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Comparative studies on nonstandard methods and constructive methods

Research Project

Project/Area Number 13640098
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field General mathematics (including Probability theory/Statistical mathematics)
Research InstitutionTohoku University

Principal Investigator

TANAKA Kasuyuki  Tohoku University, Graduate School of Science, Professor, 大学院・理学研究科, 教授 (70188291)

Co-Investigator(Kenkyū-buntansha) AKAMA Yohji  Tohoku University, Graduate School of Science, Associate Professor, 大学院・理学研究科, 助教授 (30272454)
TAKEDA Masayoshi  Tohoku University, Graduate School of Science, Professor, 大学院・理学研究科, 教授 (30179650)
MORITA Yasuo  Tohoku University, Graduate School of Science, Professor, 大学院・理学研究科, 教授 (20011653)
YAMAZAK Takeshi  Osaka Prefecture University, Dept.of Math.and Info.Sciences, Research Assistant, 総合科学部, 助手 (30336812)
Project Period (FY) 2001 – 2003
Project Status Completed (Fiscal Year 2003)
Budget Amount *help
¥1,700,000 (Direct Cost: ¥1,700,000)
Fiscal Year 2003: ¥800,000 (Direct Cost: ¥800,000)
Fiscal Year 2002: ¥900,000 (Direct Cost: ¥900,000)
Keywordssecond order arithmetic / WKL_0 / nonstandard method / WKLo
Research Abstract

Second order arithmetic was first introduced by D.Hilbert around 1920's to set a firm foundation of analysis. Among many different formalizations of second order arithmetic, RCA_0 and WKL_0 are particularly important with respect to Hilbert's program. Tanaka and Yamazaki, in cooperation with Simpson, proved that one can eliminate, the compactness argument from proofs for a certain sort of theorems in WKL_0, namely WKL_0 is conservative over RCA_0 for the formulas in a certain form. They also showed that the bounded dependent choice scheme, which is often used in the computation of real numbers, can not be eliminated likewise. Thus, Tanaka and Yamazaki constructed an appropriate truth definition for the real number system via the quantifier elimination method, by which one can manipulate the reals freely in RCA_0. By sophisticating this argument, Tanaka, jointly with Sakamoto, could prove Hilbert's Nullstellensatz within RCA_0. Yamazaki and Sakamoto also studied uniform versions of WKL_0 and other statements within higher order arithmetic. Yamazaki investigated the logical strength of completeness theorems for intuitionistic logic.

Report

(4 results)
  • 2003 Annual Research Report   Final Research Report Summary
  • 2002 Annual Research Report
  • 2001 Annual Research Report
  • Research Products

    (31 results)

All Other

All Publications (31 results)

  • [Publications] S.Simpson, K.Tanaka, T.Yamazaki: "Some conservation results on weak Koenig's lemma"Ann.Pure Appl.Logic. 118. 87-114 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] N.Sakamoto, K.Tanaka: "The strong soundness theorem for real closed fields and Hilbert's Nullstellensatz in second order arithmetic"Archive for Math.Logic. 43-3. 337-349 (2004)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] K.Tanaka, T.Yamazaki: "Manipulating the reals in RCA_0"S.Simpson(ed.),Reverse Mathematics 2001. (印刷中). (2004)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] T.Yamazaki: "Reverse mathematics and weak systems of O-1 strings for feasible analysis"S.Simpson(ed.),Reverse Mathematics 2001. (印刷中). (2004)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] N.Sakamoto, T.Yamazaki: "Uniform versions of some axioms of second order arithmetic"Math Logic Quarterly. (印刷中). (2004)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] T.Yamazaki: "Reverse mathematics and completeness theorems for intuitionistic logic"Notre Dame J.Formal Logic. 42,2001. 143-148 (2003)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] 田中一之: "数の体系と超準モデル"裳華房. 284 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] S.Simpson, K.Tanaka, T.Yamazaki: "Some conservation results on weak Koenig's lemma"Ann.Pure Appl.Logic. 118. 87-114 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] N.Sakamoto, K.Tanaka: "The strong soundness theorem for real closed fields and Hilbert's Nullstellensatz in second order arithmetic"Archive for Math.Logic. 43-3. 337-349 (2004)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] K.Tanaka, T.Yamazaki (S.Simpson(ed)): "Manipulating the reals in RCA_0"Reverse Mathematics 2001. (to appear). (2004)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] T.Yamazaki (S.Simpson(ed.)): "Reverse mathematics and weak systems of 0-1 strings for feasible analysis"Reverse Mathematics 2001. (to appear). (2004)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] N.Sakamoto, T.Yamazaki: "Uniform versions of some axioms of second order arithmetic"Math Logic Quarterly. (to appear). (2004)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] T.Yamazaki: "Reverse mathematics and completeness theorems for intuitionistic logic"Notre Dame J.Formal Logic. 42-2001. 143-148 (2003)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] K.Tanaka: "Logic of Arithmetic-Formal systems and Non-standard Models-(in Japanese)"Shokabo. 284 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] 田中 一之: "超算術的選択公理HACとその仲間たち"京大数理解析研 講究録. 1301. 79-83 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] N.Sakanioto, K.Tanaka: "The strong soundness theorem for real closed fields and Hilbert's Nullstellensatz in second order arithmetic"Archive for Math.Logic. (印刷中).

