Archive for Mathematical Logic

, Volume 56, Issue 5–6, pp 541–553 | Cite as

Minimal elementary end extensions

  • James H. SchmerlEmail author


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.


Minimal end extensions Coded sets MacDowell–Specker Theorem 

Mathematics Subject Classification

03C62 03H15 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    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)Google Scholar
  2. 2.
    Knight, J.F.: Omitting types in set theory and arithmetic. J. Symb. Log. 41, 25–32 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Kossak, R., Schmerl, J.H.: The Structure of Models of Peano Arithmetic. Oxford University Press, Oxford (2006)CrossRefzbMATHGoogle Scholar
  4. 4.
    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)Google Scholar
  5. 5.
    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)Google Scholar
  6. 6.
    Phillips, R.G.: A minimal extension that is not conservative. Mich. Math. J. 21, 27–32 (1974)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Schmerl, J.H.: Subsets coded in elementary end extensions. Arch. Math. Log. 53, 571–581 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Simpson, S.G.: Subsystems of Second Order Arithmetic, Perspectives in Logic, 2nd edn. Cambridge University Press, Cambridge (2009)CrossRefGoogle Scholar
  9. 9.
    Simpson, S.G., Smith, R.L.: Factorization of polynomials and \(\Sigma ^0_1\) induction. Ann. Pure Appl. Log. 31, 289–306 (1986)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.University of ConnecticutStorrsUSA

Personalised recommendations