Gödel’s interpretation of intuitionism

Gödel regarded the Dialectica interpretation as giving constructive content to intuitionism, which otherwise failed to meet reasonable conditions of constructivity. He founded his theory of primitive recursive functions, in which the interpretation is given, on the concept of computable function of finite type. I will 1) criticize this foundation, 2) propose a quite different one, and 3) note that essentially the latter foundation also underlies the Curry-Howard type theory and hence Heyting’s intuitionistic conception of logic. Thus the Dialectica interpretation (in so far as its aim was to give constructive content to intuitionism) is superfluous. In the set of notes [1938a] for an informal lecture, Gödel refers to a hierarchy of constructive or finitary systems, the lowest level of which he called finitary number theory and is, in fact, primitive recursive arithmetic (PRA). He mentions three constructive extensions of this system: the intuitionism of Brouwer and Heyting (the “modal-logical route”), concerning which he is quite critical, the use of transfinite induction, inspired by Gentzen’s consistency proof for PA, and the use of functions of finite or even transfinite type over the type N of natural numbers. The latter extension seems to be mentioned here for the first time in connection with constructive extensions of finitary number theory. He ranks it the most satisfactory of these extensions ∗Versions of this paper were presented at a joint workshop on Hilbert at CarnegieMellon University and the University of Pittsburgh in June 1998, at a workshop on foundations of mathematics at UCLA in May 2005 and at the IMLA ’05 workshop [Intuitionistic Modal Logic and Applications, of course] in Chicago. I received helpful comments on all of these occasions. As for the history of the idea of a hierarchy of types, I know only of these examples: The hierarchy of transfinite types in which the natural numbers constitute the lowest type