{
  "id": "1503.07577",
  "version": "v1",
  "published": "2015-03-25T23:20:00.000Z",
  "updated": "2015-03-25T23:20:00.000Z",
  "title": "Definability and almost disjoint families",
  "authors": [
    "Asger Tornquist"
  ],
  "categories": [
    "math.LO"
  ],
  "abstract": "We show that there are no infinite maximal almost disjoint (\"mad\") families in Solovay's model, thus solving a long-standing problem posed by A.D.R. Mathias in 1967. We also give a new proof of Mathias' theorem that no analytic infinite almost disjoint family can be maximal, and show more generally that if Martin's Axiom holds at $\\kappa<2^{\\aleph_0}$, then no $\\kappa$-Souslin infinite almost disjoint family can be maximal. Finally we show that if $\\aleph_1^{L[a]}<\\aleph_1$, then there are no $\\Sigma^1_2[a]$ infinite mad families.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-03-25T23:20:00.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "03E05",
      "03E15",
      "03E35",
      "03E45",
      "03E50"
    ],
    "keywords": [
      "disjoint family",
      "definability",
      "infinite mad families",
      "martins axiom holds",
      "souslin infinite"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}