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On bar recursion of types 0 and 1
Published online by Cambridge University Press: 12 March 2014
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For general information on bar recursion the reader should consult the papers of Spector [8], where it was introduced, Howard [2] and Tait [11]. In this note we shall prove that the terms of Gödel's theory T(in its extensional version of Spector [8]) are closed under the rule BR0,1 of bar recursion of types 0 and 1. Our method of proof is based on the notion of an infinite term introduced by Tait [9]. The main tools of the proof are (i) the normalization theorem for (notations for) infinite terms and (ii) valuation functionals. Both are elaborated in [6]; for brevity some familiarity with this paper is assumed here. Using (i) and (ii) we reduce BR0,1 to ξ-recursion with ξ < ε0. From this the result follows by work of Tait [10], who gave a reduction of 2ξ-recursion to ξ-recursion at a higher type. At the end of the paper we discuss a perhaps more natural variant of bar recursion introduced by Kreisel in [4].
Related results are due to Kreisel (in his appendix to [8]), who obtains results which imply, using the reduction given by Howard [2] of the constant of bar recursion of type τ to the rule of bar recursion of type (0 → τ) → τ, that T is not closed under the rule of bar recursion of a type of level ≥ 2, to Diller [1], who gave a reduction of BR0,1 to ξ-recursion with ξ bounded by the least ω-critical number, and to Howard [3], who gave an ordinal analysis of the constant of bar recursion of type 0. I am grateful to H. Barendregt, W. Howard and G. Kreisel for many useful comments and discussions.
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- Copyright © Association for Symbolic Logic 1979
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