{
  "id": "2006.12111",
  "version": "v1",
  "published": "2020-06-22T10:07:15.000Z",
  "updated": "2020-06-22T10:07:15.000Z",
  "title": "Well ordering principles and $Π^1_4$-statements: a pilot study",
  "authors": [
    "Anton Freund"
  ],
  "categories": [
    "math.LO"
  ],
  "abstract": "In previous work, the author has shown that $\\Pi^1_1$-induction along $\\mathbb N$ is equivalent to a suitable formalization of the statement that every normal function on the ordinals has a fixed point. More precisely, this was proved for a representation of normal functions in terms of J.-Y. Girard's dilators, which are particularly uniform transformations of well orders. The present paper works on the next type level and considers uniform transformations of dilators, which are called $2$-ptykes. We show that $\\Pi^1_2$-induction along $\\mathbb N$ is equivalent to the existence of fixed points for all $2$-ptykes that satisfy a certain normality condition. Beyond this specific result, the paper paves the way for the analysis of further $\\Pi^1_4$-statements in terms of well ordering principles.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-06-22T10:07:15.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "03B30",
      "03D65",
      "03F15",
      "03F35"
    ],
    "keywords": [
      "ordering principles",
      "pilot study",
      "normal function",
      "fixed point",
      "equivalent"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}