Proof-theoretic investigations on set theories

• 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: TOSHIYASU Arai (ARAI Toshiyasu) 千葉大学, 大学院・理学研究科, 教授 (40193049)
• Project Period (FY): 2008 – 2012
• Project Status: Completed (Fiscal Year 2012)
• Budget Amount: ¥9,100,000 (Direct Cost: ¥7,000,000、Indirect Cost: ¥2,100,000); Fiscal Year 2012: ¥1,560,000 (Direct Cost: ¥1,200,000、Indirect Cost: ¥360,000); Fiscal Year 2011: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000); Fiscal Year 2010: ¥1,560,000 (Direct Cost: ¥1,200,000、Indirect Cost: ¥360,000); Fiscal Year 2009: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000); Fiscal Year 2008: ¥3,380,000 (Direct Cost: ¥2,600,000、Indirect Cost: ¥780,000)
• Keywords: 数学基礎論 / 証明の長さ / 集合論の証明論 / 証明論 / 集合論

Research Abstract

I have primarily concerned with proof-theoretic investigations of axioms in set theories. Set theory is the basis of mathematics, and I have investigated proofs in set theory in terms of iterations: the number of iterations and operations. The axiom in set theory is a collection of propositions which are supposed to hold on sets. On the other side proving propositions is an indispensable action in mathematics including set theory. I have acquired knowledge of such foundational objects.

