Project/Area Number  09640299 
Research Category 
GrantinAid for Scientific Research (C)

Section  一般 
Research Field 
General mathematics (including Probability theory/Statistical mathematics)

Research Institution  Kanagawa University 
Principal Investigator 
ABE Yoshihiro Kanagawa University, Mathematics, Associated Professor, 工学部, 助教授 (10159452)

CoInvestigator(Kenkyūbuntansha) 
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)

Project Fiscal Year 
1997 – 1998

Project Status 
Completed(Fiscal Year 1998)

Budget Amount *help 
¥1,400,000 (Direct Cost : ¥1,400,000)
Fiscal Year 1998 : ¥600,000 (Direct Cost : ¥600,000)
Fiscal Year 1997 : ¥800,000 (Direct Cost : ¥800,000)

Keywords  compact cardinal / continuum hypothesis / normal ultrafilter / forcing / indescribability / bounded ideal / partition property / precipitous / コンパクト基数 / 連続体仮説 / 正規極大フィルター / 強制 / boundedイデアル / 分割の性質 / Precipitous / seminormal ideal / clused unbounded filter / diamond principle / very weak square principle / elementary submodel 
Research Abstract 
We investigated a K<@D1+@>D1C.C.(]SY.disubstituted right.[)Kstrategically 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 nonreflecting stationary sets to P_klambda (3)Forcing a nonregular ultrafilter on P_kK^<+++> with K supercompact. Combinatorics : (1)Combinatorial characterization of PIiindescribability 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.
