We continue the study of the Galvin property from Benhamou, Garti, and Shelah (2023, Proceedings of the American Mathematical Society 151, 1301–1309) and Benhamou (2023, Saturation properties in canonical inner models, submitted). In particular, we deepen the connection between certain diamond-like principles and non-Galvin ultrafilters. We also show that any Dodd sound non p-point ultrafilter is non-Galvin. We use these ideas to formulate what appears to be the optimal large cardinal hypothesis implying the existence of a non-Galvin ultrafilter, improving on a result from Benhamou and Dobrinen (2023, Journal of Symbolic Logic, 1–34). Finally, we use a strengthening of the Ultrapower Axiom to prove that in all the known canonical inner models, a $\kappa $-complete ultrafilter has the Galvin property if and only if it is an iterated sum of p-points.

Let M be a short extender mouse. We prove that if $E\in M$ and $M\models $“E is a countably complete short extender whose support is a cardinal $\theta $ and $\mathcal {H}_\theta \subseteq \mathrm {Ult}(V,E)$”, then E is in the extender sequence $\mathbb {E}^M$ of M. We also prove other related facts, and use them to establish that if $\kappa $ is an uncountable cardinal of M and $\kappa ^{+M}$ exists in M then $(\mathcal {H}_{\kappa ^+})^M$ satisfies the Axiom of Global Choice. We prove that if M satisfies the Power Set Axiom then $\mathbb {E}^M$ is definable over the universe of M from the parameter $X=\mathbb {E}^M\!\upharpoonright \!\aleph _1^M$, and M satisfies “Every set is $\mathrm {OD}_{\{X\}}$”. We also prove various local versions of this fact in which M has a largest cardinal, and a version for generic extensions of M. As a consequence, for example, the minimal proper class mouse with a Woodin limit of Woodin cardinals models “$V=\mathrm {HOD}$”. This adapts to many other similar examples. We also describe a simplified approach to Mitchell–Steel fine structure, which does away with the parameters $u_n$.

A short core model induction proof of $\mathsf {AD}^{L(\mathbb {R})}$ from $\mathsf {TD} + \mathsf {DC}_{\mathbb {R}}$.

We construct divergent models of $\mathsf {AD}^+$ along with the failure of the Continuum Hypothesis ($\mathsf {CH}$) under various assumptions. Divergent models of $\mathsf {AD}^+$ play an important role in descriptive inner model theory; all known analyses of HOD in $\mathsf {AD}^+$ models (without extra iterability assumptions) are carried out in the region below the existence of divergent models of $\mathsf {AD}^+$. Our results are the first step toward resolving various open questions concerning the length of definable prewellorderings of the reals and principles implying $\neg \mathsf {CH}$, like $\mathsf {MM}$, that divergent models shed light on, see Question 5.1.

There is a balance between the amount of (weak) indestructibility one can have and the amount of strong cardinals. It’s consistent relative to large cardinals to have lots of strong cardinals and all of their degrees of strength are weakly indestructible. But this necessitates the destructibility of the partially strong cardinals. Guaranteeing the indestructibility of the partially strong cardinals is shown to be harder. In particular, this work establishes an equiconsistency between:

1. 1. a proper class of cardinals that are strong reflecting strongs; and

2. 2. weak indestructibility for (κ+2)-strength for all cardinals κ in the presence of a proper class of strong cardinals.

1. 3. weak indestructibility for all degrees of strength for a proper class of strong cardinals.

One direction of the equiconsistency of (1) and (2) is proven using forcing and the other using core model techniques from inner model theory. Additionally, connections between weak indestructibility and the reflection properties associated with Woodin cardinals are discussed, and similar results are derived for supercompacts and supercompacts reflecting supercompacts.

Abstract prepared by James Holland.

