{
  "id": "1608.04913",
  "version": "v1",
  "published": "2016-08-17T10:02:45.000Z",
  "updated": "2016-08-17T10:02:45.000Z",
  "title": "When an Equivalence Relation with All Borel Classes will be Borel Somewhere?",
  "authors": [
    "William Chan",
    "Menachem Magidor"
  ],
  "categories": [
    "math.LO"
  ],
  "abstract": "In $\\mathsf{ZFC}$, if there is a measurable cardinal with infinitely many Woodin cardinals below it, then for every equivalence relation $E \\in L(\\mathbb{R})$ on $\\mathbb{R}$ with all $\\mathbf{\\Delta}_1^1$ classes and every $\\sigma$-ideal $I$ on $\\mathbb{R}$ so that the associated forcing $\\mathbb{P}_I$ of $I^+$ $\\mathbf{\\Delta}_1^1$ subsets is proper, there exists some $I^+$ $\\mathbf{\\Delta}_1^1$ set $C$ so that $E \\upharpoonright C$ is a $\\mathbf{\\Delta}_1^1$ equivalence relation. In $\\mathsf{ZF} + \\mathsf{DC} + \\mathsf{AD}_\\mathbb{R} + V = L(\\mathscr{P}(\\mathbb{R}))$, for every equivalence relation $E$ on $\\mathbb{R}$ with all $\\mathbf{\\Delta}_1^1$ classes and every $\\sigma$-ideal $I$ on $\\mathbb{R}$ so that the associated forcing $\\mathbb{P}_I$ is proper, there is some $I^+$ $\\mathbf{\\Delta}_1^1$ set $C$ so that $E \\upharpoonright C$ is a $\\mathbf{\\Delta}_1^1$ equivalence relation.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-08-17T10:02:45.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "equivalence relation",
      "borel classes",
      "woodin cardinals"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}