{
  "id": "1510.07759",
  "version": "v1",
  "published": "2015-10-27T03:14:28.000Z",
  "updated": "2015-10-27T03:14:28.000Z",
  "title": "Scott ranks of models of a theory",
  "authors": [
    "Matthew Harrison-Trainor"
  ],
  "comment": "36 pages",
  "categories": [
    "math.LO"
  ],
  "abstract": "The Scott rank of a countable structure is a measure, coming from the proof of Scott's isomorphism theorem, of the complexity of that structure. The Scott spectrum of a theory (by which we mean a sentence of $\\mathcal{L}_{\\omega_1 \\omega}$) is the set of Scott ranks of countable models of that theory. In $ZFC + PD$ we give a descriptive-set-theoretic classification of the sets of ordinals which are the Scott spectrum of a theory: they are particular $\\boldsymbol{\\Sigma}^1_1$ classes of ordinals. Our investigation of Scott spectra leads to the resolution (in $ZFC$) of a number of open problems about Scott ranks. We answer a question of Montalb\\'an by showing, for each $\\alpha < \\omega_1$, that there is a $\\Pi^{\\mathtt{in}}_2$ theory with no models of Scott rank less than $\\alpha$. We also answer a question of Knight and Calvert by showing that there are computable models of high Scott rank which are not computably approximable by models of low Scott rank. Finally, we answer a question of Sacks and Marker by showing that $\\delta^1_2$ is the least ordinal $\\alpha$ such that if the models of a computable theory $T$ have Scott rank bounded below $\\omega_1$, then their Scott ranks are bounded below $\\alpha$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-10-27T03:14:28.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "03D45",
      "03C57"
    ],
    "keywords": [
      "scott spectrum",
      "scotts isomorphism theorem",
      "low scott rank",
      "high scott rank",
      "open problems"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 36,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv151007759H"
    }
  }
}