RIMS workshop on Set Theory 2025, titles and abstracts

Ultrafilters on countable sets have become of great importance in infinite combinatorics. A non-principal ultrafilter U is called a P-point if every countable subset of U has a pseudointersection in U. Ultrafilters of this special type have been extensively studied in set theory, topology, and other mathematical disciplines. The existence of P-points is independent of the axioms of ZFC, and interaction of P-ultrafilters and forcing is of great interest in set theory of the reals.

The tutorial will provide an overview of the phenomenon of destroying and preserving P-points in various generic extensions. The question of existence of P-points in some models will be explored in some detail. In particular, we may cover a streamlined proof of non-existence of P-points in Silver type models (Chodounsky, Guzman), a construction of P-ultrafilters using pathways which solves the question of existence of P-points in the random model (Dow, Guzman), and time permitting a recent method for forcing ultrafilters to be Tukey equivalent (Cancino, Zapletal).

We show that a multiset is the same as an artinian tree, where ar- tinian tree is a tree without infinite linearly ordered subsets and also that a set is the same as a rigid artinian tree. We define an addition and a multiplication on them extending those on ordinals.
This is a joint work with Diego A. Mejía.

We will present a dichotomy statement for transitive lists. At the level of $\lambda^+$, this dichotomy implies that, among other things, every $\lambda^+$-Aronszajn tree is special, that every $\lambda^+$-tower in $(\mathcal{P}(\lambda),\subseteq^*)$ is Hausdorff, and that there are no $\lambda^+$-Souslin lower semi-lattices. We will also show that at the level of $\aleph_1$ this dichotomy follows from $\mathrm{MA}(\aleph_1)$, while at the level of $\aleph_2$, assuming the existence of a weakly compact cardinal, it is consistent with $\mathrm{ZFC}+\mathrm{CH}$. This is joint work with Roy Shalev and Stevo Todorcevic.

Let us denote $SC(\omega_1, \kappa)$ iff there exists a strong chain consisting of subsets of $\omega_1$ ordered by $\kappa$. Koszmider forced $SC(\omega_1,\omega_2)$ by a ccc poset assuming Jensen’s square principle and negated $SC(\omega_1,\omega_2)$ under the Chang’s Conjecture in [K]. Recently, Aspero and Rodriguez forced $SC(\omega_1,\omega_3)$ in [AR]. Their construction involves a proper poset whose side conditions consist of two types of elementary substructures.

We present a proper forcing construction for the weaker result $SC(\omega_1,\omega_2)$ different than [M], aiming to reflect key aspects of Aspero and Rodriguez. Our present poset involves symmetric systems that consist of a single type of elementary substructures, fast functions, and bubbly paths. In addition to a well-known notion of isomorphic two elementary substructures, we introduce that of similar two finite unions of elementary substructures.

Of these, fast functions come for free—that is, they emerge naturally without additional constraints or constructions, and bubbly paths are sort of overkill with the chains of shorter length. Nevertheless, we wish to retain a hint of Aspero and Rodriguez’s approach and illustrate how the construction can be naturally extended from $SC(\omega_1,\omega_2)$ into $SC(\omega_1,\omega_3)$. However, it remains open, if we can go further, say, into $SC(\omega_1, \omega_4)$.

[AR] D. Aspero, C.G. Rodriguez, Side Conditions of Models of Two Types and High Forcing Axioms, 2024.
https://ueaeprints.uea.ac.uk/id/eprint/97643/

[K] P. Koszmider, On the Existence of Strong Chains in $P(\omega_1)/ {\rm Fin}$, J. Symbolic Logic 63 (1998), no. 3, 1055-1062.

[M] T. Miyamoto, A Strong Chain with Side Condition Method, a note, 2024.
https://nanzan-u.repo.nii.ac.jp/records/2001253

We investigate forcing axioms for posets satisfying chain conditions stronger than precaliber $\aleph_1$. As fragments of Martin’s Axiom, we study their consequences and mutual separations, using the uniformization of ladder system colorings as a main tool.

In [1], Shelah defined a family of non-principal ultrafilters on $\omega$ that seem to be, in some sense, very far from P-points. We attempt to axiomatize such ultrafilters in the following definition: We say that $\mathcal{U}$ an ultrafilter on $\omega$ is a Shelah ultrafilter if the following holds:

> For every $\alpha<\omega_1$, there is and ideal $\mathcal{I}$ isomorphic to $fin^\alpha$ such that $\mathcal{I}\subseteq\mathcal{U}^*$.
> Let $\mathcal{I}$ be an analytic ideal. If $\mathcal{I}\cap\mathcal{U}=\emptyset$, then there are $\alpha<\omega _1$ and an ideal $\mathcal{J}$ such that $\mathcal{J}$ is Kat\v{e}tov equivalent to $fin^\alpha$ and $\mathcal{I}\subseteq\mathcal{J}\subseteq\mathcal{U}^*$.

