2013 Fiscal Year Final Research Report
Reverse Mathematics in Constructive Set Theory
| Project/Area Number |
23540130
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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 | Japan Advanced Institute of Science and Technology |
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Principal Investigator |
ISHIHARA HAJIME 北陸先端科学技術大学院大学, 情報科学研究科, 教授 (10211046)
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| Project Period (FY) |
2011 – 2013
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| Keywords | 数理論理学 / 構成的数学 / 逆数学 / 集合論 |
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| Research Abstract |
Some open problems in constructive reverse mathematics, such as the monotone completeness theorem, the binary expansion theorem and the intermediate value theorem have been partially solved. The monotone completeness theorem is equivalent to LPO, a weak induction axiom and a kind of countable choice, and the binary expansion theorem and the intermediate value theorem are equivalent to versions of WKL with some convexity conditions on trees. Those results also hold in a subsystem of the constructive set theory.
A method of interpreting a set theory by interpreting it into the theory of operation APP, as an intermediate theory, which has an interpretation in to a theory of elementary analysis has been proposed. Axioms in APP which are sufficient to interpret the axioms of empty set, pair, infinity and a weak separation have been given. An extensive investigation on an axiom in APP which is sufficient to interpret the axiom of extensionality has been carried out.
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