Reverse Mathematics and Models of Arithmetic

• Project/Area Number: 19740042
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Single-year Grants
• Research Field: General mathematics (including Probability theory/Statistical mathematics)
• Research Institution: Tohoku University
• Principal Investigator: YAMAZAKI Takeshi 東北大学, 大学院・理学研究科, 准教授 (30336812)
• Project Period (FY): 2007 – 2010
• Project Status: Completed (Fiscal Year 2010)
• Budget Amount: ¥3,490,000 (Direct Cost: ¥2,800,000、Indirect Cost: ¥690,000); Fiscal Year 2010: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2009: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000); Fiscal Year 2008: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2007: ¥500,000 (Direct Cost: ¥500,000)
• Keywords: 逆数学 / 二階算術 / オメガモデル / 再帰理論 / 帰納的関数論 / オメガ-モデル / ランダム / オメガーモデル / 超準モデル / 弱ケーニッヒの補題

Research Abstract

Reverse mathematics is the research program to classify mathematical theorems according to which set existence axioms are needed to prove them. In this poin of view, we got the following three kinds of results: (1) a new method to construct omaga-model and its application to conservativity result; (2) new notions of randomness and new systems corresponding to them; (3) new reverse mathematical results on Discrete Mathematics and Ring Theory.

Research Products

• [Journal Article] Does truth-table of linear norm reduce the on-query tautologies to a random oracle? (2008)
  • Author(s): M. Kumabe, T. Suzuki, T. Yamazaki
  • Journal Title: Arch. Math. Logic 47 Volume: no. 2 Pages: 159-180
  • Peer Reviewed
• [Journal Article] Does truth-table of linear norm reduce the one-query tautologies to a random oracle? (2008)
  • Author(s): M. Kumabe, T. Suzuki, T. Yamazaki
  • Journal Title: Arch. Math. Logic 47,no.2 Pages: 159-180
  • Peer Reviewed
• [Presentation] Computable Ring Theory and Reverse Mathematics (2010)
  • Author(s): T. Yamazaki
  • Organizer: Nanyang Technological University Logic Seminar
  • Place of Presentation: シンガポール
  • Year and Date: 2010-12-10
• [Presentation] Computable Ring Theory and Reverse Mathematics (2010)
  • Author(s): 山崎武
  • Organizer: Nanyang Technological University Logic Seminar
  • Place of Presentation: Nanyang Technological University(シンガポール)
  • Year and Date: 2010-12-10
• [Presentation] How to define randomness (2009)
  • Author(s): T. Yamazaki
  • Organizer: Xian University of Technology
  • Place of Presentation: 中国
  • Year and Date: 2009-10-23
• [Presentation] Topics on Conservation Results (2009)
  • Author(s): 山崎武
  • Organizer: Sendai logic and Philosophy seminar
  • Place of Presentation: 松島
  • Year and Date: 2009-02-24
• [Presentation] Topics on Conservation Results (2009)
  • Author(s): 山崎武
  • Organizer: Sendai logic and Philosophy seminar
  • Place of Presentation: 仙台, 松島
  • Year and Date: 2009-02-24
• [Presentation] 二階算術における代数学の展開 (2007)
  • Author(s): 佐藤隆, 山崎武, 横山啓太
  • Organizer: 日本数学会秋季総合分科会
  • Place of Presentation: 東北大学
  • Year and Date: 2007-09-24
• [Presentation] More on the partial randomness (2007)
  • Author(s): C. G. Liu, 山崎武
  • Organizer: 日本数学会秋季総合分科会
  • Place of Presentation: 東北大学
  • Year and Date: 2007-09-24
• [Presentation] 二階算術における代数学の展開 (2007)
  • Author(s): 佐藤隆, 山崎武, 横山啓太
  • Organizer: 日本数学階秋期総合分科会
  • Place of Presentation: 東北大学
  • Year and Date: 2007-09-24
• [Presentation] More on the partial randomness (2007)
  • Author(s): C.G. Liu, 山崎武
  • Organizer: 日本数学階秋期総合分科会
  • Place of Presentation: 東北大学
  • Year and Date: 2007-09-24
• [Book] ゲーデルと20世紀の論理学3不完全性定理と算術の体系 (2007)
  • Author(s): 田中一之, 他
  • Publisher: 東京大学出版会
• [Remarks] ・学会以外での発表 1. T. Yamazaki How to define randomness. Xi'an University of Technology, 2009年10月23日
• [Remarks] ・主催した会議 1.証明論研究集会 2009 (福島), 2010 年2月 2. Sendai Logic Spring Workshop 2008,(東北大学), 2008 年2 月