{
  "id": "2404.10455",
  "version": "v1",
  "published": "2024-04-16T10:52:30.000Z",
  "updated": "2024-04-16T10:52:30.000Z",
  "title": "Berkeley Cardinals and Vopěnka's Principle",
  "authors": [
    "Marwan Salam Mohammd"
  ],
  "comment": "24 pages",
  "categories": [
    "math.LO"
  ],
  "abstract": "We introduce \"$n$-choiceless\" supercompact and extendible cardinals in Zermelo-Fraenkel set theory without the Axiom of Choice. We prove relations between these cardinals and Vop\\v{e}nka's Principle similar to those of Bagaria's work in his papers \"$C^{(n)}$-Cardinals\" and \"More on the Preservation of Large Cardinals Under Class Forcing.\" We use these relations to characterize Berkeley cardinals in terms of a restricted form of Vop\\v{e}nka's Principle. Finally, we establish the equiconsistency of the \"$n$-choiceless\" extendible cardinals with their original counterparts, and study the consistency strength of other relevant theories.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-04-16T10:52:30.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "vopěnkas principle",
      "extendible cardinals",
      "zermelo-fraenkel set theory",
      "original counterparts",
      "consistency strength"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 24,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}