Research of formal provability by means of the investigation of incompleteness theorems

2023 Fiscal Year Final Research Report

• Project/Area Number: 19K14586
• Research Category: Grant-in-Aid for Early-Career Scientists
• Allocation Type: Multi-year Fund
• Review Section: Basic Section 12030:Basic mathematics-related
• Research Institution: Kobe University (2020-2023); Kisarazu National College of Technology (2019)
• Principal Investigator: Kurahashi Taishi 神戸大学, システム情報学研究科, 准教授 (10738446)
• Project Period (FY): 2019-04-01 – 2024-03-31
• Keywords: 数学基礎論 / 数理論理学 / 不完全性定理 / 様相論理 / 証明可能性論理 / 形式的算術 / 算術のモデル / 形式的証明可能性

Outline of Final Research Achievements

The notion of provability in formal systems is studied through the analysis of various aspects of the incompleteness theorems. The main purpose of this research is to explore the essence of the nature of the notion of formalized provability, and to understand its structure. By studying the first incompleteness theorem, the second incompleteness theorem, provability predicates through modal logic, and basic properties of modal arithmetic and semi-classical arithmetic, I could derive several results about the situation surrounding the incompleteness theorems.

Free Research Field

数学基礎論・数理論理学

Academic Significance and Societal Importance of the Research Achievements

不完全性定理の周辺に関する分析を行うことで形式的証明の構造を理解するという研究は，これまでにも様々な角度から行われている．こうした先行研究を踏まえながらも，独自の視点を常に持ち，内容的にも，また技術的にもそれなりに意義のある研究成果を得ることができたと感じている．本研究により，形式的証明可能性の概念の理解に向けた，いくつかの本質的な前進ができたといえる．