2006 Fiscal Year Final Research Report Summary
Application of ideals for Godel's Program
| Project/Area Number |
17540110
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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, Associate Professor, 大学院情報科学研究科, 助教授 (90281063)
ABE Yoshihiro Kanagawa University, School of Engineering, Professor, 工学部, 教授 (10159452)
SHIOYA Masahiro Tsukuba University, Institute of Mathematics, Lecturer, 数学系, 講師 (30251028)
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| Project Period (FY) |
2005 – 2006
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| Keywords | set theory |
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| Research Abstract |
Let X be a subset of P_κλ. For each ordinal α【less than or equal】λ, let X_α = {s∈X:sup(s) = α}.
We say that X is skinny if ∀α【less than or equal】λ(|X_α|<2^<|α|>) holds. Furthermore X is said to be skinnier if ∀α【less than or equal】λ(|X_α|<a) and skinniest if ∀α【less than or equal】λ(|X_α|【less than or equal】1). We prove the following :
For a strong limit cardinal λ, a stationary subset X has a skinny stationary subset if NS_<κλ>|X is either precipitous or 2^λ saturated. On the other hand, Shelah proved that there is no skinny stationary subset of P_κλ if λ is a singular strong limit cardinal. Thus we obtained the following theorem :
Theorem
If λ, is a singular strong limit cardinal, then NS_<κλ> is nowhere precipitous and nowhere 2^λ saturated.
We proved that the existence of a skinnier stationary subset X of P_κλ, and λ^<<λ>=λ imply ◇_λ(E_X) where E_X = {sup(s):s∈X}. Clearly "skinny" and "skinnier" are equivalent under GCH. Therefore, assuming GCH ; if NS_<κλ> is precipitous or is 2^λ saturated, then ◇_λ(F) holds for every stationary subset F of E^λ_<<K>={α<λ:cf (α)<κ}.
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