Minimal elementary end extensions. (English) Zbl 1417.03238
Summary: 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.
References:
| [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) · Zbl 0209.30801 |
| [2] | Knight, J.F.: Omitting types in set theory and arithmetic. J. Symb. Log. 41, 25-32 (1976) · Zbl 0328.02039 · doi:10.1017/S0022481200051719 |
| [3] | Kossak, R., Schmerl, J.H.: The Structure of Models of Peano Arithmetic. Oxford University Press, Oxford (2006) · Zbl 1101.03029 · doi:10.1093/acprof:oso/9780198568278.001.0001 |
| [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) · Zbl 0288.02030 |
| [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) · Zbl 0278.02043 |
| [6] | Phillips, R.G.: A minimal extension that is not conservative. Mich. Math. J. 21, 27-32 (1974) · Zbl 0288.02030 · doi:10.1307/mmj/1029001205 |
| [7] | Schmerl, J.H.: Subsets coded in elementary end extensions. Arch. Math. Log. 53, 571-581 (2014) · Zbl 1298.03099 · doi:10.1007/s00153-014-0381-z |
| [8] | Simpson, S.G.: Subsystems of Second Order Arithmetic, Perspectives in Logic, 2nd edn. Cambridge University Press, Cambridge (2009) · Zbl 1181.03001 · doi:10.1017/CBO9780511581007 |
| [9] | Simpson, S.G., Smith, R.L.: Factorization of polynomials and \[\Sigma^0_1\] Σ10 induction. Ann. Pure Appl. Log. 31, 289-306 (1986) · Zbl 0603.03019 · doi:10.1016/0168-0072(86)90074-6 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
