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

Andreas Weiermann
The Journal of Symbolic Logic
Vol. 63, No. 4 (Dec., 1998), pp. 1348-1370
DOI: 10.2307/2586654
Stable URL: http://www.jstor.org/stable/2586654
Page Count: 23
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How is It That Infinitary Methods can be Applied to Finitary Mathematics? Gödel's T: A Case Study
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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 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: ε0 → 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 ωn+2-descent recursive functions. The derivation lengths of type one functionals from Tn (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 ∈ T0 is primitive recursive, thus any type one functional a in T0 defines a primitive recursive function. Similarly we also obtain (reobtain?) a full classification of T1 in terms of multiple recursion. As proof-theoretic corollaries we reobtain the classification of the IΣn+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 ($\varepsilon_0) \vdash \Pi^0_2$ - Refl(PA) and PRA + PRWO ($\omega_{n+2}) \vdash \Pi^0_2$ - Refl(I Σn+1), hence PRA + PRWO($\epsilon_0) \vdash$ Con(PA) and PRA + PRWO($\omega_{n+2}) \vdash$ Con(IΣn+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.

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