## Journal of Symbolic Logic

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

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

#### 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

**Permanent link to this document**

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

**Mathematical Reviews number (MathSciNet)**

MR1665727

**Zentralblatt MATH identifier**

0928.03066

**JSTOR**

links.jstor.org

#### Citation

Weiermann, Andreas. How is It That Infinitary Methods can be Applied to Finitary Mathematics? Godel's T: A Case Study. J. Symbolic Logic 63 (1998), no. 4, 1348--1370. https://projecteuclid.org/euclid.jsl/1183745635