{
  "abstract": "We study fixed points in abstract provability structures (APSs), which were introduced by Beklemishev and Shamkanov as an order-theoretic framework for studying Gödel's second incompleteness theorem (G2). For one-variable APS terms built from the provability and refutability operations $\\Box$ and $\\boxtimes$, we investigate the existence and uniqueness of fixed points in terms of their degree. The degree of a term is the number of occurrences of $\\boxtimes$. We prove that every term of degree two has a fixed point and establish a strict hierarchy among the fixed-point properties of the terms $\\boxtimes\\Box^k v$. We further show that suitable levels of this hierarchy guarantee the existence and uniqueness of fixed points for arbitrary one-variable APS terms of positive degree. We also obtain several G2-like non-refutability results, some of which depend only on whether the degree is even or odd. Finally, we consider APSs based on meet-semilattices. Under an additional condition, we obtain stronger fixed-point results and show that a parametrized fixed-point property implies an abstract form of Löb's theorem.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.29914",
  "author_total": 2,
  "authors": [
    {
      "name": "Tsubasa Kumasaka"
    },
    {
      "name": "Taishi Kurahashi"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/97jqhmamhbyv9.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.29914",
  "fulltext_url": null,
  "id": "97jqhmamhbyv9",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.29914",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.LO"
    ],
    [
      "openalex",
      "1703|Advanced Algebra and Logic"
    ]
  ],
  "openalex": "W7204919599",
  "pdf_url": "https://arxiv.org/pdf/2608.29914",
  "pmcid": null,
  "pmid": null,
  "published": "2026-08-30",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/97jqhmamhbyv9.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:48Z",
      "native_id": "2608.29914",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.29914"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.LO",
      "subfield": "logic-foundations"
    }
  ],
  "title": "Fixed points in abstract provability structures",
  "title_text": "Fixed points in abstract provability structures",
  "type": "preprint",
  "updated_at": "2026-10-30T12:17:48Z",
  "url": "https://research.draconitic.com/paper/97jqhmamhbyv9",
  "venue": "arXiv",
  "versions": []
}