Constructive zermelo-fraenkel set theory, power set, and the calculus of constructions

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 of CZF plus 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 in CZF plus full separation. This paper is concerned with the strength of CZF augmented by the power set axiom, CZFP. It will be shown that it is of the same strength as Power Kripke-Platek set theory, KP(P), as well as a certain system of type theory, MLVP, which is a calculus of constructions with one universe. The reduction of CZFP to KP(P) uses 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 of KP(P) to CZFP employs techniques from ordinal analysis which, when combined with a special double negation interpretation that respects extensionality, also show that KP(P) can be reduced to CZF with the negative power set axiom. As CZF augmented by the latter axiom can be interpreted in MLVP and this type theory has a types-as-classes interpretation in CZFP, the circle will be completed.