End extensions of models of arithmetic (Q1203781)

An iteration of the arithmetized completeness theorem is used to prove that, for every nonstandard \(M \models \text{PA}\) and every complete theory \(T \supseteq \text{PA}\), \(M\) has an end extension \(N \models T\) iff all sets representable in \(T\) are in the standard system of \(M\) and \(T \cap \Pi_ 1 \subseteq \text{Th} (M)\). Two applications of this result are given. The first is a short proof of Wilkie's theorem that for every model of PA there is a diophantine equation having no solution in that model but having a solution in some end extension of that model. The second is a very short proof of the MacDowell-Specker theorem (this proof is due to Kaufmann).