Predicativity beyond Gamma_0

TL;DR

This paper challenges the traditional view that predicative reasoning is limited to Gamma_0, proposing a new method that extends the range of predicatively accessible ordinals beyond this boundary.

Contribution

It introduces a novel, principled approach to accessing larger ordinals predicatively, surpassing the previously assumed Gamma_0 limit.

Findings

Veblen ordinal (0) is predicatively provable

Larger ordinals than Gamma_0 are accessible via the new method

Critiques of previous arguments supporting the Gamma_0 limit are provided

Abstract

We reevaluate the claim that predicative reasoning (given the natural numbers) is limited by the Feferman-Schutte ordinal Gamma_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 \phi_{\Omega^\omega}(0), and larger ordinals, are predicatively provable.