Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

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



Continuous first order logic and local stability

Authors: Itaï Ben Yaacov and Alexander Usvyatsov
Journal: Trans. Amer. Math. Soc. 362 (2010), 5213-5259
MSC (2000): Primary 03C90, 03C45
Published electronically: May 17, 2010
MathSciNet review: 2657678
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We develop continuous first order logic, a variant of the logic described by Chang and Keisler (1966). We show that this logic has the same power of expression as the framework of open Hausdorff cats, and as such extends Henson's logic for Banach space structures. We conclude with the development of local stability, for which this logic is particularly well-suited.

References [Enhancements On Off] (What's this?)

  • [BBH] Itaï Ben Yaacov, Alexander Berenstein, and C. Ward Henson, Model-theoretic independence in the Banach lattices $ L^p(\mu)$, Israel Journal of Mathematics, to appear, arXiv:0907.5273.
  • [Bena] Itaï Ben Yaacov, Definability of groups in $ \aleph_0$-stable metric structures, Journal of Symbolic Logic, to appear, arXiv:0802.4286.
  • [Benb] -, On perturbations of continuous structures, Journal of Mathematical Logic, to appear, arXiv:0802.4388.
  • [Ben03] -, Positive model theory and compact abstract theories, Journal of Mathematical Logic 3 (2003), no. 1, 85-118. MR 1978944 (2004c:03040)
  • [Ben05] -, Uncountable dense categoricity in cats, Journal of Symbolic Logic 70 (2005), no. 3, 829-860. MR 2155268 (2006g:03063)
  • [Ben08] -, Topometric spaces and perturbations of metric structures, Logic and Analysis 1 (2008), no. 3-4, 235-272. MR 2448260 (2009k:03056)
  • [Ben09] -, Modular functionals and perturbations of Nakano spaces, Journal of Logic and Analysis 1:1 (2009), 1-42. MR 2491238 (2010c:03024)
  • [BU07] Itaï Ben Yaacov and Alexander Usvyatsov, On $ d$-finiteness in continuous structures, Fundamenta Mathematicæ 194 (2007), 67-88. MR 2291717 (2007m:03080)
  • [CK66] C. C. Chang and H. Jerome Keisler, Continuous model theory, Princeton University Press, 1966. MR 0231708 (38:36)
  • [Fre04] D. H. Fremlin, Measure theory volumne 3: Measure algebras, Torres Fremlin, 25 Ireton Road, Colchester CO3 3AT, England, 2004,
  • [HI02] C. Ward Henson and José Iovino, Ultraproducts in analysis, Analysis and Logic (Catherine Finet and Christian Michaux, eds.), London Mathematical Society Lecture Notes Series, no. 262, Cambridge University Press, 2002. MR 1967834 (2004d:03001)
  • [KM81] Jean-Louis Krivine and Bernard Maurey, Espaces de Banach stables, Israel Journal of Mathematics 39 (1981), no. 4, 273-295. MR 636897 (83a:46030)
  • [Mac73] Hilton Vieira Machado, A characterization of convex subsets of normed spaces, Kōdai Mathematical Seminar Reports 25 (1973), 307-320. MR 0326359 (48:4703)
  • [Pil96] Anand Pillay, Geometric stability theory, Oxford Logic Guides, vol. 32, The Clarendon Press, Oxford University Press, New York, 1996, Oxford Science Publications. MR 1429864 (98a:03049)
  • [SU] Saharon Shelah and Alexander Usvyatsov, Model theoretic stability and categoricity for complete metric spaces, preprint.

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 03C90, 03C45

Retrieve articles in all journals with MSC (2000): 03C90, 03C45

Additional Information

Itaï Ben Yaacov
Affiliation: Université de Lyon, Université Lyon 1, Institut Camille Jordan, CNRS UMR 5208, 43 boulevard du 11 novembre 1918, F-69622 Villeurbanne Cedex, France

Alexander Usvyatsov
Affiliation: Department of Mathematics, University of California, Los Angeles, Box 951555, Los Angeles, California 90095-1555
Address at time of publication: Centro de Matemática e Aplicações Fundamentais, Universidade de Lisboa, Av. Prof. Gama Pinto, 2, 1649-003 Lisboa, Portugal

Keywords: Continuous logic, metric structures, stability, local stability
Received by editor(s): November 28, 2005
Received by editor(s) in revised form: October 18, 2007, and June 2, 2008
Published electronically: May 17, 2010
Additional Notes: The research of the first author was supported by NSF grant DMS-0500172
The authors would like to thank C. Ward Henson for stimulating discussions, and Sylvia Carlisle and Eric Owiesny for a careful reading of the manuscript
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society