Definable sets and expansions of models of Peano arithmetic (Q1102945)

Let T denote the fragment of the second order arithmetic \(A_ 2\) consisting of the \(\Delta^ 1_ 1\)-comprehension scheme and the \(\Sigma^ 1_ 1\)-axiom of choice (together with the induction and extensionality axioms). By a classical result of Barwise and Schlipf a nonstandard model M of PA is expandable to a model of T iff M is recursively saturated. The expansion of a recursively saturated model M in the above result is given by the family of subsets of M which are definable over M with parameters. Murawski generalizes this using a notion of nonstandard definability for nonstandard countable models of PA which have full inductive satisfaction classes (such models must be recursively saturated, but not every countable recursively saturated model has such a class). It is shown that if I is an nonstandard initial segment (satisfying a certain closure property) of a countable model M and S is a full inductive satisfaction class of M, then the family of subsets of M which are definable in M (in the sense of S) by formulas from I from an expansion of M to a model of T. Satisfaction classes which differ on formulas from I provide incompatible expansions, and since every countable model with a full inductive satisfaction class has an elementary extension with continuum many such classes which do not agree on formulas from I it follows that there are countable models with rich families of noncompatible expansions to models of T. Other constructions using nonstandard definability are also considered. There seems to be a gap in the proof of theorem 12 (in which existence of extensions with many satisfaction classes is proven), namely it is not quite clear why the constructed model \(M_ 1\) should be countable.