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Borel structures and Borel theories

Published online by Cambridge University Press:  12 March 2014

Greg Hjorth  and
André Nies
Greg Hjorth
Affiliation:
Department of Mathematics and Statistics, The University of Melbourne, Melbourne, Australia
André Nies
Affiliation:
Department of Computer Science, Auckland University, Auckland, New Zealand, E-mail: [email protected]
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Abstract

We show that there is a complete, consistent Borel theory which has no “Borel model” in the following strong sense: There is no structure satisfying the theory for which the elements of the structure are equivalence classes under some Borel equivalence relation and the interpretations of the relations and function symbols are uniformly Borel.

We also investigate Borel isomorphisms between Borel structures.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 76 , Issue 2 , June 2011 , pp. 461 - 476
Copyright
Copyright © Association for Symbolic Logic 2011

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References

REFERENCES

[1]Becker, H. and Kechris, A.S., The descriptive set theory of Polish group actions, Cambridge University Press, 1996.CrossRefGoogle Scholar
[2]Harrington, L. and Shelah, S., Counting equivalence classes for co-κ-Souslin equivalence relations, Logic Colloquium '80 (Prague, 1980), Studies in Logic and the Foundations of Mathematics, vol. 108, North-Holland, Amsterdam, 1982, pp. 147–152.Google Scholar
[3]Hjorth, G., Borel equivalence relations, Handbook on set theory (Foreman, M. and Kanamori, A., editors), Springer, to appear, pp. 297–332.Google Scholar
[4]Hjorth, G., Khoussainov, B., Montalban, A., and Nies, A., From automatic structures to Borel structures, Proceedings of the 19th IEEE symposium on Logic in Computer Science, Lecture Notes in Computer Science, IEEE Computer Society, 2008, pp. 110–119.Google Scholar
[5]Kechris, A. S., Classical descriptive set theory, Graduate Texts in Mathematics, vol. 156, Springer-Verlag, New York, 1995.CrossRefGoogle Scholar
[6]Lindenstrauss, J. and Tzafriri, L., Classical Banach spaces I, Springer, 1996.Google Scholar
[7]Malitz, J., The Hanf number for complete lω1,ω sentences, The syntax and semantics of infinitary languages (Barwise, J., editor), Lecture Notes in Mathematics, vol. 72, Springer, 1968, pp. 166–181.CrossRefGoogle Scholar
[8]Montalbán, A. and Nies, A., Borel structures: a brief survey, Proceedings of the workshop on effective models of the uncountable, 2009, to appear.Google Scholar
[9]Steinhorn, C. I., Borel structures and measure and category logics, Model-theoretic logics (Barwise, J. and Feferman, S., editors), Perspectives in Mathematical Logic, Springer, New York, 1985, pp. 579–596.Google Scholar