# Blog Archives

# Topic Archive: large cardinals

# Regular Jónsson cardinals

For an infinite cardinal $\lambda$, we say that $\lambda$ is Jónsson if it satisfies the square bracket partition property $\lambda\to[\lambda]^{<\omega}_\lambda$. This means that, for every coloring $F\colon [\lambda]^{<\omega}\to\lambda$, there is some set $H\in [\lambda]^\lambda$ with the property that $\mathrm{ran}(F\upharpoonright [H]^{<\omega})\subsetneq\lambda$. It is rather well known that the consistency strength of "there is a Jónsson cardinal" lies above "$0^\sharp$ exists", but below the existence of a measurable. However, the question of what sorts of cardinals can be Jónsson has turned out to be rather difficult. The goal of this talk is to sketch a simplified proof of the result (due to Shelah) that if $\lambda$ is the least regular Jónsson cardinal, then $\lambda$ must be $\lambda\times\omega$-Mahlo. The advantage of the proof presented here is that the machinery employed can be easily generalized, and time permitting I would like to discuss how one might attempt to prove that such a cardinals must be greatly Mahlo.

# The uniqueness of $\mathbb R$-supercompactness measures in ZFC

${\rm AD}_{\mathbb R}$ is a strengthening of the determinacy axiom that states that all games on the real numbers are determined. It is a Theorem of Solovay that under ${\rm ZF}+{\rm AD}_{\mathbb R}$ there is a fine, countably complete and normal filter on $P_{\omega_1}(\mathbb R)$, so $\omega_1$ is $\mathbb R$-supercompact. The exact consistency strength of the theory ${\rm ZF}+ {\rm AD}+$“$\omega_1$ is $\mathbb{R}$-supercompact” is, however, weaker than the one of ${\rm ZF}+{\rm AD}_{\mathbb R}$.

One central interest of Inner Model Theory is to construct/find canonical models for theories extending ${\rm ZF}$. A natural question is, then, whether there is a canonical model for the theory ${\rm ZF}+ {\rm AD}+$“$\omega_1$ is $\mathbb{R}$-supercompact”.

In this talk, we will discuss the consistency strength and minimal models of this theory. We will discuss the proof of the uniqueness of minimal models of this theory, under various appropriate hypotheses. And time permitting we will discuss the proof of the result that under ${\rm ZFC}$ there is at most one minimal model of this theory. This is joint work with Nam Trang.

# On the non-existence and definability of mad families

By an old result of Mathias, there are no mad families in the Solovay model constructed by the Levy collapse of a Mahlo cardinal. By a recent result of Törnquist, the same is true in the classical model of Solovay as well. In a recent paper, we show that ZF+DC+”there are no mad families” is actually equiconsistent with ZFC. I’ll present the ideas behind the proof in the first part of the talk.

In the second part of the talk, I’ll discuss the definability of maximal eventually different families and maximal cofinitary groups. In sharp contrast with mad families, it turns out that Borel MED families and MCGs can be constructed in ZF. Finally, I’ll present a general problem in Borel combinatorics whose solution should explain the above difference between mad and maximal eventually different families, and I’ll show how large cardinals must be involved in such a solution.

This is joint work with Saharon Shelah.

# Set-theoretic geology and the downward-directed grounds hypothesis: part II

I will continue presenting Toshimichi Usuba’s recent proof of the strong downward-directed grounds hypothesis. See the main abstract at Set-theoretic geology and the downward directed ground hypothesis.

See my blog post about this talk.

# Set-theoretic geology and the downward-directed grounds hypothesis

Forcing is often viewed as a method of constructing larger models extending a given model of set theory. The topic of set-theoretic geology inverts this perspective by investigating how the current set-theoretic universe $V$ might itself have arisen as a forcing extension of an inner model. Thus, an inner model $W\subset V$ is a ground of $V$ if we can realize $V=W[G]$ as a forcing extension of $W$ by some $W$-generic filter $G\subset\mathbb Q\in W$. Reitz had inquired in his dissertation whether any two grounds of $V$ must have a common deeper ground. Fuchs, myself and Reitz introduced the downward-directed grounds hypothesis, which asserts a positive answer, even for any set-indexed collection of grounds, and we showed that this axiom has many interesting consequences for set-theoretic geology.

