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Set-theoretic study of measures

Research Project

Project/Area Number 16540094
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 InstitutionUniversity of Tsukuba

Principal Investigator

SHIOYA Masahiro  University of Tsukuba, Graduate School of Pure and Applied Sciences, Instructor, 大学院数理物質科学研究科, 講師 (30251028)

Co-Investigator(Kenkyū-buntansha) ABE Yoshihiro  Kanagawa University, Department of Engineering, Professor, 工学部, 教授 (10159452)
TSUBOI Akito  University of Tsukuba, Graduate School of Pure and Applied Sciences, Professor, 大学院数理物質科学研究科, 教授 (30180045)
MATSUBARA Yo  Nagoya University, Graduate School of Information Sciences, Professor, 大学院情報科学研究科, 教授 (30242788)
塚田 信高  筑波大学, 大学院・数理物質科学研究科, 助手 (50015559)
Project Period (FY) 2004 – 2006
Project Status Completed (Fiscal Year 2006)
Budget Amount *help
¥2,700,000 (Direct Cost: ¥2,700,000)
Fiscal Year 2006: ¥900,000 (Direct Cost: ¥900,000)
Fiscal Year 2005: ¥900,000 (Direct Cost: ¥900,000)
Fiscal Year 2004: ¥900,000 (Direct Cost: ¥900,000)
KeywordsMathematical logic / Set theory / Infinitary combinatorics / Large cardinal / Forcing / Filter / Saturation / Stationary set / 公理的集合論 / 生成超巾 / 飽和イデアル / 定常集合 / 超フィルタ / 反映原理 / イデアル
Research Abstract

The crucial fact that separates the uncountable from the countable is that the notion of stationariness can be defined for subsets of uncountable sets. Roughly speaking, stationariness is a combinatorial counterpart of having a positive measure.
The combinatorics of P_κλ investigates that notion in its greatest generality. Here κ denotes a regular uncountable cardinal and λ is a cardinal at least κ.
When κ, is a λ-supercompact cardinal, a normal ultrafilter on P_κλ is an important object to study. It has been an open question whether P_κλ carries a normal ultrafilter with property χ if κ, is λ-supercompact. In [3] we constructed one with a property slightly weaker than this. The proof shows in fact that the Generalized Continuum Hypothesis implies the existence of an ultrafilter with property χ. The latter result should be useful in relative consistency results.
In joint work with Shelah [2] we showed that the principle of Stationary Reflection in P_κλ fails if κ is greater than ω1 and λ is greater than κ. Moreover we have made progress in applying the principle of Stationary Reflection. In [4] we gave two extensions of Shelah's result that the Diamond principle holds in P_<ω1>λ In [5] we extended the well-known result of Foreman-Magidor-Shelah that the club filter on P_<ω1>λ can be presaturated.

Report

(4 results)
  • 2006 Annual Research Report   Final Research Report Summary
  • 2005 Annual Research Report
  • 2004 Annual Research Report
  • Research Products

    (26 results)

All 2007 2006 2005 Other

All Journal Article (23 results) Book (3 results)

  • [Journal Article] Nonstandard models that are definable in models of Peano Arithmetic2007

    • Author(s)
      Akito Tsuboi, Kazuma Ikeda
    • Journal Title

      Math. Log. Quart. 53

      Pages: 27-37

    • Related Report
      2006 Annual Research Report
  • [Journal Article] Nonreflecting stationary sets in Puλ2006

    • Author(s)
      Saharon Shelah, Masahiro shioya
    • Journal Title

      Adv. Math. 199

      Pages: 185-191

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Au ultrafilter with property σ2006

    • Author(s)
      Masahiro Shioya
    • Journal Title

      Proc. Amer. Math. Soc. 134

      Pages: 1819-1821

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Nonreflecting stationary sets in P_κλ.2006

    • Author(s)
      Saharon Shelah, Masahiro Shioya
    • Journal Title

      Adv.Math. 199

      Pages: 185-191

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] An ultrafilter with property σ.2006

    • Author(s)
      Masahiro Shioya
    • Journal Title

      Proc.Amer.Math.Soc. 134

      Pages: 1819-1821

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Nonreflecting stationary sets in Pud2006

    • Author(s)
      Saharon Shelan, Masahiro Shioya
    • Journal Title

      Adv. Math. 199

      Pages: 185-191

    • Related Report
      2006 Annual Research Report
  • [Journal Article] An ultra filter with property 62006

