This paper appears in the Archive for Mathematical Logic. It has been published first online here http://link.springer.com/article/10.1007/s00153-014-0410-y and will soon appear in print.

This is an adaptation of many of the results from the second chapter of my dissertation into a journal article. You can read the preprint here.

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 equal to $\sup\{j(f)(\kappa) \ |\ f: \kappa \to \kappa \}$. Some of the other cardinals analyzed include the super-high-jump cardinals, almost-high-jump cardinals, Shelah-for-supercompactness cardinals, Woodin-for-supercompactness cardinals, 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.

– Vopenka cardinals are the same as Woodin-for-supercompactness cardinals.

– There are no excessively hypercompact cardinals.

Furthermore, I prove some results relating high-jump cardinals to forcing, as well as analyzing Laver functions for super-high-jump cardinals.

UPDATE (2/25/15):

Kentaro Sato wrote to me recently and informed me that he has proved the result on the equivalence of Woodin-for-supercompactness cardinals and Vopenka cardinals in his paper,

Kentaro Sato
Double helix in large large cardinals and iteration of elementary embeddings
Annals of Pure and Applied Logic
Volume 146, Issues 2–3, May 2007, Pages 199–236

What I call Woodin-for-supercompactness, he calls 0-W-huge in Def. A.1. In Cor. A.7, he shows this to be equivalent to what he calls 1-W-strong, which is also called 2-fold Woodin in Def. 9.1. In Cor. 10.6, he shows that 2-fold Woodin and 1-fold Vopenka are equivalent, with an even stronger statement, Thm. 10.5, the equivalence on the filter level.

I was not aware that Sato had proven this fact at the time that I wrote my paper.

My own modest contribution regarding the equivalence was to provide a different proof using a different theoretical framework, and to highlight the result using a different vocabulary.