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Proof-theoretic investigations on set theories

Research Project

Project/Area Number 20340020
Research Category

Grant-in-Aid for Scientific Research (B)

Allocation TypeSingle-year Grants
Section一般
Research Field General mathematics (including Probability theory/Statistical mathematics)
Research InstitutionChiba 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 *help
¥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.

Report

(7 results)
  • 2012 Annual Research Report   Final Research Report ( PDF )
  • 2011 Annual Research Report
  • 2010 Annual Research Report   Self-evaluation Report ( PDF )
  • 2009 Annual Research Report
  • 2008 Annual Research Report
  • Research Products

    (68 results)

All 2013 2012 2011 2010 2009 2008 Other

All Journal Article (21 results) (of which Peer Reviewed: 21 results) Presentation (39 results) (of which Invited: 2 results) Book (2 results) Remarks (6 results)

  • [Journal Article] A sneak preview of proof theory of ordinals2012

    • Author(s)
      Toshiyasu Arai
    • Journal Title

      Annals of the Japan Association for Philosophy of Science

      Volume: 20 Pages: 29-47

    • Related Report
      2012 Annual Research Report 2012 Final Research Report 2011 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Provably $¥Delta_{2}$ and weakly descending chains2012

    • Author(s)
      Toshiyasu Arai
    • Journal Title

      Proceedings of the 11th Asian Logic Conference, World Scientific

      Pages: 1-21

    • Related Report
      2012 Final Research Report
    • Peer Reviewed
  • [Journal Article] Provably $\Delta_{2}$ and weakly descending chains2012

    • Author(s)
      Toshiyasu Arai
    • Journal Title

      Proceedings of the 11th Asian Logic Conference

      Volume: 1 Pages: 1-21

    • Related Report
      2012 Annual Research Report 2011 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Quick cut-elimination for strictly positive cuts2011

    • Author(s)
      Toshiyasu Arai
    • Journal Title

      Annals of Pure and Applied Logic

      Volume: 162 Pages: 807-815

    • Related Report
      2012 Final Research Report 2011 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Exact bounds on epsilon processes2011

    • Author(s)
      Toshiyasu Arai
    • Journal Title

      Archive for Mathematical Logic

      Volume: 50 Pages: 445-458

    • Related Report
      2012 Final Research Report 2011 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Nested PLS2011

    • Author(s)
      Toshiyasu Arai
    • Journal Title

      Archive for Mathematical Logic

      Volume: 50 Pages: 395-409

    • Related Report
      2012 Final Research Report 2011 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Wellfoundedness proofs by means of non-monotonic inductive definitions II: first order operators2010

    • Author(s)
      Toshiyasu Arai
    • Journal Title

      Annals of Pure and Applied Logic

      Volume: 162 Pages: 107-143

    • Related Report
      2012 Final Research Report
    • Peer Reviewed
  • [Journal Article] Intuitionistic fixed point theories over Heyting arithmetic2010

    • Author(s)
      Toshiyasu Arai
    • Journal Title

      Proofs, Categories and Computations, College Publications

      Pages: 1-14

    • Related Report
      2012 Final Research Report
    • Peer Reviewed
  • [Journal Article] Wellfoundedness proofs by means of non-monotonic inductive definitions II : first order operators2010

    • Author(s)
      Toshiyasu Arai
    • Journal Title

      Annals of Pure and Applied Logic vol.162

      Pages: 107-143

    • Related Report
      2010 Self-evaluation Report
    • Peer Reviewed
  • [Journal Article] Intuitionistic fixed point theories over Heyting arithmetic2010

    • Author(s)
      Toshiyasu Arai
    • Journal Title

      Proofs, Categories and Computations

      Pages: 1-14

    • Related Report
      2010 Annual Research Report 2010 Self-evaluation Report
    • Peer Reviewed
  • [Journal Article] Wellfoundedness proofs by means of non-monotonic inductive definitions II : first order operators2010

    • Author(s)
      Toshiyasu Arai
    • Journal Title

      Annals of Pure and Applied Logic

      Volume: 162 Pages: 107-143

    • Related Report
      2010 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Iterating the recursively Mahlo operations2009

