Let V be a set of number-theoretical functions. We define a notion of V -realizability for predicate formulas in such a way that the indices of functions in V are used for interpreting the implication and the universal quantifier. In this article, we prove that Intuitionistic Predicate Calculus is sound with respect to the semantics of V -realizability if and only if some natural conditions for V hold.

Let V be a set of number-theoretical functions. We define a notion of absolute V -realizability for predicate formulas and sequents in such a way that the indices of functions in V are used for interpreting the implication and the universal quantifier. In this article, we prove that Basic Predicate Calculus is sound with respect to the semantics of absolute V -realizability if V satisfies some natural conditions.