A Solovay function for the least 1-inconsistent subtheory of PA
Paula Henk, V. Yu. Shavrukov

Abstract:
We consider the theory IΣμ, where μ is the least x such that IΣx together with all true Π1-sentences is inconsistent. We show that the provability logic of IΣμ coincides with that of PA, i.e. the Gödel–Löb provability logic GL. While arithmetical completeness of GL with respect to PA is established by using a single Solovay function, our proof for IΣμ relies on a uniformly defined infinite sequence of such functions.