1988 Fiscal Year Final Research Report Summary
| Project/Area Number |
61302010
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| Research Category |
Grant-in-Aid for Co-operative Research (A)
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| Allocation Type | Single-year Grants |
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| Research Field |
General mathematics (including Probability theory/Statistical mathematics)
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| Research Institution | Science University of Tokyo (1987-1988)
Kyushu University (1986) |
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Principal Investigator |
UESU Tadahiro Science University of Tokyo ・ Professor, 理学部, 教授 (60015550)
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| Co-Investigator(Kenkyū-buntansha) |
SHIRAI Kohio Shizuoka University ・ Assistant Professor, 理学部, 助教授 (70077915)
UMEZAWA Toshio Shizuoka University ・ Professor, 理学部, 教授 (40021919)
EDA Katsuya Tsukuba University ・ Assistant Professor, 数学系, 助教授 (90015826)
MOTOHASHI Nobuyoshi Tsukuba University ・ Professor, 数学系, 教授 (70015874)
NISHIMURA Toshio Tsukuba University ・ Professor, 数学系, 教授 (00015471)
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| Project Period (FY) |
1986 – 1988
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| Keywords | Mathematical Logic / Foundation of Mathematics / Proof Thevy / Set Thevy / Model Thevy / Construction Mathematics / Recursin Thery / 超準解析 |
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| Research Abstract |
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.
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