Predicative Functionals and an Interpretation of ID^_

In 1958 Gödel published his Dialectica interpretation, which reduces classical arithmetic to a quantifier-free theory T axiomatizing the primitive recursive functionals of finite type. Here we extend Gödel’s T to theories Pn of “predicative” functionals, which are defined using MartinLöf’s universes of transfinite types. We then extend Gödel’s interpretation to the theories of arithmetic inductive definitions ÎDn, so that each ÎDn is interpreted in the corresponding Pn. Since the strengths of the theories ÎDn are cofinal in the ordinal Γ0, as a corollary this analysis provides an ordinal-free characterization of the <Γ0-recursive functions.