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Thorn-forking in continuous logic

Published online by Cambridge University Press:  12 March 2014

Clifton Ealy  and
Isaac Goldbring
Clifton Ealy
Affiliation:
Western Illinois University, Department of Mathematics, 476 Morgan Hall, 1 University Circle, Macomb, IL 61455, USA, E-mail: [email protected]
Isaac Goldbring
Affiliation:
University of California, Los Angeles, Department of Mathematics, 520 Portola Plaza, Box 951555, Los Angeles, CA 90095-1555, USA, E-mail: [email protected], URL: www.math.ucla.edu/~isaac
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Abstract

We study thorn forking and rosiness in the context of continuous logic. We prove that the Urysohn sphere is rosy (with respect to finitary imaginaries), providing the first example of an essentially continuous unstable theory with a nice notion of independence. In the process, we show that a real rosy theory which has weak elimination of finitary imaginaries is rosy with respect to finitary imaginaries, a fact which is new even for discrete first-order real rosy theories.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 77 , Issue 1 , March 2012 , pp. 63 - 93
Copyright
Copyright © Association for Symbolic Logic 2012

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References

REFERENCES

[1] Adler, H., A geometric introduction to forking and thorn-forking, Journal of Mathematical Logic, vol. 9 (2009), no. 1, pp. 1–20.CrossRefGoogle Scholar
[2] Yaacov, I. Ben, Simplicity in compact abstract theories, Journal of Mathematical Logic, vol. 3 (2003), no. 2, pp. 163–191.CrossRefGoogle Scholar
[3] Yaacov, I. Ben, Continuous and random Vapnik-Chervonenkis classes, Israel Journal of Mathematics, vol. 173 (2009), pp. 309–333.CrossRefGoogle Scholar
[4] Yaacov, I. Ben, On theories of random variables, submitted.Google Scholar
[5] Yaacov, I. Ben, Berenstein, A., Henson, C. W., and Usvyatsov, A., Model theory for metric structures, Model theory with applications to algebra and analysis, London Mathematical Society Lecture Note Series (350), vol. 2, Cambridge University Press, Cambridge, 2008, pp. 315–427.CrossRefGoogle Scholar
[6] Yaacov, I. Ben and Keisler, H.J., Randomizations of models as metric structures, Conftuentes Mathematici, vol. 1 (2009), no. 2, pp. 197–223.CrossRefGoogle Scholar
[7] Yaacov, I. Ben and Usvyatsov, A., Continuous first order logic and local stability, Transactions of the American Mathematical Society, vol. 362 (2010), no. 10, pp. 5213–5259.CrossRefGoogle Scholar
[8] Berenstein, A. and Villaveces, A., Hilbert spaces with generic predicates, submitted.Google Scholar
[9] Carlisle, S., Model theory of ℝ-trees, Ph.D. thesis, University of Illinois at Urbana-Champaign, 2009.Google Scholar
[10] Ealy, C. and Onshuus, A., Characterizing rosy theories, this Journal, vol. 72 (2007), no. 3, pp. 919–940.Google Scholar
[11] Henson, C. W. and Tellez, H., Algebraic closure in continuous logic, Revista Colombiana de Matemáticas, vol. 41 (2001), pp. 279–285.Google Scholar
[12] Keisler, H. Jerome, Randomizing a model, Advances in Mathematics, vol. 143 (1999), no. 1, pp. 124–158.CrossRefGoogle Scholar
[13] Melleray, J., Topology of the isometry group of the Urysohn space, Fundamenta Mathematicae, vol. 207 (2010), no. 3, pp. 273–287.CrossRefGoogle Scholar
[14] Onshuus, A., Properties and consequences of thorn-independence, this Journal, vol. 71 (2006), no. 1, pp. 1–21.Google Scholar
[15] Poizat, B., A course in model theory: An introduction to contemporary mathematical logic, Universitext, Springer-Verlag, New York, 2000.CrossRefGoogle Scholar
[16] Usvyatsov, A., Generic separable metric structures, Topology and its Applications, vol. 155 (2008), no. 14, pp. 1607–1617.CrossRefGoogle Scholar
[17] Wagner, F., Simple theories, Kluwer Academic Publishers, Dordrecht, 2000.CrossRefGoogle Scholar