Minimal elementary end extensions

Abstract

Suppose that \({\mathcal M}\models \mathsf{PA}\) and \({\mathfrak X} \subseteq {\mathcal P}(M)\). If \({\mathcal M}\) has a finitely generated elementary end extension \({\mathcal N}\succ _\mathsf{end} {\mathcal M}\) such that \(\{X \cap M : X \in {{\mathrm{Def}}}({\mathcal N})\} = {\mathfrak X}\), then there is such an \({\mathcal N}\) that is, in addition, a minimal extension of \({\mathcal M}\) iff every subset of M that is \(\Pi _1^0\)-definable in \(({\mathcal M}, {\mathfrak X})\) is the countable union of \(\Sigma _1^0\)-definable sets.