{
  "id": "1905.11146",
  "version": "v1",
  "published": "2019-05-27T11:43:39.000Z",
  "updated": "2019-05-27T11:43:39.000Z",
  "title": "Expansions of the $p$-adic numbers that interprets the ring of integers",
  "authors": [
    "Nathanaël Mariaule"
  ],
  "categories": [
    "math.LO"
  ],
  "abstract": "Let $\\widetilde{\\mathbb{Q}_p}$ be the field of $p$-adic numbers in the language of rings. In this paper we consider the theory of $\\widetilde{\\mathbb{Q}_p}$ expanded by two predicates interpreted by multiplicative subgroups $\\alpha^\\mathbb{Z}$ and $\\beta^\\mathbb{Z}$ where $\\alpha, \\beta\\in\\mathbb{N}$ are multiplicatively independent. We show that the theory of this structure interprets Peano arithmetic if $\\alpha$ and $\\beta$ have positive $p$-adic valuation. If either $\\alpha$ or $\\beta$ has zero valuation we show that the theory of $(\\widetilde{\\mathbb{Q}_p}, \\alpha^\\mathbb{Z}, \\beta^\\mathbb{Z})$ does not interpret Peano arithmetic. In that case we also prove that the theory is decidable iff the theory of $(\\widetilde{\\mathbb{Q}_p}, \\alpha^\\mathbb{Z}\\cdot \\beta^\\mathbb{Z})$ is decidable.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-05-27T11:43:39.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "03C65",
      "03C10"
    ],
    "keywords": [
      "adic numbers",
      "expansions",
      "structure interprets peano arithmetic",
      "interpret peano arithmetic",
      "zero valuation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}