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.
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.