 
Summary: Predicative Functionals and
an Interpretation of ID<
Jeremy Avigad
December 22, 1997
Abstract
In 1958 GĻodel published his Dialectica interpretation, which reduces
classical arithmetic to a quantifierfree theory T axiomatizing the prim
itive recursive functionals of finite type. Here we extend GĻodel's T to
theories Pn of "predicative" functionals, which are defined using Martin
LĻof's universes of transfinite types. We then extend GĻodel's interpretation
to the theories of arithmetic inductive definitions IDn, so that each IDn
is interpreted in the corresponding Pn. Since the strengths of the theories
IDn are cofinal in the ordinal 0, as a corollary this analysis provides an
ordinalfree characterization of the <0recursive functions.
1 Introduction
1.1 Background
In 1958, GĻodel [18] published what is now known as the Dialectica interpretation
of arithmetic, consisting of a quantifierfree theory T and interpretation of Peano
Arithmetic (PA) in that theory. T allows for the definition of functionals of
