RIMS Set Theory Workshop 2023
Large Cardinals and the Continuum
From Tue Oct 24, 9:45 to Fri Oct 27, 16:15
At Room 110, Research Institute of Mathematical Science,
Kyoto University, Kyoto Japan.

1Timetable

Tue Oct 24
9:45–10:00 Opening
10:00–11:00 Sakaé Fuchino
11:00–11:15 break
11:15–12:00 Gunter Fuchs
12:00–13:30 lunch break
13:30–14:15 Yo Matsubara*
14:15–15:15 Joan Bagaria (mini course 1)
15:15–15:30 break
15:30–16:15 Takehiko Gappo*
16:15–17:00 Andrés Felipe Uribe-Zapata*Thu Oct 26
9:15–10:00 Nam Trang*
10:00–10:45 Catalina Torres
10:45–11:00 break
11:00–12:00 Joan Bagaria (mini course 3)
12:00–13:30 lunch break
13:30–14:15 Kenta Tsukuura
14:15–15:00 Takashi Yamazoe
15:00–15:15 break
15:15–16:00 Maxwell Simon Levine*
16:00–16:45 Stefan Hoffelner*

Wed Oct 25 (see Note 3. below)
9:15–10:00 Yosuke Kasai*
10:00–10:45 Francesco Parente
10:45–11:00 break
11:00–12:00 Joan Bagaria (mini course 2)
12:00–13:30 lunch break
13:30–14:15 Tatsuya Goto
14:15–15:00 Miguel A. Cardona*
15:00–16:00 discussion and short talks

Fri Oct 279:15–10:00 Russell Stetson
10:00–10:45 Philipp Lücke*10:45–11:00 break
11:00–12:00 Joan Bagaria (mini course 4)
12:00–13:30 lunch break
13:30–14:15 Rihito Takase
14:15–15:00 Diego Mejia
15:00–15:15 break
15:15–16:00 Toshimichi Usuba
16:00–16:15 Closing■Notes on timetable

1. All times are in JST (Japan Standard Time, UTC + 9:00).
2. * marks talks via Zoom (in-person talks are also going to be broadcast via Zoom).
3. On Wed 25, we have to leave the lecture room by 16:00 at the latest, due to the evening program.

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Detailed program

Mini course
Joan Bagaria (ICREA)
Large cardinals as reflection properties of the set-theoretic universe
Abstract. This will be a mini course consisting of four one-hour lectures on large cardinals, seen as
reflection properties of the set-theoretic universe. The first lecture will introduce and give a general
overview of Structural Reflection Principles, which characterize many of the best known large cardinals
in terms of Reflection. The second lecture will present some recent results, obtained in joint work withPhilipp Lücke, on new Structural Reflection Principles that yield new large cardinals lying on the upperridges and beyond the current large-cardinal hierarchy, and also on Structural Reflection Principles
characterizing large cardinals below the first measurable cardinal. The third and fourth lectures will be
devoted to very recent joint work with Gabriel Goldberg which shows how to incorporate the reflection
properties of large cardinals into their combinatorial definitions in terms of measures. In particular, ourresults provide new purely combinatorial characterizations of some large cardinals, such as extendible
cardinals, in terms of Reflecting Measures.

