相关论文: Extensions with the approximation and cover proper…

Many of the most common reverse Easton iterations found in the large cardinal context, such as the Laver preparation, admit a gap at some small delta in the sense that they factor as P*Q, where P has size less than delta and Q is forced to…

Assume ZF (without the Axiom of Choice). Let $j:V_\varepsilon\to V_\delta$ be a non-trivial $\in$-cofinal $\Sigma_1$-elementary embedding, where $\varepsilon,\delta$ are limit ordinals. We prove some restrictions on the constructibility of…

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…

Assume ZF($j$) and there is a Reinhardt cardinal, as witnessed by the elementary embedding $j:V\to V$. We investigate the linear iterates $(N_{\alpha},j_{\alpha})$ of $(V,j)$, and their relationship to $(V,j)$, forcing and definability,…

An elementary embedding $j:M\rightarrow N$ between two inner models of ZFC is cardinal preserving if $M$ and $N$ correctly compute the class of cardinals. We look at the case $N=V$ and show that there is no nontrivial cardinal preserving…

The mantle is the intersection of all ground models of $V$. We show that if there exists an extendible cardinal then the mantle is a ground model of $V$.

Assuming that there is no inner model with a Woodin cardinal, we obtain a characterization of $\lambda$-tall cardinals in extender models that are iterable. In particular we prove that in such extender models, a cardinal $\kappa$ is a tall…

In the first part of this paper, we explore the possibility for a very large cardinal $\kappa$ to carry a $\kappa$-complete ultrafilter without Galvin's property. In this context, we prove the consistency of every ground model…

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…

Superstrong cardinals are never Laver indestructible. Similarly, almost huge cardinals, huge cardinals, superhuge cardinals, rank-into-rank cardinals, extendible cardinals, 1-extendible cardinals, 0-extendible cardinals, weakly superstrong…

We show relative to strong hypotheses that patterns of compact cardinals in the universe, where a compact cardinal is one which is either strongly compact or supercompact, can be virtually arbitrary. Specifically, we prove if V is a model…

The landmark Levy-Solovay Theorem limits the kind of large cardinal embeddings that can exist in a small forcing extension. Here I announce a generalization of this theorem to a broad new class of forcing notions. One consequence is that…

We strengthen a result of Bagaria and Magidor~\cite{MR3152715} about the relationship between large cardinals and torsion classes of abelian groups, and prove that (1) the \emph{Maximum Deconstructibility} principle introduced in…

Suppose $\kappa$ is $\lambda$-supercompact witnessed by an elementary embedding $j:V\rightarrow M$ with critical point $\kappa$, and further suppose that $F$ is a function from the class of regular cardinals to the class of cardinals…

We show that many large cardinal notions can be characterized in terms of the existence of certain elementary embeddings between transitive set-sized structures, that map their critical point to the large cardinal in question. In…

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…

For each natural number $n$, let $C^{(n)}$ be the closed and unbounded proper class of ordinals $\alpha$ such that $V_\alpha$ is a $\Sigma_n$ elementary substructure of $V$. We say that $\kappa$ is a \emph{$C^{(n)}$-cardinal} if it is the…

The existence of End Elementary Extensions of models M of ZFC is related to the ordinal height of M, according to classical results due to Keisler, Morley and Silver. In this paper, we further investigate the connection between the height…

We show that Weak Vop\v{e}nka's Principle, which is the statement that the opposite category of ordinals cannot be fully embedded into the category of graphs, is equivalent to the large cardinal principle Ord is Woodin, which says that for…

We show that supercompactness and strong compactness can be equivalent even as properties of pairs of regular cardinals. Specifically, we show that if V models ZFC + GCH is a given model (which in interesting cases contains instances of…