{
  "id": "math/9308220",
  "version": "v1",
  "published": "1993-08-15T00:00:00.000Z",
  "updated": "1993-08-15T00:00:00.000Z",
  "title": "Consequences of arithmetic for set theory",
  "authors": [
    "Lorenz Halbeisen",
    "Saharon Shelah"
  ],
  "journal": "J. Symbolic Logic 59 (1994), 30--40",
  "categories": [
    "math.LO"
  ],
  "abstract": "In this paper, we consider certain cardinals in ZF (set theory without AC, the Axiom of Choice). In ZFC (set theory with AC), given any cardinals C and D, either C <= D or D <= C. However, in ZF this is no longer so. For a given infinite set A consider Seq(A), the set of all sequences of A without repetition. We compare |Seq(A)|, the cardinality of this set, to |P(A)|, the cardinality of the power set of A. What is provable about these two cardinals in ZF? The main result of this paper is that ZF |- for all A: |Seq(A)| not= |P(A)| and we show that this is the best possible result. Furthermore, it is provable in ZF that if B is an infinite set, then |fin(B)|<|P(B)|, even though the existence for some infinite set B^* of a function f from fin(B^*) onto P(B^*) is consistent with ZF.",
  "revisions": [
    {
      "version": "v1",
      "updated": "1993-08-15T00:00:00.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "set theory",
      "infinite set",
      "arithmetic",
      "consequences",
      "cardinality"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "1993math......8220H"
    }
  }
}