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• W2025478275 abstract "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 ε 0 -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 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 ω +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 < 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 I Σ 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(ε 0 ) ⊢ Π 2 0 − Refl(PA) and PRA + PRWO( ω n +2 ) ⊢ Π 2 0 − Refl( I Σ n +1 ), hence PRA + PRWO(ε 0 ) ⊢ Con( PA ) and PRA + PRWO( ω n +2 ) ⊢ Con( I Σ 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." @default.
• W2025478275 created "2016-06-24" @default.
• W2025478275 creator A5056618682 @default.
• W2025478275 date "1998-12-01" @default.
• W2025478275 modified "2023-09-27" @default.
• W2025478275 title "How is it that infinitary methods can be applied to finitary mathematics? Gödel's T: a case study" @default.
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• W2025478275 doi "https://doi.org/10.2307/2586654" @default.
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