Characterizing the elementary recursive functions by a fragment of Gödel's T

Abstract

Let T be Godel's system of primitive recursive functionals of finite type in a combinatory logic formulation. Let \(T^{\star}\) be the subsystem of T in which the iterator and recursor constants are permitted only when immediately applied to type 0 arguments. By a Howard-Schutte-style argument the \(T^{\star}\)-derivation lengths are classified in terms of an iterated exponential function. As a consequence a constructive strong normalization proof for \(T^{\star}\) is obtained. Another consequence is that every \(T^{\star}\)-representable number-theoretic function is elementary recursive. Furthermore, it is shown that, conversely, every elementary recursive function is representable in \(T^{\star}\).