Research Products

• [Journal Article] A sneak preview of proof theory of ordinals (2012)
  • Author(s): Toshiyasu Arai
  • Journal Title: Annals of the Japan Association for Philosophy of Science Volume: 20 Pages: 29-47
  • Peer Reviewed
• [Journal Article] Provably $￥Delta_{2}$ and weakly descending chains (2012)
  • Author(s): Toshiyasu Arai
  • Journal Title: Proceedings of the 11th Asian Logic Conference, World Scientific Pages: 1-21
  • Peer Reviewed
• [Journal Article] Provably $\Delta_{2}$ and weakly descending chains (2012)
  • Author(s): Toshiyasu Arai
  • Journal Title: Proceedings of the 11th Asian Logic Conference Volume: 1 Pages: 1-21
  • Peer Reviewed
• [Journal Article] Quick cut-elimination for strictly positive cuts (2011)
  • Author(s): Toshiyasu Arai
  • Journal Title: Annals of Pure and Applied Logic Volume: 162 Pages: 807-815
  • Peer Reviewed
• [Journal Article] Exact bounds on epsilon processes (2011)
  • Author(s): Toshiyasu Arai
  • Journal Title: Archive for Mathematical Logic Volume: 50 Pages: 445-458
  • Peer Reviewed
• [Journal Article] Nested PLS (2011)
  • Author(s): Toshiyasu Arai
  • Journal Title: Archive for Mathematical Logic Volume: 50 Pages: 395-409
  • Peer Reviewed
• [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 Volume: 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, College Publications Pages: 1-14
  • Peer Reviewed
• [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] 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 Volume: 162 Pages: 107-143
  • 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, College Publications Pages: 21-35
  • 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] A new function algebra of EXPTIME functions by safe nested recursion (2008)
  • Author(s): Toshiyasu Arai and Naohi Eguchi
  • Journal Title: ACM Transactions on Computational Logic Volume: 10 Pages: 1-18
  • Peer Reviewed
• [Journal Article] Non-elementary speed-ups in logic calculi (2008)
  • Author(s): Toshiyasu Arai
  • Journal Title: Mathematical Logic Quarterly Volume: 6 Pages: 629-640
  • Peer Reviewed
• [Journal Article] Toshiyasu Arai and Noriko H. Arai, Polynomial-size Frege proofs of Bollob￥'as' theorem on the trace of sets (2008)
  • Author(s): Akihiro Nozaki
  • Journal Title: Proceedings of the Japan Academy, Ser. A Volume: 84 Pages: 159-161
  • 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
• [Journal Article] Polynomial-size Frege proofs of Bollob\'{a}s' theorem on the trace of sets (2008)
  • Author(s): A.Nozaki, T.Arai, N.H.Arai
  • Journal Title: Proceedings of the Japan Academy, Ser.A 84 Pages: 159-161
  • Peer Reviewed
• [Journal Article] Non-elementary speed-ups in logic calculi (2008)
  • Author(s): T.Arai
  • Journal Title: Mathematical Logic Quarterly 6 Pages: 629-640
  • Peer Reviewed
• [Journal Article] Proof theory of weak compactness
  • Author(s): Toshiyasu Arai
  • Journal Title: Journal of Mathematical Logic Volume: (掲載決定)
  • Peer Reviewed
• [Presentation] Predicatively computable functions on hereditarily finite sets (2013)
  • Author(s): Toshiyasu Arai
  • Organizer: the 3rd Workshop on Proof Theory and Rewriting
  • Place of Presentation: 金沢県立美術館
  • Year and Date: 2013-05-06
• [Presentation] $￥Pi^{1}_{n}$-indescribabilities in proof theory (2013)
  • Author(s): Toshiyasu Arai
  • Organizer: Computability Theory and Foundations of Mathematics
  • Place of Presentation: 東京工業大学
  • Year and Date: 2013-02-19
• [Presentation] $\Pi^{1}_{n}$-indescribabilities in proof theory (2013)
  • Author(s): Toshiyasu Arai
  • Organizer: Computability Theory and Foundations of Mathematics
  • Place of Presentation: 東京工業大学
  • Invited
• [Presentation] Predicatively computable functions on hereditarily finite sets (2013)
  • Author(s): Toshiyasu Arai
  • Organizer: 3rd Workshop on Proof Theory and Rewriting
  • Place of Presentation: 石川県立美術館
  • Invited
• [Presentation] Proof theory of set theories beyond ZF (2012)
  • Author(s): Toshiyasu Arai
  • Organizer: Stanford Logic Seminar
  • Place of Presentation: Stanford University, USA
  • Year and Date: 2012-09-04
• [Presentation] Proof theoretic bounds of set theories (2012)
  • Author(s): Toshiyasu Arai
  • Organizer: Logic Coloquium 2012
  • Place of Presentation: University of Manchester, UK
  • Year and Date: 2012-07-12
• [Presentation] Bounds on provability in set theories (2012)
  • Author(s): Toshiyasu Arai
  • Organizer: Workshop on proof theory and computability theory 2012
  • Place of Presentation: 晴海グランドホテル
  • Year and Date: 2012-02-21
• [Presentation] Searching witnesses of $￥Sigma^{0}_{2}$-formulas in proofs (2011)
  • Author(s): Toshiyasu Arai
  • Organizer: 東北 大学ロジックセミナー
  • Place of Presentation: 東北大学
  • Year and Date: 2011-12-26
• [Presentation] Searching witnesses of $\Sigma^{0}_{2}$-formulas in proofs (2011)
