{
  "id": "math/0610306",
  "version": "v2",
  "published": "2006-10-10T00:22:06.000Z",
  "updated": "2007-11-01T20:46:11.000Z",
  "title": "Reals n-generic relative to some perfect tree",
  "authors": [
    "Bernard A. Anderson"
  ],
  "comment": "12 pages. Updated to final form (a few details added, minor errors corrected)",
  "journal": "Journal of Symbolic Logic, volume 73, June 2008, pages 401-411.",
  "categories": [
    "math.LO"
  ],
  "abstract": "We say that a real X is n-generic relative to a perfect tree T if X is a path through T and for all Sigma^0_n (T) sets S, there exists a number k such that either X|k is in S or for all tau in T extending X|k we have tau is not in S. A real X is n-generic relative to some perfect tree if there exists such a T. We first show that for every number n all but countably many reals are n-generic relative to some perfect tree. Second, we show that proving this statement requires ZFC^- + ``There exist infinitely many iterates of the power set of the natural numbers''. Third, we prove that every finite iterate of the hyperjump, O^(n), is not 2-generic relative to any perfect tree and for every ordinal alpha below the least lambda such that sup_{beta < lambda} (beta th admissible) = lambda, the iterated hyperjump O^(alpha) is not 5-generic relative to any perfect tree. Finally, we demonstrate some necessary conditions for reals to be 1-generic relative to some perfect tree.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2007-11-01T20:46:11.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "03D99"
    ],
    "keywords": [
      "perfect tree",
      "reals n-generic relative",
      "beta th",
      "necessary conditions",
      "ordinal alpha"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 12,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2006math.....10306A"
    }
  }
}