{
  "id": "2306.06471",
  "version": "v1",
  "published": "2023-06-10T15:47:08.000Z",
  "updated": "2023-06-10T15:47:08.000Z",
  "title": "Arrow's theorem, ultrafilters, and reverse mathematics",
  "authors": [
    "Benedict Eastaugh"
  ],
  "comment": "21 pages",
  "categories": [
    "math.LO"
  ],
  "abstract": "This paper initiates the reverse mathematics of social choice theory, studying Arrow's impossibility theorem and related results including Fishburn's possibility theorem and the Kirman--Sondermann theorem within the framework of reverse mathematics. We formalise fundamental notions of social choice theory in second-order arithmetic, yielding a definition of countable society which is tractable in $\\mathsf{RCA}_0$. We then show that the Kirman--Sondermann analysis of social welfare functions can be carried out in $\\mathsf{RCA}_0$. This approach yields a proof of Arrow's theorem in $\\mathsf{RCA}_0$, and thus in $\\mathrm{PRA}$, since Arrow's theorem can be formalised as a $\\Pi^0_1$ sentence. Finally we show that Fishburn's possibility theorem for countable societies is equivalent to $\\mathsf{ACA}_0$ over $\\mathsf{RCA}_0$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-06-10T15:47:08.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "03B30",
      "03F35",
      "91B12",
      "91B14"
    ],
    "keywords": [
      "reverse mathematics",
      "arrows theorem",
      "fishburns possibility theorem",
      "social choice theory",
      "ultrafilters"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 21,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}