density of infinite-coinfinite sets (see [2]). In this talk, we will present some relations between these cardinals and known cardinal characteristics.

References:
[1] Jörg Brendle, Lorenz J. Halbeisen, Lukas Daniel Klausner, Marc Lischka, and Saharon Shelah, Halfway new cardinal characteristics, Annals of Pure and Applied Logic 174 (2023), no. 9, 103303.
[2] Christina Brech, Jörg Brendle, and Márcio Telles, Density cardinals, 2024.
[3] Barnabás Farkas, Lukas Daniel Klausner, and Marc Lischka, More on halfway new cardinal characteristics, The Journal of Symbolic Logic (2023), 1–16.

http://www2.kobe-u.ac.jp/~fuchino/kobe-set-theory-seminar/files/Catalina_Talk_Abstract.pdf

The set theorist is familiar with the idea that filters are generalised points. Topos theorists push this question further by asking: what is a generalised space?

This talk will focus on one such answer: LT-Topologies (“Lawvere-Tierney Topologies”). In the 1980s, Hyland introduced the Effective Topos, and showed that the Turing Degrees embed into its poset of LT-Topologies. Progress was slow until 2013, where Lee-van Oosten showed that this topological framework reveals a meaningful interaction between combinatorics and computable complexity. In ongoing work in progress with T. Kihara, we push this idea much further, establishing a surprising connection: the partial order on LT-topologies restricted to filters on N is strictly refined by the Rudin-Keisler and Katetov Order. This means, for instance, we have a framework for using Turing complexity to probe the complexity of filters on N (in fact, any upper set in N). Depending on time, we plan to discuss the relative coarseness of this LT-order, as well as our original motivation of connecting the LT-topologies to Keisler’s Order on first-order theories.

The covering number of a family of sets of reals has been studied extensively with a focus on the null ideal and the meager ideal in set theory of reals. In this talk, we consider covering numbers of families, which are not ideals, of sets avoiding sets of a particular form. In particular, I will show that the covering number of the family of sets that avoid arithmetic progressions of length 3 in $\mathbb{R}$ is $\aleph_0$. I will also mention that the axiom of choice cannot be dropped for the proof.

Although Stability Theory for First Order logic has been widely developed in a relatively "set-theory free" way (for some deep reasons), the newer developments of Stability Theory outside that context (in Abstract Elementary Classes, but also in some Infinitary Logics) have had quite interesting interactions with set theory (in the form of dichotomies between diamond-like situations vs forcing axioms, in the form of use of large cardinal hypotheses to improve the model-theoretic behavior, or in the form of very intricate, and interesting, combinatorial principles).

I will describe some of these interactions.

For classical characterized subgroups of the circle (obtained for the ideal Fin), Eggleston's dichotomy states that the characterized subgroup corresponding to an arithmetic sequence is countable if the sequence is b-bounded and of cardinality c if the sequence is otherwise. For natural density ideal the dichotomy breaks down. We try to find out a specific property of ideals which seems to play a prominent role in enforcing the dichotomy.

Based on the work of Shelah (2000) and Kellner, Shelah, and Tănasie (2019), we developed a general forcing (and iteration) theory with finitely additive measures. This not only simplifies the proof of the results in their work but leads to new applications in the context of the real line. In this talk, I present this general theory, some of its properties, and explain how it generalizes iterations with ultrafilters, like the one of Goldstern, Mejía, and Shelah (2016), and those present in the work of Yamazoe (2024). This is a joint work with Andres Uribe-Zapata and Miguel Cardona, available at https://arxiv.org/abs/2406.09978

I will present some results on strong combinatorial properties of MAD families, like (strong) tightness, the Shelah-Steprans property and the raving property. these properties are related to indestructibility of MAD families under some forcing notions. this is joint work with Osvaldo Guzmán, Michael Hrušák, and Dilip Raghavan.

