Complexity of Gödel’s T in λ-Formulation

Abstract

Let T be Gödel’s system of primitive recursive functionals of finite type in the λ-formulation. We define by constructive means using recursion on nested multisets a multivalued function I from the set of terms of T into the set of natural numbers such that if a term a reduces to a term b and if a natural number I(a) is assigned to a then a natural number I(b) can be assigned to b such that I(a) > I(b). The construction of I is based on Howard’s 1970 ordinal assignment for T and Weiermann’s 1996 treatment of T in the combinatory logic version. As a corollary we obtain an optimal derivation length classification for the λ-formulation of T and its fragments. Compared with Weiermann’s 1996 exposition this article yields solutions to several non-trivial problems arising from dealing with λ-terms instead of combinatory logic terms.