  • Author(s): Toshiyasu Arai
  • Organizer: 東北大学ロジックセミナー
  • Place of Presentation: 東北大学(招待講演)
  • Year and Date: 2011-12-26
• [Presentation] はじめての数学基礎論 (2011)
  • Author(s): 新井敏康
  • Organizer: 日本数学会
  • Place of Presentation: 信州大学(招待講演)
  • Year and Date: 2011-09-28
• [Presentation] Two contributions in proof theory on sets (2011)
  • Author(s): Toshiyasu Arai
  • Organizer: Workshop on proof theory and theory of computing,2011
  • Place of Presentation: 首都大学
  • Year and Date: 2011-09-13
• [Presentation] Two contributions in proof theory on sets (2011)
  • Author(s): Toshiyasu Arai
  • Organizer: Workshop on proof theory and theory of computing 2011
  • Place of Presentation: 首都大学東京
  • Year and Date: 2011-09-13
• [Presentation] Lifting up the proof theory to the countables, Kobe Colloquium on Logic (2011)
  • Author(s): Toshiyasu Arai
  • Organizer: Statistics and Informatics
  • Place of Presentation: 神戸大学
  • Year and Date: 2011-05-30
• [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] Quick cut-elimination in intuitionistic logic calculi (2011)
  • Author(s): Toshiyasu Arai
  • Organizer: Workshop on Proof Theory and Computability Theory
  • Place of Presentation: 秋保温泉
  • Year and Date: 2011-02-25
• [Presentation] Quick cut-elimination in intuitionistic logic calculi (2011)
  • Author(s): Toshiyasu Arai
  • Organizer: Workshop on Proof Theory and Computability Theory
  • Place of Presentation: 岩沼屋
  • Year and Date: 2011-02-23
• [Presentation] Search problems in $T^{2}_{2}$ (2011)
  • Author(s): Toshiyasu Arai
  • Organizer: 東北大学ロジックセミナー
  • Place of Presentation: 東北大学
  • Year and Date: 2011-01-21
• [Presentation] Search problems in $T^{2}-{2}$ (2011)
  • Author(s): Toshiyasu Arai
  • Organizer: 東北大学ロジックセミナー
  • Place of Presentation: 東北大学
  • Year and Date: 2011-01-21
• [Presentation] はじめての数学基礎論,日本数学会企画特別講演 (2011)
  • Author(s): 新井敏康
  • Organizer: 平成23年秋季総合分科会
• [Presentation] $￥omega_{1}$の証明論 (2010)
  • Author(s): 新井敏康
  • Organizer: 神奈川大学集合論セミナー
  • Place of Presentation: 神奈川大学
  • Year and Date: 2010-11-13
• [Presentation] $\omega_{1}$の証明論 (2010)
  • Author(s): 新井敏康
  • Organizer: 神奈川大学数学教室談話会
  • Place of Presentation: 神奈川大学
  • Year and Date: 2010-11-13
• [Presentation] Wellfoundedness proofs and the bounds on provability in set theories (2010)
  • Author(s): Toshiyasu Arai
  • Organizer: Oberseminar, LMU M￥"unchen
  • Place of Presentation: M￥"unchen, Germany
  • Year and Date: 2010-09-22
• [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: 慶應大学
  • Year and Date: 2010-05-18
• [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] 集合論の証明論へ (2010)
  • Author(s): 新井敏康
  • Organizer: 神奈川大学集合論セミナー
  • Place of Presentation: 神奈川大学
  • Year and Date: 2010-02-22
• [Presentation] 集合論の証明論へ (2010)
  • Author(s): 新井敏康
  • Organizer: 神奈川大学数学教室談話会
  • Place of Presentation: 神奈川大学
  • Year and Date: 2010-02-22
• [Presentation] Mini-course on proof theory (2010)
  • Author(s): Toshiyasu Arai
  • Organizer: LMU M￥"unchen
  • Place of Presentation: M￥"unchen, Germany
• [Presentation] Mini-course on proof theory (2010)
  • Author(s): Toshiyasu Arai
  • Organizer: Oberseminar
  • Place of Presentation: LMU Muenchen
• [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, Singapore
  • Year and Date: 2009-07-25
• [Presentation] In germ,Leeds Symposium on Proof Theory and Constructivism (2009)
  • Author(s): Toshiyasu Arai
  • Organizer: EPSRC Workshop
  • Place of Presentation: Leeds University, UK
  • Year and Date: 2009-07-06
• [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] In germ (2009)
  • Author(s): Toshiyasu Arai
  • Organizer: Leeds Symposium on 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
• [Presentation] Intuitionistic fixed point theories (2009)
  • Author(s): 新井敏康
  • Organizer: 日本数学会年会
  • Place of Presentation: 東京大学
  • Year and Date: 2009-05-26
• [Presentation] Polysize Frege proofs of Bollob￥'as' theorem (2009)
  • Author(s): 野崎 昭弘 , 新井 敏康 , 新井 紀子
  • Organizer: 日本数学会年会
  • Place of Presentation: 東京大学
  • Year and Date: 2009-05-26
• [Presentation] Polysize Frege proofs of Bollob\'{a}s' theorem (2009)
  • Author(s): 野崎昭弘, 新井敏康, 新井紀子
  • Organizer: 日本数学会年会
  • Place of Presentation: 東京大学
  • Year and Date: 2009-03-26
• [Presentation] Proofs and sets (2008)
  • Author(s): Toshiyasu Arai
  • Organizer: International Workshop on Constructivism: Logic and Mathematics
  • Place of Presentation: 金沢
  • Year and Date: 2008-05-28
• [Presentation] Proofs and sets (2008)
  • Author(s): 新井敏康
  • Organizer: International Workshop on Constructivism, Logic and Mathematics
  • Place of Presentation: 金沢(日本)
  • Year and Date: 2008-05-28
• [Book] 数学基礎論 (2011)
  • Author(s): 新井敏康
  • Total Pages: 550
  • Publisher: 岩波書店
• [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]
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