In this talk, we will construct a model where Shelah ultrafilters exist. If time permits, we’ll analyze some of their combinatorial properties.

This is joint work with Osvaldo Guzmán and David Chodounský.

[1] Shelah, S. Nice $\aleph_1$ generated non-P-points, Part I. Mathematical Logic Quarterly, 2023, vol. 69, pp 117–129.

We present a method for consistently constructing a tree whose square is sharply different from the original. Specifically, we construct an $\aleph_1$-tree $T$ whose interval topology $X_T$ is perfectly normal, but $(X_T)^2$ is not even countably metacompact.
This is joint work with Assaf Rinot and Shira Yadai.

PFA implies that the Diamond Principle at the continuum ($\Diamond_{2^{\aleph_0}}$) holds because of a fairly trivial reason. If the continuum is a weakly inaccessible cardinal $\kappa$, the Diamond Principle at the continuum may fail even if we have $2^{<\kappa}=\kappa$.

In this talk, we consider a generic version of the Laver diamond similar to the generalized Laver diamond introduced by Matteo Viale. The generic Laver diamond at $\kappa$ implies $\Diamond_\kappa$, and its stronger version at $\kappa_{refl}$ ($:=\max\{\aleph_2,2^{\aleph_0}\}$) even implies Laver generic Large cardinal axiom. We examine the consistency of generic Laver diamond and its variations at $\kappa_{refl}$.

In [S1],[S2] Steel proposed an axiom system of set-generic multiverse and showed that the class

$\mathcal{MV}^{M,\mathbb{G}}_{\mathsf{ST}}:=\{N\,: N$ is a ground of $M[\mathbb{G}\restriction\alpha]$ for some $\alpha\in \mathsf{On}^M\}$

is a modelof his axiom system. Here $M$ is a countable transitive model of $\mathsf{ZFC}$ and $\mathbb{G}$ is a $(M,Col(\omega,<\mathsf{On}^M))$-generic filter.

In this talk, we show that under the axiom (scheme) Super-$C^{(\infty)}$-Laver generic Large Cardinal Axiom for all posets and for hyperhugeness (Super-$C^{(\infty)}$-$\mathsf{LgLCAA}$ for hyperhuge, for short), the class

$\mathcal{MV}:=\{{V_\kappa}^{\overline{\mathsf{W}}}[\mathbb{G}]\,:\mathbb{G}$ is a $(\overline{\mathsf{W}},\mathbb{P})$-generic filter $\in\mathsf{V}$ for some $\mathbb{P}\in{V_\kappa}^{\overline{\mathsf{W}}} \,\}$

is also a model of the axiom system. Here, $\kappa={2^{\aleph_{\mathsf{ST}}}}^{\mathsf{V}}$, and $\overline{\mathsf{W}}$ is the bedrock to the universe $\mathsf{V}$ which exists under the Super-$C^{(\infty)}$-$\mathsf{LgLCAA}$ for hyperhuge [FU].

While Steel's $\mathcal{MV}^{M,\mathbb{G}}_{\mathsf{ST}}$ is rather a miniature (or even toy) model of the multiverse and the theory of its generating element $M$ may be totally unrelated to that of $\mathsf{V}$, our $\mathcal{MV}$ is more directly connected to the “real” multiverse.

Note also that $\kappa$ as above is a super-$C^{(\infty)}$-hyperhuge cardinal in $\overline{\mathsf{W}}$ [FU] and thus ${V_\kappa}^{\overline{\mathsf{W}}}\prec
\overline{\mathsf{W}}$, and ${V_\kappa}^{\overline{\mathsf{W}}}[\mathbb{G}]\prec
\overline{\mathsf{W}}[\mathbb{G}]$ holds for $\mathbb{G}$ as above. Because of this, and since the Super-$C^{(\infty)}$-$\mathsf{LgLCAA}$ for hyperhuge implies Maximality Principle for all posets and for parameters from $\mathcal{H}(\kappa)$, first-order properties of generic extensions of $\mathsf{V}$ are reflected down to elements of $\mathcal{MV}$. Note also that “properties of generic extensions of $\mathsf{V}$” is just a reformulation of something like “$\|\!\!{-}\!\!\!{-}_{\mathbb{P}}\mbox{“}\,\varphi\,\mbox{”}$ for some $\mathbb{P}$” in $\mathsf{V}$ while $\mathbb{G}$s in the defintion of $\mathcal{MV}$ are real sets in $\mathsf{V}$ (real in both senses of the word, since each of such $\mathbb{G}$s is a hereditarily countable set in $\mathsf{V}$!).

