Predicativity beyond Γ0

We reevaluate the claim that predicative reasoning (given the natural numbers) is limited by the Feferman-Schütte ordinal Γ 0. First we comprehensively criticize the arguments that have been offered in support of this position. Then we analyze predicativism from first principles and develop a general method for accessing ordinals which is predicatively valid according to this analysis. We find that the Veblen ordinal φ Ω ω (0), and larger ordinals, are predicatively provable. The precise delineation of the extent of predicative reasoning is possibly one of the most remarkable modern results in the foundations of mathematics. Building on ideas of Kreisel [27, 28], Feferman [10] and Schütte [40, 41] independently identified a countable ordinal Γ 0 and argued that it is the smallest predicatively non-provable ordinal. (Throughout, I take " predicative " to mean " predicative given the natural numbers " .) This conclusion has become the received view in the foundations community , with reference [10] in particular having been cited with approval in virtually every discussion of predicativism for the past forty years. Γ 0 is now commonly referred to as " the ordinal of predicativity ". Some recent publications which explicitly make this assertion This achievement is notable both for its technical sophistication and for the insight it provides into an important foundational stance. Although predicativism is out of favor now, at one time it was advocated by such luminaries as Poincaré, Russell, and Weyl. (Historical overviews are given in [18] and [35].) Its central principle — that sets have to be " built up from below " — is, on its face, reasonable and attractive. With its rejection of a metaphysical set concept, predicativism also provides a cogent resolution of the set-theoretic paradoxes and is more in line with the positivistic aspect of modern analytic philosophy than are the essentially platonic views which have become mathematically dominant. Undoubtedly one of the main reasons predicativism was not accepted by the general mathematical public early on was its apparent failure to support large portions of mainstream mathematics. However, we now know that the bulk of core mathematics can in fact be developed in predicative systems [15, 43], and the limitation identified by Feferman and Schütte is probably now a primary reason, possibly the primary reason, for predicativism's nearly universal unpopularity. There do exist important mainstream theorems which are known to in various senses require provability of Γ 0 …