{
  "id": "2307.07209",
  "version": "v1",
  "published": "2023-07-14T07:58:15.000Z",
  "updated": "2023-07-14T07:58:15.000Z",
  "title": "Degrees of the finite model property: the antidichotomy theorem",
  "authors": [
    "Guram Bezhanishvili",
    "Nick Bezhanishvili",
    "Tommaso Moraschini"
  ],
  "categories": [
    "math.LO"
  ],
  "abstract": "A classic result in modal logic, known as the Blok Dichotomy Theorem, states that the degree of incompleteness of a normal extension of the basic modal logic $\\sf K$ is $1$ or $2^{\\aleph_0}$. It is a long-standing open problem whether Blok Dichotomy holds for normal extensions of other prominent modal logics (such as $\\sf S4$ or $\\sf K4$) or for extensions of the intuitionistic propositional calculus $\\mathsf{IPC}$. In this paper, we introduce the notion of the degree of finite model property (fmp), which is a natural variation of the degree of incompleteness. It is a consequence of Blok Dichotomy Theorem that the degree of fmp of a normal extension of $\\sf K$ remains $1$ or $2^{\\aleph_0}$. In contrast, our main result establishes the following Antidichotomy Theorem for the degree of fmp for extensions of $\\mathsf{IPC}$: each nonzero cardinal $\\kappa$ such that $\\kappa \\leq \\aleph_0$ or $\\kappa = 2^{\\aleph_0}$ is realized as the degree of fmp of some extension of $\\mathsf{IPC}$. We then use the Blok-Esakia theorem to establish the same Antidichotomy Theorem for normal extensions of $\\sf S4$ and $\\sf K4$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-07-14T07:58:15.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "finite model property",
      "antidichotomy theorem",
      "normal extension",
      "blok dichotomy theorem",
      "main result establishes"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}