2004 Fiscal Year Final Research Report Summary
Large cardinal properties of ideals
| Project/Area Number |
15540115
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
General mathematics (including Probability theory/Statistical mathematics)
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| Research Institution | Nagoya University |
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Principal Investigator |
MATSUBARA Yo Nagoya University, Graduate School of information Science, Professor, 大学院・情報科学研究科, 教授 (30242788)
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| Co-Investigator(Kenkyū-buntansha) |
YOSHINOBU Yasuo Nagoya University, Graduate School of information Science, Research Associate, 大学院・情報科学研究科, 助手 (90281063)
ABE Yoshihiro Kanagawa University, Department of Engineering, Professor, 工学部, 講師 (10159452)
SHIOYA Masahiro Tsukuba University, Department of mathematics, Assistant Professor, 数学系, 教授 (30251028)
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| Project Period (FY) |
2003 – 2004
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| Keywords | axiomatic set theory |
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| Research Abstract |
For X⊆P_κλ, if |{t∈X:sup(t)= δ}|<2^<|δ|> holds for every δ<λ, then we say that X is skinny. Let NS_<κλ> denote the non-stationary ideal over P_κλ. For X⊆P_κλ define the ideal NS_<κλ>|X as follows ; ∀Y⊆P_κλ(Y∈NS_<κλ>|X⇔X∩Y∈NS_<κλ>).
Y.Matsubara proved that if NS_<κλ>|X is precipitous then X has a skinny stationary subset of X・Previously Y.Matsubara and S.Shelah proved that if λ is a strong limit singular cardinal, then there is no skinny stationary subset of P_κλ.Therefore we can conclude that if λ is a strong limit singular cardinal, then NS_<κλ>|X cannot be precipitous for every X⊆P_κλ.
We also proved that assuming GCH below λ the existence of skinny stationary subset of P_κλ is equivalent to the diamond principle on {α<λ|cf(α)<κ}. Using this we can prove that under GCH the precipitousness of NS_<κλ> implies the diamond principle on every stationary subset of {α<λ|cf(α)<κ}. Therefore under GCH, the precipitousness of NS_<κλ> implies the ideal NS_λ|A cannot be saturated for every A⊆{α<λ|cf(α)<κ}. Here NS_λ denotes the non-stationary ideal over λ and NS_λ|A denotes the ideal over λ generated by NS_λ and λ-A・
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