In this talk, I will review the characterization of Maximality Principles in terms of Recurrence Axioms, and prove the implication of Maximality Principles from the existence of a strong variation of Laver-generic large cardinal (a super-$C^{(\infty)}$ $\mathcal{P}$-Laver gen. ultrahuge cardinal). The presentation is going to use both the blackboads and the slides: RIMS2024-set-theory-fuchino-pf.pdf

Recently, in [UZ23], a formalization of the concept of a probability tree was introduced with the aim of proving certain results in forcing theory using finitely additive measures. In particular, these trees were used to show that random forcing is $\sigma$-$\mathrm{FAM}$-linked (see [MU24]), a linkedness notion that emerged from studying the cofinality of $\mathrm{cov}(\mathcal{N})$ (see [Sh00] and [CMU24]). Probability trees also played a fundamental role in the limit step of the general theory of iterated forcing with finitely additive measures developed in [CMU24]. In this talk, we will present the formalization of the notion of probability tree, exploring some characterizations and basic properties. We will also prove that there exists a connection between probability trees of infinite height and the real line. Finally, we will show some applications of probability trees in the study of cardinal invariants. This is joint work with Diego A. Mejía and Carlos M. Parra-Londoño.

References:
[CMUZ24] Miguel~A. Cardona, Diego~A. Mejía, and Andrés~F. Uribe-Zapata. A general theory of iterated forcing using finitely additive measures. Preprint, arXiv:2406.09978, 2024.
[She00] Saharon Shelah. Covering of the null ideal may have countable cofinality. Fund. Math., 166(1-2):109--136, 2000.
[MU24] Diego A. Mejía and Andrés F. Uribe-Zapata. The measure algebra adding $\theta$-many random reals is $\theta$-FAM-linked. Topology and its Applications. To appear, arXiv:2312.13443, 2024.
[UZ23] Andrés F. 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://sites.google.com/view/andres-uribe-afuz/publications.

Given a MAD family $A$, we consider the ideal $I_A$ it generates and explore the behavior of the cardinal invariants associated with the quotient algebra $P(\omega)/I_A$. In this talk, I will shed light on this topic, demonstrating the relationships of these cardinal invariants with some classical ones. Subsequently, I will discuss the construction of forcing-indestructible partitions within this algebra. This talk is based on ongoing joint work with Michael Hrušák.

In work with Vera Fischer, Sy Friedman, and Diana Montoya (2018), we introduced the notion of coherent systems of FS iterations to force constellations of Cichoń's diagram. Afterward, I developed a detailed theory of this iteration technique and presented further models including singular values (2019), resulting in recent applications to strong measure zero sets (with M. Cardona) and slalom numbers (with M. Cardona, V. Gavalová, M. Repický and J. Šupina).
In a series of two talks, I will present this iteration theory and the mentioned applications, dedicating part I to the iteration theory and (hopefully) its applications to Cichoń's diagram.

The paper [1] is a streamlined framework on slalom numbers, i.e., cardinal invariants employing (generalized) slaloms in their definitions. It includes well-known localization and anti-localization cardinals as particular cases. We explain the concept of streamlined exposition in [1] and connections to selection principles, highlight the main results, and pose open questions.

[1] Cardona M.A., Gavalová V., Mejía D.A., Repický M., and Šupina J., Slalom numbers, submitted, arXiv:2406.19901.

An infinite series of real numbers is conditionally convergent if it converges, but the sums of the positive and of the negative terms are both divergent. How many infinite subsets of the naturals are necessary such that every conditionally convergent series has a subseries given by one of our infinite subsets that is divergent? The answer to this question is known as the subseries number ß, and was isolated as a cardinal characteristic of the continuum by Brendle, Brian and Hamkins.
In this talk we will consider several variants of the subseries number, where we restrict our attention to infinite subsets of the naturals that are also coinfinite. Due to this change, we may consider subseries produced by infinite coinfinite subsets of the naturals that remain convergent, producing various closely related cardinal characteristics of the continuum.

Let $\mathcal{E}$ denote the $\sigma$-ideal generated by closed null sets on $\mathbb{R}$. We show that the uniformity and the covering of $\mathcal{E}$ can be added to Cichoń's maximum with distinct values, more specifically, it is consistent that $\aleph_1<\mathrm{add}(\mathcal{N})<\mathrm{cov}(\mathcal{N})<\mathfrak{b}<\mathrm{non}(\mathcal{E})<\mathrm{non}(\mathcal{M})<\mathrm{cov}(\mathcal{M})<\mathrm{cov}(\mathcal{E})<\mathfrak{d}<\mathrm{non}(\mathcal{N})<\mathrm{cof}(\mathcal{N})<2^{\aleph_0}$ holds. We use the method ``ultrafilter-limits for intervals'', which was introduced by Mejía in his RIMS Kôkyûroku paper last year. The speaker will present how the method works and why the separation is forced.

