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2007 Fiscal Year Final Research Report Summary

Ideals on P_κλ considered from the point of view of the relation with large cardinals axioms and the generalized continuum hypothesis

Research Project

Project/Area Number 18540143
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 InstitutionKanagawa University

Principal Investigator

ABE Yoshihiro  Kanagawa University, Faculty of Engineering, Professor (10159452)

Co-Investigator(Kenkyū-buntansha) KAMO Shizuo  University of Osaka Prefecture, Faculty of Science, Professor (30128764)
SHIOYA Masahiro  University of Tsukuba, Graduate School of Pure and Applied Sciences, Fulltime Lecturer (30251028)
FUCHINO Sakae  Chubu University, Faculty of Engineering, Professor (30292098)
Project Period (FY) 2006 – 2007
KeywordsP_κλ / partition property / ineffability / stationary reflection / saturated ideal / fiorcing / club filter / diamond principle
Research Abstract

(1) We proved that |Pκλ|=λ if κ is mildly λ ineffable and cf(λ)〓κ.
(2) We showed the club filter on Pκλ is weakly presaturated under the Reflection principle.
(3) We showed n is mildly A ineffable Pκ(2^<λ<κ>) has the weak partition property.
(4) We proved every ineffbale subset of Pκλ has the partition property if no cardinal in (κ, λ] is II^2_1-indescribable.
(5) Starting from a model of a huge cardinal we built a simple forcing model in which ω_1 carries an ω_2 saturated ideal.
(6) We extended a theorem of Shelah on the diamond principle in Pω_1λ(◇ω_, λ,) to Pκλ as follows:
(a) ◇_<κ, λ> holds if 2^ω=2^κ with κ<λ
(b) Building upon a model of a supercompact cardinal, we gave a forcing model in which 2^ω=2^κ whereas holds whenever κ< λ.
(7) We got some independence results for cofinality, additivity number, and covering number of the ideal I_f.

  • Research Products

    (35 results)

All 2008 2007 2006 Other

All Journal Article (11 results) (of which Peer Reviewed: 5 results) Presentation (21 results) Book (3 results)

  • [Journal Article] Stationary reflection and the club filter2007

    • Author(s)
      Masahiro Shioya
    • Journal Title

      Journal of Mathematical Society of Japan 59

      Pages: 1045-1065

    • Description
      「研究成果報告書概要(和文)」より
    • Peer Reviewed
  • [Journal Article] An ultrafilter with the property σ2006

    • Author(s)
      Masahiro Shioya
    • Journal Title

      Proceedings of the American Mathematical Society 134

      Pages: 1819-1821

    • Description
      「研究成果報告書概要(和文)」より
    • Peer Reviewed
  • [Journal Article] An ultrafilter with the property δ2006

    • Author(s)
      Masahiro Shioya
    • Journal Title

      Proceedings of the American Mathematical Society 134

      Pages: 1819-1821

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] The cardinal coefficients of the ideal I_f

    • Author(s)
      Shizuo Kamo(with Noboru Osuga)
    • Journal Title

      Archive for Mathematical Logic (to appear)

    • Description
      「研究成果報告書概要(和文)」より
    • Peer Reviewed
  • [Journal Article] Diamonds on P_kλ

    • Author(s)
      Masahiro Shioya
    • Journal Title

      Computational Prospects of Infinity (to appear)

    • Description
      「研究成果報告書概要(和文)」より
    • Peer Reviewed
  • [Journal Article] A proof of Shelah's covering theorem for P_kλ

    • Author(s)
      Masahiro Shioya
    • Journal Title

      Asian Journal of Mathematics (to appear)

    • Description
      「研究成果報告書概要(和文)」より
    • Peer Reviewed
  • [Journal Article] A new saturated filter

    • Author(s)
      Masahiro Shioya
    • Journal Title

      数理解析研究所考究録 (to appear)

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] The cardinal coefficients of the ideal I_f

    • Author(s)
      Shizuo Karno
    • Journal Title

      Archive for Mathematical Logic (in press)(to appear)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Diamonds on ρ_kλ

