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Non-standard models of arithmetic and the incompleteness theorems

Research Project

Project/Area Number 24540125
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeMulti-year Fund
Section一般
Research Field General mathematics (including Probability theory/Statistical mathematics)
Research InstitutionKobe 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
Project Status Completed (Fiscal Year 2014)
Budget Amount *help
¥4,550,000 (Direct Cost: ¥3,500,000、Indirect Cost: ¥1,050,000)
Fiscal Year 2014: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000)
Fiscal Year 2013: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
Fiscal Year 2012: ¥1,820,000 (Direct Cost: ¥1,400,000、Indirect Cost: ¥420,000)
Keywords不完全性定理 / 超準モデル / 算術 / 様相論理 / 逆理 / 数学基礎論 / 算術の超準モデル / 矛盾許容型論理
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.

Report

(4 results)
  • 2014 Annual Research Report   Final Research Report ( PDF )
  • 2013 Research-status Report
  • 2012 Research-status Report
  • Research Products

    (7 results)

All 2015 2014 2013 2012

All Journal Article (1 results) (of which Peer Reviewed: 1 results) Presentation (5 results) Book (1 results)

  • [Journal Article] On proofs of the incompleteness theorems based on Berry's paradox by Vopenka, Chaitin, and Boolos2012

    • Author(s)
      Makoto Kikuchi, Taishi Kurahashi, Hiroshi Sakai
    • Journal Title

      Mathematical Logic Quarterly

      Volume: 57 Issue: 4-5 Pages: 307-316

    • DOI

      10.1002/malq.201110067

    • Related Report
      2012 Research-status Report
    • Peer Reviewed
  • [Presentation] Σn 定義可能な算術の不完全性定理2015

    • Author(s)
      倉橋太志,菊池誠
    • Organizer
      日本数学会2015年度年会
    • Place of Presentation
      明治大学駿河台キャンパス(東京都・千代田区)
    • Year and Date
      2015-03-22
    • Related Report
      2014 Annual Research Report
  • [Presentation] 状況理論としてのチャンネル理論2014

    • Author(s)
      菊池誠
    • Organizer
      科学基礎論学会秋の研究例会
    • Place of Presentation
      東京大学駒場キャンパス(東京都・目黒区)
    • Year and Date
      2014-11-01
    • Related Report
      2014 Annual Research Report
  • [Presentation] 不完全性定理の構成的性質について2014

    • Author(s)
      菊池誠,渕野昌
    • Organizer
      日本数学会2014年度秋季総合分科会
    • Place of Presentation
      広島大学東広島キャンパス(広島県・東広島市)
    • Year and Date
      2014-09-27
    • Related Report
      2014 Annual Research Report
  • [Presentation] 算術の超準モデルにおける定理と証明について2014

    • Author(s)
      菊池誠,倉橋太志
    • Organizer
      日本数学会2014年度秋季総合分科会
    • Place of Presentation
      広島大学東広島キャンパス(広島県・東広島市)
    • Year and Date
      2014-09-27
    • Related Report
      2014 Annual Research Report
  • [Presentation] 嘘つき型の矛盾による不完全性定理の証明について2013

    • Author(s)
      菊池誠,倉橋太志
    • Organizer
      日本数学会2013年度秋季総合分科会
    • Place of Presentation
      愛媛大学
    • Related Report
      2013 Research-status Report
  • [Book] 不完全性定理2014

    • Author(s)
      菊池誠
    • Total Pages
      368
    • Publisher
      共立出版
    • Related Report
      2014 Annual Research Report

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Published: 2013-05-31   Modified: 2019-07-29  

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