{
  "id": "1909.10603",
  "version": "v1",
  "published": "2019-09-23T20:18:06.000Z",
  "updated": "2019-09-23T20:18:06.000Z",
  "title": "Incompleteness and Jump Hierarchies",
  "authors": [
    "Patrick Lutz",
    "James Walsh"
  ],
  "comment": "10 pages",
  "categories": [
    "math.LO"
  ],
  "abstract": "This paper is an investigation of the relationship between G\\\"odel's second incompleteness theorem and the well-foundedness of jump hierarchies. It follows from a classic theorem of Spector's that the relation $\\{(A,B) \\in \\mathbb{R}^2 : \\mathcal{O}^A \\leq_H B\\}$ is well-founded. We provide an alternative proof of this fact that uses G\\\"odel's second incompleteness theorem instead of the theory of admissible ordinals. We then derive a semantic version of the second incompleteness theorem, originally due to Mummert and Simpson, from this result. Finally, we turn to the calculation of the ranks of reals in this well-founded relation. We prove that, for any $A\\in\\mathbb{R}$, if the rank of $A$ is $\\alpha$, then $\\omega_1^A$ is the $(1 + \\alpha)^{\\text{th}}$ admissible ordinal. It follows, assuming suitable large cardinal hypotheses, that, on a cone, the rank of $X$ is $\\omega_1^X$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-09-23T20:18:06.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "jump hierarchies",
      "second incompleteness theorem",
      "admissible ordinal",
      "assuming suitable large cardinal hypotheses",
      "semantic version"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 10,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}