{
  "id": "1908.00280",
  "version": "v1",
  "published": "2019-08-01T09:03:00.000Z",
  "updated": "2019-08-01T09:03:00.000Z",
  "title": "A note on ordinal exponentiation and derivatives of normal functions",
  "authors": [
    "Anton Freund"
  ],
  "categories": [
    "math.LO"
  ],
  "abstract": "Michael Rathjen and the present author have shown that $\\Pi^1_1$-bar induction is equivalent to (a suitable formalization of) the statement that every normal function has a derivative, provably in $\\mathbf{ACA_0}$. In this note we show that the base theory can be weakened to $\\mathbf{RCA_0}$. Our argument makes crucial use of a normal function $f$ with $f(\\alpha)\\leq 1+\\alpha^2$ and $f'(\\alpha)=\\omega^{\\omega^\\alpha}$. We will also exhibit a normal function $g$ with $g(\\alpha)\\leq 1+\\alpha\\cdot 2$ and $g'(\\alpha)=\\omega^{1+\\alpha}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-08-01T09:03:00.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "03F15",
      "03F35",
      "03B30"
    ],
    "keywords": [
      "normal function",
      "ordinal exponentiation",
      "derivative",
      "michael rathjen",
      "base theory"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}