Reflecting on Inaccessible J\'onsson Cardinals

Abstract

Given an inaccessible J\'onsson cardinal $\lambda$, a sequence of results due to Shelah from Cardinal Arithmetic and Sh413 tell us that $\lambda$ must be at least $\lambda\times\omega$-Mahlo. We may then ask ourselves whether we can improve this bound, and unfortunately not much progress has been made on this question since Sh413. Part of the problem may be that the proofs involved are rather complicated and make use of somewhat ad hoc machinery. In order to remedy this, we present a survey and new proofs of these results regarding the Mahlo degree of an inaccessible J\'onsson cardinal with the hope of making this material accessible.