Non-standard models of arithmetic and the incompleteness theorems
| Project/Area Number |
24540125
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Multi-year Fund |
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| Section | 一般 |
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| Research Field |
General mathematics (including Probability theory/Statistical mathematics)
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| Research Institution | Kobe University |
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Principal Investigator |
KIKUCHI Makoto 神戸大学, システム情報学研究科, 准教授 (60273801)
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| Project Period (FY) |
2012-04-01 – 2015-03-31
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| Project Status |
Completed (Fiscal Year 2014)
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| Budget Amount *help |
¥4,550,000 (Direct Cost: ¥3,500,000、Indirect Cost: ¥1,050,000)
Fiscal Year 2014: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000)
Fiscal Year 2013: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
Fiscal Year 2012: ¥1,820,000 (Direct Cost: ¥1,400,000、Indirect Cost: ¥420,000)
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| Keywords | 不完全性定理 / 超準モデル / 算術 / 様相論理 / 逆理 / 数学基礎論 / 算術の超準モデル / 矛盾許容型論理 |
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| Outline of Final Research Achievements |
Firstly, we showed relationships between some proofs of the incompleteness theorems based on Berry's paradox. Then, we defined the concept of liar-type paradox by using modal logic and showed the existence of the arithmetical independent statements based on liar-type paradoxes. Thirdly, we investigated the set of theorems of Peano arithmetic on non-standard models of arithmetic on which Peano arithmetic is inconsistent, and showed the definability of complete theories of arithmetic on such models. At last, we gave generalizations of the incompleteness theorems, and indicated a new relationship between formalization of mathematical proofs and Hilbert's program.
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Report
(4 results)
Research Products
(7 results)
