Choice principles, the bar rule and autonomously iterated comprehension schemes in analysis

Abstract

In [10] Friedman showed that (-AC) is a conservative extension of (-CA)<ε0for-sentences wherei= min(n+ 2, 4), i.e.,i= 2, 3, 4 forn= 0, 1, 2 +m. Feferman [5], [7] and Tait [11], [12] reobtained this result forn= 0, 1 and even with (-DC) instead of (-AC). Feferman and Sieg established in [9] the conservativeness of (-DC) over (-CA)<ε0for-sentences (ias above) for alln. In each paper, different methods of proof have been used. In particular, Feferman and Sieg showed how to apply familiar proof-theoretical techniques by passing through languages with Skolem functionals.In this paper we study the same choice principles in the presence of theBar Rule(BR), which permits one to infer the scheme of transfinite induction on a primitive recursive relation ≺ when it has been proved that ≺ is wellfounded. The main result (Theorem 1 below) characterizes (-DC) + (BR) as a conservative extension of a system of the autonomously iterated-comprehension axiom for-sentences (idepending onnas above). Forn= 0 this has been proved by Feferman in the form that (-DC) + (BR) is a conservative extension of (-CA)<Γ0; this was first done in [8] by use of the Gödel functional interpretation for the stronger systemZω+μ+ (QF-AC) + (BR) and then more recently by the simpler methods of [9]. Jäger showed how the latter methods could also be used to obtain the general result of Theorem 1 below.