Solovay's theorem cannot be simplified

Annals of Pure and Applied Logic 112 (1):27-41 (2001)
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→∞.
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DOI 10.1016/S0168-0072(01)00095-1
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Extensions of Some Theorems of Gödel and Church.Barkley Rosser - 1936 - Journal of Symbolic Logic 1 (3):87-91.

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