{
  "id": "1906.04351",
  "version": "v1",
  "published": "2019-06-11T02:10:05.000Z",
  "updated": "2019-06-11T02:10:05.000Z",
  "title": "Bounds on Scott Ranks of Some Polish Metric Spaces",
  "authors": [
    "William Chan"
  ],
  "categories": [
    "math.LO"
  ],
  "abstract": "If $\\mathcal{N}$ is a proper Polish metric space and $\\mathcal{M}$ is any countable dense submetric space of $\\mathcal{N}$, then the Scott rank of $\\mathcal{N}$ in the natural first order language of metric spaces is countable and in fact at most $\\omega_1^{\\mathcal{M}} + 1$, where $\\omega_1^{\\mathcal{M}}$ is the Church-Kleene ordinal of $\\mathcal{M}$ (construed as a subset of $\\omega$) which is the least ordinal with no presentation on $\\omega$ computable from $\\mathcal{M}$. If $\\mathcal{N}$ is a rigid Polish metric space and $\\mathcal{M}$ is any countable dense submetric space, then the Scott rank of $\\mathcal{N}$ is countable and in fact less than $\\omega_1^{\\mathcal{M}}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-06-11T02:10:05.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "scott rank",
      "countable dense submetric space",
      "natural first order language",
      "rigid polish metric space",
      "proper polish metric space"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}