{
  "id": "1909.09100",
  "version": "v1",
  "published": "2019-09-19T17:17:30.000Z",
  "updated": "2019-09-19T17:17:30.000Z",
  "title": "The Sigma_1-definable universal finite sequence",
  "authors": [
    "Joel David Hamkins",
    "Kameryn J. Williams"
  ],
  "comment": "20 pages",
  "categories": [
    "math.LO"
  ],
  "abstract": "We introduce the $\\Sigma_1$-definable universal finite sequence and prove that it exhibits the universal extension property amongst the countable models of set theory under end-extension. That is, (i) the sequence is $\\Sigma_1$-definable and provably finite; (ii) the sequence is empty in transitive models; and (iii) if $M$ is a countable model of set theory in which the sequence is $s$ and $t$ is any finite extension of $s$ in this model, then there is an end extension of $M$ to a model in which the sequence is $t$. Our proof method grows out of a new infinitary-logic-free proof of the Barwise extension theorem, by which any countable model of set theory is end-extended to a model of $V=L$ or indeed any theory true in a suitable submodel of the original model. The main theorem settles the modal logic of end-extensional potentialism, showing that the potentialist validities of the models of set theory under end-extensions are exactly the assertions of S4. Finally, we introduce the end-extensional maximality principle, which asserts that every possibly necessary sentence is already true, and show that every countable model extends to a model satisfying it.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-09-19T17:17:30.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "03H05",
      "03E40",
      "03E45"
    ],
    "keywords": [
      "set theory",
      "countable model",
      "universal extension property",
      "end-extensional maximality principle",
      "proof method grows"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 20,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}