Reverse Mathematics in Constructive Set Theory

构造性集合论中的逆向数学

• 批准号: 23540130
• 负责人: ISHIHARA HAJIME
• 金额: $ 3.24万
• 依托单位: Japan Advanced Institute of Science and Technology
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2011
• 资助国家: 日本
• 起止时间: 2011 至 2013
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-23540130/
• 关键词: 数理論理学; 構成的数学; 逆数学; 集合論; 数学基礎論

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.

构造性逆数学中的一些公开问题，如单调完备性定理、二元展开定理和介值定理，得到了部分解决。单调完备性定理等价于一个弱归纳公理和一种可数选择，二元展开定理和介值定理等价于树上具有某些凸性条件的WKL。这些结果在构造性集合论的一个子系统中也是成立的。提出了一种通过将集合论解释成运算论APP来解释集合论的方法，APP是一种中间理论，它对初等分析理论有解释作用。给出了APP中的公理，这些公理足以解释空集公理、对公理、无穷公理和弱分离公理。对APP中的一个公理进行了广泛的研究，该公理足以解释可拓性公理。

项目成果

Some conservative extension results of classical logic over intuitionistic logic

经典逻辑对直觉逻辑的一些保守推广结果

• 发表时间: 2012
• 作者: S. Kakihara;A. Ohara and T. Tsuchiya;Hajime Ishihara
• 通讯作者: Hajime Ishihara

Logic@JAIST

逻辑@JAIST

A monad in the combinatory algebras

组合代数中的单子

• 发表时间: 2014
• 作者: Peter Aczel;Hajime Ishihara;Takako Nemoto and Yasushi Sangu;I. Kubo;Hajime Ishihara and Tatsuji Kawai;Nobuhiro Asai;Hajime Ishihara;Nobuhiro Asai;Hajime Ishihara
• 通讯作者: Hajime Ishihara

Inifinitary propositional theories and set-generated classes

不定式命题理论和集合生成类

• 发表时间: 2012
• 作者: S. Kakihara;A. Ohara and T. Tsuchiya;Hajime Ishihara;Y. Nagahata;Y. Nagahata;N. Asai;Jun Sekine;Takafumi Kanamori and Atsumi Ohara;Y. Nagahata;Hajime Ishihara
• 通讯作者: Hajime Ishihara

Completeness and cocompleteness of the categories of basic pairs and concrete spaces

基本对和具体空间范畴的完备性和共完备性

• 期刊: Math. Structures Comput. Sci
• 作者: Peter Aczel;Hajime Ishihara;Takako Nemoto and Yasushi Sangu;I. Kubo;Hajime Ishihara and Tatsuji Kawai
• 通讯作者: Hajime Ishihara and Tatsuji Kawai