{
  "id": "2305.06676",
  "version": "v1",
  "published": "2023-05-11T09:23:47.000Z",
  "updated": "2023-05-11T09:23:47.000Z",
  "title": "Does $\\mathsf{DC}$ imply $\\mathsf{AC}_ω$, uniformly?",
  "authors": [
    "Alessandro Andretta",
    "Lorenzo Notaro"
  ],
  "comment": "23 pages, to be submitted",
  "categories": [
    "math.LO"
  ],
  "abstract": "The Axiom of Dependent Choice $\\mathsf{DC}$ and the Axiom of Countable Choice $\\mathsf{AC}_\\omega$ are two weak forms of the Axiom of Choice that can be stated for a specific set: $\\mathsf{DC}(X)$ asserts that any total binary relation on $X$ has an infinite chain, while $\\mathsf{AC}_\\omega (X)$ asserts that any countable collection of nonempty subsets of $X$ has a choice function. It is well-known that $\\mathsf{DC} \\Rightarrow \\mathsf{AC}_\\omega$. We study for which sets and under which hypotheses $\\mathsf{DC}(X) \\Rightarrow \\mathsf{AC}_\\omega (X)$, and then we show it is consistent with $\\mathsf{ZF}$ that there is a set $A \\subseteq \\mathbb{R}$ for which $\\mathsf{DC} (A)$ holds, but $\\mathsf{AC}_\\omega (A)$ fails.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-05-11T09:23:47.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "03E25",
      "03E35",
      "03E40"
    ],
    "keywords": [
      "total binary relation",
      "specific set",
      "choice function",
      "dependent choice",
      "weak forms"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 23,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}