{
  "id": "1203.2715",
  "version": "v2",
  "published": "2012-03-13T05:07:23.000Z",
  "updated": "2012-03-29T18:16:30.000Z",
  "title": "The Non-Axiomatizability of O-Minimality",
  "authors": [
    "Alex Rennet"
  ],
  "comment": "7 pages",
  "categories": [
    "math.LO"
  ],
  "abstract": "Fix a language L extending the language of real closed fields by at least one new predicate or function symbol. Call an L-structure R pseudo-o-minimal if it is (elementarily equivalent to) an ultraproduct of o-minimal structures. We show that for any recursive list of L-sentences \\Lambda, there is a real closed field R satisfying \\Lambda, which is not pseudo-o-minimal. In particular, there are locally o-minimal, definably complete real closed fields which are not pseudo-o-minimal. This answers negatively a question raised by Schoutens, and shows that the theory consisting of those L-sentences true in all o-minimal L-structures, called the theory of o-minimality (for L), is not recursively axiomatizable.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2012-03-29T18:16:30.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "03C64",
      "03C20"
    ],
    "keywords": [
      "o-minimality",
      "non-axiomatizability",
      "definably complete real closed fields",
      "pseudo-o-minimal",
      "o-minimal structures"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 7,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1203.2715R"
    }
  }
}