E-mail: jch258@scarletmail.rutgers.edu

URL: https://sites.math.rutgers.edu/~jch258/assets/dis.pdf

Given an uncountable cardinal $\kappa $, we consider the question of whether subsets of the power set of $\kappa $ that are usually constructed with the help of the axiom of choice are definable by $\Sigma _1$-formulas that only use the cardinal $\kappa $ and sets of hereditary cardinality less than $\kappa $ as parameters. For limits of measurable cardinals, we prove a perfect set theorem for sets definable in this way and use it to generalize two classical nondefinability results to higher cardinals. First, we show that a classical result of Mathias on the complexity of maximal almost disjoint families of sets of natural numbers can be generalized to measurable limits of measurables. Second, we prove that for a limit of countably many measurable cardinals, the existence of a simply definable well-ordering of subsets of $\kappa $ of length at least $\kappa ^+$ implies the existence of a projective well-ordering of the reals. In addition, we determine the exact consistency strength of the nonexistence of $\Sigma _1$-definitions of certain objects at singular strong limit cardinals. Finally, we show that both large cardinal assumptions and forcing axioms cause analogs of these statements to hold at the first uncountable cardinal $\omega _1$.

We establish an equiconsistency between (1) weak indestructibility for all $\kappa +2$-degrees of strength for cardinals $\kappa $ in the presence of a proper class of strong cardinals, and (2) a proper class of cardinals that are strong reflecting strongs. We in fact get weak indestructibility for degrees of strength far beyond $\kappa +2$, well beyond the next inaccessible limit of measurables (of the ground model). One direction is proven using forcing and the other using core model techniques from inner model theory. Additionally, connections between weak indestructibility and the reflection properties associated with Woodin cardinals are discussed. This work is a part of my upcoming thesis [7].

We show that under $\mathsf {BMM}$ and “there exists a Woodin cardinal,$"$ the nonstationary ideal on $\omega _1$ cannot be defined by a $\Pi _1$ formula with parameter $A \subset \omega _1$. We show that the same conclusion holds under the assumption of Woodin’s $(\ast )$-axiom. We further show that there are universes where $\mathsf {BPFA}$ holds and $\text {NS}_{\omega _1}$ is $\Pi _1(\{\omega _1\})$-definable. Lastly we show that if the canonical inner model with one Woodin cardinal $M_1$ exists, there is a generic extension of $M_1$ in which $\text {NS}_{\omega _1}$ is saturated and $\Pi _1(\{ \omega _1\} )$-definable, and $\mathsf {MA_{\omega _1}}$ holds.

The stable core, an inner model of the form $\langle L[S],\in , S\rangle $ for a simply definable predicate S, was introduced by the first author in [8], where he showed that V is a class forcing extension of its stable core. We study the structural properties of the stable core and its interactions with large cardinals. We show that the $\operatorname {GCH} $ can fail at all regular cardinals in the stable core, that the stable core can have a discrete proper class of measurable cardinals, but that measurable cardinals need not be downward absolute to the stable core. Moreover, we show that, if large cardinals exist in V, then the stable core has inner models with a proper class of measurable limits of measurables, with a proper class of measurable limits of measurable limits of measurables, and so forth. We show this by providing a characterization of natural inner models $L[C_1, \dots , C_n]$ for specially nested class clubs $C_1, \dots , C_n$, like those arising in the stable core, generalizing recent results of Welch [29].

We prove several consistency results concerning the notion of $\omega $ -strongly measurable cardinal in $\operatorname {\mathrm {HOD}}$ . In particular, we show that is it consistent, relative to a large cardinal hypothesis weaker than $o(\kappa ) = \kappa $ , that every successor of a regular cardinal is $\omega $ -strongly measurable in $\operatorname {\mathrm {HOD}}$ .

We put in print a classical result that states that for most purposes, there is no harm in assuming the existence of saturated models in model theory. The presentation is aimed for model theorists with only basic knowledge of axiomatic set theory.

