Functional interpretation and the existence property

Stephen C. Kleene [7] introduced the notion of realisability for analysing the relation between intuitionism and the theory of recursive functions. In particular he was able to show that intuitionistic number theory is closed under a (weak) Church rule: If ∀x∃yA(x, y) is provable, then there is a recursive function f such that for every natural number n, A(n, f(n) ) is provable. Using this it was observed (later on) that the theory has the existence property: For any provable sentence ∃xA(x) there is a number n such that A(n) is provable. Before realisability was discovered Kurt Gödel worked with ideas concerning a functional interpretation [2, 4]. One of the applications motivating Gödel was akin to Kleene’s, namely: