{
  "id": "2011.14833",
  "version": "v1",
  "published": "2020-11-30T14:33:49.000Z",
  "updated": "2020-11-30T14:33:49.000Z",
  "title": "Connectedness in structures on the real numbers: o-minimality and undecidability",
  "authors": [
    "Alfred Dolich",
    "Chris Miller",
    "Alex Savatovsky",
    "Athipat Thamrongthanyalak"
  ],
  "categories": [
    "math.LO"
  ],
  "abstract": "We initiate an investigation of structures on the set of real numbers having the property that path components of definable sets are definable. All o\\nobreakdash-\\hspace{0pt}minimal structures on $(\\mathbb{R},<)$ have the property, as do all expansions of $(\\mathbb{R},+,\\cdot,\\mathbb{N})$. Our main analytic-geometric result is that any such expansion of $(\\mathbb{R},<,+)$ by boolean combinations of open sets (of any arities) either is o\\nobreakdash-\\hspace{0pt}minimal or defines an isomorph of $(\\mathbb N,+,\\cdot\\,)$. We also show that any given expansion of $(\\mathbb{R}, <, +,\\mathbb{N})$ by subsets of $\\mathbb{N}^n$ ($n$ allowed to vary) has the property if and only if it defines all arithmetic sets. Variations arise by considering connected components or quasicomponents instead of path components.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-11-30T14:33:49.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "real numbers",
      "structures",
      "connectedness",
      "o-minimality",
      "undecidability"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}