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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Les modèles dénombrables d’une théorie ayant des fonctions de Skolem
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by Daniel Lascar PDF
Trans. Amer. Math. Soc. 268 (1981), 345-366 Request permission

Abstract:

Let $T$ be a countable complete theory having Skolem functions. We prove that if all the types over finitely generated models are definable (this is the case for example if $T$ is stable), then either $T$ has ${2^{{\aleph _0}}}$ countable models or all its models are homogeneous. The proof makes heavy use of stability techniques.
References
  • Daniel Lascar, Sur les théorie convexes “modèles complètes”, C. R. Acad. Sci. Paris Sér. A 278 (1974), 1001–1004 (French). MR 349376
  • —, Généralisation de l’ordre de Rudin-Keisler aux types d’une theorie, Colloq. Internat. C.N.R.S., No. 249, Clermont-Ferrand, 1975, pp. 73-81. —, Les modèles dénombrables d’une théorie superstable ayant des fonctions de Skolem, C. R. Acad. Sci. Paris Sér. A 289 (1979), 655-658.
  • Daniel Lascar and Bruno Poizat, An introduction to forking, J. Symbolic Logic 44 (1979), no. 3, 330–350. MR 540665, DOI 10.2307/2273127
  • Saharon Shelah, Classification theory and the number of nonisomorphic models, Studies in Logic and the Foundations of Mathematics, vol. 92, North-Holland Publishing Co., Amsterdam-New York, 1978. MR 513226
  • Saharon Shelah, End extensions and numbers of countable models, J. Symbolic Logic 43 (1978), no. 3, 550–562. MR 503792, DOI 10.2307/2273531
  • R. L. Vaught, Denumerable models of complete theories, Infinitistic Methods (Proc. Sympos. Foundations of Math., Warsaw, 1959), Pergamon, Oxford; Państwowe Wydawnictwo Naukowe, Warsaw, 1961, pp. 303–321. MR 0186552
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Additional Information
  • © Copyright 1981 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 268 (1981), 345-366
  • MSC: Primary 03C15; Secondary 03C45
  • DOI: https://doi.org/10.1090/S0002-9947-1981-0632533-X
  • MathSciNet review: 632533