    • Author(s)
      Masahiro Shioya
    • Journal Title

      Proc. Awer. Math. Soc 134

      Pages: 1819-1821

    • Related Report
      2006 Annual Research Report
  • [Journal Article] Nonreflecting stationary sets in Pκλ2006

    • Author(s)
      Shelah, Saharon
    • Journal Title

      Advances in Math ematics 199・1

      Pages: 185-191

    • Related Report
      2005 Annual Research Report
  • [Journal Article] An ultrafilter with property σ2006

    • Author(s)
      Shioya, Masahiro
    • Journal Title

      Proceedings of Amer.Math.Soc 134・6

      Pages: 1819-1821

    • Related Report
      2005 Annual Research Report
  • [Journal Article] Partitioning pairs of uncountable sets2005

    • Author(s)
      Masahiro Shioya
    • Journal Title

      Lecture Notes in Logic 19

      Pages: 350-364

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Notes on subtlety and ineffability in Puλ2005

    • Author(s)
      Yoshihiro Abe
    • Journal Title

      Arch. Math. Logic 44

      Pages: 619-631

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Partitioning pairs of uncountable sets.2005

    • Author(s)
      Masahiro Shioya
    • Journal Title

      Lecture Notes in Logic 19

      Pages: 350-364

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Notes on subtlety and ineffability in P_κλ.2005

    • Author(s)
      Yoshihiro Abe
    • Journal Title

      Arch.Math.Logic 44

      Pages: 619-631

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Partitioning pairs of uncountable sets2005

    • Author(s)
      Shioya, Masahiro
    • Journal Title

      Lecture Notes in Logic 19

      Pages: 350-364

    • Related Report
      2005 Annual Research Report
  • [Journal Article] Diamonds on Puλ

    • Author(s)
      Masahiro Shioya
    • Journal Title

      Proc. of Comput. Prospects of Infinity (to appear)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Stationary reflection and the club filter

    • Author(s)
      Masahiro Shioya
    • Journal Title

      J. Math. Soc. Japan (to appear)

    • NAID

      10019989322

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Diamonds on P_κλ.

    • Author(s)
      Masahiro Shioya
    • Journal Title

      Proceedings of Computational Prospects of Infinity (to appear)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Stationary reflection and the club filter.

    • Author(s)
      Masahiro Shioya
    • Journal Title

      J.Math.Soc.Japan (to appear)

    • NAID

      10019989322

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Diamonds on Pud

    • Author(s)
      Masahiro Shioya
    • Journal Title

      Proc. Computational Prospects of Infinity (to appear)

    • Related Report
      2006 Annual Research Report
  • [Journal Article] Stationary reflection and the club filter

    • Author(s)
      Masahiro Shioya
    • Journal Title

      Jour. Math. Soc. Japan (to appear)

    • NAID

      10019989322

    • Related Report
      2006 Annual Research Report
  • [Journal Article] Partitioning pairs of uncountable sets

    • Author(s)
      塩谷真弘
    • Journal Title

      Proceedings of LC2000 (発表予定)

    • Related Report
      2004 Annual Research Report
  • [Journal Article] An ultrafilter with property

    • Author(s)
      塩谷真弘
    • Journal Title

      Proc.Amer.Math.Soc. (発表予定)

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Nonreflecting stationary sets in P_κλ

    • Author(s)
      Saharon Shelah, 塩谷真弘
    • Journal Title

      Advances in Math. (発表予定)

    • Related Report
      2004 Annual Research Report
  • [Book] ケ"ーテ"ルと20世紀の論理学 22006

    • Author(s)
      坪井明人
    • Total Pages
      286
    • Publisher
      東京大学出版会
    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Book] Godel and Logic in the Twentieth Century, 2. (in Japanese)2006

    • Author(s)
      Akito Tsuboi
    • Total Pages
      286
    • Publisher
      University of Tokyo Press
    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Book] ゲーデルと20世紀の論理学・完全性定理とモデル理論2006

    • Author(s)
      田中一之, 坪井明人, 野本和幸
    • Total Pages
      286
    • Publisher
      東京大学出版会
    • Related Report
      2006 Annual Research Report

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Published: 2004-04-01   Modified: 2016-04-21  

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