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. This is a joint work with Jörg Brendle and Michael Hrušák.

We present slalom numbers with ideals on the natural numbers as parameters and show the effect of Cohen reals on them. As applications, we show how they behave in models where many classical cardinal characteristics of the continuum are separated. This is a joint work with Miguel Cardona, Viera Gavalová, Jaroslav Šupina, and Miroslav Repický.

Solovay proved the following lemma: for every ccc ideal, every countable model $N$ and every Solovay set $A$ over $N$, there is a Borel set $B$ coded in $N$ such that for every $I$-generic real $x$ over $N$, it holds that $x \in B \iff x \in S$. In this talk, I will show that the converse of Solovay’s lemma in a sense. That is, for every nowhere ccc ideal, every countable model $N$, there is a Solovay set $A$ over $N$ such that for every Borel set $B$ coded in $N$, there is an $I$-generic real $x$ over $N$ such that $x \in B \triangle S$.

Many classical cardinal invariants of the continuum are defined in terms of the quotient $P(\omega) \,/\, fin$, and they can be naturally redefined for quotients of the form $P(\omega) \,/\, I$ where $I$ is an ideal on $\omega$. A fair amount of research on such cardinals has been done when $I$ is Borel, but very little is known for interesting non-Borel ideals, e.g. for the ideal $I(A)$ generated by some mad family $A$ on omega. In my talk, i will discuss the splitting number $s(A)$ of $P(\omega) \,/\, I(A)$. The main result says that consistently there is a mad family $A$ such that $s(A) < s$. This is joint work with Hiroaki Minami.

It is well known that Laver forcing preserves the Lebesgue outer measure. In this talk, the speaker will generalize the proof and show that Laver forcing preserves that a set of real numbers has positive Hausdorff measure.

The speaker will also touch on additivity of Hausdorff measures as a related topic.

In this talk I shall give proofs of the following theorems:

Theorem A. If $\kappa$ is ccc-generically supercompact and $X$ is a locally countably compact non-metrizable space then there is a non-metrizable subspace $Y$ of $X$ of cardinality $<\kappa$.
Theorem B. Assume that $(\mathcal{P},\mathcal{H}(\kappa))$-RcA holds. Then for any $\mathbb{P}\in\mathcal{P}$ and for any $\Sigma_1$-formula $\varphi$ and $a\in\mathcal{H}(\kappa)$, if $\Vdash_{\mathbb{P}}\varphi(a)$ then $\varphi(a)$ holds.
Theorem C. Assume that $(\mathcal{P},\mathcal{H}(\aleph_1))$-RcA$^+$ holds. Then for any $\mathbb{P}\in\mathcal{P}$ s.t. $\Vdash_{\mathbb{P}}$BFA$_{\aleph_1}(\mathcal{P})$, we have $\mathcal{H}(\aleph_1)^{\mathsf{V}}\prec_{\Sigma_2}\mathcal{H}({\aleph_1})^{\mathsf{V}^{\mathbb{P}}}$.

In Theorems A and B, $\kappa$ can be $2^{\aleph_0}$. In Theorem A, the ccc-generically supercompactness implies that $\kappa$ is at least weakly inaccessible. $(\mathcal{P},\mathcal{H}(\kappa))$-RcA$^+$ is known to be equivalent with the Maximality Principle MP$(\mathcal{P},\mathcal{H}(\kappa))$.

Theorem A answers a question asked in my seminar talk on May 1. Theorems B, C are obtained in connection with recent zoom discussions with T. Gappo of TU Vienna. Note that by Bagaria's Absoluteness Theorem, Theorem B implies that BFA$_{<\kappa}(\mathcal{P})$ follows from $(\mathcal{P},\mathcal{H}(\kappa))$-RcA. For $\mathcal{P}=$ ccc posets this has been proved by Leibman.

In the second talk I shall prove a theorem which improves both Theorem B and Theorem C.

Chu transforms satisfying a density condition have recently emerged as a powerful tool to compare abstract logics. In this talk, we will introduce and analyse inconsistency-flow formulae, which are infinitary formulae in a two-sorted language with an added predicate to express inconsistency. We establish a preservation theorem for inconsistency-flow formulae under dense Chu transforms. As a result, we characterize exactly what is preserved by the sublogic relation between abstract logics. Finally, we discuss further applications to topological spaces and graph colourings. This is joint work in progress with Mirna Džamonja.

