{
  "id": "1501.01022",
  "version": "v1",
  "published": "2015-01-05T22:09:48.000Z",
  "updated": "2015-01-05T22:09:48.000Z",
  "title": "Incomparable $ω_1$-like models of set theory",
  "authors": [
    "Gunter Fuchs",
    "Victoria Gitman",
    "Joel David Hamkins"
  ],
  "comment": "15 pages. Commentary concerning this article can be made at http://jdh.hamkins.org/incomparable-omega-one-like-models-of-set-theory",
  "categories": [
    "math.LO"
  ],
  "abstract": "We show that the analogues of the Hamkins embedding theorems, proved for the countable models of set theory, do not hold when extended to the uncountable realm of $\\omega_1$-like models of set theory. Specifically, under the $\\diamondsuit$ hypothesis and suitable consistency assumptions, we show that there is a family of $2^{\\omega_1}$ many $\\omega_1$-like models of ZFC, all with the same ordinals, that are pairwise incomparable under embeddability; there can be a transitive $\\omega_1$-like model of ZFC that does not embed into its own constructible universe; and there can be an $\\omega_1$-like model of PA whose structure of hereditarily finite sets is not universal for the $\\omega_1$-like models of set theory.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-01-05T22:09:48.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "03Cxx"
    ],
    "keywords": [
      "set theory",
      "incomparable",
      "suitable consistency assumptions",
      "hamkins embedding theorems",
      "hereditarily finite sets"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 15,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv150101022F"
    }
  }
}