{
  "id": "math/9209205",
  "version": "v1",
  "published": "1992-09-15T00:00:00.000Z",
  "updated": "1992-09-15T00:00:00.000Z",
  "title": "Perfect sets of random reals",
  "authors": [
    "Jörg Brendle",
    "Haim Judah"
  ],
  "categories": [
    "math.LO"
  ],
  "abstract": "We discuss the relationship between perfect sets of random reals, dominating reals, and the product of two copies of the random algebra B. Recall that B is the algebra of Borel sets of 2^omega modulo the null sets. Also given two models M subseteq N of ZFC, we say that g in omega^omega cap N is a dominating real over M iff forall f in omega^omega cap M there is m in omega such that forall n geq m (g(n) > f(n)); and r in 2^omega cap N is random over M iff r avoids all Borel null sets coded in M iff r is determined by some filter which is B-generic over M. We show that there is a ccc partial order P which adds a perfect set of random reals without adding a dominating real, thus answering a question asked by the second author in joint work with T. Bartoszynski and S. Shelah some time ago. The method of the proof of this result yields also that B times B does not add a dominating real. By a different argument we show that B times B does not add a perfect set of random reals (this answers a question that A. Miller asked during the logic year at MSRI).",
  "revisions": [
    {
      "version": "v1",
      "updated": "1992-09-15T00:00:00.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "perfect set",
      "random reals",
      "dominating real",
      "ccc partial order",
      "borel null sets"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "1992math......9205B"
    }
  }
}