Metamathematical Properties of Intuitionistic Set Theories with Choice Principles

This paper is concerned with metamathematical properties of intuitionistic set theories with choice principles. It is proved that the disjunction property, the numerical existence property, Church’s rule, and several other metamathematical properties hold true for Constructive Zermelo-Fraenkel Set Theory and full Intuitionistic Zermelo-Fraenkel augmented by any combination of the principles of Countable Choice, Dependent Choices and the Presentation Axiom. Also Markov’s principle may be added. Moreover, these properties hold effectively. For instance from a proof of a statement ∀n ∈ ω ∃m ∈ ω φ(n,m) one can effectively construct an index e of a recursive function such that ∀n ∈ ω φ(n, {e}(n)) is provable. Thus we have an explicit method of witness and program extraction from proofs involving choice principles. As for the proof technique, this paper is a continuation of [32]. [32] introduced a selfvalidating semantics for CZF that combines realizability for extensional set theory and truth. MSC:03F50, 03F35