Hausdorff measures are measures that can be used to more finely measure Lebesgue null sets and each of them has a parameter called a gauge function. Hausdorff measures create a hierarchy between the ideal of strong measure zero sets and the ideal of Lebesgue null sets. This property is the same as that of the Yorioka ideals introduced by Teruyuki Yorioka.

Shizuo Kamo and Noboru Osuga proved that many covering numbers of the Yorioka ideals can be separated. Moreover Lukas Daniel Klausner and Diego Mejia did the same thing for uniformity numbers. The speaker used this to show that many covering numbers and many uniformity numbers of Hausdorff measure zero ideals can also be separated respectively.

On the other hand, not much is known about additivity numbers and cofinality numbers of Hausdorff measure zero ideals. The only known fact is the one David Fremlin showed that states for every $0 < r < 1$, the additivity (resp. cofinality) of Hausdorff measure zero ideals with respect to the gauge function $x \mapsto x^r$ is equal to the additivity (resp. cofinality) of Lebesgue null ideals. In this talk, generalizing Fremlin’ result, the speaker will show that additivity numbers of Hausdorff measure zero ideals are greater than or equal to $\mathrm{add}(\mathsf{null})$ and cofinality numbers of Hausdorff measure zero ideals are less than or equal to $\mathrm{cof}(\mathsf{null})$ under a mild assumption on gauge functions.

Hamburger's Problem (named after the topologist P. Hamburger) asks whether the following statement is consist:

(HH): If $X$ is a first countable non-metrizable topological space, then there is a non-metrizable subspace $Y$ of $X$ of cardinality $<\aleph_2$.

Hamburger's Problem has been one of the test problems in search of set-theoretic reflection principles.
Though this problem is still open, we give a positive solution of a variation of this problem asking the consistency of the following statement which we will call Generic Hamburger's Problem.

(GHH)$_{\mathcal{P},<\kappa}$: If $X$ is a first countable $\mathcal{P}$-indestructibly non-metrizable topological space, then there is a $\mathcal{P}$-indestructibly non-metrizable subspace $Y$ of $X$ of cardinality $<\kappa$.

Here, $\mathcal{P}$ is a class of posets and $\kappa$ is typically the cardinal $\aleph_2$. A topological space $X$ is said to be $\mathcal{P}$-indestructibly non-metrizable if $\Vdash_P$“$X$ is non-metrizable” for all $P\in\mathcal{P}$.

Theorem. Suppose $\kappa$ is $\mathcal{P}$-generic supercompact and the Recurrence Axiom $(\mathcal{P},\kappa)$-RcA holds. Then (GHH)$_{\mathcal{P},<\kappa}$ holds.

In the talks, we will prove the theorem and show that the combination of the assumptions of the theorem is consistent under a very large large cardinal for some natural classes $\mathcal{P}$ of posets with $\kappa=\kappa_{refl}:=\max\ssetof{2^{\aleph_0},\aleph_2}$.

It is easy to see that given an infinite-coinfinite subset $A$ of the natural numbers $\mathbb{N}$ and any real $r$ in $[0,1]$ there is permutation $\pi$ of $\mathbb{N}$ such that the density of $\pi[A]$ is $r$. There is also a permutation such that $\pi[A]$ does not have asymptotic density. This is somewhat analogous to Riemann's rearrangement theorem. In analogy to the rearrangement number $\mathfrak{rr}$ we define the asymptotic density number $\mathfrak{dd}$ as the smallest cardinality of a family $P$ of permutations such that for every infinite-coinfinite $A$ with asymptotic density there is $\pi$ in $P$ such that $d(\pi[A])$ is distinct from $d(A)$, i.e. either $\pi[A]$ does not have asymptotic density or its asymptotic density differs from that of $A$. We present a number of results on $\mathfrak{dd}$ and its relatives. This is ongoing joint work with Christina Brech and Márcio Telles.

Cruz Chapital, Goto and Hayashi investigated game-theoretic variants of cardinal invariants and proved that $\mathfrak{s}_{\mathrm{game}^{*}}^\mathrm{I}$, which is a number related to splitting families, can be separated from the standard splitting number $\mathfrak{s}$. In my talk I will introduce a new game-theoretic number $\mathfrak{s}_{\mathrm{game}^{**}}^\mathrm{I}$ by slightly changing the rule of the game and show that the slight change is crucial in the sense that the new number can be separated from $\mathfrak{s}_{\mathrm{game}^{*}}^\mathrm{I}$.

