Research of formal provability by means of the analysis of provability predicates

2015 Fiscal Year Final Research Report

• Project/Area Number: 26887045
• Research Category: Grant-in-Aid for Research Activity Start-up
• Allocation Type: Single-year Grants
• Research Field: Foundations of mathematics/Applied mathematics
• Research Institution: Kisarazu National College of Technology
• Principal Investigator: Kurahashi Taishi 木更津工業高等専門学校, その他部局等, 講師 (10738446)
• Project Period (FY): 2014-08-29 – 2016-03-31
• Keywords: 数理論理学 / 数学基礎論 / 不完全性定理 / 形式的算術 / 証明可能性 / 可証性述語 / 算術の超準モデル

Outline of Final Research Achievements

I studied the notions of proofs and provability in formal systems by investigating provability predicates.
1. Syntactical approach: We extended the first and second incompleteness theorems to arithmetically definable theories of arithmetic. This research is a joint work with Professor Kikuchi Makoto (Kobe University). Also I defined the notion that a theory has the Sigma_n disjunction property, and we revealed several properties of this notion.
2. Semantical approach: I studied the structure of proofs in nonstandard models of arithmetic by investigating provability predicates. By this research, our understanding of the structure of proofs in nonstandard models having a proof of 0=1 was deepened. This is also a joint work with Professor Makoto Kikuchi.

Free Research Field

数理論理学