We investigate iterating the construction of $C^{*}$, the L-like inner model constructed using first order logic augmented with the “cofinality $\omega $” quantifier. We first show that $\left (C^{*}\right )^{C^{*}}=C^{*}\ne L$ is equiconsistent with $\mathrm {ZFC}$, as well as having finite strictly decreasing sequences of iterated $C^{*}$s. We then show that in models of the form $L[U]$ we get infinite decreasing sequences of length $\omega $, and that an inner model with a measurable cardinal is required for that.

We show that in many extender models, e.g., the minimal one with infinitely many Woodin cardinals or the minimal with a Woodin cardinal that is a limit of Woodin cardinals, there are no generic embeddings with critical point $\omega _1$ that resemble the stationary tower at the second Woodin cardinal. The meaning of “resemble” is made precise in the paper (see Definition 0.3).

We show that after forcing with a countable support iteration or a finite product of Sacks or splitting forcing over L, every analytic hypergraph on a Polish space admits a $\mathbf {\Delta }^1_2$ maximal independent set. This extends an earlier result by Schrittesser (see [25]). As a main application we get the consistency of $\mathfrak {r} = \mathfrak {u} = \mathfrak {i} = \omega _2$ together with the existence of a $\Delta ^1_2$ ultrafilter, a $\Pi ^1_1$ maximal independent family, and a $\Delta ^1_2$ Hamel basis. This solves open problems of Brendle, Fischer, and Khomskii [5] and the author [23]. We also show in ZFC that $\mathfrak {d} \leq \mathfrak {i}_{cl}$ , addressing another question from [5].

In providing a good foundation for mathematics, set theorists often aim to develop the strongest theories possible and avoid those theories that place undue restrictions on the capacity to possess strength. For example, adding a measurable cardinal to $ZFC$ is thought to give a stronger theory than adding $V=L$ and the latter is thought to be more restrictive than the former. The two main proponents of this style of account are Penelope Maddy and John Steel. In this paper, I’ll offer a third account that is intended to provide a simple analysis of restrictiveness based on the algebraic concept of retraction in the category of theories. I will also deliver some results and arguments that suggest some plausible alternative approaches to analyzing restrictiveness do not live up to their intuitive motivation.

Let P be a forcing notion and $G\subseteq P$ its generic subset. Suppose that we have in $V[G]$ a $\kappa{-}$ complete ultrafilter1 , 2 W over $\kappa $ . Set $U=W\cap V$ .

We investigate the mathematics of a model of the human mind which has been proposed by the psychologist Jens Mammen. Mathematical realizations of this model consists of what the first author (A.T.) has called Mammen spaces, where a Mammen space is a triple $(U,\mathcal S,\mathcal C)$, where U is a non-empty set (“the universe”), $\mathcal S$ is a perfect Hausdorff topology on U, and $\mathcal C\subseteq \mathcal P(U)$ together with $\mathcal S$ satisfy certain axioms.

We refute a conjecture put forward by Hoffmann-Jørgensen, who conjectured that the existence of a “complete” Mammen space implies the Axiom of Choice, by showing that in the first Cohen model, in which ZF holds but AC fails, there is a complete Mammen space. We obtain this by proving that in the first Cohen model, every perfect topology can be extended to a maximal perfect topology.

On the other hand, we also show that if all sets are Lebesgue measurable, or all sets are Baire measurable, then there are no complete Mammen spaces with a countable universe.

Further, we investigate two new cardinal invariants $\mathfrak u_{M}$ and $\mathfrak u_{T}$ associated with complete Mammen spaces and maximal perfect topologies, and establish some basic inequalities that are provable in ZFC. Then we show $\mathfrak u_{M}=\mathfrak u_{T}=2^{\aleph _{0}}$ follows from Martin’s Axiom, and, contrastingly, we show that $\aleph _{1}=\mathfrak u_{M}=\mathfrak u_{T}<2^{\aleph _{0}}=\aleph _{2}$ in the Baumgartner–Laver model.

Finally, consequences for psychology are discussed.

