Berkeley Inner Model Theory Conference

Abstracts

A mutual stationarity property at the level of almost linear iterations (Dominik Adolf).

Set Theory seems to lack intermediate properties. By this we mean properties above the level of linear iterations or the sharp for an inner model with a strong cardinal ('zero-pistol'), but below a Woodin cardinal. In this talk we will introduce a natural property from the realm of mutual stationarity that we will show requires non-linear iterations, but is (presumably) below the sharp for proper class many strong cardinals ('zero-handgrenade').

We believe that there should be an entire class of properties equivalent to models with increasingly complex overlaps. This line of research is part of an (informal) project of the presenter and Omer Ben-Neria.

Cofinalities of ultrafilters (Tom Benhamou).

Consider an ultrafilter ordered by reversed inclusion or reversed inclusion modulo bounded. The study of this type of posets has been of recent interest both on countable sets and large cardinals due to its relation to several open problems concerning ultrafilter combinatorics. After an overview of the subject, we will present a clean characterization in Mitchell-Steel's L[E] of the class of ultrafilters for which the corresponding poset has maximal cofinal complexity (i.e. Tukey-maximal). We will then present a new technique for forcing ultrafilters with controlled complexity, along with several applications. If time permits, we will also demonstrate how to use the theory developed in the previous results to express combinatorial notions such as the character of an ultrafilter ch(U) and the pseudo-intersection of an ultrafilter p(U) in terms of the ultrapower.

On the influence of inner model theory on forcing and ultrafilters (Omer Ben-Neria).

Inner model theory has helped shape the development of forcing in several important ways—for example, by providing ground model assumptions that facilitate certain forcing constructions, and by offering tools to analyze central generic objects.

A key instance of this influence is Schindler's work 'Iterates of the Core Model', which has played a central role in consistency results involving large cardinals and the structure of ultrafilters. I will begin with a discussion of this theorem and some of its applications over the past two decades, focusing in particular on its recent reappearance in joint work with Eyal Kaplan on non-normal ultrafilters over measurable cardinals.

I will then turn to another way in which fine-structural methods contribute to this area. Building on work of Foreman, Magidor, and Zeman—who introduced techniques for iterating certain distributive forcings—we have recently developed new iteration methods to handle more complex forcing constructions. I will outline these recent developments and applications.

Independent Families on $\kappa < \Theta$ (Stewart Brown).

W. Chan, S. Jackson, and N. Trang analyzed the existence (or lack thereof) of maximally almost disjoint (mad) families on cardinals $\kappa$ under $\text{AD}^+$. We give a brief overview of these results, discuss the non-existence of maximally independent families (mif's) under $\text{AD}^+$, and contrast the methods for proving the non-existence of mad families and mif's. This is work done under the supervision of Nam Trang.

On the interplay between fragments of SRP, saturation and diagonal reflection (Gunter Fuchs).

In prior work, I introduced fragments of Todorcevic's strong reflection principle (SRP) associated in a natural way to forcing classes more restrictive than stationary set preserving forcing. The fragment associated to the subcomplete forcings, while retaining many crucial consequences of SRP, is compatible with CH, and even diamond. Surprisingly, the weaker principle, when paired with CH, has more far reaching consequences in terms of diagonal reflection than the original, stronger principle. The subcomplete fragment of SRP together with CH contradicts the saturation of the nonstationary ideal, and the pattern that seems to emerge is that pairing this fragment of SRP with stronger failures of saturation sets the stage for stronger principles of diagonal reflection.

This is joint work with Hiroshi Sakai.

Determinacy of games of fixed countable length (Takehiko Gappo).

We discuss the large cardinal strength of determinacy of games on natural numbers of length $\omega \cdot \alpha$ with payoff in ${<}\omega^2\text{-}\Pi^1_1$ for arbitrary countable ordinal $\alpha$. This is joint work with Juan P. Aguilera.

Filter extension games on mini supercompactness measures (Victoria Gitman).

