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HIGHER INFINITY AXIOMS AND RELATED PROPOSITIONS OF VARIOUS FIELD OF MATHEMATICS

数学各领域的高等无穷公理及相关命题

基本信息

批准号：
09440078
负责人：
KAKUDA Yuzuru
金额：
$ 4.1万
依托单位：
KOBE UNIVERSITY
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (B)
财政年份：
1997
资助国家：
日本
起止时间：
1997 至 1998
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-09440078/
关键词：
ideal precipitous proper forcing axiom Cohen real large cardinals precipitous コーエン実数飽和的 precipitousness 中間論理 Proper foking axiom 分配律 Hyting Algebra inner modee

项目摘要

For properties of ideals on sets, the core of the research project, we obtained the following results by Matsubara, Abe and their cooperators ;1. They investigated the relation between ideals with higher infinity properties and infinite conbinatorics like t square principle, and obtained several results concerning it,2. They got the fact that the non-stationary ideals on Pκ(λ) cannot be precipitous under the certain assumption on the cardinality arithmetics.3. Abe and Shioya succeeded to characterize the fixed point of elementary embeddings defined by regular ultra filter, and showed that their result cannot be extended to the general case of uniform ultrafilters by using the forcing method.For forcing method, Miyamoto showed that the weak part of PFA is equiconsistent to the existence of some large cardinal.For properties on subsets of the reals, Blendle showed that a set of Cohen reals is either meager or empty Fuchino introduced the axiom concerning the coloring reals by using ordinals, and showed that the axiom is a generalization of the axiom introduced by Juhasz, Szentnikosse, Soukup. He also shoed the axiom holds in various models of set theory.For the application of axiomatic set theory to other mathematics,1.Eda showed that the fundamental group of the topological space subtracted lines and planes from the 3 dimensional Eucridian space is isomorphic to the subgroup of the fundamental group of Hawaiian earring.2. Kakuda showed that the propositional infinite conjunction and disjunction are enough for introducing the quantifier of the "existence of fictions", and succeeded to formulate the system of infinitesimals.4. Kakuda started to develop the mathematical formulation of the structural systems with accumulated hierarchies. This was inspired by the channel theory of Barwise, and it might be applicable not only to information theory, but also other fields, for example, the design theory of engineering.

对于本研究项目的核心——集合上理想的性质，松原、阿部及其合作者得到了以下成果： 1．他们研究了具有更高无穷大性质的理想与t平方原理等无穷组合之间的关系，并得到了一些相关结果；2．他们得出这样的事实：在基数算法的一定假设下，Pκ(λ)上的非平稳理想不可能是陡峭的。 3． Abe 和 Shioya 成功地刻画了由正则超滤波器定义的基本嵌入的不动点，并表明他们的结果不能通过使用强制方法扩展到均匀超滤波器的一般情况。对于强制方法，Miyamoto 表明 PFA 的弱部分与某些大基数的存在性等一致。对于实数子集的属性，Blendle 表明一组 Cohen 实数为或薄或空 Fuchino 用序数引入了有关着色实数的公理，并表明该公理是 Juhasz、Szentnikosse、Soukup 引入的公理的推广。他还将该公理应用于集合论的各种模型中。对于公理集合论在其他数学中的应用，1.Eda证明了从3维欧几里得空间中减去线和平面的拓扑空间的基本群与夏威夷耳环基本群的子群是同构的。2．角田证明命题的无限合取和析取足以引入“虚构的存在”的量词，并成功地阐述了无穷小系统。 4．角田开始开发具有累积层次结构系统的数学公式。这是受到Barwise通道理论的启发，它可能不仅适用于信息论，还适用于其他领域，例如工程设计理论。