Journal of Symbolic Logic

How is It That Infinitary Methods can be Applied to Finitary Mathematics? Godel's T: A Case Study

Andreas Weiermann

Abstract

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 Godel's system T of primitive recursive functionals of finite types by constructing an $\varepsilon_0$-recursive function [ ]$_0$: T $\rightarrow \omega$ so that a reduces to b implies [a]$_0 > [b]_0$. The construction of [ ]$_0$ is based on a careful analysis of the Howard-Schutte treatment of Godel's T and utilizes the collapsing function $\psi: \varepsilon_0 \rightarrow \omega$ 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 Godel'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 T$_n$ 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 T$_n$.) As a corollary of the main theorem of this paper we obtain (reobtain?) optimal bounds for the T$_n$-derivation lengths in terms of $\omega_{n+2}$-descent recursive functions. The derivation lengths of type one functionals from T$_n$ (hence also their computational complexities) are classified optimally in terms of $< \omega_{n+2}$-descent recursive functions. In particular we obtain (reobtain?) that the derivation lengths function of a type one functional a $\in T_0$ 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 I$\Sigma_{n+1}$-provably recursive functions. Taking advantage from our finitistic and constructive treatment of the terms of Godel's T we reobtain additionally (without employing continuous cut elimination techniques) that PRA + PRWO ($\varepsilon_0) \vdash \Pi^0_2$ - Refl(PA) and PRA + PRWO ($\omega_{n+2}) \vdash \Pi^0_2$ - Refl(I $\Sigma_{n+1}$), hence PRA + PRWO($\epsilon_0) \vdash$ Con(PA) and PRA + PRWO($\omega_{n+2}) \vdash$ Con(I$\Sigma_{n+1})$. 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.

Article information

Source
J. Symbolic Logic, Volume 63, Issue 4 (1998), 1348-1370.

Dates
First available in Project Euclid: 6 July 2007

https://projecteuclid.org/euclid.jsl/1183745635

Mathematical Reviews number (MathSciNet)
MR1665727

Zentralblatt MATH identifier
0928.03066

JSTOR