  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.
