{
  "id": "2104.10341",
  "version": "v1",
  "published": "2021-04-21T03:42:45.000Z",
  "updated": "2021-04-21T03:42:45.000Z",
  "title": "On the absoluteness of $\\aleph_1$-freeness",
  "authors": [
    "Daniel Herden",
    "Alexandra V. Pasi"
  ],
  "categories": [
    "math.GR",
    "math.LO"
  ],
  "abstract": "$\\aleph_1$-free groups, abelian groups for which every countable subgroup is free, exhibit a number of interesting algebraic and set-theoretic properties. In this paper, we give a complete proof that the property of being $\\aleph_1$-free is absolute; that is, if an abelian group $G$ is $\\aleph_1$-free in some transitive model $\\textbf{M}$ of ZFC, then it is $\\aleph_1$-free in any transitive model of ZFC containing $G$. The absoluteness of $\\aleph_1$-freeness has the following remarkable consequence: an abelian group $G$ is $\\aleph_1$-free in some transitive model of ZFC if and only if it is (countable and) free in some model extension. This set-theoretic characterization will be the starting point for further exploring the relationship between the set-theoretic and algebraic properties of $\\aleph_1$-free groups. In particular, this paper will demonstrate how proofs may be dramatically simplified using model extensions for $\\aleph_1$-free groups.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-04-21T03:42:45.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "free groups",
      "abelian group",
      "absoluteness",
      "transitive model",
      "model extension"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}