Abstract

In this paper we consider three potential simplifications to a result of Solovay’s concerning the Turing degrees of nonstandard models of arbitrary completions of first-order Peano Arithmetic (PA). Solovay characterized the degrees of nonstandard models of completions T of PA, showing that they are the degrees of sets X such that there is an enumeration R ≤T X of an “appropriate” Scott set and there is a family of functions (tn)n∈ω, ∆0 n(X) uniformly in n, such that lim tn(s) s→∞.