    • Related Report
      2003 Annual Research Report
  • [Publications] K.Tanaka, T.Yamazaki: "Manipulating the reals in RCAo"S.Siinpson(ed.), Reverse Mathematics 2001. (印刷中).

    • Related Report
      2003 Annual Research Report
  • [Publications] T.Yamazaki: "Reverse mathematics and weak systems of 0-1 strings for feasible analysis"S.Simpson(ed.), Reverse Mathematics 2001. (印刷中).

    • Related Report
      2003 Annual Research Report
  • [Publications] N.Sakamoto, T.Yamazaki: "Uniform versions of some axioms of second order arithmetic"Math Logic Quarterly. (印刷中).

    • Related Report
      2003 Annual Research Report
  • [Publications] T.Yamazaki: "Reverse mathematics and completeness theorems for intuitionistic logic"Notre Dame J.Formal Logic. 42,2001. 143-148 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] 田中一之, 鈴木登志雄: "数学のロジックと集合論"培風館. 240 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] S.Simpson, K.Tanaka, T.Yamazaki: "Some conservation results on weak Koenig's lemma"Ann. Pure Appl.Logic. 118. 87-114 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] T.Yamazaki: "Reverse mathematics and completeness theorems for intuitionistic logic"Notre Dame J. Formal Logic. (印刷中).

    • Related Report
      2002 Annual Research Report
  • [Publications] 田中 一之: "超算術的選択公理HACとその仲間たち"京大数理解析研講究録. (印刷中).

    • Related Report
      2002 Annual Research Report
  • [Publications] T.Yamazaki: "Reverse mathematics and weak systems of 0-1 strings for feasible analysis"Reverse Mathematics 2001. (印刷中).

    • Related Report
      2002 Annual Research Report
  • [Publications] 田中 一之: "数の体系と超準モデル"裳華房. 270 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] S.simpson, K.Tanaka, T.Yamazaki: "Some conservation results on weak Konig'slemma"Ann. Pure Applied Logic. (印刷中). (2002)

    • Related Report
      2001 Annual Research Report
  • [Publications] 坂本伸幸, 田中一之: "2階算術における実数と複素数"数理解析研究所構究録 圏論と証明論の新たな融合を目指して. 1217. 98-120 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] Yohji Akama: "Limiting Partial Combinatory Algebras towanls infinitary Lambola calculus and classical logic"Proc. of Comp.Sci.Logic, Lec.Notes in Comp.Sci. 2142. 399-413 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] 赤間 陽二: "Limiting Partial Combinatory Algebras"数理解析研究所構究録 圏論と証明論の新たな融合を目指して. 1217. 1-22 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] 田中 一之: "数の体系と超準モデル"裳華房. 270 (2002)

    • Related Report
      2001 Annual Research Report

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Published: 2002-04-01   Modified: 2016-04-21  

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