Refinements of provability and consistency principles for the second incompleteness theorem

2025/07/01 by Taishi Kurahashi, Kurahashi, Taishi · 2 citations
Computer Science · #Advanced Algebra and Logic #FOS: Mathematics #Logic (math.LO) #Logic, Reasoning, and Knowledge #Logic, programming, and type systems

paper · pdf · doi:10.48550/arxiv.2507.00955

openalex publication_date 2025/07/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

This paper continues the author's previous study \citeKura20, showing that several weak principles inspired by non-normal modal logic suffice to derive various refined forms of the second incompleteness theorem. Among the main results of the present paper, we show that the set \E,C, D3\ suffices to establish the unprovability of the consistency statement ¬ Pr_T(\ulcorner 0=1 \urcorner). We also prove that the set \E^U, \mathbfCB_∃\ yields formalized Σ₁-completeness.

Citations

• Arithmetical completeness for some extensions of the pure logic of necessitation
• The finite frame property of some extensions of the pure logic of necessitation
• The provability logic of all provability predicates
• Arithmetical completeness theorems for monotonic modal logics
• Rosser provability and the second incompleteness theorem
• Provability interpretations of modal logic

Cited by

• Provability interpretation of non-normal modal logics having neighborhood semantics
• Modal logical aspects of provability predicates and consistency statements

Related

• Completeness theorems for modal logic in second-order arithmetic
• Arithmetical completeness theorems for monotonic modal logics
• Modal logical aspects of provability predicates and consistency statements
• Provability interpretation of non-normal modal logics having neighborhood semantics
• Some abstract versions of Gödel's second incompleteness theorem based on non-classical logics