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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Extendible sets in Peano arithmetic

Author: Stuart T. Smith
Journal: Trans. Amer. Math. Soc. 316 (1989), 337-367
MSC: Primary 03C62; Secondary 03H15
MathSciNet review: 946223
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Abstract: Let $ \mathcal{A}$ be a structure and let $ U$ be a subset of $ \vert\mathcal{A}\vert$. We say $ U$ is extendible if whenever $ \mathcal{B}$ is an elementary extension of $ \mathcal{A}$, there is a $ V \subseteq \vert\mathcal{B}\vert$ such that $ (\mathcal{A},U) \prec (\mathcal{B},V)$. Our main results are: If $ \mathcal{M}$ is a countable model of Peano arithmetic and $ U$ is a subset of $ \vert\mathcal{M}\vert$, then $ U$ is extendible iff $ U$ is parametrically definable in $ \mathcal{M}$. Also, the cofinally extendible subsets of $ \vert\mathcal{M}\vert$ are exactly the inductive subsets of $ \vert\mathcal{M}\vert$. The end extendible subsets of $ \vert\mathcal{M}\vert$ are not completely characterized, but we show that if $ \mathcal{N}$ is a model of Peano arithmetic of arbitrary cardinality and $ U$ is any bounded subset of $ \mathcal{N}$, then $ U$ is end extendible.

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  • [BS] J. Barwise and J. Schlipf, An introduction to recursively saturated and resplendent models, J. Symbolic Logic 41 (1976), 531-536. MR 0403952 (53:7761)
  • [CK] C. C. Chang and H. J. Keisler, Model theory, North-Holland, Amsterdam, 1973.
  • [G] H. Gaifman, A note on models and submodels of arithmetic (Conference in Mathematical Logic, London 1970; W. Hodges, ed.), Lecture Notes in Math., vol. 255, Springer-Verlag, Heidelberg, 1972, pp. 128-144. MR 0419215 (54:7247)
  • [Ka] M. Kaufmann, A rather classless model, Proc. Amer. Math. Soc. 62 (1977), 330-333. MR 0476498 (57:16058)
  • [KS] M. Kaufmann and J. Schmerl, Saturation and simple extensions of models of Peano Arithmetic, Ann. Pure Appl. Logic 27 (1984), 109-136. MR 763736 (85j:03051)
  • [Ki] L. Kirby, Initial segments of models of arithmetic, Thesis, Manchester University, 1977.
  • [KP] L. Kirby and J. Paris, Initial segments of models of Peano's axioms (Set Theory' and Hierarchy Theory V, Bierutowice 1976), Lecture Notes in Math., vol. 619, Springer-Verlag, Heidelberg, 1977, pp. 211-226. MR 0491157 (58:10423)
  • [Ko] H. Kotlarski, On cofinal extensions of models of arithmetic, J. Symbolic Logic 40 (1983), pp. 253-262. MR 704082 (85d:03129)
  • [MS] R. MacDowell and E. Specker, Modelle der Arithmetik, Infinitistic Methods (Proceedings of the Symposium on the Foundations of Mathematics, Warsaw 1959), Pergamon Press, London, 1961, pp. 257-263. MR 0152447 (27:2425)
  • [Sch] J. Schmerl, Recursively saturated, rather classless models of Peano arithmetic (Logic Year 1979-80: The University of Connecticut; M. Lerman. J. Schmerl, and R. Shore, eds.), Lecture Notes in Math., vol. 859, Springer-Verlag, Heidelberg, 1981, pp. 268-282. MR 619874 (83b:03039)
  • [Smo1] C. Smorynski, Cofinal extensions of nonstandard models of arithmetic, Notre Dame J. Formal Logic 22 (1981), 133-144. MR 611481 (82g:03063)
  • [Smo2] -, Recursively saturated nonstandard models of arithmetic, J. Symbolic Logic 46 (1981), 259-286. MR 613281 (82i:03045)
  • [Smo3] -, Back-and-forth inside a recursively saturated model of arithmetic (Logic Colloquium '80; D. van Dalen, D. Lascar, and J. Smiley, eds.), Studies in Logic and the Foundations of Mathematics, Vol. 108, North-Holland, Amsterdam, 1982, pp. 273-278. MR 673798 (83m:03041)
  • [SS] C. Smorynski and J. Stavi, Cofinal extension preserves recursive saturation (Model Theory of Algebra and Arithmetic: Proceedings, Karpacz, Poland, 1979; L. Pacholski, J. Wierzejewski, and A. J. Wilkie, eds.), Lecture Notes in Math., vol. 834, Springer-Verlag, Heidelberg, 1980, pp. 338-345. MR 606792 (82g:03062)

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