Related papers: Measures in Mice

We construct a model of the form $L[A,U]$ that exhibits the simplest structural behavior of $\sigma$-complete ultrafilters in a model of set theory with a single measurable cardinal $\kappa$ , yet satisfies $2^\kappa = \kappa^{++}$. This…

Large cardinals arising from the existence of arbitrarily long end elementary extension chains over models of set theory are studied here. In particular, we show that the large cardinals obtained that way (`Unfoldable cardinals') behave as…

This paper explores several topics related to Woodin's HOD conjecture. We improve the large cardinal hypothesis of Woodin's HOD dichotomy theorem from an extendible cardinal to a strongly compact cardinal. We show that assuming there is a…

The consistency of the theory $\mathsf{ZF} + \mathsf{AD}_{\mathbb{R}} + {}$``every set of reals is universally Baire'' is proved relative to $\mathsf{ZFC} + {}$``there is a cardinal that is a limit of Woodin cardinals and of strong…

Let GCH hold and let $j:V\longrightarrow M$ be a definable elementary embedding such that $crit(j)=\kappa$, $^{\kappa}M\subseteq M$ and $\kappa^{++}=\kappa_{M}^{++}$. H. Woodin proved that there is a cofinality preserving generic extension…

We study connections between definability in generalized descriptive set theory and large cardinals, under ZFC. We show that if $\kappa$ is a limit of measurables then there is no wellorder of a subset of $P(\kappa)$ of length…

We prove a result concerning elementary embeddings of the set-theoretic universe into itself (Reinhardt embeddings) and functions on ordinals that "eventually dominate" such embeddings. We apply that result to show the existence of…

Let $M$ be a short extender mouse. We prove that if $E\in M$ and $M$ satisfies "$E$ 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…

The solution to fine tuning is one of the principal motivations for supersymmetry. However constraints on the parameter space of the Minimal Supersymmetric Standard Model (MSSM) suggest it may also require fine tuning (although to a much…

We define a weak iterability notion that is sufficient for a number of arguments concerning $\Sigma_1$-definability at uncountable regular cardinals. In particular we give its exact consistency strength firstly in terms of the second…

We make use of some observations on the core model, for example assuming $V=L [ E ]$, and that there is no inner model with a Woodin cardinal, and $M$ is an inner model with the same cardinals as $V$, then $V=M$. We conclude in this latter…

In this paper we explore a connection between descriptive set theory and inner model theory. From descriptive set theory, we will take a countable, definable set of reals, A. We will then show that A is equal to the reals of M, where M is a…

This brief survey comes from the slides of a seminar I gave to philosophy of mathematics students. I will present some different characterizations of Woodin cardinals, including the one obtained by Ernest Schimmerling in [6]. I will try to…

Let $C\subseteq M$ be stably embedded in a structure $\cM=(M;\dots)$. We consider {\em Fubini measures} on the subcategory $\Def(C)$ of the category $\Def(\cM)$ of definable sets in $\cM$, with ``Fubini" signaling good behaviour in…

We answer a question of Moore by building a forcing extension satisfying measuring together with CH. The construction works over any model of ZFC and can be described as a forcing iteration with countable structures as side conditions and…

We show, assuming the consistency of one measurable cardinal, that it is consistent for there to be exactly kappa+ many normal measures on the least measurable cardinal kappa. This answers a question of Stewart Baldwin. The methods…

We introduce exacting cardinals and a strengthening of these, ultraexacting cardinals. These are natural large cardinals defined equivalently as weak forms of rank-Berkeley cardinals, strong forms of J\'onsson cardinals, or in terms of…

All spaces below are $T_0$ and crowded (i.e. have no isolated points). For $n \le \omega$ let $M(n)$ be the statement that there are $n$ measurable cardinals and $\Pi(n)$ ($\Pi^+(n)$) that there are $n+1$ (0-dimensional $T_2$) spaces whose…

We define a new inner model C2(omega) based on the fragment of second order logic in which second order variables range over countable subsets of the domain. We compare C2(omega) to the previously studied inner model C(aa). We argue that…

We determine the consistency strength of determinacy for projective games of length $\omega^2$. Our main theorem is that $\boldsymbol\Pi^1_{n+1}$-determinacy for games of length $\omega^2$ implies the existence of a model of set theory with…