    • Author(s)
      Masahiro Shioya
    • Journal Title

      Proceedings of Computational Prospects of infinity (in press)(to appear)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] A proof of Shelah's covering theorem for ρ_kλ

    • Author(s)
      Masahiro Shioya
    • Journal Title

      Asian Journal of Mathematics (in press)(to appear)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] A new saturated filter

    • Author(s)
      Masahiro Shioya
    • Journal Title

      Surikaisekikenkyusho Kokyuroku (in press)(to appear)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Presentation] 定常集合の組合せ論2008

    • Author(s)
      塩谷 真弘
    • Organizer
      日本数学会年会(数学基礎論・歴史分科会特別講演)
    • Place of Presentation
      近畿大学
    • Year and Date
      2008-03-25
    • Description
      「研究成果報告書概要(和文)」より
  • [Presentation] Combinatorics on stationary sets2008

    • Author(s)
      Masahiro Shioya
    • Organizer
      Annual Meeting of Mathematical Society of Japan
    • Place of Presentation
      Kinki University
    • Year and Date
      2008-03-25
    • Description
      「研究成果報告書概要(欧文)」より
  • [Presentation] Ineffability and the partition property on P_kλ2008

    • Author(s)
      Yoshihiro Abe
    • Organizer
      神戸数理論理学セミナー
    • Place of Presentation
      神戸大学
    • Year and Date
      2008-03-05
    • Description
      「研究成果報告書概要(和文)」より
  • [Presentation] Ineffability and the partition property on P_κλ2008

    • Author(s)
      Yoshihiro Abe
    • Organizer
      Kobe Mathematical Logic Seminar
    • Place of Presentation
      Kobe University
    • Year and Date
      2008-03-05
    • Description
      「研究成果報告書概要(欧文)」より
  • [Presentation] 集合論のブール値モデル2007

    • Author(s)
      加茂 静夫
    • Organizer
      数学基礎論サマースクール
    • Place of Presentation
      静岡大学
    • Year and Date
      20070905-07
    • Description
      「研究成果報告書概要(和文)」より
  • [Presentation] Boolean-valued model of set theory2007

    • Author(s)
      Shizuo Kamo
    • Organizer
      Summer School of Foundation of Mathematics
    • Place of Presentation
      Shizuoka University
    • Year and Date
      20070905-07
    • Description
      「研究成果報告書概要(欧文)」より
  • [Presentation] 強制法入門2007

    • Author(s)
      渕野 昌
    • Organizer
      数学基礎論サマースクール
    • Place of Presentation
      静岡大学
    • Year and Date
      20070904-06
    • Description
      「研究成果報告書概要(和文)」より
  • [Presentation] Introduction to forcing2007

    • Author(s)
      Sakae Fuchino
    • Organizer
      Summer School of Foundation of Mathematics
    • Place of Presentation
      Shizuoka University
    • Year and Date
      20070904-06
    • Description
      「研究成果報告書概要(欧文)」より
  • [Presentation] A new model with a saturated filter2007

    • Author(s)
      Masahiro Shioya
    • Organizer
      公理的集合論と集合論的位相空間論
    • Place of Presentation
      京都大学数理解析研究所
    • Year and Date
      2007-11-28
    • Description
      「研究成果報告書概要(和文)」より
  • [Presentation] On the coincidence of ineffability with partition property2007

    • Author(s)
      阿部 吉弘
    • Organizer
      日本数学会秋季総合分科会
    • Place of Presentation
      東北大学
    • Year and Date
      2007-09-24
    • Description
      「研究成果報告書概要(和文)」より
  • [Presentation] On the coincidence of ineffability with partition property2007

    • Author(s)
      Yoshihiro Abe
    • Organizer
      Autumn Meeting of Mathematical Society of Japan
    • Place of Presentation
      Tohoku University
    • Year and Date
      2007-09-24
    • Description
      「研究成果報告書概要(欧文)」より
  • [Presentation] From projective algebra, κ-metrizable space to Cohen model2007

