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2005 Fiscal Year Final Research Report Summary

Applications of pcf to ideals on P_κλ and infinitary combinatorics, and independence proof

Research Project

Project/Area Number 16540127
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, 工学部, 教授 (10159452)

Co-Investigator(Kenkyū-buntansha) KAMO Shizuo  University of Osaka Prefecture, Faculty of Science, Professor, 理学部, 教授 (30128764)
SHIOYA Masahiro  University of Tsukuba, Institute of Mathematics, Full-time Lecture, 数理物質科学研究科, 講師 (30251028)
Project Period (FY) 2004 – 2005
KeywordsP_κλ / partition property / supercompact / ineffability / stationary reflection / forcing / nonstationary ideal / presaturation
Research Abstract

(1)We give a simple proof for Kamo's theorem that there exists a normal ultrafilter on P_κλ if κ is λ supercompact.
(2)We show the ideal of the subsets of P_κλ」a without the weak partition property is not κ^+ saturated.
(3)We showed the minimal ideal on P_κλ with the partition property is not λ^+ saturated.
(4)We proved the minimal normal ideal on P_κλ with the partition priperty is a proper extension of the ideal of noncompletely ineffable subsets.
(5)We showed stationary reflection principle for P_κλ fails if ω_1<κ<λ.
(6)We present a forcing model in which, for every regular μ【less than or equal】κ, the restriction of the nonstationary ideal on P_μκ to {x:cf(sup(x))=ω} is weakly presaturated.
(7)We show that the restriction of the nonstationary ideal on P_κ to {x:cf(sup(x))=ω} is weakly presaturated under the σ stationary reflection in P_κ2^2^2^<<κ>.
(8)We give a correct and simple proof for Shelah's theorem that the minimal cardinality of unbounded subset of P_κλ is equal to that of stationary subset.
(9)We prove the existence of a diamond sequence for P_κλ provided that 2^ω=2^<<κ>, and present a forcing model such that P_<ω2>λ carries a diamond sequence whereas 2^ω<2^<ω_1>.
(10)We showed that the additivity of the ideal Ι_f) is not, greater than the bounded number, and the cofinality of Ι_f) is not smaller than the dominating number.
(11)We present a function f∈^ωω such that T_f is not an ideal.

  • Research Products

    (12 results)

All 2006 2005 Other

All Journal Article (12 results)

  • [Journal Article] An ultrafilter with preperty 62006

    • Author(s)
      Masahiro Shioya
    • Journal Title

      Proceedings of the American Mathematical Society 134.6

      Pages: 1819-1821

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Non reflecting stationary sets in P_kλ2006

    • Author(s)
      Masahiro Shioya
    • Journal Title

      Advance in Mathematics 199.1

      Pages: 185-191

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] An ultrafilter with 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(欧文)」より
  • [Journal Article] Nonreflecting stationary sets in P_κλ2006

    • Author(s)
      Masahiro Shioya
    • Journal Title

      Advances in Mathematics 199

      Pages: 185-191

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Notes on subtlety and ineffability in P_kλ2005

    • Author(s)
      Yoshiniro abe
    • Journal Title

      Archive for Mathematical Logic 44.5

      Pages: 619-631

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Partitioning pairs of uncountable sets2005

    • Author(s)
      Masahiro Shioya
    • Journal Title

      Logic Colloquium 200,Lecture Notes in Logic 19

      Pages: 350-364

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Notes on subtlety and ineffability in P_κλ2005

    • Author(s)
      Yoshihiro Abe
    • Journal Title

      Archive for Mathematical Logic 44

      Pages: 619-631

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

    • Author(s)
      Masahiro Shioya
    • Journal Title

      Logic Colloquium 2000, Lecture Notes in Logic 19

      Pages: 350-364

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

    • Author(s)
      Masahiro Shioya
    • Journal Title

      Proceedings of Computational Prospects of Infinity to appear

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

    • Author(s)
      Masahiro Shioya
    • Journal Title

      Journal of Mathematical Society of Japan to appear

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

    • Author(s)
      Masahiro Shioya
    • Journal Title

      Journal of Mathematical Society of Japan to appear

    • Description
      「研究成果報告書概要(欧文)」より

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Published: 2007-12-13  

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