{
  "id": "1307.7387",
  "version": "v1",
  "published": "2013-07-28T17:57:22.000Z",
  "updated": "2013-07-28T17:57:22.000Z",
  "title": "The large cardinals between supercompact and almost-huge",
  "authors": [
    "Norman Lewis Perlmutter"
  ],
  "categories": [
    "math.LO"
  ],
  "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}",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-07-28T17:57:22.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "03E55"
    ],
    "keywords": [
      "large cardinals",
      "supercompact cardinal",
      "woodin-for-supercompactness cardinals",
      "super-high-jump cardinals",
      "results relating high-jump cardinals"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1307.7387P"
    }
  }
}