{
  "id": "1902.05890",
  "version": "v1",
  "published": "2019-02-15T17:16:03.000Z",
  "updated": "2019-02-15T17:16:03.000Z",
  "title": "Mice with finitely many Woodin cardinals from optimal determinacy hypotheses",
  "authors": [
    "Sandra Müller",
    "Ralf Schindler",
    "W. Hugh Woodin"
  ],
  "comment": "121 pages",
  "categories": [
    "math.LO"
  ],
  "abstract": "We prove the following result which is due to the third author. Let $n \\geq 1$. If $\\boldsymbol\\Pi^1_n$ determinacy and $\\Pi^1_{n+1}$ determinacy both hold true and there is no $\\boldsymbol\\Sigma^1_{n+2}$-definable $\\omega_1$-sequence of pairwise distinct reals, then $M_n^\\#$ exists and is $\\omega_1$-iterable. The proof yields that $\\boldsymbol\\Pi^1_{n+1}$ determinacy implies that $M_n^\\#(x)$ exists and is $\\omega_1$-iterable for all reals $x$. A consequence is the Determinacy Transfer Theorem for arbitrary $n \\geq 1$, namely the statement that $\\boldsymbol\\Pi^1_{n+1}$ determinacy implies $\\Game^{(n)}(<\\omega^2 - \\boldsymbol\\Pi^1_1)$ determinacy.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-02-15T17:16:03.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "optimal determinacy hypotheses",
      "woodin cardinals",
      "determinacy implies",
      "determinacy transfer theorem",
      "pairwise distinct reals"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 121,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}