{
  "id": "2109.09678",
  "version": "v1",
  "published": "2021-09-20T16:47:00.000Z",
  "updated": "2021-09-20T16:47:00.000Z",
  "title": "An incompleteness theorem via ordinal analysis",
  "authors": [
    "James Walsh"
  ],
  "categories": [
    "math.LO"
  ],
  "abstract": "We present an analogue of G\\\"{o}del's second incompleteness theorem for systems of second-order arithmetic. Whereas G\\\"{o}del showed that sufficiently strong theories that are $\\Pi^0_1$-sound and $\\Sigma^0_1$-definable do not prove their own $\\Pi^0_1$-soundness, we prove that sufficiently strong theories that are $\\Pi^1_1$-sound and $\\Sigma^1_1$-definable do not prove their own $\\Pi^1_1$-soundness. Our proof does not involve the construction of a self-referential sentence but rather relies on ordinal analysis.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-09-20T16:47:00.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "03F15",
      "03F40"
    ],
    "keywords": [
      "ordinal analysis",
      "sufficiently strong theories",
      "second incompleteness theorem",
      "second-order arithmetic",
      "self-referential sentence"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}