| 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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| 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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