Contributed talks
Tue Oct 24■10:00–11:00 Sakaé Fuchino (Kobe Univ.) Resurrection and Recurrence
Abstract. For a class P of posets and a set A the Recurrence Axiom for P and A ((P, A)-RcA, for short)
is the following statement expressible in an axiom scheme in the language of ZFC: For any L∈ -formulaϕ = ϕ(x) and a ∈ A, if P ϕ(a) for a P ∈ P, then there is a ground W of the universe V such thata ∈ W and W |= ϕ(a).
For an iterable class P of posets, Maximality Principle for P and A is equivalent to a slight strengthening of (P, A)-RcA while (P, A)-RcA is equivalent to a set-generic version of S.Friedman’s Inner ModelHypothesis for these P and A.
Laver generic large cardinal axioms do not imply Maximality Principle if the notion the large cardinalcorresponding to the generic large cardinal axiom is consistent with the existence of stationarily manyinaccessible cardinal above it.
We introduce the notion of tightly super C (∞) -P-Laver gen. hyperhuge cardinal, and show that theexistence of a tightly super C (∞) -P-Laver gen. hyperhuge cardinal (Tightly Super C (∞) -P-Laver gen. hyperhuge Axiom) implies MP(P, H(κref l )) (or MP(P, H(2ℵ0 )) in case of P = all posets). The consistencyof the Tightly Super C (∞) -P-Laver gen. Hyperhuge Axiom follows from a 2-huge cardinal.
We extend Usuba’s theorem on the existence of the minimal ground and obtain as a corollary that theexistence of tightly P-Laver gen. hyperhuge cardinal κ (without “super C (∞) ” ) implies the existence ofthe minimal ground W and that κ is hyperhuge in W. This implies the equiconsistency of the Tightly
P-Laver gen. Hyperhuge Axiom with the existence of hyperhuge cardinal, as well as the equiconsistencyof the Tightly Super C (∞) -P-Laver gen. Hyperhuge Axiom with the statement that 2ℵ0 is super C (∞) hyperhuge in the minimal ground (for appropriate P).
In a previous work we have shown that the Tightly P-Laver gen. Ultrahuge Axiom implies strong formsof Resurrection Axioms. Since the Tightly P-Laver gen. Ultrahuge Axiom follows from the Tightly PLaver gen. Hyperhuge Axiom, we see that the Tightly Super C (∞) -P-Laver gen. Hyperhuge Axiom for
P = semi-proper posets implies almost all preferable auxiliary “axioms” of set theory.
We shall also discuss the impact of these results to the Continuum Problem.
This is a joint work with Toshimichi Usuba.
■11:15–12:00 Gunter Fuchs (CUNY)CH

On a fragment of the Strong Reflection Principle consistent with2

Abstract. Subcomplete forcing was introduced by Jensen, and the associated forcing axiom is consistentwith CH, and even the diamond principle, while otherwise having many of the known consequences ofMartin’s Maximum. One notable exception is the strong reflection principle, SRP. However, I isolate anatural fragment of SRP which follows from the subcomplete forcing axiom and relates to this forcing
axiom very much as SRP relates to MM.
■13:30–14:15 Yo Matsubara (Chubu Univ.)

work with Toshimichi Usuba)

Countable Stationary Towers and Reflection Principles (Joint■15:30–16:15 Takehiko Gappo (TU Wien) Chang-type models of determinacy■16:15–17:00 Andrés Felipe Uribe-Zapata (TU Wien)

A general theory of iterated forcing with finitelyadditive measures
Abstract. Saharon Shelah in 2000 introduced an iterated forcing method using finitely additive measures
to prove that, consistently, cov(N ) has countable cofinality. In this talk, we will present a generalizationof this method and a new separation of the left-hand of Cichoń ’s diagram with cov(N ) singular.
Wed Oct 25■9:15–10:00 Yosuke Kasai (Shizuoka Univ.) Dowker spaces which are defined from ♣AD sequences, andthose normality in squares■10:00–10:45 Francesco Parente (Kobe Univ.) Universality properties of forcingAbstract. There have been several attempts in the literature to account for the central role of forcing inmodern set-theoretic practice. For instance, forcing axioms and generic absoluteness results single out
forcing as a canonical method for independence proofs in set theory. In this talk, we complement andsupport this view by investigating forcing as a tool to construct universal models. Namely, we look attheories of initial segments of the universe and show that any model of a sufficiently rich fragment of thosetheories can be embedded into a model constructed by forcing. Our results rely on the model-theoretic
properties of good ultrafilters, which recently returned to prominence in connection with Malliaris andShelah’s work on Keisler’s order. This is a joint work with Matteo Viale.
■13:30–14:15 Tatsuya Goto (Kobe Univ.)

Games related to splitting families
Abstract. Both cardinal invariants of the continuum and infinite games are important topics in set theory.
We connect the two fields by studying variants of well-known cardinal invariants using infinite games.
Although we have considered various games, in this talk, we focus on games related to splitting families
called splitting games and splitting* games. We can consider two cardinal invariants for each of these
two games: the minimum cardinality of a family for which Player I has no winning strategy and theminimum cardinality of a family for which Player II has a winning strategy. Most of our results areequalities and inequalities in ZFC, but include a proof of the consistency of the strict inequality. This isa joint work with Jorge Antonio Cruz Chapital and Yusuke Hayashi.
■14:15–15:00 Miguel A. Cardona (Pavol Jozef Šafárik Univ.)

