Minimality and completions of PA (Q2758069)

Let \(T\) be a completion of Peano Arithmetic. The results of the paper concern possible relationships between Turing degrees of \(T\), the degrees of sets representable in \(T\), and degrees of enumerations of standard systems of models of \(T\). They show that, in general, there are no minimal degrees in various naturally defined families of degrees of structures. The main theorem: For any completion \(T\) of PA there is a completion \(S\) of strictly smaller Turing degree, and such that \(T\) and \(S\) represent the same sets. It is shown that we can additionally require that the jumps of \(S\) and \(T\) have the same Turing degree. There are several applications. The best one, which implies all the others, says that any countable nonstandard model has an isomorphic copy with a strictly smaller Turing degree.