{
  "id": "1712.00769",
  "version": "v1",
  "published": "2017-12-03T13:47:31.000Z",
  "updated": "2017-12-03T13:47:31.000Z",
  "title": "Non-uniformizable sets with countable cross-sections on a given level of the projective hierarchy",
  "authors": [
    "Vladimir Kanovei",
    "Vassily Lyubetsky"
  ],
  "comment": "Within a few weeks, the Russian text will be replaced by an English translation",
  "categories": [
    "math.LO"
  ],
  "abstract": "A generic extension of $L$, the constructible universe, is defined, in which it is true for a given $n\\ge2$ that there exists a non-ROD-uniformizable lightface $\\varPi^1_n$ set in $\\mathbb R\\times\\mathbb R$ with all vertical cross-sections being Vitali classes (hence, countable sets of reals), and in the same time every boldface $\\bf\\Sigma^1_n$ set with countable cross-sections is uniformizable by a boldface $\\bf\\Delta^1_{n+1}$ set. Thus it is true in this model that the ROD-uniformization for sets with countable cross-sections first fails exactly at a given projective level.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-12-03T13:47:31.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "03E15",
      "03E35",
      "03E47"
    ],
    "keywords": [
      "projective hierarchy",
      "non-uniformizable sets",
      "countable cross-sections first fails",
      "generic extension",
      "vitali classes"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}