Several classical smaller large cardinals $\kappa$, among them weakly compact, ineffable, and Ramsey cardinals, can be characterized by the existence of measure-like filters on $\kappa$-sized families of subsets of $\kappa$. It suffices to consider families that arise as $P(\kappa)^M$ for some $\kappa$-sized $\in$-model of a sufficiently large fragment of set theory. Isolating patterns in the properties of these filters has led to a better understanding of the classical large cardinals and the definition of new ones. Holy and Schlicht introduced the filter extension games of length $1$ up to $\kappa^+$ in which the first player plays an increasing sequence of these $\kappa$-sized models and the second player responds by choosing an increasing sequence of filters for them. They, and later Nielsen and Welch, used the existence of a winning strategy for one of the players in these games to characterize (or imply) classical large cardinals and define new ones. The existence of the strategy for the second player can be used to characterize (or imply) versions of generic measurability, including, as shown by Foreman, Magidor, and Zeman, the existence of precipitous ideals.

We generalize the filter extension games to the two-cardinal setting by considering corresponding filters on $\lambda$-sized families of subsets of $P_\kappa(\lambda)$ (arising as $P(P_\kappa(\lambda))^M$ for $\lambda$-sized $\in$-models). We show that the existence of a winning strategy for the second player in these games characterizes (or implies) several large cardinal notions in the neighborhood of a supercompact, including, nearly $\lambda$-supercompact, completely $\lambda$-ineffable and $\lambda$-$\Pi^1_n$-indescribable cardinals. We also connect the existence of winning strategies for the second player in these games with various notions of generic supercompactness, including the existence of precipitous ideals. This is joint work with Tom Benhamou.

More periodicity from Reinhardt cardinals (Gabriel Goldberg).

We show that if \(\kappa\) is an indescribable cardinal far above a Reinhardt cardinal, then for all \(n < \omega\), the \(\Sigma^1_{2n+1}\) subsets of \(V_{\kappa+1}\) have the separation property. This follows from a periodicity theorem for the \(\Pi^1_{n}\)-indescribability filters, joint with Nai-Chung Hou, and ideas from joint work with Tom Benhamou on the structure of normal filters in the constructible universe. We will conclude with some speculation about the structure of indescribable cardinals under AD and the more general problem of transferring theorems about Reinhardt cardinals to the AD context.

Forcing and generic absoluteness in ZF (Daisuke Ikegami).

This research was originally motivated by the following question: Is there a model of ZF (without the Axiom of Choice) where one cannot change the truth value of any first-order statement in set theory via forcing? Woodin proved that there is no such a model of ZF. His arugments led us wonder the following question: What kind of forcings preserve cardinals in ZF? Related to this question, Karagila and Schweber showed that there is a model of ZF where for some ccc forcing \(\mathbb P\), \(\mathbb P\) collapses \(\omega_1\). In this talk, we introduce norrowness and uniform narrowness as properties of posets and show the following: 1) both narrowness and uniform narrowness are equivalent to ccc in ZFC, 2) all the narrow forcings and uniformly narrow forcings preserve cardinals in ZF, and 3) every uniform iteration of narrow forcings with finite support is uniformly narrow in ZF. As applications of the above result, we discuss certain generic absoluteness principles in ZF which imply all uncountable cardinals are singular. This is joint work with Philipp Schlicht.

Regularity, primeness, and partition arguments (Steve Jackson).

For sets $X$, $Y$, we introduce the notions of the set $X$ being $Y$-regular. We also introduce the notion of a set $X$ being prime and of being strongly prime. We study these notions in models of determinacy as well as introduce some new partition type arguments. For example, we show that $(\omega_\omega)^\omega$ is prime, fully ordinal regular, and $\mathcal{P}(\kappa)$-regular for $\kappa<\omega_\omega$. We also show $(\omega_n)^{<\omega_2}$ is ordinal regular and get a result about successors of $\kappa$ in $\text{HOD}_f$ for $f \in \kappa^\kappa$ at partition cardinals. This is joint work with William Chan and Nam Trang.

Applications of the fusion technique (Eyal Kaplan).

The analysis of ultrafilters and ultrapower embeddings in forcing extensions often proceeds in two main steps. First, one identifies the possible restrictions of an ultrapower embedding from the generic extension back to the ground model. Second, one investigates the possible lifts of such restricted embedding to ultrapower embeddings of the extension.

A recent example appears in a joint work with Omer Ben-Neria, where we demonstrated the consistency of the Ultrapower Axiom with the failure of GCH at a measurable cardinal. The forcing construction integrated several techniques, including Schindler’s result regarding iterates of the core model, fine-structure-based forcing, coding generic information and fusion arguments.

