{
  "id": "2108.06632",
  "version": "v1",
  "published": "2021-08-15T00:14:20.000Z",
  "updated": "2021-08-15T00:14:20.000Z",
  "title": "$I$-regularity, determinacy, and $\\infty$-Borel sets of reals",
  "authors": [
    "Daisuke Ikegami"
  ],
  "categories": [
    "math.LO"
  ],
  "abstract": "We show under $\\sf{ZF} + \\sf{DC} + \\sf{AD}_{\\mathbb{R}}$ that every set of reals is $I$-regular for any $\\sigma$-ideal $I$ on the Baire space $\\omega^{\\omega}$ such that $\\mathbb{P}_I$ is proper. This answers the question of Khomskii. We also show that the same conclusion holds under $\\sf{ZF} + \\sf{DC} + \\sf{AD}^+$ if we additionally assume that the set of Borel codes for $I$-positive sets is $\\mathbf{\\Delta}^2_1$. If we do not assume $\\sf{DC}$, the notion of properness becomes obscure as pointed out by Asper\\'{o} and Karagila. Using the notion of strong properness similar to the one introduced by Bagaria and Bosch, we show under $\\sf{ZF} + \\sf{DC}_{\\mathbb{R}}$ without using $\\sf{DC}$ that every set of reals is $I$-regular for any $\\sigma$-ideal $I$ on the Baire space $\\omega^{\\omega}$ such that $\\mathbb{P}_I$ is strongly proper assuming every set of reals is $\\infty$-Borel and there is no $\\omega_1$-sequence of distinct reals. In particular, the same conclusion holds in a Solovay model.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-08-15T00:14:20.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "03E15",
      "03E60",
      "28A05"
    ],
    "keywords": [
      "borel sets",
      "regularity",
      "baire space",
      "conclusion holds",
      "determinacy"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}