Ordinal Analysis of Weak Theories

Michael Rathjen, University of Leeds. Part of the proofs, constructions and computations seminar series.

We fill an apparent gap in the literature by giving a short and self-contained proof that the ordinal of the theory RCA_0 + WO(\sigma) is \sigma^\omega, for any ordinal \sigma satisfying \omega \cdot \sigma = \sigma (e.g., \omega^\omega, \omega^{\omega^\omega}, \varepsilon_0). Theories of the form RCA_0 + WO(\sigma) are of interest in Proof Theory and Reverse Mathematics because of their connections to a number of well-investigated combinatorial principles related to various subsystems of arithmetic.