2013/07/28 by Norman Lewis Perlmutter, Perlmutter, Norman Lewis
Mathematics · #03E55 #Advanced Topology and Set Theory #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Logic (math.LO) #Rings, Modules, and Algebras
paper · pdf · doi:10.48550/arxiv.1307.7387
openalex publication_date 2013/07/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 → M such that M is closed under sequences of length sup\setj(f)(κ) \st f: κ→ κ. 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. \beginitemize \item \Vopenka cardinals are the same as Woodin-for-supercompactness cardinals. \item There are no excessively hypercompact cardinals. \enditemize Furthermore, I prove some results relating high-jump cardinals to forcing, as well as analyzing Laver functions for super-high-jump cardinals. \keywordshigh-jump cardinals \and \Vopenka cardinals \and Woodin-for-supercompactness cardinals \and hypercompact cardinals \and forcing and large cardinals \and Laver functions