Project/Area Number  10640099 
Research Category 
GrantinAid for Scientific Research (C).

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

Research Institution  Kitami Institute of Technology 
Principal Investigator 
SAKAE Kitami Kitami Institute of Technology, faculty of engineering, Prof., 工学部, 教授 (30292098)

CoInvestigator(Kenkyūbuntansha) 
TAKAHASHI Makoto Kobe University, faculty of human development, Ass. Prof., 人間発達学部, 助教授 (50154860)
ABE Yoshihiro Kitami Institute of Technology, faculty of engineering, Associate Prof., 工学部, 助教授 (10159452)
SANNAMI Atsushi Kitami Institute of Technology, faculty of engineering, Professor, 工学部, 教授 (30154157)
KADA Masaru Kitami Institute of Technology, faculty of engineering, Assistant, 工学部, 教授 (00312447)
BRENDLE Jorg Kobe University, graduate school of sci. and engin., Associate Prof., 自然科学研究科, 助教授 (70301851)

Project Fiscal Year 
1998 – 1999

Project Status 
Completed(Fiscal Year 1999)

Budget Amount *help 
¥3,800,000 (Direct Cost : ¥3,800,000)
Fiscal Year 1999 : ¥1,700,000 (Direct Cost : ¥1,700,000)
Fiscal Year 1998 : ¥2,100,000 (Direct Cost : ¥2,100,000)

Keywords  weak FreeseNation property / Cohen models / complete Boolean algebras / supercompact cardinal / Chang's conjecture / P(ω) / 完備ブール代数 / Chang'sconjecture / Ρ(ω) / Cohen model / Open Coloring Axiom / forcing axioms / homogeneity principle / Cichon diagram / Cohen モデル / ブール代数 / 基数不変量 
Research Abstract 
The main achievement of this project was that we could establish the theory of so called the weak FreeseNation property (WFN) of partial orderings. The study of WFN began with FuchinoKoppelbergShelah paper in 1996. In FuchinoSoukup 1997 we then realized that WFN is highly settheoretic in terms of independency results and consistency strength often involved in such independency. In the framework of our project we could solve all the open problems posed in the FuchinoSoukup paper mentioned above, and several other problems which came to our mind as natural questions during this research. For some results we obtaind in which some consistency strength of large cardinals is involved, exact equiconsistency remained to be determined. Though this problem seems to be very difficult if not possible, since the large cardinals employed in some of these concistency results are so large that there is no inner model theory known for them. One of the knowledge we obtained through the study of WFN
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was that the assumption of WFN of (P(ω)), ⊆) captures a lot of the combinatorial properties of Cohen models so that this assumption could be regarded as one of the natural axioms for Cohen models. Quite recently Istvan Juhasz and Kenneth Kunen introduced the property which they called SEP which generalizes both WFN and CH* of Juhasz, Soukup and Szentmiklossy. The last principle was also known as a natural axiom valid in Cohen models. In the new light of SEP, Juhasz and Kunen could obtain some very interesting results on WFN and CH*, and on their relations to CィイD1sィエD1(κ) of Juhasz, Soukup and Szentmiklossy, and to HP(κ) of D Fuchino and Brendle. It seems that this new setting with SEP and its variations offers a bunch of new problems and gives possibilities to recast known results on the principles mentioned above which were studied sofar rather separately to each other, so that they can be gradually put together to build a unified theory. In the framework of the project we could also obtain some other results in Pκ (λ)combinatorics, settheoretic analysis and set theory of reals as well. Less
