{
  "id": "2110.12051",
  "version": "v2",
  "published": "2021-10-22T20:34:54.000Z",
  "updated": "2025-05-13T15:48:23.000Z",
  "title": "Varsovian models II",
  "authors": [
    "Grigor Sargsyan",
    "Ralf Schindler",
    "Farmer Schlutzenberg"
  ],
  "comment": "110 pages. Corrected acknowledgements, corrected/updated references and affiliation of 3rd author",
  "categories": [
    "math.LO"
  ],
  "abstract": "Assume the existence of sufficent large cardinals. Let $M_{\\mathrm{sw}n}$ be the minimal iterable proper class $L[E]$ model satisfying \"there are $\\delta_0<\\kappa_0<\\ldots<\\delta_{n-1}<\\kappa_{n-1}$ such that the $\\delta_i$ are Woodin cardinals and the $\\kappa_i$ are strong cardinals\". Let $M=M_{\\mathrm{sw}2}$. We identify an inner model $\\mathscr{V}_2^M$ of $M$, which is a proper class model satisfying \"there are 2 Woodin cardinals\", and is iterable both in $V$ and in $M$, and closed under its own iteration strategy. The construction also yields significant information about the extent to which $M$ knows its own iteration strategy. We characterize the universe of $\\mathscr{V}_2^M$ as the mantle and the least ground of $M$, and as $\\mathrm{HOD}^{M[G]}$ for $G\\subseteq\\mathrm{Coll}(\\omega,\\lambda)$ being $M$-generic with $\\lambda$ sufficiently large. These results correspond to facts already known for $M_{\\mathrm{sw}1}$, and the proofs are an elaboration of those, but there are substantial new issues and new methods used to handle them.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2025-05-13T15:48:23.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "03E45",
      "03E55",
      "03E40"
    ],
    "keywords": [
      "varsovian models",
      "iteration strategy",
      "woodin cardinals",
      "yields significant information",
      "minimal iterable proper class"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 110,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}