1998 Fiscal Year Final Research Report Summary
Independence proof concerning compact cardinals, continuum hypotbesis, and normal ultrafilters
| Project/Area Number |
09640299
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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 | Kanagawa University |
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Principal Investigator |
ABE Yoshihiro Kanagawa University, Mathematics, Associated Professor, 工学部, 助教授 (10159452)
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| Co-Investigator(Kenkyū-buntansha) |
FUCHINO Sakae Kitami Institute of Technology, System Engineering, Professor, 工学部, 教授 (30292098)
SHIOYA Masahiro Tsukuba University, Mathematics, Assistant Professor, 数学系, 助手 (30251028)
KAMO Shizuo University of Osaka Prefecture, Mathematics, Professor, 総合科学部, 教授 (30128764)
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| Project Period (FY) |
1997 – 1998
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| Keywords | compact cardinal / continuum hypothesis / normal ultrafilter / forcing / indescribability / bounded ideal / partition property / precipitous |
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| Research Abstract |
We investigated a K<@D1+@>D1-C.C.(]SY.di-substituted right.[)K-strategically closed forcing adding arbitrary nunber of closed unbounded subsets of K.A Lottery preparation followed by this forcing preserves the strong compactness of K.In the resulting model the restricions of all normal ultrafilters to the ground model coincide.
We know a symmetric model in which the genemlized continuum hypothesis holds below a strongly compact cardinal K and fails at K.However it turned out establishing the same thing together with the axiom of choice is far more difficult than had been expected. It is not clear whether the strong compactness of K is preserved when we force the the axion of choice on the symmetric model.
Several facts and new technique are founded as in the folllowing.
Forcing : (1)Adding many closed unbounded sets (2)Adding non-reflecting stationary sets to P_klambda (3)Forcing a non-regular ultrafilter on P_kK^<+++> with K supercompact.
Combinatorics : (1)Combinatorial characterization of PIi-indescribability in P_klambda.
(2)Bounded ideal may or may not have the partition property.
(3)Bounded ideal is not precipitous
It is expected that further research makes the relation between the partition property and ineffability of P_klambda clear to bring great progress in the combinatorics of P_klambda.
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