Lifschitz realizability for intuitionistic Zermelo–Fraenkel set theory

Abstract

A variant of realizability for Heyting arithmetic which validates Church’s thesis with uniqueness condition, but not the general form of Church’s thesis, was introduced by Lifschitz (Proc Am Math Soc 73:101–106, 1979). A Lifschitz counterpart to Kleene’s realizability for functions (in Baire space) was developed by van Oosten (J Symb Log 55:805–821, 1990). In that paper he also extended Lifschitz’ realizability to second order arithmetic. The objective here is to extend it to full intuitionistic Zermelo–Fraenkel set theory, IZF. The machinery would also work for extensions of IZF with large set axioms. In addition to separating Church’s thesis with uniqueness condition from its general form in intuitionistic set theory, we also obtain several interesting corollaries. The interpretation repudiates a weak form of countable choice, AC _ω,ω , asserting that a countable family of inhabited sets of natural numbers has a choice function. AC _ω,ω is validated by ordinary Kleene realizability and is of course provable in ZF. On the other hand, a pivotal consequence of AC _ω,ω , namely that the sets of Cauchy reals and Dedekind reals are isomorphic, remains valid in this interpretation. Another interesting aspect of this realizability is that it validates the lesser limited principle of omniscience.