A graph $G=(V,E)$ is highly connected if it remains connected after deleting fewer than $|V|$-many vertices. We shall be reporting on joint work with Bergfalk-Shelah and Shelah-Zhang concerning the natural weakening of the Ramsey type theorem on colouring edges of a complete graph requiring that there be a monochromatic highly connected subgraph. We focus on the case of countably many colours.

In 1964, Keisler introduced $\kappa$-good ultrafilters, which can be characterized as those ultrafilters which produce $\kappa$-saturated ultrapowers. The problem of finding an analogous characterization for ultrafilters on Boolean algebras has been considered by Mansfield (1971), Benda (1974), and Balcar and Franěk (1982), who proposed and compared different notions of “goodness” for such ultrafilters. In the first part of my talk, I shall outline the different definitions introduced in the literature and show that they are in fact all equivalent, thus providing a complete characterization of those ultrafilters which produce $\kappa$-saturated Boolean ultrapowers. In the second part of the talk, I shall present a joint work with Matteo Viale, which is available at arXiv:2310.11691. The aim of our project is to study the universality properties of forcing. More precisely, we shall prove that, for many interesting signatures, every model of the universal theory of an initial segment of the universe can be embedded into a model constructed by forcing. To achieve this goal, we build good ultrafilters on forcing notions such as the Lévy collapsing algebra and Woodin's stationary tower.

In [1] were introduced, among many other new cardinal invariants, two variants of the splitting number: the cofinally bisecting number and the bisecting number. In this talk, we will present some consistency results about these cardinals obtained using the Hechler and dual Hechler forcing. References:

[1] Brendle, Jörg and Halbeisen, Lorenz and Klausner, Lukas Daniel and Lischka, Marc and Shelah, Saharon. (2023). Halfway New Cardinal Characteristics. Annals of Pure and Applied Logic 174, no. 9, 103303.
[2] Valderrama, David. (2024). Cardinal invariants related to density. Available at https://arxiv.org/abs/2401.09649

I will report recent progress in the study of the cardinal characteristics associated with the strong measure zero ideal and with some other additive ideals on the reals, particularly the null-additive and meager-additive ideals. The results are joint with Jörg Brendle and Miguel Cardona.

In my talk, I want to present results from an ongoing project with Omer Ben-Neria (Jerusalem) that studies definable closed unbounded subsets of uncountable cardinals, focusing on large cardinals and singular cardinals. Our results show that, for certain collections of formulas and classes of parameters, important combinatorial properties of the given cardinal are reflected in the structural features of the collection of all closed unbounded sets definable in the given way. For example, the assumption that the first limit cardinal $\aleph_\omega$ is Jónsson can be shown to have various non-trivial implications on the collection of closed unbounded subsets of $\aleph_\omega$ that can be defined by $\Sigma_1$-formulas with parameters in $H(\aleph_\omega)\cup\{\aleph_\omega\}$, and these consequences can then be used to restrict the class of possible models of set theory in which $\aleph_\omega$ is Jónsson.

For a long time it had been an open problem whether all the ten cardinal characteristics in Cichoń's diagram can be separated simultaneously. Recently, this problem was positively solved by constructing such a simultaneous separation model, which is called Cichoń's maximum. The aim of our study is to add another cardinal characteristic to Cichoń's maximum and for this purpose we focus on evasion number $\mathfrak{e}$. We showed that $\mathfrak{e}$ can be added to Cichoń's maximum. In other words, we can construct a model where all the ten cardinal characteristics in Cichoń's diagram and $\mathfrak{e}$ are totally separated in the following order $\aleph_1<\mathrm{add}(\mathcal{N})<\mathrm{cov}(\mathcal{N})<\mathfrak{b}<\mathfrak{e}<\mathrm{non}(\mathcal{M})<\mathrm{cov}(\mathcal{M})<\mathfrak{d}<\mathrm{non}(\mathcal{N})<\mathrm{cof}(\mathcal{N})<2^{\aleph_0}$.

Recall that a forcing is ssp if it preserves stationarity of stationary subsets of $\omega_1$, and it is proper if it preserves stationarity of all ground-model stationary subsets of $[\kappa]^\omega$, for every uncountable $\kappa$.

