 Intuitionistic Zermelo–Fraenkel set theory

Full intuitionistic Zermelo–Fraenkel set theory uses intuitionistic first-order logic, extensionality, pairing, union, infinity, full separation, power set, collection and epsilon induction. Collection is stronger than replacement intuitionistically. This is distinct from predicative constructive ZF, which restricts separation and replaces full power set. The full theory supports a Boolean-valued inner universe over intuitionistic set theory, giving the relative consistency of classical ZF.