Continuous first order logic and local stability

Ben Yaacov, Itaï; Usvyatsov, Alexander

doi:10.1090/S0002-9947-10-04837-3

Continuous first order logic and local stability
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by Itaï Ben Yaacov and Alexander Usvyatsov PDF

Trans. Amer. Math. Soc. 362 (2010), 5213-5259 Request permission

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

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.
Itaï Ben Yaacov, Definability of groups in $\aleph _0$-stable metric structures, Journal of Symbolic Logic, to appear, arXiv:0802.4286.
—, On perturbations of continuous structures, Journal of Mathematical Logic, to appear, arXiv:0802.4388.
Itay Ben-Yaacov, Positive model theory and compact abstract theories, J. Math. Log. 3 (2003), no. 1, 85–118. MR 1978944, DOI 10.1142/S0219061303000212
Itay Ben-Yaacov, Uncountable dense categoricity in cats, J. Symbolic Logic 70 (2005), no. 3, 829–860. MR 2155268, DOI 10.2178/jsl/1122038916
Itaï Ben Yaacov, Topometric spaces and perturbations of metric structures, Log. Anal. 1 (2008), no. 3-4, 235–272. MR 2448260, DOI 10.1007/s11813-008-0009-x
Itaï Ben Yaacov, Modular functionals and perturbations of Nakano spaces, J. Log. Anal. 1 (2009), Paper 1, 42. MR 2491238, DOI 10.4115/jla.2009.1.1
Itaï Ben Yaacov and Alexander Usvyatsov, On $d$-finiteness in continuous structures, Fund. Math. 194 (2007), no. 1, 67–88. MR 2291717, DOI 10.4064/fm194-1-4
Chen-chung Chang and H. Jerome Keisler, Continuous model theory, Annals of Mathematics Studies, No. 58, Princeton University Press, Princeton, N.J., 1966. MR 0231708
D. H. Fremlin, Measure theory volumne 3: Measure algebras, Torres Fremlin, 25 Ireton Road, Colchester CO3 3AT, England, 2004, http://www.essex.ac.uk/maths/staff/fremlin/mt3.2004/index.htm.
C. Ward Henson and José Iovino, Ultraproducts in analysis, Analysis and logic (Mons, 1997) London Math. Soc. Lecture Note Ser., vol. 262, Cambridge Univ. Press, Cambridge, 2002, pp. 1–110. MR 1967834
J.-L. Krivine and B. Maurey, Espaces de Banach stables, Israel J. Math. 39 (1981), no. 4, 273–295 (French, with English summary). MR 636897, DOI 10.1007/BF02761674
Hilton Vieira Machado, A characterization of convex subsets of normed spaces, K\B{o}dai Math. Sem. Rep. 25 (1973), 307–320. MR 326359
Anand Pillay, Geometric stability theory, Oxford Logic Guides, vol. 32, The Clarendon Press, Oxford University Press, New York, 1996. Oxford Science Publications. MR 1429864
Saharon Shelah and Alexander Usvyatsov, Model theoretic stability and categoricity for complete metric spaces, preprint.