Project/Area Number  06302014 
Research Category 
GrantinAid for Cooperative Research (A).

Research Field 
General mathematics (including Probability theory/Statistical mathematics)

Research Institution  Nagoya University 
Principal Investigator 
篠田 寿一 名古屋大学, 助教授
SHINODA Juichi Nagoya University, Graduate School of Human Informatics, Associate Professor, 大学院・人間情報学研究科, 助教授 (30022685)

CoInvestigator(Kenkyūbuntansha) 
TANAKA Kazuyuki Tohoku University, Department of Mathematics, Associate Professor, 理学部, 助教授 (70188291)
KAKUDA Yuzuru Kobe University, Faculty of Engineering, Professor, 工学部, 教授 (50031365)
MATSUBARA Yo Nagoya University, School of Informatics and Sciences, Associate Professor, 情報文化学部, 助教授 (30242788)
YASUMOTO Masahiro Nagoya University, Graduate School of Polymathematics, Associate Professor, 大学院・多元数理科学研究科, 助教授 (10144114)
OZAWA Masanao Nagoya University, School of Informatics and Sciences, Professor, 情報文化学部, 教授 (40126313)

Project Fiscal Year 
1994 – 1995

Project Status 
Completed(Fiscal Year 1995)

Budget Amount *help 
¥5,100,000 (Direct Cost : ¥5,100,000)
Fiscal Year 1995 : ¥2,500,000 (Direct Cost : ¥2,500,000)
Fiscal Year 1994 : ¥2,600,000 (Direct Cost : ¥2,600,000)

Keywords  recursive function / computational complexity / Booleanvaled model / axiomatic set theory / nonstandard analysis / 帰納的関数 / 計算量 / ブール値モデル / 公理的集合論 / 超準解析 / 巨大基数 
Research Abstract 
The detail of the research results obtained in this project is to be published as a report in Japanese. Summary of several results is as follows. (1) On computational complexity of functions and their graphs, it is shown that there are continuaously many functions of polynomial growth rate which are not polynomial time computable from their graphs. (2) On generalized Kolmogorov complexity, it is shown that there are continuaously many sets which are sparse but not self Pprintable. (3) The theory of Booleanvalued models on nonstandard models of Peano Arithmetic is established. As an application, the theory I Sigma_0 plus Pigeon Hole Principle does not prove the proposition Count. (4) Every function which dominates all arithmeical functions has higher degree than a generic degree. There is a function such that its degree is a minimal upper bound of the arithmetical degrees and any functon of degree below its degree is dominated by an arithmetical function. (5) It is shown that from a given supersutructure in a Booleanvaled model of set theory a nonstandard universe can be condtructed so that the forcing method is applicable to nonstandard analysis. Using this method, a uniform incomplete ultrapower of reals is constructed. (6) By improving the formalized Berry's paradox, a new proof of the Godel first incompleteness theorem is obtained, and from it the Godel second incomplete theorem is deduced modeltheoretically. Also, a new proof of the Godel second incompleteness theorem is given based on the Kolmogorov complexity. (7) Assuming V = L,it is shown that the Kleene degrees of II^1_ sets are nondistributive.
