{
  "id": "1712.02880",
  "version": "v1",
  "published": "2017-12-07T22:47:50.000Z",
  "updated": "2017-12-07T22:47:50.000Z",
  "title": "Universal classes near $\\aleph_1$",
  "authors": [
    "Marcos Mazari Armida",
    "Sebastien Vasey"
  ],
  "comment": "12 pages",
  "categories": [
    "math.LO"
  ],
  "abstract": "Shelah has provided sufficient conditions for an $L_{\\omega_1, \\omega}$-sentence $\\psi$ to have arbitrarily large models and for a Morley-like theorem to hold of $\\psi$. These conditions involve structural and set-theoretic assumptions on all the $\\aleph_n$'s. Using tools of Boney, Shelah, and the second author, we give assumptions on $\\aleph_0$ and $\\aleph_1$ which suffice when $\\psi$ is restricted to be universal: $\\mathbf{Theorem}$ Assume $2^{\\aleph_{0}} < 2 ^{\\aleph_{1}}$. Let $\\psi$ be a universal $\\L_{\\omega_{1}, \\omega}$-sentence. - If $\\psi$ is categorical in $\\aleph_{0}$ and $1 \\leq I(\\psi, \\aleph_{1}) < 2 ^{\\aleph_{1}}$, then $\\psi$ has arbitrarily large models and categoricity of $\\psi$ in some uncountable cardinal implies categoricity of $\\psi$ in all uncountable cardinals. - If $\\psi$ is categorical in $\\aleph_1$, then $\\psi$ is categorical in all uncountable cardinals. The theorem generalizes to the framework of $L_{\\omega_1, \\omega}$-definable tame abstract elementary classes with primes.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-12-07T22:47:50.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "03C48",
      "03C45",
      "03C52",
      "03C55",
      "03C75"
    ],
    "keywords": [
      "universal classes",
      "arbitrarily large models",
      "definable tame abstract elementary classes",
      "uncountable cardinal implies categoricity",
      "set-theoretic assumptions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 12,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}