The large cardinals between supercompact and almost-huge

Abstract

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 point of an elementary embedding $j: V \to M$ such that $M$ is closed under sequences of length $\sup\set{j(f)(\kappa) \st f: \kappa \to \kappa}$. Some of the other cardinals analyzed include the super-high-jump cardinals, almost-high-jump cardinals, Shelah-for-supercompactness cardinals, Woodin-for-supercompactness cardinals, \Vopenka\ cardinals, hypercompact cardinals, and enhanced supercompact cardinals. I organize these cardinals in terms of consistency strength and implicational strength. I also analyze the superstrong cardinals, which are weaker than supercompact cardinals but are related to high-jump cardinals. Two of my most important results are as follows. \begin{itemize} \item \Vopenka\ cardinals are the same as Woodin-for-supercompactness cardinals. \item There are no excessively hypercompact cardinals. \end{itemize} Furthermore, I prove some results relating high-jump cardinals to forcing, as well as analyzing Laver functions for super-high-jump cardinals. \keywords{high-jump cardinals \and \Vopenka\ cardinals \and Woodin-for-supercompactness cardinals \and hypercompact cardinals \and forcing and large cardinals \and Laver functions}