{
  "id": "1801.10420",
  "version": "v1",
  "published": "2018-01-31T12:29:04.000Z",
  "updated": "2018-01-31T12:29:04.000Z",
  "title": "The $\\mathsf{HOD}$ Hypothesis and a supercompact cardinal",
  "authors": [
    "Yong Cheng"
  ],
  "comment": "Published in Mathematical Logic Quarterly. 63, No. 5, 462-472 (2017)",
  "categories": [
    "math.LO"
  ],
  "abstract": "In this paper, we prove that: if $\\kappa$ is supercompact and the $\\mathsf{HOD}$ Hypothesis holds, then there is a proper class of regular cardinals in $V_{\\kappa}$ which are measurable in $\\mathsf{HOD}$. Woodin also proved this result. As a corollary, we prove Woodin's Local Universality Theorem. This work shows that under the assumption of the $\\mathsf{HOD}$ Hypothesis and supercompact cardinals, large cardinals in $\\mathsf{V}$ are reflected to be large cardinals in $\\mathsf{HOD}$ in a local way, and reveals the huge difference between $\\mathsf{HOD}$-supercompact cardinals and supercompact cardinals under the $\\mathsf{HOD}$ Hypothesis.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-01-31T12:29:04.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "03E55",
      "03E99"
    ],
    "keywords": [
      "supercompact cardinal",
      "large cardinals",
      "woodins local universality theorem",
      "hypothesis holds",
      "proper class"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}