Controlling the uniformity of the ideal
generated by the Fσ measure zero subsets of the reals
Abstract. Denote by E the ideal generated by the Fσ measure zero subsets of the reals and let non(E) =min{|A| : A ⊆ R and A ∈/ E}, the uniformity of E. In [BS92], Bartoszyński and Shelah proved that
random forcing does not increase non(E). Motivated by Bartoszyński’s and Shelah’s proof, the following
question was raised by the speaker:
Question. [[Car23]] Does the well-known eventually different real forcing (denoted by E) keep non(E)small?
The speaker and Mejía proved that E increases non(E), which gives a negative answer to [[Car23]].
This led them to the conclusion that the notion of σ-Fr-linked introduced by Mejía [Mej19] does not helpto force non(E) small. So they wondered:
Question. [C. and Mejía] Does FAM-limits help to force non(E) small?
Inspired by the notion of σ-FAM-linked introduced by the speaker, Mejía and Uribe-Zapata [CMUZ23]
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(see also [UZ23]), the speaker and Mejía proved that σ-FAM-linked posets help to force non(E) small,which gives a positive answer to [C. and Mejía]. Moreover, this provides a proof to Kellner’s, Tănasie’s,and Shelah’s conjecture that “E does not have FAM-limits” [KST19].
The aim of this talk is to show that σ-FAM-linked posets help to force non(E) small.
References[BS92][Car23][CMUZ23][KST19][Mej19][UZ23]

Tomek Bartoszynski and Saharon Shelah. Closed measure zero sets. Annals of Pure and Applied
Logic, 58(2):93–110, 1992.
Miguel A. Cardona. A friendly iteration forcing that the four cardinal characteristics of E can bepairwise different. Colloq. Math., 173:123–157, 2023.
Miguel A. Cardona, Diego A. Mejía, and Andrés F. Uribe-Zapata. A general theory of iterated
forcing using finitely additive measures. In prepration, 2023.
Jakob Kellner, Saharon Shelah, and Anda Tănasie. Another ordering of the ten cardinal characteristics in Cichoń’s diagram. Comment. Math. Univ. Carolin., 60(1):61–95, 2019.
Diego A. Mejía. Matrix iterations with vertical support restrictions. In Proceedings of the 14thand 15th Asian Logic Conferences, pages 213–248. World Sci. Publ., Hackensack, NJ, 2019.
Andrés Uribe-Zapata. Iterated forcing with finitely additive measures: applications of probability
to forcing theory. Master’s thesis, Universidad Nacional de Colombia, sede Medellín, 2023.

Thu Oct 26■9:15–10:00 Nam Trang (Univ. North Texas) Dense ideals on ω1 and strong axioms of determinacy
Abstract. I will present some new results regarding the equiconsistency of strong ideals and strong
determinacy theories. Ideals and determinacy are two foundational frameworks in set theory and theseresults give the bi-interpretability between one framework and the other (up to a certain complexity). In
particular, one of our theorems implies the theories “CH + there is an ω1 -dense ideal on ω1 ” and “ADR +Θ is regular” are equiconsistent. This settles a well-known conjecture by Woodin and completes a major
research direction in descriptive inner model theory that started by R. Ketchersid in his thesis some 25years ago. I will give background and history that lead up to these results as well as future researchalong this direction. Results presented are part of joint work with D.Adolf, G.Sargsyan, T.Wilson, andM.Zeman.
■10:00–10:45 Catalina Torres (Univ. Barcelona)

Hyperstationary subsets of Pκ λAbstract. We explore the concept of n-stationarity and n-stationary reflection on Pκ (λ) as introduced byHiroshi Sakai and a modified version of the same. We investigate which results from Bagaria’ s article“Derived Topologies on Ordinals and Stationary Reflection” can be translated to the context of Pκ λ.
■13:30–14:15 Kenta Tsukuura (Hosei Univ.)

The Semiproperness of Namba forcings in Prikry-type extensions
Abstract. The semiproperness of Namba forcing describes some kind of compactness. We show that
Namba forcing over µ+ is not semiproper in the extensions by Prikry-type forcings over a measurable
cardinal µ. We also study the semiproperness of saturated ideals around ℵω+1 .
■14:15–15:00 Takashi Yamazoe (Kobe Univ.)