    • Author(s)
      Toshiyasu Arai
    • Journal Title

      Proceedings of the thirteenth International Congress of Logic Methodology, Philosophy of Science, College Publications

      Pages: 21-35

    • Related Report
      2012 Final Research Report
    • Peer Reviewed
  • [Journal Article] Iterating the recursively Mahlo operations2009

    • Author(s)
      Toshiyasu Arai
    • Journal Title

      Proceedings of the thirteenth International Congress of Logic Methodology, Philosophy of Science

      Pages: 21-35

    • Related Report
      2010 Self-evaluation Report 2009 Annual Research Report
    • Peer Reviewed
  • [Journal Article] A new function algebra of EXPTIME functions by safe nested recursion2008

    • Author(s)
      Toshiyasu Arai and Naohi Eguchi
    • Journal Title

      ACM Transactions on Computational Logic

      Volume: 10 Pages: 1-18

    • Related Report
      2012 Final Research Report
    • Peer Reviewed
  • [Journal Article] Non-elementary speed-ups in logic calculi2008

    • Author(s)
      Toshiyasu Arai
    • Journal Title

      Mathematical Logic Quarterly

      Volume: 6 Pages: 629-640

    • Related Report
      2012 Final Research Report
    • Peer Reviewed
  • [Journal Article] Toshiyasu Arai and Noriko H. Arai, Polynomial-size Frege proofs of Bollob¥'as' theorem on the trace of sets2008

    • Author(s)
      Akihiro Nozaki
    • Journal Title

      Proceedings of the Japan Academy, Ser. A

      Volume: 84 Pages: 159-161

    • Related Report
      2012 Final Research Report
    • Peer Reviewed
  • [Journal Article] Non-elementary speed-ups in logic calculi2008

    • Author(s)
      Toshiyasu Arai
    • Journal Title

      Mathematical Logic Quarterly vol.6

      Pages: 629-640

    • Related Report
      2010 Self-evaluation Report
    • Peer Reviewed
  • [Journal Article] Polynomial-size Frege proofs of Bollob\'as' theorem on the trace of sets2008

    • Author(s)
      Akihiro Nozaki, Toshiyasu Arai, Noriko Honda Arai
    • Journal Title

      Prcceedings of the Japan Academy, Ser.A vol.84

      Pages: 159-161

    • Related Report
      2010 Self-evaluation Report
    • Peer Reviewed
  • [Journal Article] Polynomial-size Frege proofs of Bollob\'{a}s' theorem on the trace of sets2008

    • Author(s)
      A.Nozaki, T.Arai, N.H.Arai
    • Journal Title

      Proceedings of the Japan Academy, Ser.A 84

      Pages: 159-161

    • Related Report
      2008 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Non-elementary speed-ups in logic calculi2008

    • Author(s)
      T.Arai
    • Journal Title

      Mathematical Logic Quarterly 6

      Pages: 629-640

    • Related Report
      2008 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Proof theory of weak compactness

    • Author(s)
      Toshiyasu Arai
    • Journal Title

      Journal of Mathematical Logic

      Volume: (掲載決定)

    • Related Report
      2012 Final Research Report
    • Peer Reviewed
  • [Presentation] Predicatively computable functions on hereditarily finite sets2013

    • 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
    • Related Report
      2012 Final Research Report
  • [Presentation] $¥Pi^{1}_{n}$-indescribabilities in proof theory2013

    • 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
    • Related Report
      2012 Final Research Report
  • [Presentation] $\Pi^{1}_{n}$-indescribabilities in proof theory2013

    • Author(s)
      Toshiyasu Arai
    • Organizer
      Computability Theory and Foundations of Mathematics
    • Place of Presentation
      東京工業大学
    • Related Report
      2012 Annual Research Report
    • Invited
  • [Presentation] Predicatively computable functions on hereditarily finite sets2013

    • Author(s)
      Toshiyasu Arai
    • Organizer
      3rd Workshop on Proof Theory and Rewriting
    • Place of Presentation
      石川県立美術館
    • Related Report
      2012 Annual Research Report
    • Invited
  • [Presentation] Proof theory of set theories beyond ZF2012

