Related papers: Characterizing large cardinals through Neeman's pu…
Assuming the P-ideal dichotomy, we attempt to isolate those cardinal characteristics of the continuum that are correlated with two well-known consequences of the proper forcing axiom. We find a cardinal invariant $\mathfrak{x}$ such that…
We introduce and study a new type of compactness principle for strong logics that, roughly speaking, infers the consistency of a theory from the consistency of its small fragments in certain outer models of the set-theoretic universe. We…
A cardinal $\lambda$ satisfies a property P robustly if, whenever $\mathbb{Q}$ is a forcing poset and $|\mathbb{Q}|^+ < \lambda$, $\lambda$ satisfies P in $V^{\mathbb{Q}}$. We study the extent to which certain reflection properties of large…
In this paper, we study the notion of a generically extendible cardinal, which is a generic version of an extendible cardinal. We prove that the generic extendibility of $\omega_1$ or $\omega_2$ has small consistency strength, but that of a…
A ccc-generically supercompact cardinal $\kappa$ can be smaller than or equal to the continuum. On the other hand, such a cardinal $\kappa$ still satisfies diverse largeness properties, like that it is a stationary limit of ccc-generically…
Exacting and ultraexacting cardinals are large cardinal numbers compatible with the Zermelo-Fraenkel axioms of set theory, including the Axiom of Choice. In contrast with standard large cardinal notions, their existence implies that the…
We introduce a hierarchy of large cardinals between weakly compact and measurable cardinals, that is closely related to the Ramsey-like cardinals introduced by Victoria Gitman, and is based on certain infinite filter games, however also has…
We give a detailed proof of the properties of the usual Prikry type forcing notion for turning a measurable cardinal into $\aleph_\omega$.
Let omega be the first infinite ordinal (or the set of all natural numbers) with the usual order <. In section 1 we show that, assuming the consistency of a supercompact cardinal, there may exist an ultrapower of omega, whose cardinality is…
Boolean ultrapowers extend the classical ultrapower construction to work with ultrafilters on any complete Boolean algebra, rather than only on a power set algebra. When they are well-founded, the associated Boolean ultrapower embeddings…
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 $\text{ZFC}_2$…
For a free ultrafilter U on omega we study several cardinal characteristics which describe part of the combinatorial structure of U. We provide various consistency results; e.g. we show how to force simultaneously many characters and many…
This paper contributes to the theory of large cardinals beyond the Kunen inconsistency, or choiceless large cardinal axioms, in the context where the Axiom of Choice is not assumed. The first part of the paper investigates a periodicity…
We introduce bounded category forcing axioms for well-behaved classes $\Gamma$. These are strong forms of bounded forcing axioms which completely decide the theory of some initial segment of the universe $H_{\lambda_\Gamma^+}$ modulo…
We study the consistency and consistency strength of various configurations concerning the cardinal characteristics $\mathfrak{s}_\theta,\mathfrak{p}_\theta,\mathfrak{g}_\theta,\mathfrak{r}_\theta,\mathfrak{t}_\theta$ at uncountable regular…
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…
Assuming an abstract comparison principle called the Ultrapower Axiom, which is motivated by the comparison process of inner model theory and generalizes the statement that the Mitchell order is linear on normal ultrafilters, we…
Remarkable cardinals were introduced by Schindler, who showed that the existence of a remarkable cardinal is equiconsistent with the assertion that the theory of $L(\mathbb R)$ is absolute for proper forcing. Here, we study the…
We study methods to obtain the consistency of forcing axioms, and particularly higher forcing axioms. We first force over a model with a supercompact cardinal $\theta>\kappa$ to get the consistency of the forcing axiom for $\kappa$-strongly…
In this paper we investigate some properties of forcing which can be considered "nice" in the context of singularizing regular cardinals to have an uncountable cofinality. We show that such forcing which changes cofinality of a regular…