Proof-theoretic investigations on set theories

2010 Fiscal Year Self-evaluation Report

• Project/Area Number: 20340020
• Research Category: Grant-in-Aid for Scientific Research (B)
• Allocation Type: Single-year Grants
• Section: 一般
• Research Field: General mathematics (including Probability theory/Statistical mathematics)
• Research Institution: Chiba University (2009-2012); Kobe University (2008)
• Principal Investigator: ARAI Toshiyasu Chiba University, 大学院・理学研究科, 教授 (40193049)
• Project Period (FY): 2008 – 2012
• Keywords: 数学基礎論

Research Abstract

従来の帰納的巨大順序数の公理系の帰納的順序数による証明論的研究に含まれる技法と構成方法を精査し、それらが当該の帰納的巨大順序数がそれの帰納的類似物たる巨大基数の証明論的解析に転用できることを、スコーレム関数を証明図の解析と順序数の表示の双方に導入することで示していく。これは、対象とする巨大基数が大きくなればなるほど技術的に困難になるであろうが、帰納的巨大順序数の公理系の帰納的順序数による証明論的研究における経験から考えて乗り越えることができない障壁ではない。更に、構成可能性公理V=L と矛盾する巨大基数(例:可測基数)へと研究対象を広げていく。

Research Products

• [Journal Article] Wellfoundedness proofs by means of non-monotonic inductive definitions II : first order operators (2010)
  • Author(s): Toshiyasu Arai
  • Journal Title: Annals of Pure and Applied Logic vol.162 Pages: 107-143
  • Peer Reviewed
• [Journal Article] Intuitionistic fixed point theories over Heyting arithmetic (2010)
  • Author(s): Toshiyasu Arai
  • Journal Title: Proofs, Categories and Computations Pages: 1-14
  • Peer Reviewed
• [Journal Article] Iterating the recursively Mahlo operations (2009)
  • Author(s): Toshiyasu Arai
  • Journal Title: Proceedings of the thirteenth International Congress of Logic Methodology, Philosophy of Science Pages: 21-35
  • Peer Reviewed
• [Journal Article] Non-elementary speed-ups in logic calculi (2008)
  • Author(s): Toshiyasu Arai
  • Journal Title: Mathematical Logic Quarterly vol.6 Pages: 629-640
  • Peer Reviewed
• [Journal Article] Polynomial-size Frege proofs of Bollob\'as' theorem on the trace of sets (2008)
  • Author(s): Akihiro Nozaki, Toshiyasu Arai, Noriko Honda Arai
  • Journal Title: Prcceedings of the Japan Academy, Ser.A vol.84 Pages: 159-161
  • Peer Reviewed
• [Presentation] Lifting up the proof theory to the countables (2011)
  • Author(s): Toshiyasu Arai
  • Organizer: Kobe Colloquium on Logic, Statistics and Informatics
  • Place of Presentation: 神戸大学
  • Year and Date: 2011-03-30
• [Presentation] Wellfoundedness proofs and the bounds on provability in set theories (2010)
  • Author(s): Toshiyasu Arai
  • Organizer: Oberseminar
  • Place of Presentation: LMU Muenchen
  • Year and Date: 2010-09-22
• [Presentation] Search problems in bounded arithmetics (2010)
  • Author(s): Toshiyasu Arai
  • Organizer: A Proof Theory Workshop
  • Place of Presentation: 慶應義塾大学
  • Year and Date: 2010-03-18
• [Presentation] In germ, Leeds Symposium on (2009)
  • Author(s): Toshiyasu Arai
  • Organizer: Proof Theory and Constructivism
  • Place of Presentation: University of Leeds
  • Year and Date: 2009-07-06
• [Presentation] Provably $\Delta^{0}_{2}$ and weakly descending chains of ordinals (2009)
  • Author(s): Toshiyasu Arai
  • Organizer: Eleventh Asian Logic Conference
  • Place of Presentation: National University of Singapore
  • Year and Date: 2009-06-25
• [Book] 計算とは何か (2009)
  • Author(s): 新井紀子・新井敏康
  • Total Pages: 224
  • Publisher: 東京図書
• [Remarks]
  • URL: http://researchmap.jp/tosarai/