Reference.
[FU] Sakaé Fuchino, and Toshimichi Usuba, On recurrence axioms, Annals of Pure and Applied Logic, Vol.176, (10), (2025).
[S1] John. R. Steel, Gödel’s program, in: J. Kennedy (ed.), Interpreting Gödel: Critical Essays. Cambridge, UK: Cambridge University Press (2014).

Goldstern’s principle (for a pointclass $\Gamma$) states for every monotone family $\langle A_x : x \in \omega^\omega \rangle$ (in $\Gamma$) of Lebesgue null sets, the union $\bigcup_{x \in \omega^\omega} A_x$ is also Lebesgue null. The speaker have showed various results on this topic. In this talk, we would like to introduce approaches to 4 open problems regarding this principle. The first problem is whether Goldstern’s priniciple for the pointclass of all subsets is consistent with large continuum $2^{\aleph_0} \ge \aleph_3$. The second is about Goldstern’s principle in the Mathias model. The third is about the Hausdorff measure version of Goldstern’s principle. The last problem is whether Goldstern’s principle for the pointclass of all subsets implies the Borel conjecture. For each of them, we have a partial result, which we will present.

This is partially joint work with Martin Goldstern.

We will introduce a combinatorial structure called Multiple Pathway, which can be used to perform complex recursive constructions. The existence of a multiple Pathway implies that there are (strong) P-points and Gruff ultrafilters. The talk will be an introduction to this topic. This is a joint work with Alan Dow.

Graph coloring is one of the central topics in combinatorics. Komjáth studied the list chromatic number of infinite graphs and introduced a variant called the restricted list chromatic number. In this talk, we introduce stationary list coloring, a new list-coloring property. We study this property and discuss its relationships with other coloring properties. In particular, we focus on its behavior under the Generalized Continuum Hypothesis (GCH) and on the monotonicity of variants of list coloring.

We show that the Axiom of Real Determinacy ($\mathsf{AD}_{\mathbb{R}}$) and the Axiom of Real Blackwell Determinacy ($\mathsf{Bl}$-$\mathsf{AD}_{\mathbb{R}}$) are equivalent in $\mathsf{ZF}$+$\mathsf{DC}$. While we do not know if $\mathsf{AD}_{\mathbb{R}}$ and $\mathsf{Bl}$-$\mathsf{AD}_{\mathbb{R}}$ are equivalent without assuming $\mathsf{DC}$, we show that $\mathsf{ZF}$+$\mathsf{AD}_{\mathbb{R}}$ and $\mathsf{ZF}$+$\mathsf{AC}_{\omega} (\mathbb{R})$+$\mathsf{Bl}$-$\mathsf{AD}_{\mathbb{R}}$ are equiconsistent. This is joint work with W. Hugh Woodin.

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 Lebesgue 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 meager sets.

In this talk, I present Isomorphism Theorems for other ideals on Polish spaces, such as the ideal generated by $F_\sigma$-measure zero sets, the ideal of strong measure zero sets, and more.

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

We study cardinal invariants of the quotient Boolean algebra $\mathcal{P}(\omega)/\mathcal{I}(\mathcal{A}))$ where $\mathcal{I}(\mathcal{A})$ is the ideal generated by a mad family $\mathcal{A}$. Using a matrix iteration of ccc posets we shall show the consistency of $\mathfrak{b}=\mathfrak{a}=\mathfrak{s}(\mathcal{A})=\kappa\mathfrak{r}=\kappa$ for regular cardinal $\kappa < \lambda$.
This is joint work with Jörg Brendle.

Justin T. Moore in 2005 introduced the closed subtree property $(\mathsf{csp})$ for Aronszajn trees. Also, Moore showed that if $\mathsf{MA}_{\aleph_{1}}$ holds, then every Aronszajn tree has $\mathsf{csp}$. This property can be seen as a generalization of the club-embeddability into all subtrees of an Aronszajn tree in terms of the negation of the weak principle.
We proved that every Suslin tree has $\mathsf{csp}$, and that the existence of a special Aronszajn tree without $\mathsf{csp}$ is consistent. We introduced a property $\mathsf{E}^{\ast}\text{-}\mathsf{ccc}$ for forcing notions, and showed that every $\mathsf{E}^{\ast}\text{-}\mathsf{ccc}$ forcing preserves subtrees with no uncountable closed subtrees.

