Mathematical Logic

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We organized five working groups. the groups exchanged informations each other, and opened meetings on several occasions. In those meetings, researchers discussed and announced results of their researches.In the followings, we list up the main results of the groups. Group I(Proof theory): The relations between subsystems of arithmetic with weak inductive definitions and ordinals were explained. Several results concerned with refrection pronciples and Paris-Harrington principles were obtained. Generalized built-up systems of fundamental sequences were proposed.Group II(Set theory and model theory): A set theory with modality and a set theory with intensionality were proposed. Assuming existence of several large cardinals, interesting abelian groups were constructed. Several properties of -stable rings were explained.Group III(Constructive mathematics): A theory of Fuzzy computability was made up. A consistency proof of Beeson's system PRS was given. A two-storied theory of transfinite mechanisms was proposed.Group IV(Theory of logical structures): It was shown that modal operators are interpretable by quantifiers in intermediate predlcate logics. The relation between the syntax and the semantics of logics without a part of structural rules, -calculus, and categorical grammars were uniformly discussed.Group V(Theory of non-standard universes): In a nonfinitary logic, an interpretation of infinitesimal caculus was obtained. Several properties of iterated polynomials in nonstandard analysis were explained. For Hilbert irreducibility theorem, a condition for that Z-If is finite was given, and it was shown that the bound is given by a polynomial. A nonstandard set theory, in which there are models of the five nonstandard set theories NST, IST, NSTE, NS2 and *NST, respectively, was proposed.