{
  "id": "1812.00960",
  "version": "v1",
  "published": "2018-12-03T18:33:15.000Z",
  "updated": "2018-12-03T18:33:15.000Z",
  "title": "An Easton-like Theorem for Zermelo-Fraenkel Set Theory with the Axiom of Dependent Choice",
  "authors": [
    "Anne Fernengel",
    "Peter Koepke"
  ],
  "categories": [
    "math.LO"
  ],
  "abstract": "We show that in the theory ZF + DC + for every cardinal {\\lambda}, the set of infinite subsets of {\\lambda} is well-ordered (i.e., Shelah's AX4), the {\\theta}-function measuring the surjective size of the powersets P({\\kappa}) can take almost arbitrary values on any set of uncountable cardinals. This complements our results from [FK16], where we prove that in ZF (without DC), any possible behavior of the {\\theta}-function can be realized; and answers a question of Shelah in [She16], where he emphasizes that ZF + DC + AX4 is a reasonable theory, where much of set theory and combinatorics is possible.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-12-03T18:33:15.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "zermelo-fraenkel set theory",
      "dependent choice",
      "easton-like theorem",
      "theory zf",
      "infinite subsets"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}