In this paper, we give an axiomatization of the ordinal number system, in the style of Dedekind’s axiomatization of the natural number system. The latter is based on a structure $(N,0,s)$ consisting of a set N, a distinguished element $0\in N$ and a function $s\colon N\to N$ . The structure in our axiomatization is a triple $(O,L,s)$ , where O is a class, L is a class function defined on all s-closed ‘subsets’ of O, and s is a class function $s\colon O\to O$ . In fact, we develop the theory relative to a Grothendieck-style universe (minus the power set axiom), as a way of bringing the natural and the ordinal cases under one framework. We also establish a universal property for the ordinal number system, analogous to the well-known universal property for the natural number system.

We analyze the hereditarily ordinal definable sets $\operatorname {HOD} $ in $M_n(x)[g]$ for a Turing cone of reals x, where $M_n(x)$ is the canonical inner model with n Woodin cardinals build over x and g is generic over $M_n(x)$ for the Lévy collapse up to its bottom inaccessible cardinal. We prove that assuming $\boldsymbol \Pi ^1_{n+2}$ -determinacy, for a Turing cone of reals x, $\operatorname {HOD} ^{M_n(x)[g]} = M_n(\mathcal {M}_{\infty } | \kappa _{\infty }, \Lambda ),$ where $\mathcal {M}_{\infty }$ is a direct limit of iterates of $M_{n+1}$ , $\delta _{\infty }$ is the least Woodin cardinal in $\mathcal {M}_{\infty }$ , $\kappa _{\infty }$ is the least inaccessible cardinal in $\mathcal {M}_{\infty }$ above $\delta _{\infty }$ , and $\Lambda $ is a partial iteration strategy for $\mathcal {M}_{\infty }$ . It will also be shown that under the same hypothesis $\operatorname {HOD}^{M_n(x)[g]} $ satisfies $\operatorname {GCH} $ .

A set of reals is universally Baire if all of its continuous preimages in topological spaces have the Baire property. ${\sf Sealing}$ is a type of generic absoluteness condition introduced by Woodin that asserts in strong terms that the theory of the universally Baire sets cannot be changed by set forcings. The ${\sf Largest\ Suslin\ Axiom}$ ( ${\sf LSA}$ ) is a determinacy axiom isolated by Woodin. It asserts that the largest Suslin cardinal is inaccessible for ordinal definable surjections. Let ${\sf LSA}$ - ${\sf over}$ - ${\sf uB}$ be the statement that in all (set) generic extensions there is a model of $\sf {LSA}$ whose Suslin, co-Suslin sets are the universally Baire sets. We outline the proof that over some mild large cardinal theory, $\sf {Sealing}$ is equiconsistent with $\sf {LSA}$ - $\sf {over}$ - $\sf {uB}$ . In fact, we isolate an exact theory (in the hierarchy of strategy mice) that is equiconsistent with both (see Definition 3.1). As a consequence, we obtain that $\sf {Sealing}$ is weaker than the theory “ $\sf {ZFC}$ + there is a Woodin cardinal which is a limit of Woodin cardinals.” This significantly improves upon the earlier consistency proof of $\sf {Sealing}$ by Woodin. A variation of $\sf {Sealing}$ , called $\sf {Tower \ Sealing}$ , is also shown to be equiconsistent with $\sf {Sealing}$ over the same large cardinal theory. We also outline the proof that if V has a proper class of Woodin cardinals, a strong cardinal, and a generically universally Baire iteration strategy, then $\sf {Sealing}$ holds after collapsing the successor of the least strong cardinal to be countable. This result is complementary to the aforementioned equiconsistency result, where it is shown that $\sf {Sealing}$ holds in a generic extension of a certain minimal universe. This theorem is more general in that no minimal assumption is needed. A corollary of this is that $\sf {LSA}$ - $\sf {over}$ - $\sf {uB}$ is not equivalent to $\sf {Sealing}$ .