{
  "id": "1811.08514",
  "version": "v1",
  "published": "2018-11-20T22:20:18.000Z",
  "updated": "2018-11-20T22:20:18.000Z",
  "title": "Tanaka's Theorem Revisited",
  "authors": [
    "Saeideh Bahrami"
  ],
  "comment": "15 pages",
  "journal": "Archive for Mathematical Logic (2020)",
  "doi": "10.1007/s00153-020-00720-z",
  "categories": [
    "math.LO"
  ],
  "abstract": "Tanaka (1997) proved a powerful generalization of Friedman's self-embedding theorem that states that given a countable nonstandard model $(\\mathcal{M},\\mathcal{A})$ of the subsystem $\\mathrm{WKL}_{0}$ of second order arithmetic, and any element $m$ of $\\mathcal{M}$, there is a self-embedding $j$ of $(\\mathcal{M},\\mathcal{A})$ onto a proper initial segment of itself such that $j$ fixes every predecessor of $m$. Here we extend Tanaka's work by establishing the following results for a countable nonstandard model $(\\mathcal{M},\\mathcal{A})$ of $\\mathrm{WKL}_{0} $ and a proper cut $\\mathrm{I}$ of $\\mathcal{M}$: Theorem A. The following conditions are equivalent: (a) $\\mathrm{I}$ is closed under exponentiation. (b) There is a self-embedding $j$ of $(\\mathcal{M},\\mathcal{A})$ onto a proper initial segment of itself such that $I$ is the longest initial segment of fixed points of $j$. Theorem B. The following conditions are equivalent: (a) $\\mathrm{I}$ is a strong cut of $\\mathcal{M} $ and $\\mathrm{I}\\prec _{\\Sigma _{1}}\\mathcal{M}.$ (b) There is a self-embedding $j$ of $(\\mathcal{M},\\mathcal{A})$ onto a proper initial segment of itself such that $\\mathrm{I} $ is the set of all fixed points of $j$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-11-20T22:20:18.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "03H15",
      "03F35",
      "03C62"
    ],
    "keywords": [
      "proper initial segment",
      "tanakas theorem",
      "countable nonstandard model",
      "second order arithmetic",
      "extend tanakas work"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "publisher": "Springer"
    },
    "note": {
      "typesetting": "TeX",
      "pages": 15,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}