{
  "id": "1609.05411",
  "version": "v1",
  "published": "2016-09-18T01:10:08.000Z",
  "updated": "2016-09-18T01:10:08.000Z",
  "title": "Determinacy from strong compactness of $ω_1$",
  "authors": [
    "Nam Trang",
    "Trevor Wilson"
  ],
  "categories": [
    "math.LO"
  ],
  "abstract": "In the absence of the Axiom of Choice, the \"small\" cardinal $\\omega_1$ can exhibit properties more usually associated with large cardinals, such as strong compactness and supercompactness. For a local version of strong compactness, we say that $\\omega_1$ is $X$-strongly compact (where $X$ is any set) if there is a fine, countably complete measure on $\\mathcal{P}_{\\omega_1}(X)$. Working in $\\mathsf{ZF} + \\mathsf{DC}$, we prove that the $\\mathcal{P}(\\omega_1)$-strong compactness and $\\mathcal{P}(\\mathbb{R})$-strong compactness of $\\omega_1$ are equiconsistent with $\\mathsf{AD}$ and $\\mathsf{AD}_\\mathbb{R} + \\mathsf{DC}$ respectively, where $\\mathsf{AD}$ denotes the Axiom of Determinacy and $\\mathsf{AD}_\\mathbb{R}$ denotes the Axiom of Real Determinacy. The $\\mathcal{P}(\\mathbb{R})$-supercompactness of $\\omega_1$ is shown to be slightly stronger than $\\mathsf{AD}_\\mathbb{R} + \\mathsf{DC}$, but its consistency strength is not computed precisely. An equiconsistency result at the level of $\\mathsf{AD}_\\mathbb{R}$ without $\\mathsf{DC}$ is also obtained.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-09-18T01:10:08.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "03E45",
      "03E15",
      "03E60"
    ],
    "keywords": [
      "strong compactness",
      "large cardinals",
      "supercompactness",
      "equiconsistency result",
      "countably complete measure"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}