    • Author(s)
      Sakae Fuchino
    • Organizer
      Autumn Meeting of Mathematical Society of Japan
    • Place of Presentation
      Tohoku University
    • Year and Date
      2007-09-24
    • Description
      「研究成果報告書概要(欧文)」より
  • [Presentation] Kunen's saturated ideal revisited2007

    • Author(s)
      Masahiro Shioya
    • Organizer
      Singular Cardinal Combinatorics and Inner Models
    • Place of Presentation
      University of Florida
    • Year and Date
      2007-03-06
    • Description
      「研究成果報告書概要(和文)」より
  • [Presentation] Simple construction of saturated ideals along Kunen's argument2007

    • Author(s)
      塩谷 真弘
    • Organizer
      数学教室談話会
    • Place of Presentation
      神奈川大学
    • Year and Date
      2007-02-20
    • Description
      「研究成果報告書概要(和文)」より
  • [Presentation] Simple construction of saturated ideals along Kunen's argument2007

    • Author(s)
      Masahiro Shioya
    • Organizer
      Mathematical Seminar
    • Place of Presentation
      Kanagawa University
    • Year and Date
      2007-02-20
    • Description
      「研究成果報告書概要(欧文)」より
  • [Presentation] Forcing notionの性質Eに関して2007

    • Author(s)
      加茂 静夫
    • Organizer
      公理的集合論研究集会
    • Place of Presentation
      静岡大学
    • Year and Date
      2007-02-17
    • Description
      「研究成果報告書概要(和文)」より
  • [Presentation] On the property of forcing notion E2007

    • Author(s)
      Shizuo Kamo
    • Organizer
      Axiomatic Set Theory Meeting
    • Place of Presentation
      Shizuoka University
    • Year and Date
      2007-02-17
    • Description
      「研究成果報告書概要(欧文)」より
  • [Presentation] Almost disjoint families on large underlying sets2006

    • Author(s)
      渕野 昌
    • Organizer
      実数の集合論と反復強制法の相互関係
    • Place of Presentation
      京都大学数理解析研究所
    • Year and Date
      2006-10-18
    • Description
      「研究成果報告書概要(和文)」より
  • [Presentation] 射影代数、κ-距離付け可能空間から、コーエンモデルへ2006

    • Author(s)
      渕野 昌
    • Organizer
      日本数学会秋季総合分科会(数学基礎論・歴史分科会特別講演)
    • Place of Presentation
      東北大学
    • Year and Date
      2006-10-18
    • Description
      「研究成果報告書概要(和文)」より
  • [Presentation] A new model with a saturated filter2006

    • Author(s)
      Masahiro Shioya
    • Organizer
      Axiomatic Set Theory and Set-Theoretic Topology
    • Place of Presentation
      Surikaisekikenkyusho
    • Year and Date
      2006-10-18
    • Description
      「研究成果報告書概要(欧文)」より
  • [Presentation] Almost disjoint families on large underlying sets2006

    • Author(s)
      Sakae Fuchino
    • Organizer
      Relationship between Set Theory of Reals and Iterated Forcing
    • Place of Presentation
      Surikaisekikenkyusho
    • Year and Date
      2006-10-18
    • Description
      「研究成果報告書概要(欧文)」より
  • [Book] 岩波数学辞典第4版(104-「巨大基数」pp.296-299)2007

    • Author(s)
      阿部 吉弘
    • Total Pages
      1976+32
    • Publisher
      岩波書店
    • Description
      「研究成果報告書概要(和文)」より
  • [Book] 岩波数学辞典第4版(239-「選択公理」pp.716-717)2007

    • Author(s)
      阿部 吉弘
    • Total Pages
      1976+32
    • Publisher
      岩波書店
    • Description
      「研究成果報告書概要(和文)」より
  • [Book] 岩波数学辞典第4版(95-「強制法」pp.247-260)2007

    • Author(s)
      加茂 静夫
    • Total Pages
      1976+32
    • Publisher
      岩波書店
    • Description
      「研究成果報告書概要(和文)」より

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Published: 2010-02-04  

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