Strong properness is a strengthening of the notion of properness that was introduced by Mitchell and naturally comes up in the study of proper forcings constructed using side conditions.

These forcings are characterized by the existence of many conditions that are in a particular strong way generic for a model $M$ and this makes all such strongly proper forcings have a distinctive behaviour, as well as certain nice properties that fail for general proper forcings.

I will discuss joint work with my thesis supervisor Boban Velickovic in which we define a related strengthening of the notion of ssp-ness, that gives rise to a class of so-called strongly ssp forcings, which extends the class of strongly proper forcings. We will discuss examples and general properties of these strongly ssp forcings.

Generalized indiscernibles can be built in first-order theories by generalizing the combinatorial Ramsey’s Theorem to classes with more structure, which is an active area of study. Trying to do the same for infinitely theories (in the guise of Abstract Elementary Classes) requires generalizing the Erdos-Rado Theorem instead. We discuss various results about generalizations of the Erdos-Rado Theorem and techniques (including large cardinals and forcing) to build generalized indiscernible. 　

We study the effect of iterated forcing on determinacy principles. The guiding question is: which countable support iterations of proper forcings preserve projective determinacy? Building on work of Judah, Shelah and Goldstern on iterations of Suslin forcings, we study iteration theorems for new variants of properness to show that projective determinacy is highly robust. For instance, it is preserved, level by level, by countable support iterations of virtually all known simply definable proper forcings that add reals. A sample application of this technique shows that the combination of the Borel conjecture with analytic determinacy is consistent, assuming the latter is consistent. This is joint work with Jonathan Schilhan and Johannes Schürz. 　

An ideal $I$ on $\omega_1$ is said to be $\omega_1$-dense if the partial order on $(P(\omega_1)/I)^+$ induced by inclusion has a dense subset of size $\omega_1$. Woodin asked whether large cardinals imply that there is a stationary set preserving forcing which forces “$NS_{\omega_1}$ is $\omega_1$-dense”. We answer this positively by showing that there is a forcing axiom $QM$ which implies that the nonstationary ideal on $\omega_1$ is $\omega_1$-dense and then force $QM$ from a supercompact limit of supercompact cardinals. Our strategy is motivated by the implication $MM^{++}\Rightarrow (*)$ proven by Asperó-Schindler. In fact, we have that $QM$ implies the version of $(*)$ for Woodin's $ℚ_{max}$.

A (definable) class $\cal P$ of posets is said to be iterable if ① $\cal P$ is closed with respect to forcing equivalence (i.e. if $ℙ\in\cal P$ and $ℙ\sim ℙ'$ then $ℙ'\in\cal P$ ), ② closed wrt restriction (i.e. if $ℙ\in\cal P$ then $ℙ\restriction \mbox{𝕡}\in\cal P$ for any $\mbox{𝕡}\in ℙ$ ), and, ③ for any $ℙ\in\cal P$ and $ℙ$-name $\dot{ℚ}$, $⫦_{ℙ}“{\dot{ℚ}\in\cal P}\,”$ implies $ℙ\ast\dot{ℚ}\in\cal P$.

For an iterable class $\cal P$ of posets, a cardinal $\kappa$ is said to be $\cal P$-Laver-generically supercompact if, for any $\lambda\geq\kappa$ and $ℙ\in\cal P$, there is a $ℙ$-name $\dot{ℚ}$ with $⫦_{ℙ}“{\dot{ℚ}\in\cal P}\,”$ such that, for $({\sf V},ℙ\ast\dot{ℚ})$-generic $ℍ$, there are $j$, $M\subseteq{\sf V}[ℍ]$ with

(a)　$j:{\sf V}\prec_\kappa{M}$,　　(b)　$j(\kappa)>\lambda$, and　　 (c)　 $ℙ$, $ℙ\ast\dot{ℚ}$, $ℍ$, $j''{\lambda}\in M$.

$\kappa$ is said to be tightly $\cal P$-Laver-generically supercompact if additionally (d)　$|ℙ\ast\dot{ℚ}|\leq j(\kappa)$　holds.

Similarly, we can also define (tightly) $\cal P$-Laver-generic versions of super almost-huge, superhuge, and ultrahuge cardinals.

In [ II ] it is shown that the existence of $\cal P$-Laver-gen. supercompact cardinal (tightly $\cal P$-Laver gen. superhuge in the case $\cal P$ $=$ ccc posets) for a reasonable $\cal P$ highlights the situations with the continuum being ① $\aleph_1$, ② $\aleph_2$ or ③ very large.

