{
  "id": "2406.18341",
  "version": "v1",
  "published": "2024-06-26T13:36:15.000Z",
  "updated": "2024-06-26T13:36:15.000Z",
  "title": "Partially-elementary end extensions of countable models of set theory",
  "authors": [
    "Zachiri McKenzie"
  ],
  "comment": "23 pages. This is a later draft of arXiv:2201.04817",
  "categories": [
    "math.LO"
  ],
  "abstract": "Let $\\mathsf{KP}$ denote Kripke-Platek Set Theory and let $\\mathsf{M}$ be the weak set theory obtained from $\\mathsf{ZF}$ by removing the collection scheme, restricting separation to $\\Delta_0$-formulae and adding an axiom asserting that every set is contained in a transitive set ($\\mathsf{TCo}$). A result due to Kaufmann shows that every countable model, $\\mathcal{M}$, of $\\mathsf{KP}+\\Pi_n\\textsf{-Collection}$ has a proper $\\Sigma_{n+1}$-elementary end extension. Here we show that there are limits to the amount of the theory of $\\mathcal{M}$ that can be transferred to the end extensions that are guaranteed by Kaufmann's Theorem. Using admissible covers and the Barwise Compactness Theorem, we show that if $\\mathcal{M}$ is a countable model $\\mathsf{KP}+\\Pi_n\\textsf{-Collection}+\\Sigma_{n+1}\\textsf{-Foundation}$ and $T$ is a recursive theory that holds in $\\mathcal{M}$, then there exists a proper $\\Sigma_n$-elementary end extension of $\\mathcal{M}$ that satisfies $T$. We use this result to show that the theory $\\mathsf{M}+\\Pi_n\\textsf{-Collection}+\\Pi_{n+1}\\textsf{-Foundation}$ proves $\\Sigma_{n+1}\\textsf{-Separation}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-06-26T13:36:15.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "03C62",
      "03H99",
      "03C70"
    ],
    "keywords": [
      "partially-elementary end extensions",
      "countable model",
      "denote kripke-platek set theory",
      "weak set theory",
      "collection scheme"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 23,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}