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The large cardinals between supercompact and almost-huge

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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.

  • Vopěnka 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.

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Authors and Affiliations

  1. Department of Mathematics, Engineering, and Computer Science, LaGuardia Community College, 31-10 Thomson Avenue, Room E218, Long Island City, NY, 11101, USA

    Norman Lewis Perlmutter

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  1. Norman Lewis Perlmutter
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Correspondence to Norman Lewis Perlmutter.

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Perlmutter, N.L. The large cardinals between supercompact and almost-huge. Arch. Math. Logic 54, 257–289 (2015). https://doi.org/10.1007/s00153-014-0410-y

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