A Note on the Ordinal Analysis of \mathbf RCA_0 + \mathrm WO(\mathbf σ ) RCA 0 + WO ( σ )

We fill an apparent gap in the literature by giving a short and self-contained proof that the ordinal of the theory \(\mathbf {RCA}_0 + \mathrm {WO}(\sigma )\) 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 \(\mathbf {RCA}_0 + \mathrm {WO}(\sigma )\) 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.