{
  "id": "1912.12435",
  "version": "v1",
  "published": "2019-12-28T10:26:19.000Z",
  "updated": "2019-12-28T10:26:19.000Z",
  "title": "A choice-free cardinal equality",
  "authors": [
    "Guozhen Shen"
  ],
  "comment": "12 pages",
  "categories": [
    "math.LO"
  ],
  "abstract": "For a cardinal $\\mathfrak{a}$, let $\\mathrm{fin}(\\mathfrak{a})$ be the cardinality of the set of all finite subsets of a set which is of cardinality $\\mathfrak{a}$. It is proved without the aid of the axiom of choice that for all infinite cardinals $\\mathfrak{a}$ and all natural numbers $n$, \\[ 2^{\\mathrm{fin}(\\mathfrak{a})^n}=2^{[\\mathrm{fin}(\\mathfrak{a})]^n}. \\] On the other hand, it is proved that the following statement is consistent with $\\mathsf{ZF}$: there exists an infinite cardinal $\\mathfrak{a}$ such that \\[ 2^{\\mathrm{fin}(\\mathfrak{a})}<2^{\\mathrm{fin}(\\mathfrak{a})^2}<2^{\\mathrm{fin}(\\mathfrak{a})^3}<\\dots<2^{\\mathrm{fin}(\\mathrm{fin}(\\mathfrak{a}))}. \\]",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-12-28T10:26:19.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "03E10",
      "03E25"
    ],
    "keywords": [
      "choice-free cardinal equality",
      "infinite cardinal",
      "finite subsets",
      "natural numbers",
      "cardinality"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 12,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}