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Fixed points in abstract provability structures

  • Tsubasa Kumasaka
  • Taishi Kurahashi
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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 boxtimesBoxk 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.

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Cite

BibTeXRIS
BibTeX
@misc{kumasaka2026fixed,
  title = {{Fixed points in abstract provability structures}},
  author = {Tsubasa Kumasaka and Taishi Kurahashi},
  howpublished = {arXiv},
  year = {2026},
  month = aug,
  doi = {10.48550/arxiv.2608.29914},
  eprint = {2608.29914},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.29914},
}
RIS
TY  - UNPB
TI  - Fixed points in abstract provability structures
AU  - Tsubasa Kumasaka
AU  - Taishi Kurahashi
PY  - 2026
DA  - 2026/08/30
PB  - arXiv
DO  - 10.48550/arxiv.2608.29914
UR  - https://arxiv.org/abs/2608.29914
AB  - 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 boxtimesBox^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.
ER  -