I shall give a complete detailed account of Toshimichi Usuba’s recent proof of the strong downward-directed grounds hypothesis. This breakthrough result answers what had been for ten years the central open question in the area of set-theoretic geology and leads immediately to numerous consequences that settle many other open questions in the area, as well as to a sharpening of some of the central concepts of set-theoretic geology, such as the fact that the mantle coincides with the generic mantle and is a model of ZFC. I shall also present Usuba’s related result that if there is a hyper-huge cardinal, then there is a bedrock model, a smallest ground. I find this to be a surprising and incredible result, as it shows that large cardinal existence axioms have consequences on the structure of grounds for the universe.

See my blog post about this talk.

# Normal Measures and Tall Cardinals

I will discuss the number of normal measures a non-$(\kappa + 2)$-strong tall cardinal $\kappa$ can carry, paying particular attention to the cases where $\kappa$ is either the least measurable cardinal or the least measurable limit of strong cardinals. This is joint work with James Cummings.

# Killing measurable and supercompact cardinals softly

This talk follows the theme of killing-them-softly between set-theoretic universes. The main theorems in this theme show how to force to reduce the large cardinal strength of a cardinal to a specified desired degree, for a variety of large cardinals including inaccessible, Mahlo, measurable and supercompact. The killing-them-softly theme is about both forcing and the gradations in large cardinal strength. This talk will focus on measurable and supercompact cardinals, and follows the larger theme of exploring interactions between large cardinals and forcing which is central to modern set theory.

# Joint Laver diamonds

Say that a collection of Laver functions is jointly Laver if the functions can guess their targets simultaneously using just a single elementary embedding between them. In this talk we shall examine the notion of jointness in the simplest case of measurable cardinals, giving both equiconsistency results for the existence of large jointly Laver families and separating the existence of small such families from large ones. We shall also comment on how these results transfer to larger large cardinals, such as supercompact and strong cardinals, and, perhaps, how the notion of jointness may be interpreted for guessing principles not connected with large cardinals.

# Singular in V, regular and non-measurable in HOD

Getting a model where $\kappa$ is singular in $V$ but measurable in ${\rm HOD}$ is somewhat straightforward however ensuring that $\kappa$ is regular but not measurable in ${\rm HOD}$ is a surprisingly more difficult problem. Magidor navigated around the issues and I will present his result starting with one measurable. His technique can be extended for set many cardinals.

# Woodin’s AD-conjecture and local Reinhardt cardinals

We will discuss Woodin’s AD-conjecture, which gives a deep relationship between very large cardinals and determined sets of reals. In particular we will show that the AD-conjecture holds for the axiom I0 and that there are many interesting consequences of this fact. We will also discuss the notion of a local Reinhardt cardinal and how variations of the AD-conjecture might show that such cardinals do not exist.

# The Mitchell order for Ramsey cardinals

The usual Mitchell relation on normal measures on a measurable cardinal $\kappa$ orders the measures based on the degree of measurability that $\kappa$ retains in their respective ultrapowers. We shall examine the analogous ordering of appropriate witnessing objects for Ramsey (and Ramsey-like) cardinals. It turns out that the resulting order is well-behaved and its degrees neatly stratify the large cardinal hierarchy between Ramsey, strongly Ramsey, and super Ramsey cardinals. We also give a soft killing argument for this notion of Mitchell rank.

This is joint work with Victoria Gitman and Erin Carmody.

# Generalized Laver tables

The Laver tables are finite self-distributive algebras generated by one element that approximate the free left-distributive algebra on one generator if a rank-into-rank cardinal exists. We shall generalize the notion of a Laver table to a class of locally finite self-distributive algebraic structures with an arbitrary number of generators. These generalized Laver tables emulate algebras of rank-into-rank embeddings with an arbitrary number of generators modulo some rank. Furthermore, if there exists a rank-into-rank cardinal, then the free left-distributive algebras on any number of generators can be embedded in a canonical way into inverse limits of generalized Laver tables. As with the classical Laver tables, the reduced generalized Laver tables can be given an associative operation that is analogous to the composition of elementary embeddings and satisfies the same identities that algebras of elementary embeddings are known to satisfy. Furthermore, the notion of the critical point also holds in these generalized Laver tables as well even though generalized Laver tables are locally finite or finite. While the only classical Laver tables are the tables of cardinality $2^{n}$, the finite generalized laver tables occur much more frequently and many generalized Laver tables can be constructed from the classical Laver tables. We shall give some results that allow one to quickly compute the self-distributive operation in a certain class of generalized Laver tables.

