Solovay's theorem cannot be simplified

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, ∆n(X) uniformly in n, such that lim s→∞ tn(s) is an index for T ∩ Σn and for all s, tn(s) is an index for a subset of T ∩ Σn. The simplifications under consideration are attempts to restrict the families of functions (tn)n∈ω that appear in Solovay’s result, known henceforth as Solovay families. We show that none of these potential simplifications may be made, by proceeding as follows. First, we construct a nonstandard model A of PA such that there is no Solovay family (tn)n∈ω for Th(A) relative to A in which all the functions tn are constant. Second, for each k we construct a nonstandard model A of PA such that there is no Solovay family (tn)n∈ω for Th(A) relative to A in which all the functions tn change values at most k many times. Third, we construct a nonstandard model A of PA such that there is no Solovay family (tn)n∈ω for Th(A) relative to A with a computable function f such that for all n, tn(s) changes values at most f(n) times. Our constructions answer three questions asked by Julia Knight [7]. Our solutions make use of several consistency results that seem to be of independent interest.