{
  "id": "1804.04490",
  "version": "v1",
  "published": "2018-04-12T13:21:12.000Z",
  "updated": "2018-04-12T13:21:12.000Z",
  "title": "On Dividing by Two in Constructive Mathematics",
  "authors": [
    "Andrew Swan"
  ],
  "categories": [
    "math.LO",
    "math.CO"
  ],
  "abstract": "A classic result due to Bernstein states that in set theory with classical logic, but without the axiom of choice, for all sets $X$ and $Y$, if $X \\times 2 \\cong Y \\times 2$ then also $X \\cong Y$. We show that this cannot be done in constructive mathematics by giving some examples of toposes where it fails.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-04-12T13:21:12.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "03F50",
      "03E10",
      "03G30"
    ],
    "keywords": [
      "constructive mathematics",
      "set theory",
      "classic result",
      "bernstein states"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}