Here are the slides.

# Tree representations from very large cardinals

We will discuss the propagation of certain tree representations in the presence of very large cardinals. These tree representations give generic absoluteness results and have structural consequences in the area of generalized descriptive set theory. In fact these representations give us a method for producing models of strong determinacy axioms.

# Split Principles

In this talk I will introduce various “Split Principles”, which, if they hold, posit the existence of a sequence which in some sense “splits” any large set into two unbounded pieces. We will see that the failure of a particular split principle to hold tends to characterize some large cardinal property; in particular weak compactness, ineffability, measurability and supercompactness can each be characterized in terms of the failure of a split principle. The initial idea of a split principle came about recently in the joint work of Fuchs, Gitman, and Hamkins. The content of the talk is the result of exploring the idea further with Gunter Fuchs.

# Deciphering the HOD Dichotomy: Part II

I will continue to attempt to present my version of the notes of Woodin’s talk at the Appalachian Set Theory seminar on his paper, the HOD Dichotomy, including the HOD conjecture. Principally, I hope to present how weak extender models relate to the HOD conjecture. I will also present the initial part of new results that were inspired by the HOD conjecture, in particular, by failures of the cover property.

# Dissertation Defense: Force to change large cardinals

This will be the dissertation defense of the speaker. There will be a one-hour presentation, followed by questions posed by the dissertation committee, and afterwards including some questions posed by the general audience. The dissertation committee consists of Joel David Hamkins (supervisor), Gunter Fuchs, Arthur Apter, Roman Kossak and Philipp Rothmaler.

# Force to change large cardinal strength

Suppose $kappain V$ is a cardinal with large cardinal property $A$. In this talk, I will present several theorems which exhibit a notion of forcing $mathbb P$ such that if $Gsubseteq mathbb P$ is $V$-generic, then the cardinal $kappa$ no longer has property $A$ in the forcing extension $V[G]$, but has as many large cardinal properties below $A$ as possible. I will also introduce new large cardinal notions and degrees for large cardinal properties.

This talk is the speaker’s dissertation defense.

# The tall and measurable cardinals can coincide on a proper class

Starting from an inaccessible limit of strong cardinals, we force to construct a model containing a proper class of measurable cardinals in which the tall and measurable cardinals coincide precisely. This is joint work with Moti Gitik which extends and generalizes an earlier result of Joel Hamkins.

# News on the Solid Core

Set-theoretic geology, a line of research jointly created by Hamkins, Reitz and myself, introduced some inner models which result from inverting forcing in some sense. For example, the mantle of a model of set theory V is the intersection of all inner models of which V is an extension by set-forcing. It was an initial, naive hope that one might arrive at a model that is in some sense canonical, but one of the main results on set-theoretic geology is that this is not so: every model of set theory V has a class forcing extension V[G] so that the mantle, as computed in V[G], is V. So quite literally, the mantle of a model of set theory can be anything.

In an attempt to arrive at a concept that fits in with the general spirit of set-theoretic geology, but that stands a chance of being canonical, I defined a set to be solid if it cannot be added to an inner model by set-forcing, and I termed the union of all solid sets the “solid core”.

I will present some results on the solid core which were obtained in recent joint work with Ralf Schindler, and which show that the solid core indeed is a canonical inner model, assuming large cardinals (more precisely, if there is an inner model with a Woodin cardinal), but that it is not as canonical as one might have hoped without that assumption.

# Joint Laver diamonds

A Laver diamond for a given large cardinal $\kappa$ is a function $\ell$, defined on $\kappa$, such that $j(\ell)(\kappa)$ can take any reasonable value, where $j$ is a relevant large cardinal embedding. A sequence of such functions is called jointly Laver or a joint Laver diamond if they can be made to take any given sequence of such values at the same time via a single embedding. In the talk we will consider questions about when such sequences outright exist, when their existence is equiconsistent with and when their existence is consistency-wise strictly stronger than the large cardinal in question.