Cichoń’s maximum with evasion number
Abstract. Cichoń’s maximum is a model where all the cardinal characteristics in Cichoń’s diagram (exceptfor add(M) and cof(M)) are separated. It was first constructed by Goldstern, Kellner and Shelah, whointroduced the UF(short for ultrafilter)-limit method to keep the bounding number b small. They andMejía showed that the evasion number e can be added to Cichoń’s maximum with another value, using
the FAM(short for finitely additive measure)-limit method to keep e small.
We showed that a modified UF-limit method can also keep e small. Combining this modified UF-limit
method and the FAM-limit method, we obtained a new separation model of cardinal characteristics,
including those in Cichoń’s diagram, e and its variant numbers, and non(E), the uniformity number of
the Fσ null ideal.
■15:15–16:00 Maxwell Simon Levine (Univ. of Freiburg)

On Disjoint Stationary Sequences
Abstract. Disjoint stationary sequences were introduced by Krueger to answer questions about forcings4that add clubs through stationary sets. We will discuss a version of Mitchell forcing that adds a disjoint
stationary sequence (given a sufficient large cardinal). The benefit of this version is that it comes with an
Abraham-style projection analysis. This allows us to obtain disjoint stationary sequences on successive
cardinals, thus making progress on Krueger’s questions.
■16:00–16:45 Stefan Hoffelner (Univ. of Münster)

Forcing the separation, the reduction and the uniformization property
Abstract. The results presented can be found in my papers “Forcing the Σ13 -Separation Property”
from the Journal of Mathematical Logic and “Forcing the Π13 -Reduction property and a Failure of Π13 Uniformization” form APAL, as well as from my (still refereed) preprint “Forcing the Π1n -Uniformization
Property”
Fri Oct 27■9:15–10:00 Russell Stetson (Rutgers Univ.) Characterizing LEF and sofic groups
Abstract. A group G is said to be locally embeddable into finite groups (LEF) if for every finite subset
F ⊆ G, there exists an injection ϕ : F → Sn for some n ≥ 1 such that whenever g, h, gh ∈ F , thenϕ(gh) = ϕ(g)ϕ(h). In the group theoretic literature, LEF groups are usually characterized in termsof embeddings into ultraproducts of finite symmetric groups. It is natural to ask whether there is acharacterization in terms
Q of the more concrete notion of a reduced product of finite symmetric groups.
In more detail, let P = n≥1 Sn be the full direct product and let N be the normal subgroup of elements(πn ) ∈ P such that πn = 1 for all but finitely many n ≥ 1. Then the reduced product is the quotient
P0 = P/N .
We have shown that it is independent of ZFC whether for all groups G such that |G| ≤ 2ℵ0 , G is LEF
if and only if G embeds into P0 . We have obtained analogous results for the more general class of sofic
groups in terms of embeddings into the metric reduced product. This work is joint with Simon Thomas.
■10:00–10:45 Philipp Lücke (Univ. Hamburg)■13:30–14:15 Rihito Takase (Kobe Univ.)

Perfect Set Theorems at singular cardinals

Modal Logic of Provability and Forcing■14:15–15:00 Diego Mejia (Shizuoka Univ.)

Two-dimensional iterations with fam-limits
Abstract. The method of finite support iterations with fams (finitely additive measures) on the power setof ω was first developed by Saharon Shelah (2000) to construct a model of ZFC where the cofinality of
the covering of measure is countable. This method was expanded by Kellner, Tanăsie and Shelah (2019),and generalized in the MSc thesis of Uribe-Zapata (2023). In this talk, we present a general method oftwo-dimensional iterations with fam-limits and show consistency results about cardinal characteristics ofthe continuum. This is a joint work in progress with Miguel Cardona and Andrés Uribe-Zapata.
■15:15–16:00 Toshimichi Usuba (Waseda Univ.) Monotonicity of the ultrafilter number

Abstract. For an infinite cardinal κ, let u(κ) be the ultrafilter number at κ. Under GCH, it is clear thatthe ultrafilter number at cardinals is monotone, that is, if κ ≤ λ then u(κ) ≤ u(λ). However it is notclear if this monotonicity can fail or not. In this talk, we show that it is consistent that the monotonicityof the ultrafilter number at cardinals fails, moreover the failure has a large cardinal strength as implying
the consistency of a measurable cardinal. We also prove some results about the ultrafilter number at
successive cardinals. For instance, we show that if κ is either regular, strong limit singular, or singularwith countable cofinality, then u(κ) ≤ u(κ+ ) always holds. On the other hand, under a certain large
cardinal assumption, it is consistent that u(κ+ ) < u(κ) for some singular κ.

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