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 $${\sup\{{j(f)(\kappa)\,|\,f: \kappa \to \kappa}\}}$$ . Some of the other cardinals analyzed include the super-high-jump cardinals, almost-high-jump cardinals, Shelah-for-supercompactness cardinals, Woodin-for-supercompactness cardinals, Vopěnka 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. Among the results, I highlight the following. Furthermore, I prove some results relating high-jump cardinals to forcing, as well as analyzing Laver functions for super-high-jump cardinals.