{
  "id": "math/9911231",
  "version": "v1",
  "published": "1999-11-29T00:02:42.000Z",
  "updated": "1999-11-29T00:02:42.000Z",
  "title": "On equivalence relations Sigma_1^1-definable over H(kappa)",
  "authors": [
    "Saharon Shelah",
    "Pauli Väisänen"
  ],
  "categories": [
    "math.LO"
  ],
  "abstract": "Let kappa be an uncountable regular cardinal. Call an equivalence relation on functions from kappa into 2 Sigma_1^1-definable over H(kappa) if there is a first order sentence F and a parameter R subseteq H(kappa) such that functions f,g:kappa --> 2 are equivalent iff for some h:kappa --> 2, the structure (H(kappa),in,R,f,g,h) satisfies F, where in, R, f, g, and h are interpretations of the symbols appearing in F. All the values mu, 1 leq mu leq kappa^+ or mu=2^kappa, are possible numbers of equivalence classes for such a Sigma_1^1-equivalence relation. Additionally, the possibilities are closed under unions of <=kappa-many cardinals and products of <kappa-many cardinals. We prove that, consistent wise, these are the only restrictions under the singular cardinal hypothesis. The result is that the possible numbers of equivalence classes of Sigma_1^1-equivalence relations might consistent wise be exactly those cardinals which are in a prearranged set, provided that the singular cardinal hypothesis holds and that some necessary conditions are fulfilled.",
  "revisions": [
    {
      "version": "v1",
      "updated": "1999-11-29T00:02:42.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "03C55",
      "03E35",
      "03C75"
    ],
    "keywords": [
      "equivalence relation",
      "singular cardinal hypothesis holds",
      "equivalence classes",
      "first order sentence",
      "leq mu leq"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "1999math.....11231S"
    }
  }
}