HIGHER INFINITY AXIOMS AND RELATED PROPOSITIONS OF VARIOUS FIELD OF MATHEMATICS

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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.