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 $\mathbf{add}(\mathcal{N})$ 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. In particular, I will show that local constant evasion prediction is small in the Hechler model, using a rank argument.

We investigate hierarchies of recurrence axioms, introduced by Fuchino and Usuba, and discuss their relationship with maximality principles. Furthermore, we highlight connections with bounded forcing axioms and generic absoluteness principles. This is joint work with Sakaé Fuchino and Takehiko Gappo, available at https://doi.org/10.1017/jsl.2025.10131

An epimorphism from a linear order $A$ onto $B$ is any motontone surjective mapping $f : A \twoheadrightarrow B$. An order is called \emph{strongly surjective} if it has epimorphisms onto all its (nonempty) suborders. We will explain how to use a theorem of Moore to deduce that MA$_{\aleph_{1}}$ implies the existence of strongly surjective Countryman lines (this was asked by Soukup). Recall that Moore proved that under PFA there is a two element basis of the class of Aronszajn lines, which is given by two Countryman lines. Since the epimorphism relation gives a preorder structure to the class of linear orders which is refinement of the classical embeddability relation, it is natural to ask if there is a finite basis in this case also. We will try to give ideas of how we answered this. This involves showing that certain forcing for introducing epimorphisms are ccc. We will also give some remaining open questions.

There are multiple interpretations of what is a forcing axiom. We shall survey these interpretations and then address respective higher analogs.

The notion of subcomplete forcing was introduced by Jensen. All $\sigma$-closed forcings, Prikry forcing, Namba forcing (under CH) etc are subcomplete, and subcomplete forcings add no new reals. Also, Jensen proved that the forcing axiom for subcomplete forcings, denoted as SCFA, is consistent with CH, even with the diamond principle. In this talk, we discuss consequences of SCFA such as reflection principles. This is a (partly) joint work with Gunter Fuchs.

Combinatorial tree forcing is a generalization of Miller forcing, which extends the notion of “infinitely branching” to “ideal-positive branching” on countable sets. It is known that the idealized forcing associated with a sigma-ideal generated by closed sets of a sigma-ideal can be expressed as a form of combinatorial tree forcing. We prove that a countable support iteration of the combinatorial tree forcing associated with the meager ideal on countable sets also increases the groupwise dense number.

The perfect set dichotomy for an equivalent relation $E$ on $\mathbb{R}$ asserts that either $\mathbb{R} / E$ is well-orderable or there exists a perfect set of $E$-inequivalent reals. In this talk, we show that this dichotomy holds for each equivalent relation on $\mathbb{R}$ in the Solovay model $V({\mathbb{R}^{V[G]}})$. Furthermore, we consider a generalization of the Solovay model, which is obtained by collapsing an inaccessible cardinal to the successor cardinal of an uncountable regular cardinal $\mu$. We show that the perfect dichotomy theorem for $\mu^\mu$ holds in the generalized Solovay model.
This is a joint work with Hiroshi Sakai.

In this talk, we introduce a $\kappa$-complete fine filter $F_d$ over $\mathcal{P}\kappa\lambda$ associated with a $\mathcal{P}\kappa(2^{\lambda})$-thin list $d$, and study an anti-compactness property of $F_d$. For example, if $d$ has no cofinal branch, then $F_d$ cannot be extended to a complete ultrafilter.
As a main result, for a thin list defined by a $\square(2^{\lambda},<\aleph_1)$-sequence, we show that the associated Namba forcing $\mathrm{Nm}(\kappa,\lambda,F_d)$ is not semiproper. This result implies that if $\mathrm{Nm}(\aleph_2,F)$ is semiproper for every $\aleph_2$-complete filter $F$ over $\aleph_2$, then both $\square(\aleph_2,<\aleph_1)$ and $\square(2^{\aleph_2},<\aleph_1)$ fail.

In 2010, Blass studied many variants of evasion numbers. Using his notation, the speaker will talk about global predictions associated with ideals on $\omega$ and compare their adaptive and non-adaptive evasion numbers. This is a joint work with Aleksander Cieslak, Takehiko Gappo and Arturo Martinez-Celis.