In particular with $\cal P$ being the class of all ① σ-closed posets, ② semi-proper posets, or ③ ccc-posets, the existence of $\cal P$-Laver-gen. supercompact cardinal (or tightly $\cal P$-Laver gen. superhuge in the case $\cal P$ $=$ ccc posets) implies a double-plused version of forcing axiom for the respective $\cal P$ and strong reflection properties down to less than $\kappa_{refl}:=\max\{\aleph_2,2^{\aleph_0}\}$ compatible with the forcing axiom.

In this talk we prove that the existence of tightly $\cal P$-Laver-generically superhuge cardinal implies the boldface version of Resurrection Axiom ([Hamkins-Johnstone 1], [Hamkins-Johnstone 2] ) for $\cal P$ over ${\cal H}(\kappa_{refl})$.

We furhter show that the existence of tightly $\cal P$-Laver-generically ultrahuge cardinal implies the Unbounded Resurrection Axiom of Tsaprounis ([Tsaprounis 1]) for $\cal P$ and strong version of local maximality principle ((slightly?) stronger than the one mentioned in [Minden]).

References.

[ I ]　 S.F., A. Ottenbreit Maschio Rodrigues, and H. Sakai, Strong downward Löwenheim-Skolem theorems for stationary logics, Archive for Mathematical Logic, Vol.60, 1-2, (2021), 17–47. https://fuchino.ddo.jp/papers/SDLS-x.pdf
[ II ]　 -----, Strong downward Löwenheim-Skolem theorems for stationary logics, II — reflection down to the continuum, Archive for Mathematical Logic, Vol.60, 3-4, (2021), 495–523. https://fuchino.ddo.jp/papers/SDLS-II-x.pdf
[Hamkins-Johnstone 1]　 Joel David Hamkins, and Thomas A.Johnstone, Resurrection axioms and uplifting cardinals, Archive for Mathematical Logic, Vol.53, Iss.3-4, (2014), 463-485.
[Hamkins-Johnstone 2]　 -----, Strongly uplifting cardinals and the boldface resurrection axioms, Archive for Mathematical Logic volume 56, (2017), 1115-1133.
[Minden]　 Kaethe Minden, Combining resurrection and maximality, The Journal of Symbolic Logic, Vol. 86, No. 1, (2021), 397–414.
[Tsaprounis 1]　 Konstantinos Tsaprounis, On resurrection axioms, The Journal of Symbolic Logic, Vol.80, No.2, (2015), 587–608.
[Tsaprounis 2]　 -----, Ultrahuge cardinals, Mathematical Logic Quarterly, Vol.62, No.1-2, (2016), 1–2.

The relationship between the Axiom of Determinacy and supercompactness of $\omega_1$ has been studied by many people. In 1990's, Woodin showed that assuming the existence of a proper class of Woodin limits of Woodin cardinals, a generalized Chang model satisfies "${\rm AD}_{ℝ}$ + $\omega_1$ is supercompact."
　Recently he also showed that the regularity of $\Theta$ in the model follows from determinacy of a long game of length $\omega_1$, which is, however, still unknown to be consistent.
　Based on these results, we conjecture that the following two theories are equiconsistent:

(1) ZFC + there is a Woodin limit of Woodin cardinals.
(2) ZF + ${\rm AD}_{ℝ}$ + $\Theta$ is regular + $\omega_1$ is supercompact.

　Toward this conjecture, we construct a new model of the Axiom of Determinacy, called the Chang model over the derived model with supercompact measures.
　We then prove that it is consistent relative to a Woodin limit of Woodin cardinals that our model satisfies "${\rm AD}_{ℝ}$ + $\Theta$ is regular + $\omega_1$ is $\lt$ $\delta$-supercompact for some regular cardinal $\delta$ $\gt$ $\Theta$."
　This is joint work with Sandra Müller and Grigor Sargsyan. 　

Given a notion of forcing P, let Part(P) be the directed set of all maximal antichains of P equipped with the refinement relation. My talk will focus on the combinatorial properties of Part(P) from the point of view of Tukey reducibility. Furthermore, I shall discuss some cardinal invariants associated to Part(P), giving a complete characterization in the case of Cohen forcing and random forcing. This is a joint work in progress with Jörg Brendle. 　

