Non-standard models of arithmetic and the incompleteness theorems

2014 Fiscal Year Final Research Report

• Project/Area Number: 24540125
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: General mathematics (including Probability theory/Statistical mathematics)
• Research Institution: Kobe University
• Principal Investigator: KIKUCHI Makoto 神戸大学, システム情報学研究科, 准教授 (60273801)
• Project Period (FY): 2012-04-01 – 2015-03-31
• Keywords: 不完全性定理 / 超準モデル / 算術 / 様相論理 / 逆理

Outline of Final Research Achievements

Firstly, we showed relationships between some proofs of the incompleteness theorems based on Berry's paradox. Then, we defined the concept of liar-type paradox by using modal logic and showed the existence of the arithmetical independent statements based on liar-type paradoxes. Thirdly, we investigated the set of theorems of Peano arithmetic on non-standard models of arithmetic on which Peano arithmetic is inconsistent, and showed the definability of complete theories of arithmetic on such models. At last, we gave generalizations of the incompleteness theorems, and indicated a new relationship between formalization of mathematical proofs and Hilbert's program.

Free Research Field

数学基礎論