In this talk, I will concentrate on the fusion technique and illustrate its role in analyzing the possible lifts of a given ground model ultrapower embedding. I will then discuss several applications: a model in which there are multiple normal measures, and all of them induce the same ultrapower; A characterization of all normal measures after iterating certain Prikry-type forcings; and the consistency of the Weak Ultrapower Axiom relative to a measurable cardinal.

The talk is based on joint works with Omer Ben-Neria and Moti Gitik.

Computing the cofinality of the uB-powerset (Lukas Koschat).

In recent work, Sandra Müller and Grigor Sargsyan introduced the uB-powerset (uBp), a certain family of sets of sets of reals that gives rise to a canonical determinacy model. They showed that, assuming the existence of a proper class of Woodin cardinals and after collapsing slightly beyond a supercompact cardinal, a version of Sealing holds for L(uBp). They further conjecture that their techniques can be extended to prove an analogous theorem for a Chang-style extension of L(uBp). A crucial question in the analysis of this Chang-style extension concerns the cofinality of uBp – that is, the cofinality of the supremum of the lengths of prewellorders coded by elements of uBp.

In this talk, we present a proof that, assuming a proper class of Woodin cardinals, the cofinality of uBp is ω in the Lévy collapse of a supercompact cardinal that has another supercompact cardinal below it. Along the way, we focus on extenders on the reals and the corresponding ultrapowers of uBp. This is joint work with Müller and Sargsyan.

Derived models and PFA (Derek Levinson).

Wilson conjectured that PFA + "\(\kappa\) is a limit of Woodin cardinals" implies the derived model at $\kappa$ satisfies \(\Theta_0 < \kappa^+.\) We prove this under the additional assumption that the derived model satisfies mouse capturing and discuss some related results. This is joint work with Nam Trang.

Translating between large cardinals and iteration strategies (Sandra Müller).

My PhD thesis under Ralf's supervision focused on mice with finitely many Woodin cardinals, the projective complexity of their iteration strategies and models of levels of projective determinacy. Since then, I have been interested to connect inner models with large cardinals and models of determinacy at various levels. In this talk I will outline some classical results as well as new developments in this area.

Maximal Prikry sequences (Ernest Schimmerling).

Dodd and Jensen proved the following theorem. Assume that \(0^\dagger\) does not exist and let \(\nu\) be a singular cardinal. Suppose that \(\nu\) is regular in \(K\). Then \(K = L[U]\) where \(U\) is a normal measure over \(\nu\) in \(K\). Moreover, there is a Prikry sequence \(C\) for \(U\) such that a) every Prikry sequence for \(U\) is eventually contained in \(C\) and b) every uncountable set of ordinals is covered by a set in \(K[C] = L[U][C] = L[C]\) with the same cardinality. Mitchell and others extended aspects of this theorem in a number of ways, some of which will be recalled. The talk will be about further generalizations of the above theorem of Dodd and Jensen that were obtained in a project with Jiaming Zhang.

In addition, two forcing examples will be given to show that the new results about maximal Prikry sequences and covering are optimal, one of which was found in collaboration with Tom Benhamou. The latter example also provides a comparison of Mitchell-Steel and Jensen indexing that is enlightening in a few ways.

Towards extending Dehornoy's analysis (Philipp Schlicht).

The Bukovsky-Dehornoy phenomenon describes a surprising connection between the intersection model of an iteration of a single ultrafilter in omega steps and Prikry forcing. Several variants of this result were discovered, beginning with Dehornoy's analysis of iterations of arbitrary length and more recent results by Fuchs, Hamkins and Hayut. We obtain results towards generalising the analysis to more than one ultrafilter in joint work with Christopher Henney-Turner.

Mouse sets in L(R) (Farmer Schlutzenberg).

Given a nicely definable set $X$ of reals, it is natural to ask whether $X$ is just the set of reals of some mouse. In many instances, this is known to hold. We will discuss some newly established instances in which $X$ is the set of reals which are ordinal definable over some level of $L(\mathbb{R})$ at a certain degree of complexity. This uses joint work with Steel on correctness of mice in $L(\mathbb{R})$, combined with related work of the author on ladder mice.

