{
  "id": "1804.08152",
  "version": "v1",
  "published": "2018-04-22T18:32:39.000Z",
  "updated": "2018-04-22T18:32:39.000Z",
  "title": "Torsion-Free Abelian Groups are Consistently $a Δ^1_2$-complete",
  "authors": [
    "Saharon Shelah",
    "Douglas Ulrich"
  ],
  "comment": "21 pages",
  "categories": [
    "math.LO"
  ],
  "abstract": "Let $\\mbox{TFAG}$ be the theory of torsion-free abelian groups. We show that if there is no countable transitive model of $ZFC^- + \\kappa(\\omega)$ exists, then $\\mbox{TFAG}$ is $a \\Delta^1_2$-complete; in particular, this is consistent with $ZFC$. We define the $\\alpha$-ary Schr\\\"{o}der- Bernstein property, and show that $\\mbox{TFAG}$ fails the $\\alpha$-ary Schr\\\"{o}der-Bernstein property for every $\\alpha < \\kappa(\\omega)$. We leave open whether or not $\\mbox{TFAG}$ can have the $\\kappa(\\omega)$-ary Schr\\\"{o}der-Bernstein property; if it did, then it would not be $a \\Delta^1_2$-complete, and hence not Borel complete.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-04-22T18:32:39.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "03C15",
      "03C55",
      "20K20"
    ],
    "keywords": [
      "torsion-free abelian groups",
      "bernstein property",
      "leave open",
      "borel complete",
      "countable transitive model"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 21,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}