| Project/Area Number |
06302014
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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 | Nagoya University |
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Principal Investigator |
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)
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| Co-Investigator(Kenkyū-buntansha) |
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)
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| Project Period (FY) |
1994 – 1995
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| Project Status |
Completed (Fiscal Year 1995)
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| 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)
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| Keywords | recursive function / computational complexity / Boolean-valed model / axiomatic set theory / nonstandard analysis / 巨大基数 |
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| 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 P-printable.
(3) The theory of Boolean-valued 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 Boolean-valed 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 model-theoretically. 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.
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