Epistemology versus Ontology - Essays on the Philosophy and Foundations of Mathematics in Honour of Per Martin-Löf
Epistemology versus Ontology - Essays on the Philosophy and Foundations of Mathematics in Honour of Per Martin-Löf
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认识论与本体论 - 纪念佩尔·马丁-洛夫的数学哲学和基础论文
DOI：
10.1007/978-94-007-4435-6_15
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发表时间：
2012
期刊：
影响因子：
--
通讯作者：
Rathjen M
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文献类型：
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作者：
Rathjen M
全直觉Zermelo-Fraenkel集合论IZF是在构造性Zermelo-Fraenkel集合论CZF的基础上，通过增加全分离公理和幂集公理而得到的。CZF+全分离的强度与二阶算法相同，使用经典二阶算法中的直接可实现性解释以及二阶Heyting算法已经嵌入CZF+全分离的事实。本文讨论了幂集公理扩充的CZF的强度。它将被证明是相同的力量，作为权力Kripke-Platek集理论，以及一个特定的系统的类型理论，这是一个微积分的建设与一个宇宙。的减少使用可实现性解释，其中存在陈述的实现者为存在量词提供一组证人，而不是单个证人。的减少采用技术从序数分析，当结合一个特殊的双重否定解释，尊重外延，也表明，可以减少到CZF与负功率集公理。当CZF被后一个公理扩充后可以在中被解释，并且这个类型理论在中有一个类型即类的解释时，这个循环就完成了。
Full intuitionistic Zermelo-Fraenkel set theory,IZF, is obtained from constructive Zermelo-Fraenkel set theory,CZF, by adding the full separation axiom scheme and the power set axiom. The strength ofCZFplus full separation is the same as that of second order arithmetic, using a straightforward realizability interpretation in classical second order arithmetic and the fact that second order Heyting arithmetic is already embedded inCZFplus full separation. This paper is concerned with the strength ofCZFaugmented by the power set axiom,. It will be shown that it is of the same strength as Power Kripke–Platek set theory,, as well as a certain system of type theory,, which is a calculus of constructions with one universe. The reduction oftouses a realizability interpretation wherein a realizer for an existential statement provides a set of witnesses for the existential quantifier rather than a single witness. The reduction oftoemploys techniques from ordinal analysis which, when combined with a special double negation interpretation that respects extensionality, also show thatcan be reduced toCZFwith the negative power set axiom. AsCZFaugmented by the latter axiom can be interpreted inand this type theory has a types-as-classes interpretation in, the circle will be completed.