{
  "id": "2409.03527",
  "version": "v1",
  "published": "2024-09-05T13:36:59.000Z",
  "updated": "2024-09-05T13:36:59.000Z",
  "title": "The Kaufmann--Clote question on end extensions of models of arithmetic and the weak regularity principle",
  "authors": [
    "Mengzhou Sun"
  ],
  "comment": "18 pages",
  "categories": [
    "math.LO"
  ],
  "abstract": "We investigate the end extendibility of models of arithmetic with restricted elementarity. By utilizing the restricted ultrapower construction in the second-order context, for each $n\\in\\mathbb{N}$ and any countable model of $\\mathrm{B}\\Sigma_{n+2}$, we construct a proper $\\Sigma_{n+2}$-elementary end extension satisfying $\\mathrm{B}\\Sigma_{n+1}$, which answers a question by Clote positively. We also give a characterization of countable models of $\\mathrm{I}\\Sigma_{n+2}$ in terms of their end extendibility similar to the case of $\\mathrm{B}\\Sigma_{n+2}$. Along the proof, we will introduce a new type of regularity principles in arithmetic called the weak regularity principle, which serves as a bridge between the model's end extendibility and the amount of induction or collection it satisfies.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-09-05T13:36:59.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "03C62",
      "03F30",
      "03H15",
      "03F35"
    ],
    "keywords": [
      "weak regularity principle",
      "kaufmann-clote question",
      "arithmetic",
      "countable model",
      "end extendibility similar"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 18,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}