Inspired by Pohlers' local predicativity approach to Pure Proof Theory and Howard's ordinal analysis of bar recursion of type zero we present a short, technically smooth and constructive strong normalization proof for Gödel's system T of primitive recursive functionals of finite types by constructing an ε0-recursive function []0: T → ω so that a reduces to b implies [a]0 > [b]0. The construction of [ ]0 is based on a careful analysis of the Howard-Schütte treatment of Gödel's T and utilizes the collapsing function ψ: ε
0 → ω which has been developed by the author for a local predicativity style proof-theoretic analysis of PA. The construction of [ ]0 is also crucially based on ideas developed in the 1995 paper “A proof of strongly uniform termination for Gödel's T by the method of local predicativity” by the author. The results on complexity bounds for the fragments of T which are obtained in this paper strengthen considerably the results of the 1995 paper.
Indeed, for given n let Tn
be the subsystem of T in which the recursors have type level less than or equal to n + 2. (By definition, case distinction functionals for every type are also contained in Tn
.) As a corollary of the main theorem of this paper we obtain (reobtain?) optimal bounds for the Tn
-derivation lengths in terms of ω+2-descent recursive functions. The derivation lengths of type one functionals from Tn
(hence also their computational complexities) are classified optimally in terms of <ω
n+2 -descent recursive functions.
In particular we obtain (reobtain?) that the derivation lengths function of a type one functional a ∈ T
0 is primitive recursive, thus any type one functional a in T
0 defines a primitive recursive function. Similarly we also obtain (reobtain?) a full classification of T
1 in terms of multiple recursion.
As proof-theoretic corollaries we reobtain the classification of the IΣ
n+1-provably recursive functions. Taking advantage from our finitistic and constructive treatment of the terms of Gödel's T we reobtain additionally (without employing continuous cut elimination techniques) that PRA + PRWO(ε0) ⊢ Π2
0 − Refl(PA) and PRA + PRWO(ω
n+2) ⊢ Π2
0 − Refl(IΣ
n+1), hence PRA + PRWO(ε0) ⊢ Con(PA) and PRA + PRWO(ω
n+2) ⊢ Con(IΣ
n+1).
For programmatic reasons we outline in the introduction a vision of how to apply a certain type of infinitary methods to questions of finitary mathematics and recursion theory. We also indicate some connections between ordinals, term rewriting, recursion theory and computational complexity.