Related papers: Determinacy and Large Cardinals
I analyze the hierarchy of large cardinals between a supercompact cardinal and an almost-huge cardinal. Many of these cardinals are defined by modifying the definition of a high-jump cardinal. A high-jump cardinal is defined as the critical…
One of the main difficulty concerning the nature of the continuum is to do justice, inside the set theoretical Cantorian framework, to the classical conception (from Aristotle to Thom, via Kant, Peirce, Brentano, Husserl and Weyl) according…
It is shown that if every projective set of reals is Lebesgue measurable and has the property of Baire, if every projective set in the plane has a projective uniformization, and if Steel's K exists, then J^K_{\omega_1} \models "there are…
We introduce the notion of a critical cardinal as the critical point of sufficiently strong elementary embedding between transitive sets. Assuming the axiom of choice this is equivalent to measurability, but it is well-known that choice is…
It is shown that if there is a measurable cardinal above n Woodin cardinals and M_{n+1}^# doesn't exist then K exists. K is not fully iterable, though, but only iterable with respect to stacks of certain trees living between the Woodin…
An optimal extension of the Jensen covering lemma, within the limits imposed by Prikry forcing, is proved. If L[E] is an "iterable" weasel with no measurable cardinals, then either L[E] has "indiscernibles", or every uncountable set of…
Inspired by Zermelo's quasi-categoricity result characterizing the models of second-order Zermelo-Fraenkel set theory $\text{ZFC}_2$, we investigate when those models are fully categorical, characterized by the addition to $\text{ZFC}_2$…
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…
Cicho\'n's diagram describes the connections between combinatorial notions related to measure, category, and compactness of sets of irrational numbers. In the second part of the 2010's, Goldstern, Kellner and Shelah constructed a forcing…
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…
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…
This paper deals with different concepts for characterizing the size of mathematical objects. A game theoretic investigation and generalization of two size concepts, which can both be formulated in topological terms, is provided: the so…
n infinite two-player zero-sum game with a Borel winning set, in which the opponent's actions are monitored eventually but not necessarily immediately after they are played, is determined. The proof relies on a representation of the game as…
Justin Moore's weak club-guessing principle $\mho$ admits various possible generalizations to the second uncountable cardinal. One of them was shown to hold in ZFC by Shelah. A stronger one was shown to follow from several consequences of…
The main goal of the paper is the full proof of a cardinal inequality for a space with points $G_\delta $, obtained with the help of a long version of the Menger game. This result, which improves a similar one of Scheepers and Tall, was…
Many set theorists point to the linearity phenomenon in the hierarchy of consistency strength, by which natural theories tend to be linearly ordered and indeed well ordered by consistency strength. Why should it be linear? In this paper I…
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…
We describe an obstacle to the analysis of $\mathrm{HOD}^{L[x]}$ as a core model: Assuming sufficient large cardinals, for a Turing cone of reals $x$ there are premice $M,N$ in $\mathrm{HC}^{L[x]}$ such that the pseudo-comparison of $L[M]$…
The notion of the magnitude of a metric space was introduced by Leinster in [8] and developed in [10], [9], [11] and [16], but the magnitudes of familiar sets in Euclidean space are only understood in relatively few cases. In this paper we…
This paper establishes a number of constraints on the structure of large cardinals under strong compactness assumptions. These constraints coincide with those imposed by the Ultrapower Axiom, a principle that is expected to hold in Woodin's…