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.
Similar content being viewed by others
References
Gaifman, H.: On local arithmetical functions and their application for constructing types of Peano’s arithmetic. In: Mathematical Logic and Foundations of Set Theory (Proceedings of the International Colloquium, Jerusalem, 1968). North-Holland, Amsterdam, pp. 105–121 (1970)
Knight, J.F.: Omitting types in set theory and arithmetic. J. Symb. Log. 41, 25–32 (1976)
Kossak, R., Schmerl, J.H.: The Structure of Models of Peano Arithmetic. Oxford University Press, Oxford (2006)
MacDowell, R., Specker, E.: Modelle der Arithmetik. In: Infinitistic Methods (Proceedings of the Symposium on the Foundations of Mathematics, Warsaw, 1959). Pergamon, Oxford, Warsaw, pp. 257–263 (1961)
Phillips, R.G.: Omitting types in arithmetic and conservative extensions. In: Victoria Symposium on Nonstandard Analysis (Univ. Victoria, Victoria, B.C., 1972), Lecture Notes in Mathematics, vol. 369, pp. 195–202. Springer, Berlin (1974)
Phillips, R.G.: A minimal extension that is not conservative. Mich. Math. J. 21, 27–32 (1974)
Schmerl, J.H.: Subsets coded in elementary end extensions. Arch. Math. Log. 53, 571–581 (2014)
Simpson, S.G.: Subsystems of Second Order Arithmetic, Perspectives in Logic, 2nd edn. Cambridge University Press, Cambridge (2009)
Simpson, S.G., Smith, R.L.: Factorization of polynomials and \(\Sigma ^0_1\) induction. Ann. Pure Appl. Log. 31, 289–306 (1986)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Schmerl, J.H. Minimal elementary end extensions. Arch. Math. Logic 56, 541–553 (2017). https://doi.org/10.1007/s00153-017-0556-5
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00153-017-0556-5