{
  "id": "1909.13372",
  "version": "v1",
  "published": "2019-09-29T21:43:57.000Z",
  "updated": "2019-09-29T21:43:57.000Z",
  "title": "The model companions of set theory",
  "authors": [
    "Giorgio Venturi",
    "Matteo Viale"
  ],
  "categories": [
    "math.LO"
  ],
  "abstract": "We show that the first order theory of $H_{\\omega_1}$ is the model companion of the first order theory of the universe of sets assuming the existence of class many Woodin cardinals, and working in a signature with predicates for all universally Baire sets of reals. We also outline some basic conditions granting the model completeness of the first order theory of $H_{\\omega_2}$ and of the axiom system $\\mathsf{ZF}+V=L$ in an appropriate language.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-09-29T21:43:57.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "model companion",
      "first order theory",
      "set theory",
      "woodin cardinals",
      "universally baire sets"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}