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Borel structures and Borel theories

  • Greg Hjorth (a1) and André Nies (a2)

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.

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References

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[1]Becker, H. and Kechris, A.S., The descriptive set theory of Polish group actions, Cambridge University Press, 1996.
[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. 147152.
[3]Hjorth, G., Borel equivalence relations, Handbook on set theory (Foreman, M. and Kanamori, A., editors), Springer, to appear, pp. 297332.
[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. 110119.
[5]Kechris, A. S., Classical descriptive set theory, Graduate Texts in Mathematics, vol. 156, Springer-Verlag, New York, 1995.
[6]Lindenstrauss, J. and Tzafriri, L., Classical Banach spaces I, Springer, 1996.
[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. 166181.
[8]Montalbán, A. and Nies, A., Borel structures: a brief survey, Proceedings of the workshop on effective models of the uncountable, 2009, to appear.
[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. 579596.
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The Journal of Symbolic Logic
  • ISSN: 0022-4812
  • EISSN: 1943-5886
  • URL: /core/journals/journal-of-symbolic-logic
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