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Some dichotomy theorems for isomorphism relations of countable models

Published online by Cambridge University Press:  12 March 2014

Su Gao
Su Gao*
Affiliation:
Department of Mathematics, University of California, Los Angeles, Los Angeles, California 90024, USA
*
Department of Mathematics, California Institute of Technology, Pasadena, CA 91125, USA, E-mail: [email protected]
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Abstract

Strengthening known instances of Vaught Conjecture, we prove the Glimm-Effros dichotomy theorems for countable linear orderings and for simple trees. Corollaries of the theorems answer some open questions of Friedman and Stanley in an Lω1ω-interpretability theory. We also give a survey of this theory.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 66 , Issue 2 , June 2001 , pp. 902 - 922
Copyright
Copyright © Association for Symbolic Logic 2001

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References

REFERENCES

[Ba]Barwise, J., Admissible sets and structures: an approach to definability theory, Perspectives in mathematical logic, Springer-Verlag, Berlin, 1975.Google Scholar
[Be]Becker, H., Vaught's conjecture for complete left-invariant Polish groups, 1996, handwritten notes.Google Scholar
[BK]Becker, H. and Kechris, A. S., The descriptive set theory of Polish group actions, London Mathematical Society Lecture Notes Series, vol. 232, Cambridge University Press, Cambridge, 1996.CrossRefGoogle Scholar
[FS]Friedman, H. and Stanley, L., A Borel reducibility theory for classes of countable structures, this Journal, vol. 54 (1989), pp. 894–914.Google Scholar
[G1]Gao, S., The isomorphism relation between countable models and definable equivalence relations, Ph.D. thesis, UCLA, 1998.Google Scholar
[G2]Gao, S., A dichotomy theorem for mono-unary algebras, Fundamenta Mathematicae, vol. 163 (2000), pp. 25–37.CrossRefGoogle Scholar
[HKL]Harrington, L., Kechris, A. S., and Louveau, A., A Glimm-Effros dichotomy for Borel equivalence relations, Journal of the American Mathematical Society, vol. 3 (1990), no. 4, pp. 903–928.CrossRefGoogle Scholar
[HK]Hjorth, G. and Kechris, A. S., Analytic equivalence relations and Ulm-type classifications, this Journal, vol. 60 (1995), pp. 1273–1300.Google Scholar
[HS]Hjorth, G. and Solecki, S., Vaught's Conjecture and the Glimm-Effros property for Polish transformation groups, preprint, 1995.Google Scholar
[Ho]Hodges, W., Model theory, Cambridge University Press, Cambridge, 1993.CrossRefGoogle Scholar
[Kec]Kechris, A.S., Classical descriptive set theory, Springer-Verlag, New York, 1995.CrossRefGoogle Scholar
[Kei]Keisler, H. J., Model theory for infinitary logic, Studies in logic and foundations of mathematics, vol. 62, North-Holland, 1971.Google Scholar
[Ma]Makkai, M., An example concerning Scott heights, this Journal, vol. 46 (1981), pp. 301–318.Google Scholar
[Mo]Moschovakis, Y. N., Descriptive set theory, North-Holland, Amsterdam, 1980.Google Scholar
[Na]Nadel, M., Scott sentences and admissible sets, Annals of Mathematical Logic, vol. 7 (1974), pp. 267–294.CrossRefGoogle Scholar
[Ru]Rubin, M., Vaught's conjecture for linear orderings, Notices of the American Mathematical Society, vol. 24 (1977), A390.Google Scholar
[Sa]Sami, R., Polish group actions and the Vaught Conjecture, Transactions of the American Mathematical Society, vol. 341 (1994), pp. 335–353.CrossRefGoogle Scholar
[St]Steel, J. R., On Vaught's Conjecture, Cabal seminar 76–77 (Berlin), Lecture Notes in Mathematics, vol. 689, Springer-Verlag, 1978, pp. 193–208.CrossRefGoogle Scholar