    • Author(s)
      Toshiyasu Arai
    • Organizer
      Stanford Logic Seminar
    • Place of Presentation
      Stanford University, USA
    • Year and Date
      2012-09-04
    • Related Report
      2012 Annual Research Report 2012 Final Research Report
  • [Presentation] Proof theoretic bounds of set theories2012

    • Author(s)
      Toshiyasu Arai
    • Organizer
      Logic Coloquium 2012
    • Place of Presentation
      University of Manchester, UK
    • Year and Date
      2012-07-12
    • Related Report
      2012 Annual Research Report 2012 Final Research Report
  • [Presentation] Bounds on provability in set theories2012

    • 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
    • Related Report
      2012 Final Research Report 2011 Annual Research Report
  • [Presentation] Searching witnesses of $¥Sigma^{0}_{2}$-formulas in proofs2011

    • 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
    • Related Report
      2012 Final Research Report
  • [Presentation] Searching witnesses of $\Sigma^{0}_{2}$-formulas in proofs2011

    • 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
    • Related Report
      2011 Annual Research Report
  • [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
    • Related Report
      2011 Annual Research Report
  • [Presentation] Two contributions in proof theory on sets2011

    • 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
    • Related Report
      2012 Final Research Report
  • [Presentation] Two contributions in proof theory on sets2011

    • 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
    • Related Report
      2011 Annual Research Report
  • [Presentation] Lifting up the proof theory to the countables, Kobe Colloquium on Logic2011

    • Author(s)
      Toshiyasu Arai
    • Organizer
      Statistics and Informatics
    • Place of Presentation
      神戸大学
    • Year and Date
      2011-05-30
    • Related Report
      2012 Final Research Report
  • [Presentation] Lifting up the proof theory to the countables2011

    • 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
    • Related Report
      2010 Annual Research Report 2010 Self-evaluation Report
  • [Presentation] Quick cut-elimination in intuitionistic logic calculi2011

    • 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
    • Related Report
      2012 Final Research Report
  • [Presentation] Quick cut-elimination in intuitionistic logic calculi2011

    • Author(s)
      Toshiyasu Arai
    • Organizer
      Workshop on Proof Theory and Computability Theory
    • Place of Presentation
      岩沼屋
    • Year and Date
      2011-02-23
    • Related Report
      2010 Annual Research Report
  • [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
    • Related Report
      2012 Final Research Report
  • [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
    • Related Report
      2010 Annual Research Report
  • [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
    • Related Report
      2012 Final Research Report
  • [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
    • Related Report
      2012 Final Research Report
  • [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
    • Related Report
      2010 Annual Research Report
  • [Presentation] Wellfoundedness proofs and the bounds on provability in set theories2010

    • Author(s)
      Toshiyasu Arai
    • Organizer
      Oberseminar, LMU M¥"unchen
    • Place of Presentation
      M¥"unchen, Germany
    • Year and Date
      2010-09-22
    • Related Report
      2012 Final Research Report
  • [Presentation] Wellfoundedness proofs and the bounds on provability in set theories2010

    • Author(s)
      Toshiyasu Arai
    • Organizer
      Oberseminar
    • Place of Presentation
      LMU Muenchen
    • Year and Date
      2010-09-22
    • Related Report
      2010 Annual Research Report 2010 Self-evaluation Report
  • [Presentation] Search problems in bounded arithmetics2010

    • Author(s)
      Toshiyasu Arai
    • Organizer
      A Proof Theory Workshop
    • Place of Presentation
      慶應大学
    • Year and Date
      2010-05-18
    • Related Report
      2012 Final Research Report
  • [Presentation] Search problems in bounded arithmetics2010

    • Author(s)
      Toshiyasu Arai
    • Organizer
      A Proof Theory Workshop
    • Place of Presentation
      慶應義塾大学
    • Year and Date
      2010-03-18
    • Related Report
      2010 Self-evaluation Report 2009 Annual Research Report
  • [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
    • Related Report
      2012 Final Research Report
  • [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
    • Related Report
      2009 Annual Research Report
  • [Presentation] Mini-course on proof theory2010

