{
  "id": "1810.09195",
  "version": "v1",
  "published": "2018-10-22T12:02:17.000Z",
  "updated": "2018-10-22T12:02:17.000Z",
  "title": "More on the preservation of large cardinals under class forcing",
  "authors": [
    "Bagaria Joan",
    "Poveda Alejandro"
  ],
  "categories": [
    "math.LO"
  ],
  "abstract": "We introduce first the large-cardinal notion of $\\Sigma_n$-supercompactness as a higher-level analog of the well-known Magidor's characterization of supercompact cardinals, and show that a cardinal is $C^{(n)}$-extendible if and only if it is $\\Sigma_{n+1}$-supercompact. This yields a new characterization of $C^{(n)}$-extendible cardinals which underlines their role as natural milestones in the region of the large-cardinal hierarchy between the first supercompact cardinal and Vop\\v{e}nka's Principle ($\\rm{VP}$). We then develop a general setting for the preservation of $\\Sigma_n$-supercompact cardinals under class forcing iterations. As a result we obtain new proofs of the consistency of the GCH with $C^{(n)}$-extendible cardinals (cf.~\\cite{Tsa13}) and the consistency of $\\rm{VP}$ with the GCH (cf.~\\cite{Broo}). Further, we show that $C^{(n)}$-extendible cardinals are preserved after forcing with standard Easton class forcing iterations for any $\\Pi_1$-definable possible behaviour of the power-set function on regular cardinals, and that $\\rm{VP}$ is preserved by any definable such iteration. Finally, we show that, assuming the GCH, the class forcing iteration of Cummings-Foreman-Magidor forcing notions for forcing $\\diamondsuit_{\\kappa^+}^+$ at every $\\kappa$ (\\cite{CummingsSquare}) preserves $C^{(n)}$-extendible cardinals, hence it also preserves $\\rm{VP}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-10-22T12:02:17.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "large cardinals",
      "extendible cardinals",
      "preservation",
      "standard easton class forcing iterations",
      "first supercompact cardinal"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}