{
  "id": "1312.0670",
  "version": "v1",
  "published": "2013-12-03T00:53:58.000Z",
  "updated": "2013-12-03T00:53:58.000Z",
  "title": "Satisfaction is not absolute",
  "authors": [
    "Joel David Hamkins",
    "Ruizhi Yang"
  ],
  "comment": "34 pages. Commentary concerning this article can be made at http://jdh.hamkins.org/satisfaction-is-not-absolute",
  "categories": [
    "math.LO"
  ],
  "abstract": "We prove that the satisfaction relation $\\mathcal{N}\\models\\varphi[\\vec a]$ of first-order logic is not absolute between models of set theory having the structure $\\mathcal{N}$ and the formulas $\\varphi$ all in common. Two models of set theory can have the same natural numbers, for example, and the same standard model of arithmetic $\\langle\\mathbb{N},{+},{\\cdot},0,1,{\\lt}\\rangle$, yet disagree on their theories of arithmetic truth; two models of set theory can have the same natural numbers and the same arithmetic truths, yet disagree on their truths-about-truth, at any desired level of the iterated truth-predicate hierarchy; two models of set theory can have the same natural numbers and the same reals, yet disagree on projective truth; two models of set theory can have the same $\\langle H_{\\omega_2},{\\in}\\rangle$ or the same rank-initial segment $\\langle V_\\delta,{\\in}\\rangle$, yet disagree on which assertions are true in these structures. On the basis of these mathematical results, we argue that a philosophical commitment to the determinateness of the theory of truth for a structure cannot be seen as a consequence solely of the determinateness of the structure in which that truth resides. The determinate nature of arithmetic truth, for example, is not a consequence of the determinate nature of the arithmetic structure $\\mathbb{N}=\\{0,1,2,\\ldots\\}$ itself, but rather, we argue, is an additional higher-order commitment requiring its own analysis and justification.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-12-03T00:53:58.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "set theory",
      "natural numbers",
      "arithmetic truth",
      "determinate nature",
      "additional higher-order commitment"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 34,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1312.0670H"
    }
  }
}