The $\in$-collapse of $H(\kappa)$ (for an uncountable regular cardinal $\kappa$) is the forcing notion which consists of the finite $\in$-chains of countable elementary submodels of $H(\kappa)$, ordered by the extension of chains. The $\in$-collapse preserves Suslin trees, is Y-proper, and collapse $\kappa$ to $\aleph_1$.

In the talk, we observe two preservation results of the $\in$-collapse; one is on non-meager sets of reals, and the other is on mad families. 　

We generalize results of [T. Yorioka (2002)] and [M. Cardona (2022)] about the cofinality of the ideal of strong measure zero sets, and present some applications. This is a joint work with J. Brendle and M. Cardona. 　

Let $\mathcal{E}$ be $\sigma$-ideal of the closed measure zero sets of reals, we prove that, for $\mathcal{E}$, their associated cardinal characteristics (i.e.\ additivity, covering, uniformity and cofinality) are pairwise different. In addition, we mention some open problems related to cardinal characteristics of $\mathcal{E}$. 　

In the case of $\omega$, we have obtained many consequences on cardinal characteristics. On singular cardinals, however, the area has not been much explored yet. In this talk, we will focus on the dominating number and the unbounded number at each singular cardinal. We will prove the following three theorems.

(1) For singular $\lambda$, $\mathfrak{b}_\lambda = \mathrm{cf}(\lambda)^{+}$
(2) If $\lambda$ is a strong limit singular cardinal, then $\mathfrak{d}_\lambda = 2^{\lambda}$.
(3) $\mathfrak{d}_{\aleph_\omega} \geq \mathrm{maxpcf}(\lbrace\aleph_n \mid n \in \omega \rbrace)$.

This is a joint work with H.Sakai. 　

We will take a look at the localisation numbers on the generalised Baire space $\kappa^\kappa$ for $\kappa$ a strongly inaccessible cardinal. In particular, we will look at slaloms limited in size by a function $h:\kappa\rightarrow\kappa$, and look at the dominating h-localisation number, which is the least number of h-slaloms that are needed to localise every element of the Baire space. We show that the choice of the function h is significant (a result from arXiv:1611.08140) and we improve upon this by showing that there can be consistently $\kappa$ many do minating h-localisation numbers of different cardinalities.

⚡date changed ⚡:　

I continue the talk on May 25 and present the proofs of sevaral theorems conserning the Löwenheim-Skolem and compactness numbers including those announced in the talk (see also [slides of the talk on May 25]). 　

This talks is a kind of continuation of my talk On Downward Löwenheim-Skolem theorems of some non first-order logics on May 18, 2022 at Tokyo model theory seminar [slides in printer version].

I will present details of a proof of the characterization of supercompact cardinals in terms of Löwenheim-Skolem theorems for second-order logic, a classical result by Menahem Magidor (1971 = 昭和46年), and discuss some variations of this theorem.

Goldstern showed that the union of an real-parametrized, increasing family of Lebesgue measure zero sets has also Lebesgue measure zero provided that the sets are uniformly $\boldsymbol{\Sigma}^1_1$. Our aim is to study how much we can drop the $\boldsymbol{\Sigma}^1_1$ assumption.

In this session, we will give a proof of Goldstern's theorem and observe that in order for Goldstern's principle for families of parameterized Lebesgue measure zero sets without the condition of uniform $\boldsymbol{\Sigma}^1_1$ definability to hold, some cardinal invariant constraints is necessary. We will also discuss a combinatorial principle which follows from Goldstern's principle for $\boldsymbol{\Sigma}^1_2$ uniformly definable families.

Goldstern showed that the union of an real-parametrized, increasing family of Lebesgue measure zero sets has also Lebesgue measure zero provided that the sets are uniformly $\boldsymbol{\Sigma}^1_1$. Our aim is to study how much we can drop the $\boldsymbol{\Sigma}^1_1$ assumption.

In this session, we will prove that Goldstern's principle for families of parameterized Lebesgue measure zero sets without the condition of uniform $\boldsymbol{\Sigma}^1_1$ definability, which called $\operatorname{GP}(\mathrm{all})$, is consistent with $\mathsf{ZFC}$. Also we prove that $\operatorname{GP}(\mathrm{all})$ holds in $\mathsf{ZF}+\mathsf{AD}+\mathsf{DC}$.