{
  "id": "1904.01815",
  "version": "v1",
  "published": "2019-04-03T07:47:50.000Z",
  "updated": "2019-04-03T07:47:50.000Z",
  "title": "On supercompactness of $ω_1$",
  "authors": [
    "Daisuke Ikegami",
    "Nam Trang"
  ],
  "categories": [
    "math.LO"
  ],
  "abstract": "This paper studies structural consequences of supercompactness of $\\omega_1$ under $\\sf{ZF}$. We show that the Axiom of Dependent Choice $(\\sf{DC})$ follows from \"$\\omega_1$ is supercompact\". \"$\\omega_1$ is supercompact\" also implies that $\\sf{AD}^+$, a strengthening of the Axiom of Determinacy $(\\sf{AD})$, is equivalent to $\\sf{AD}_\\mathbb{R}$. It is shown that \"$\\omega_1$ is supercompact\" does not imply $\\sf{AD}$. The most one can hope for is Suslin co-Suslin determinacy. We show that this follows from \"$\\omega_1$ is supercompact\" and Hod Pair Capturing $(\\sf{HPC})$, an inner-model theoretic hypothesis that imposes certain smallness conditions on the universe of sets. \"$\\omega_1$ is supercompact\" on its own implies that every Suslin co-Suslin set is the projection of a determined (in fact, homogenously Suslin) set. \"$\\omega_1$ is supercompact\" also implies all sets in the Chang model have all the usual regularity properties, like Lebesgue measurability and the Baire property.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-04-03T07:47:50.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "03E55",
      "03E60",
      "03E45"
    ],
    "keywords": [
      "supercompactness",
      "paper studies structural consequences",
      "inner-model theoretic hypothesis",
      "usual regularity properties",
      "suslin co-suslin determinacy"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}