{
  "id": "1602.02633",
  "version": "v1",
  "published": "2016-02-08T16:27:21.000Z",
  "updated": "2016-02-08T16:27:21.000Z",
  "title": "Shelah's eventual categoricity conjecture in universal classes. Part II",
  "authors": [
    "Sebastien Vasey"
  ],
  "comment": "45 pages",
  "categories": [
    "math.LO"
  ],
  "abstract": "We prove that a universal class categorical in a high-enough cardinal is categorical on a tail of cardinals. As opposed to other results in the literature, we work in ZFC, do not require the categoricity cardinal to be a successor, do not assume amalgamation, and do not use large cardinals. Moreover we give an explicit bound on the \"high-enough\" threshold: $\\mathbf{Theorem}$ Let $\\psi$ be a universal $\\mathbb{L}_{\\omega_1, \\omega}$ sentence. If $\\psi$ is categorical in some $\\lambda \\ge \\beth_{\\beth_{\\omega_1}}$, then $\\psi$ is categorical in all $\\lambda' \\ge \\beth_{\\beth_{\\omega_1}}$. As a byproduct of the proof, we show that a conjecture of Grossberg holds in universal classes: $\\mathbf{Corollary}$ Let $\\psi$ be a universal $\\mathbb{L}_{\\omega_1, \\omega}$ sentence that is categorical in some $\\lambda \\ge \\beth_{\\beth_{\\omega_1}}$, then the class of models of $\\psi$ has the amalgamation property for models of size at least $\\beth_{\\beth_{\\omega_1}}$. We also establish generalizations of these two results to uncountable languages. As part of the argument, we develop machinery to transfer model-theoretic properties between two different classes satisfying a compatibility condition. This is used as a bridge between Shelah's milestone study of universal classes (which we use extensively) and a categoricity transfer of the author for abstract elementary classes that have amalgamation, are tame, and have primes over sets of the form $M \\cup \\{a\\}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-02-08T16:27:21.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "03C48",
      "03C45",
      "03C52",
      "03C55"
    ],
    "keywords": [
      "shelahs eventual categoricity conjecture",
      "universal class",
      "shelahs milestone study",
      "transfer model-theoretic properties",
      "categorical"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 45,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2016arXiv160202633V"
    }
  }
}