On Bar Recursion of Types 0 and 1

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 Godel's theory T (in its extensional version of Spector [8]) are closed under the rule BRo•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 BRo.1 to ';-recursion with'; < co. From this the result follows by work of Tait [10], who gave a reduction of 2E-recursion to ';-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 :&reisel (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 '0 to the rule of bar recursion of type (0 ~ '0) ~ '0, that T is not closed under the rule of bar recursion of a type oflevel ~ 2, to Diller [1], who gave a reduction of BRo.1 to ';-recursion with'; bounded by the least (V-critical number, and to Howard [3], who gave an ordinal analysis of the constant of bar recursion of type O. I am grateful to H. Barendregt, W. Howard and G. Kreisel for many useful comments and discussions. Recall that a functional F of type 0 ~ (0 ~ '0) ~ (J is said to be defined by (the rule of) bar recursion of type '0 from Yand functionals G, H of the proper types if