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Infinite combinatrics and its applications

无限组合及其应用

基本信息

批准号：
10640099
负责人：
SAKAE Kitami
金额：
$ 2.43万
依托单位：
Kitami Institute of Technology
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
1998
资助国家：
日本
起止时间：
1998 至 1999
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-10640099/
关键词：
weak Freese-Nation property Cohen models complete Boolean algebras supercompact cardinal Chang's conjecture P(ω)Cohen model Chang's conjecture Open Coloring Axiom forcing axioms homogeneity principle Cichon diagram Cohen モデルブール

项目摘要

The main achievement of this project was that we could establish the theory of so called the weak Freese-Nation property (WFN) of partial orderings. The study of WFN began with Fuchino-Koppelberg-Shelah paper in 1996. In Fuchino-Soukup 1997 we then realized that WFN is highly set-theoretic 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 Fuchino-Soukup 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 … More 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, set-theoretic analysis and set theory of reals as well. Less

该项目的主要成果是建立了偏序的弱自由民族性质（WFN）理论。WFN的研究始于1996年Fuchino-Koppelberg-Shelah的论文。在Fuchino-Soukup 1997中，我们意识到WFN在独立性结果和这种独立性中经常涉及的一致性强度方面是高度集理论的。在我们的项目框架内，我们可以解决上面提到的Fuchino-Soukup论文中提出的所有开放问题，以及在本研究期间作为自然问题出现在我们脑海中的其他几个问题。对于我们得到的一些涉及到大基数的相容性强度的结果，精确的等相容性还有待确定。虽然这个问题似乎是非常困难的，如果不可能的话，因为在这些简洁性结果中使用的大基数是如此之大，以至于没有已知的内模型理论。 ...更多信息（P（ω）），ω）的WFN假设捕捉了Cohen模型的许多组合性质，因此该假设可以被视为Cohen模型的自然公理之一。最近Istvan Juhasz和Kenneth Kunen介绍了他们称之为SEP的属性，它概括了Juhasz，Soukup和Szentmiklossy的WFN和CH*。最后一个原则也被称为在科恩模型中有效的自然公理。在SEP的新观点下，Juhasz和Kunen得到了关于WFN和CH* 的一些非常有趣的结果，以及它们与Juhasz、Soukup和Szentmiklossy的C_（10）D_似乎SEP及其变体的这种新设置提供了一堆新问题，并提供了根据上述原理重新构建已知结果的可能性，这些结果是彼此分开研究的，因此它们可以逐渐地放在一起以建立统一的理论。在该项目的框架内，我们还可以获得Pκ（λ）-组合学中的一些其他结果，集合论分析和实数的集合论。少