{
  "id": "2012.02306",
  "version": "v1",
  "published": "2020-12-03T22:13:25.000Z",
  "updated": "2020-12-03T22:13:25.000Z",
  "title": "The ultrafilter and almost disjointness numbers",
  "authors": [
    "Osvaldo Guzman",
    "Damjan Kalajdzievski"
  ],
  "categories": [
    "math.LO"
  ],
  "abstract": "We prove that every \\textsf{MAD }family can be destroyed by a proper forcing that preserves $P$-points. With this result, we prove that it is consistent that $\\omega_{1}=\\mathfrak{u}<\\mathfrak{a,}$ solving a nearly 20 year old problem of Shelah and a problem of Brendle. We will also present a simple proof of a result of Blass and Shelah that the inequality $\\mathfrak{u<s}$ is consistent.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-12-03T22:13:25.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "03E17",
      "03E35"
    ],
    "keywords": [
      "disjointness numbers",
      "ultrafilter",
      "consistent",
      "old problem"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}