The minimal e-degree problem in fragments of Peano arithmetic

We study the minimal enumeration degree (e-degree) problem in models of fragments of Peano arithmetic ( PA ) and prove the following results: in any model M of Σ 2 induction, there is a minimal enumeration degree if and only if M is a nonstandard model. Furthermore, any cut in such a model has minimal e-degree. By contrast, this phenomenon fails in the absence of Σ 2 induction. In fact, whether every Σ 2 cut has minimal e-degree is independent of the Σ 2 bounding principle.