A Normalizing Intuitionistic Set Theory with Inaccessible Sets

Details A Normalizing Intuitionistic Set Theory with Inaccessible Sets A Normalizing Intuitionistic Set Theory with Inaccessible Sets

2007-07-13

arxiv.org/abs/0707.1981v3

LMCS 3 (3:6) 2007

Computer Science

Logic in Computer Science

To be published in Logical Methods in Computer Science

Scientific paper

10.2168/LMCS-3(3:6)2007

We propose a set theory strong enough to interpret powerful type theories underlying proof assistants such as LEGO and also possibly Coq, which at the same time enables program extraction from its constructive proofs. For this purpose, we axiomatize an impredicative constructive version of Zermelo-Fraenkel set theory IZF with Replacement and $\omega$-many inaccessibles, which we call \izfio. Our axiomatization utilizes set terms, an inductive definition of inaccessible sets and the mutually recursive nature of equality and membership relations. It allows us to define a weakly-normalizing typed lambda calculus corresponding to proofs in \izfio according to the Curry-Howard isomorphism principle. We use realizability to prove the normalization theorem, which provides a basis for program extraction capability.

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