Hilbert's ϵ-Operator in Intuitionistic Type Theories

We investigate Hilbert's ϵ-calculus in the context of intuitionistic type theories, that is, within certain systems of intuitionistic higher-order logic. We determine the additional deductive strength conferred on an intuitionistic type theory by the adjunction of closed ϵ-terms. We extend the usual topos semantics for type theories to the ϵ-operator and prove a completeness theorem. The paper also contains a discussion of the concept of “partially defined” ϵ-term. MSC: 03B15, 03B20, 03G30.