    • Author(s)
      Toshiyasu Arai
    • Organizer
      LMU M¥"unchen
    • Place of Presentation
      M¥"unchen, Germany
    • Related Report
      2012 Final Research Report
  • [Presentation] Mini-course on proof theory2010

    • Author(s)
      Toshiyasu Arai
    • Organizer
      Oberseminar
    • Place of Presentation
      LMU Muenchen
    • Related Report
      2010 Annual Research Report
  • [Presentation] Provably $¥Delta^{0}_{2}$ and weakly descending chains of ordinals2009

    • Author(s)
      Toshiyasu Arai
    • Organizer
      Eleventh Asian Logic Conference
    • Place of Presentation
      National University of Singapore, Singapore
    • Year and Date
      2009-07-25
    • Related Report
      2012 Final Research Report
  • [Presentation] In germ,Leeds Symposium on Proof Theory and Constructivism2009

    • Author(s)
      Toshiyasu Arai
    • Organizer
      EPSRC Workshop
    • Place of Presentation
      Leeds University, UK
    • Year and Date
      2009-07-06
    • Related Report
      2012 Final Research Report
  • [Presentation] In germ, Leeds Symposium on2009

    • Author(s)
      Toshiyasu Arai
    • Organizer
      Proof Theory and Constructivism
    • Place of Presentation
      University of Leeds
    • Year and Date
      2009-07-06
    • Related Report
      2010 Self-evaluation Report
  • [Presentation] In germ2009

    • Author(s)
      Toshiyasu Arai
    • Organizer
      Leeds Symposium on Proof Theory and Constructivism
    • Place of Presentation
      University of Leeds
    • Year and Date
      2009-07-06
    • Related Report
      2009 Annual Research Report
  • [Presentation] Provably $\Delta^{0}_{2}$ and weakly descending chains of ordinals2009

    • Author(s)
      Toshiyasu Arai
    • Organizer
      Eleventh Asian Logic Conference
    • Place of Presentation
      National University of Singapore
    • Year and Date
      2009-06-25
    • Related Report
      2010 Self-evaluation Report 2009 Annual Research Report
  • [Presentation] Intuitionistic fixed point theories2009

    • 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
    • Related Report
      2012 Final Research Report 2008 Annual Research Report
  • [Presentation] Polysize Frege proofs of Bollob¥'as' theorem2009

    • 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
    • Related Report
      2012 Final Research Report
  • [Presentation] Polysize Frege proofs of Bollob\'{a}s' theorem2009

    • 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
    • Related Report
      2008 Annual Research Report
  • [Presentation] Proofs and sets2008

    • Author(s)
      Toshiyasu Arai
    • Organizer
      International Workshop on Constructivism: Logic and Mathematics
    • Place of Presentation
      金沢
    • Year and Date
      2008-05-28
    • Related Report
      2012 Final Research Report
  • [Presentation] Proofs and sets2008

    • 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
    • Related Report
      2008 Annual Research Report
  • [Book] 数学基礎論2011

    • Author(s)
      新井敏康
    • Total Pages
      550
    • Publisher
      岩波書店
    • Related Report
      2012 Final Research Report 2011 Annual Research Report
  • [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
    • Related Report
      2012 Final Research Report 2010 Self-evaluation Report 2009 Annual Research Report
  • [Remarks]

    • URL

      http://researchmap.jp/tosarai

    • Related Report
      2012 Final Research Report
  • [Remarks] Read & Researchmap

    • URL

      http://researchmap.jp/tosarai

    • Related Report
      2012 Annual Research Report
  • [Remarks]

    • URL

      http://researchmap.jp/tosarai/

    • Related Report
      2011 Annual Research Report
  • [Remarks]

    • URL

      http://researchmap.jp/tosarai/

    • Related Report
      2010 Self-evaluation Report
  • [Remarks]

    • URL

      http://researchmap.jp/tosarai/

    • Related Report
      2010 Annual Research Report
  • [Remarks]

    • URL

      http://researchmap.jp/tosarai/

    • Related Report
      2009 Annual Research Report

URL: 

Published: 2008-04-01   Modified: 2019-07-29  

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