Abstract. Extending Gödel’s Dialectica interpretation, we provide a functional interpretation of classical theories of positive arithmetic inductive definitions, reducing them to theories of finite-type functionals defined using transfinite recursion on well-founded trees. 1.

A central theme running through all the main areas of Mathematical Logic is the classification of sets, functions or theories, by means of transfinite hierarchies whose ordinal levels measure their `rank' or `complexity' in some sense appropriate to the underlying context. In Proof Theory this is manifest in the assignment of `proof theoretic ordinals' to theories, gauging their `consistency strength' and `computational power'. Ordinal-theoretic proof theory came into existence in 1936, springing forth from Gentzen's head in the course of his consistency proof of arithmetic. To put it roughly, ordinal analyses attach ordinals in a given representation system to formal theories. Though this area of mathematical logic has is roots in Hilbert's "Beweistheorie " - the aim of which was to lay to rest all worries about the foundations of mathematics once and for all by securing mathematics via an absolute proof of consistency - technical results in pro...

The context for this paper is Feferman's theory of explicit mathematics, T 0 . We address a problem that was posed in [F 82]. Let MID be the principle stating that any monotone operation on classifications has a least fixed point. The main objective of this paper is to show that T 0 + MID, when based on classical logic, also proves the existence of non-monotone inductive definitions that arise from arbitrary extensional operations on classifications. From the latter we deduce that MID, when adjoined to classical T 0 , leads to a much stronger theory than T 0 . 1 Introduction Prompted by the question of constructive justification of Spector's consistency proof for analysis, Kreisel initiated in 1963 the study of formal theories featuring inductive definitions (cf. [K 63]). Proof--theoretic investigations (cf. [BFPS 81], [F 82], [Ra 89]) of such theories have shown that the strength of monotone inductive definitions is not greater than that of positive or even accessibility inductive de...