{
  "id": "1906.11949",
  "version": "v1",
  "published": "2019-06-27T20:30:23.000Z",
  "updated": "2019-06-27T20:30:23.000Z",
  "title": "The consistency strength of long projective determinacy",
  "authors": [
    "Juan P. Aguilera",
    "Sandra Müller"
  ],
  "categories": [
    "math.LO"
  ],
  "abstract": "We determine the consistency strength of determinacy for projective games of length $\\omega^2$. Our main theorem is that $\\boldsymbol\\Pi^1_{n+1}$-determinacy for games of length $\\omega^2$ implies the existence of a model of set theory with $\\omega + n$ Woodin cardinals. In a first step, we show that this hypothesis implies that there is a countable set of reals $A$ such that $M_n(A)$, the canonical inner model for $n$ Woodin cardinals constructed over $A$, satisfies $A = \\mathbb{R}$ and the Axiom of Determinacy. Then we argue how to obtain a model with $\\omega + n$ Woodin cardinal from this. We also show how the proof can be adapted to investigate the consistency strength of determinacy for games of length $\\omega^2$ with payoff in $\\Game^\\mathbb{R} \\boldsymbol\\Pi^1_1$ or with $\\sigma$-projective payoff.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-06-27T20:30:23.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "03E45",
      "03E60",
      "03E15",
      "03E55"
    ],
    "keywords": [
      "consistency strength",
      "long projective determinacy",
      "woodin cardinal",
      "set theory",
      "first step"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}