{
  "id": "2402.07136",
  "version": "v2",
  "published": "2024-02-11T09:33:36.000Z",
  "updated": "2024-11-22T14:33:10.000Z",
  "title": "On the $Π^1_2$ consequences of $Π^1_1$-$\\mathsf{CA}_0$",
  "authors": [
    "Yudai Suzuki",
    "Keita Yokoyama"
  ],
  "categories": [
    "math.LO"
  ],
  "abstract": "In this paper, we introduce a hierarchy dividing the set $\\{\\sigma \\in \\Pi^1_2 : \\Pi^1_1$-$\\mathsf{CA}_0 \\vdash \\sigma\\}$. Then, we give some characterizations of this set using weaker variants of some principles equivalent to $\\Pi^1_1$-$\\mathsf{CA}_0$: leftmost path principle, Ramsey's theorem for $\\Sigma^0_n$ classes of $[\\mathbb{N}]^{\\mathbb{N}}$ and determinacy for $(\\Sigma^0_1)_n$ classes of $\\mathbb{N}^{\\mathbb{N}}$.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2024-11-22T14:33:10.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "consequences",
      "leftmost path principle",
      "weaker variants",
      "principles equivalent",
      "ramseys theorem"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}