A new $\Sigma^1_3$ version of Gandy-Harrington forcing (Benny Siskind).

Gandy-Harrington forcing is an important tool in effective descriptive set theory at the level of $\Pi^1_1$ and $\Sigma^1_1$ sets. For example, it was used by Harrington to give a forcing proof of Silver's perfect set theorem for $\Pi^1_1$ equivalence relations. A $\Sigma^1_3$ version of the Gandy-Harrington forcing had already been isolated and used to prove a generalization of Silver's theorem to $\Pi^1_3$ equivalence relations. Despite this success, this previously known version has some defects which seem to prevent appropriately generalizing some other results from the $\Sigma^1_1$ level to the $\Sigma^1_3$ level. We'll introduce a new $\Sigma^1_3$ version of Gandy-Harrington forcing, discuss some applications, and mention a connection to inner model theory. This is ongoing joint work with William Chan, Sandra Müller, and Farmer Schlutzenberg.

HOD mice with indiscernible Woodins (John Steel).

It is more difficult to construct a strategy mouse than it is to construct a pure extender mouse. One symptom of this is that the existence of a strategy mouse satisfying ${\sf ZFC}$ + ``there are infinitely many Woodin cardinals" implies the existence of pure extender mice having measurable limits of Woodins (and more). Until recently, it was not known whether the existence of strategy mice with Woodin limits of Woodin cardinals follows from the existence of pure extender mice with long extenders.

We shall show that it does, and that in fact the existence of a strategy mouse with indiscernible Woodins follows from the existence of a pure extender mouse with indicernible Woodins.

The proof involves showing that in the HOD of a model of $\sf{AD}_{\mathbb{R}}$, both statements are equivalent to ``there are divergent models of ${\sf{AD}}$ containing all the reals". It rests heavily on earlier work of Sargsyan in the same vein, and on joint work by Sargsyan and the author that develops a more effective process for comparing mouse pairs.

Because it goes through divergent models, the proof leaves open pretty much all similar questions on the relative strengths of large cardinal hypotheses in strategy mice and pure extender mice in the region past Woodin limits of Woodin cardinals.

Small fragments of the Martin's Maximum (Nam Trang).

We show that mild extensions of the theory \(\text{MM}(\mathfrak{c}) + \neg \square_{\omega_2}\) imply the existence of models of \(\text{AD}_{\mathbb R}\) + DC. We also discuss dichotomy theorems that "characterize" determinacy theories in terms of \(\text{MM}(\mathfrak{c}) + \neg \square_{\omega_2}\) and \(\text{MM}(\mathfrak{c}) + \neg \square(\omega_3)\). These results are likely to lead us to determining the exact consistency strength of these fragments of MM. Relevant conjectures will be provided. Part of this work is joint with M. Zeman.

Martin's Maximum and the cofinality of universally Baire sets (Taichi Yasuda)

We study the cofinality of uB (universally Baire) sets under Martin's Maximum. We shall show under Martin's Maximum and proper class many Woodin limit of Woodins, “the cofinality of uB sets is $\omega_{1}$” is equivalent to “there is no SSP forcing adding a new uB set on top”. And we discuss whether Martin's Maximum implies that the cofinality of uB sets is bigger than $\omega_{1}$. We shall show a positive instance for it.

Some better lower bounds for the consistency strengths of the failure of square principles at two small successive regular cardinals (Martin Zeman).

In the past 15 years, several results concerning successive failures of square principles at small regular cardinals have been obtained. Of primary interest here are the smallest possible cardinals where such failures can occur, and we use the core model induction to obtain lower bounds. This way, we significantly improve previously known lower bounds for the consistency strength of the following theories, in terms of combinatorial properties of \( \Theta \) in canonical models of \(\text{AD}_{\mathbb{R}}\):

• \( \mathrm{ZFC} + 2^\omega = \omega_2 + 2^{\omega_2} = \omega_3 + \lnot \square(\omega_3) + \lnot \square_{\omega_3} \)
• \( \mathrm{ZFC} + 2^\omega = \omega_2 + 2^{\omega_2} = \omega_3 + \lnot \square(\omega_3) + \lnot \square(\omega_4) \)

Work on related situations is in progress. This is joint work with Nam Trang.