Constructive Zermelo-Fraenkel Set Theory, Power Set, and the Calculus of Constructions

If the power set operation is considered as a definite operation, but the universe of all sets is regarded as an indefinite totality, we are led to systems of set theory having Power Set as an axiom but only Bounded Separation axioms and intuitionistic logic for reasoning about the universe at large. The study of subsystems of ZF formulated in intuitionistic logic with Bounded Separation but containing the Power Set axiom was apparently initiated by Pozsgay (1971, 1972) and then pursued more systematically by Tharp (1971), Friedman (1973a), and Wolf (1974). These systems are actually semi-intuitionistic as they contain the law of excluded middle for bounded formulae. Pozsgay had conjectured that his system is as strong as ZF, but Tharp and Friedman proved its consistency in ZF using a modification of Kleene’s method of realizability. Wolf established the equivalence in strength of several related systems. In the classical context, weak subsystems of ZF with Bounded Separation and Power Set have been studied by Thiele (1968), Friedman (1973b) and more recently