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MODEL THEORETIC STABILITY AND DEFINABILITY OF TYPES, AFTER A. GROTHENDIECK

Published online by Cambridge University Press:  30 December 2014

ITAÏ BEN YAACOV
ITAÏ BEN YAACOV*
Affiliation:
UNIVERSITÉ CLAUDE BERNARD – LYON 1, INSTITUT CAMILLE JORDAN, CNRS UMR 5208, 43 BOULEVARD DU 11 NOVEMBRE 1918, 69622 VILLEURBANNE CEDEX, FRANCE URL: http://math.univ-lyon1.fr/∼begnac/
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Abstract

We point out how the “Fundamental Theorem of Stability Theory”, namely the equivalence between the “non order property” and definability of types, proved by Shelah in the 1970s, is in fact an immediate consequence of Grothendieck’s “Critères de compacité” from 1952. The familiar forms for the defining formulae then follow using Mazur’s Lemma regarding weak convergence in Banach spaces.

Keywords

03C4546E15stable formulaorder propertydefinable typerelative weak compactness
Type
Articles
Information
Bulletin of Symbolic Logic , Volume 20 , Issue 4 , December 2014 , pp. 491 - 496
Copyright
Copyright © The Association for Symbolic Logic 2014 

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References

REFERENCES

Yaacov, Itaï Ben, Berenstein, Alexander, and Ferri, Stefano, Reflexive representability and stable metrics. Mathematische Zeitschrift, vol. 267 (2011), no. 1–2, pp. 129–138, doi:10.1007/s00209-009-0612-x.CrossRefGoogle Scholar
Yaacov, ItaïBen, Berenstein, Alexander, Ward Henson, C., and Usvyatsov, Alexander, Model theory for metric structures, Model theory with Applications to Algebra and Analysis, volume 2 (Chatzidakis, Zoé, Macpherson, Dugald, Pillay, Anand, and Wilkie, Alex, editors), London Mathematical Society Lecture Note Series, vol.350, Cambridge University Press, Cambridge, 2008, pp. 315–427.Google Scholar
Yaacov, ItaïBen and Tsankov, Todor, Weakly almost periodic functions, model-theoretic stability, and minimality of topological groups, preprint, arXiv:1312.7757.Google Scholar
Yaacov, ItaïBen and Usvyatsov, Alexander, Continuous first order logic and localstability. Transactions of the American Mathematical Society, vol. 362 (2010), no. 10, pp. 5213–5259, doi:10.1090/S0002-9947-10-04837-3.Google Scholar
Brezis, Haïm, Analyse fonctionnelle: Théorie et applications, Collection Mathématiques Appliquées pour la Maîtrise, Masson, Paris, 1983.Google Scholar
Grothendieck, Alexander, Critères de compacité dans les espaces fonctionnels généraux. American Journal of Mathematics, vol. 74 (1952), pp. 168–186.Google Scholar
Iovino, José, Stable Banach spaces and Banach space structures, I and II, Models, algebras, and proofs (Bogotá, 1995), Lecture Notes in Pure and Applied Mathematics, vol. 203, Dekker, New York, 1999, pp. 77–117.Google Scholar
James, Robert C., Uniformly non-square Banach spaces, Annals of Mathematics, second series, vol. 80 (1964), pp. 542–550.Google Scholar
Krivine, Jean-Louis and Maurey, Bernard, Espaces de Banach stables. Israel Journal of Mathematics, vol. 39 (1981), no. 4, pp. 273–295.Google Scholar
Megrelishvili, Michael, Fragmentability and representations of flows. Proceedings of the 17th Summer Conference on Topology and its Applications, vol. 27 (2003), pp. 497–544.Google Scholar
Pillay, Anand, An introduction to stability theory, Oxford Logic Guides, vol. 8, The Clarendon Press Oxford University Press, New York, 1983.Google Scholar
Shelah, Saharon, Classification theory and the number of nonisomorphic models, second edition, Studies in Logic and the Foundations of Mathematics, vol. 92, North-Holland, Amsterdam, 1990.Google Scholar
Whitley, Robert, An elementary proof of the Eberlein-Šmulian theorem. Mathematische Annalen, vol. 172 (1967), pp. 116–118.Google Scholar