Reverse Mathematics in Constructive Mathematics

• Project/Area Number: 19500012
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Single-year Grants
• Section: 一般
• Research Field: Fundamental theory of informatics
• Research Institution: Japan Advanced Institute of Science and Technology
• Principal Investigator: ISHIHARA Hajime Japan Advanced Institute of Science and Technology, 情報科学研究科, 准教授 (10211046)
• Co-Investigator(Kenkyū-buntansha): AKAMA Yohji 東北大学, 大学院・理学研究科, 准教授 (30272454)
• Project Period (FY): 2007 – 2009
• Project Status: Completed (Fiscal Year 2009)
• Budget Amount: ¥4,420,000 (Direct Cost: ¥3,400,000、Indirect Cost: ¥1,020,000); Fiscal Year 2009: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000); Fiscal Year 2008: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000); Fiscal Year 2007: ¥1,560,000 (Direct Cost: ¥1,200,000、Indirect Cost: ¥360,000)
• Keywords: 数理論理学 / 数学基礎論 / 構成的数学 / 逆数学

Research Abstract

本研究は、構成的逆数学のための形式体系の調査・提案・評価・改良を目的として行われた。型概念に基づいた体系、および集合概念に基づいた体系、それぞれに対して様々な原理(例えば、連続性原理)、および位相空間・一様空間の定理を対象に調査を行った。その結果、集合概念に基づいた体系としては、構成的集合論(CZF)およびその部分体系が、有力な候補であることが分かった。また、型概念に基づいた体系としては、単調完備定理を詳細に分析することにより、構成的解析体系(EL)の部分体系を構成的逆数学のための形式体系として提案した。

Research Products

• [Journal Article] Two subcategories of apartness space (2010)
  • Author(s): Hajime Ishihara
  • Journal Title: Ann. Pure Appl. Logic (in press)
  • NAID: 120004247360
  • Peer Reviewed
• [Journal Article] Kronecker's density theorem and irrational numbers in constructive reverse mathematics (2010)
  • Author(s): Hajime Ishihara, Peter Schuster
  • Journal Title: 57 Pages: 57-72
  • Peer Reviewed
• [Journal Article] Kronecker's density theorem and irrational numbers in constructive reverse mathematics (2010)
  • Author(s): Hajime Ishihara, Peter Schuster
  • Journal Title: Math.Semesterber 57 Pages: 57-72
  • Peer Reviewed
• [Journal Article] Two subcategories of apartness spaces (2010)
  • Author(s): Hajime Ishihara
  • Journal Title: Ann.Pure Appl.Logic (印刷中)
  • NAID: 120004247360
  • Peer Reviewed
• [Journal Article] A continuity principle, a version of Baire's theorem and a boundedness principle (2008)
  • Author(s): Hajime Ishihara, Peter Schuster
  • Journal Title: J. Symbolic Logic 73 Pages: 1354-1360
  • NAID: 120001746753
  • Peer Reviewed
• [Journal Article] A continuity principle, a version of Baire's theorem and a boundedness principle (2008)
  • Author(s): Hajime Ishihara, Peter Schuster
  • Journal Title: Journal of Symbolic Logic 73 Pages: 1354-1360
  • NAID: 120001746753
  • Peer Reviewed
• [Journal Article] A continuity principle,a version of Baire's theorem and a boundedness principle
  • Author(s): Hajime Ishihara and Peter Schuster
  • Journal Title: Journal of Symbolic Logic (印刷中)
  • NAID: 120001746753
  • Peer Reviewed
• [Presentation] A boundedness principle in constructive reverse mathematics, Continuity (2009)
  • Author(s): Hajime Ishihara
  • Organizer: Computablility, Constructivity: From Logic to Algorithms
  • Place of Presentation: Koln, Germany
  • Year and Date: 2009-06-17
• [Presentation] A boundedness principle in constructive reverse mathematics (2009)
  • Author(s): Hajime Ishihara
  • Organizer: Continuity, Computability, Constructivity : From Logic to Algorithms
  • Place of Presentation: Koeln(Germany)
  • Year and Date: 2009-06-17
• [Presentation] Continuous morphisms in constructive topological spaces (2009)
  • Author(s): Hajime Ishihara
  • Organizer: Leeds Symposium on Proof Theory and Constructivism
  • Place of Presentation: Leeds, UK
  • Year and Date: 2009-06-15
• [Presentation] Continuous morphisms in constructive topological spaces (2009)
  • Author(s): Hajime Ishihara
  • Organizer: Leeds Symposium on Proof Theory and Constructivism
  • Place of Presentation: University of Leeds(UK)
  • Year and Date: 2009-06-15
• [Presentation] On a set generation scheme in constructive set theory (2008)
  • Author(s): Hajime Ishihara
  • Organizer: Advances in Constructive Topology and Logical Foundations
  • Place of Presentation: Padova, Italy
  • Year and Date: 2008-10-09
• [Presentation] A continuity principle, a version of Baire's theorem and a boundedness principle (2008)
  • Author(s): Hajime Ishihara
  • Organizer: Conference on Abelian Groups and on Constructive Mathematics
  • Place of Presentation: Boca Raton, USA
  • Year and Date: 2008-05-10
• [Presentation] A continuity principle, a version of Baire's theorem and a boundedness principle (2008)
  • Author(s): Hajime Ishihara
  • Organizer: Conference on Abelian Groups and on Construetive Mathematics
  • Place of Presentation: Boca Raton, USA
• [Presentation] ω-rules and learnability (2007)
  • Author(s): Yohji Akama
  • Organizer: Computability in Europe 2007
  • Place of Presentation: Siena, Italy
  • Year and Date: 2007-06-21
• [Presentation] ω-rules and learnablility (2007)
  • Author(s): Yohji Akama
  • Organizer: Computability in Europe 2007
  • Place of Presentation: Siena
  • Year and Date: 2007-06-21
• [Presentation] Unique existence and computability in constructive reverse mathematics (2007)
  • Author(s): Hajime Ishihara
  • Organizer: Computability in Europe 2007
  • Place of Presentation: Siena, Italy
  • Year and Date: 2007-06-20
• [Presentation] Unique existence and computability in constructive reverse mathematics (2007)
  • Author(s): Hajime Ishihara
  • Organizer: Computability in Europe 2007
  • Place of Presentation: Siena
  • Year and Date: 2007-06-20
• [Presentation] Neighbourhood and quasi-apartness spaces, and separation properties (2007)
  • Author(s): Hajime Ishihara
  • Organizer: 3rd Workshop on Formal Topology
  • Place of Presentation: Padova, Italy
  • Year and Date: 2007-05-12
• [Presentation] Neighbourhood and quasi-apartness spaces,and separation properties (2007)
  • Author(s): Hajime Ishihara
  • Organizer: 3rd Workshop on Formal Topology
  • Place of Presentation: Padua
  • Year and Date: 2007-05-12
• [Remarks]
  • URL: http://www.jaist.ac.jp/~ishihara