HIGHER INFINITY AXIOMS AND RELATED PROPOSITIONS OF VARIOUS FIELD OF MATHEMATICS

对于本课题的核心部分--集合上的理想的性质，我们得到了Matsubara、Abe及其合作者的如下结果：1.他们研究了具有更高无穷性的理想与t平方原理等无限组合之间的关系，得到了与之相关的几个结果；2.他们得到了P-κ(λ上的非平稳理想在基数算术上的某些假设下不是陡峭的。Abe和Shioya成功地刻画了正则超滤子定义的初等嵌入的不动点，并证明了他们的结果不能用强迫方法推广到一般的均匀超滤子的情况；对于强迫方法，Miyamoto证明了PFA的弱部分等价于某些大基数的存在；对于实数子集的性质，Blendle证明了Cohen实数集要么是稀疏的，要么是空的。Fuchino利用序数引入了关于着色实数的公理，并证明了公理是Juhasz，Sentnikosse，Soukup引入的公理的推广。他还支持了集合论的各种模型中的公理。为了将公理集合论应用到其他数学中，1.Eda证明了从三维欧几里得空间减去直线和平面的拓扑空间的基本群与夏威夷环的基本群的子群同构。Kakuda证明命题无限合取和析取足以引入“虚构的存在”的量词，并成功地建立了无穷小系统。Kakuda开始发展具有累积等级的结构系统的数学公式。这是受到巴威斯渠道理论的启发，它可能不仅适用于信息论，也适用于其他领域，如工程设计理论。

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.