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

Abstract.

Let T be Gödel'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-Schütte-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 $T^{\star}$.

The expressive weakness of $T^{\star}$ compared to the full system T can be explained as follows: In contrast to $T$, computation steps in $T^{\star}$ never increase the nesting-depth of ${\mathcal I}_\rho$ and ${\mathcal R}_\rho$ at recursion positions.