Skip to main content
SpringerLink
Log in

Orbit cardinals: On the effective cardinalities arising as quotient spaces of the formX/G whereG acts on a Polish spaceX

  • Published:
Israel Journal of Mathematics Aims and scope Submit manuscript

Abstract

We prove an Ulm-type classification theorem for actions inL(ℝ), thereby answering a question of Becker and Kechris, and investigate the effective cardinalities which can be induced by various classes of Polish groups.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. H. Becker,Polish group actions: Dichotomies and generalized elementary embeddings, preprint, University of North Carolina, Columbia, 1997.

    Google Scholar 

  2. H. Becker and A. S. Kechris,Sets of ordinals constructible from trees on ordinals and the third Victoria Delfino problem, Contemporary Mathematics31 (1984), 13–29.

    MathSciNet  Google Scholar 

  3. H. Becker and A. S. Kechris,The descriptive set theory of Polish group actions, London Mathematical Society Lecture Note Series 232, Cambridge University Press, Cambridge, 1996.

    MATH  Google Scholar 

  4. E. Effros,Transformation groups and C *-algebras, Annals of Mathematics (2)81 (1975), 38–55.

    MathSciNet  Google Scholar 

  5. H. Friedman and L. Stanley,A Borel reducibility theory for classes of countable structures, Journal of Symbolic Logic54 (1989), 894–914.

    Article  MATH  MathSciNet  Google Scholar 

  6. S. Gao,Automorphism groups of countable structures, Journal of Symbolic Logic, to appear.

  7. A. Gregorczyk, A. Mostowski and C. Ryall-Nardzewski,Definability of sets of models of axiomatic theories, Bulletin of the Polish Academy of Sciences (series Mathematics, Astronomy, Physics)9 (1961), 163–167.

    Google Scholar 

  8. L. A. Harrington,Analytic determinacy and 0♯, Journal of Symbolic Logic43 (1978), 685–693.

    Article  MATH  MathSciNet  Google Scholar 

  9. L. A. Harrington,A powerless proof of a theorem by Silver, handwritten notes, UC Berkeley, 1976.

  10. L. A. Harrington and S. Shelah,Counting the equivalence classes for co-κ-Souslin equivalence relations inLogic Colloquium ’80, North-Holland, Amsterdam, 1982, pp. 147–152.

    Google Scholar 

  11. L. A. Harrington, A. S. Kechris and A. Louveau,A Glimm-Effros dichotomy for Borel equivalence relations, Journal of the American Mathematical Society3 (1990), 903–928.

    Article  MATH  MathSciNet  Google Scholar 

  12. G. Hjorth,A dichotomy for the definable universe, Journal of Symbolic Logic60 (1995), 1199–1207.

    Article  MATH  MathSciNet  Google Scholar 

  13. G. Hjorth,An absoluteness principle for Borel sets, Journal of Symbolic Logic63 (1998), 663–693.

    Article  MATH  MathSciNet  Google Scholar 

  14. G. Hjorth,On ℵ1 many minimal models, Journal of Symbolic Logic61 (1996), 906–919.

    Article  MATH  MathSciNet  Google Scholar 

  15. G. Hjorth,A universal Polish G-space, Topology and its Applications, to appear.

  16. G. Hjorth,Classification and orbit equivalence relations, unpublished manuscript.

  17. G. Hjorth and A. S. Kechris,Analytic equivalence relations and Ulm-type classifications, Journal of Symbolic Logic60 (1995), 1273–1300.

    Article  MATH  MathSciNet  Google Scholar 

  18. T. Jech,Set Theory, Academic Press, New York, 1978.

    Google Scholar 

  19. A. S. Kechris,The structure of Borel equivalence relations in Polish spaces, inSet Theory of the Continuum (H. Judah, W. Just and W. H. Woodin, eds.), MSRI Publication 26, Springer-Verlag, New York, 1992, pp. 75–84.

    Google Scholar 

  20. A. S. Kechris,Classical Descriptive Set Theory, Graduate Texts in Mathematics, Springer-Verlag, Berlin, 1995.

    MATH  Google Scholar 

  21. A. S. Kechris,Definable equivalence relations and Polish group actions, unpublished manuscript.

  22. H. J. Keisler,Model Theory for Infinitary Logic, North-Holland, Amsterdam, 1971.

    MATH  Google Scholar 

  23. J. Knight,A complete L ω 1ω characterizing ℵ1, Journal of Symbolic Logic42 (1977), 59–62.

    Article  MATH  MathSciNet  Google Scholar 

  24. D. A. Martin and J. R. Steel,The extent of scales in L(ℝ), inCabal Seminar 79–81, Lecture Notes in Mathematics1019 (A. S. Kechris, D. A. Martin and J. R. Steel, eds.), Springer-Verlag, Berlin, 1983, pp. 86–96.

    Chapter  Google Scholar 

  25. Y. N. Moschovakis,Descriptive Set Theory, North-Holland, Amsterdam, 1980.

    MATH  Google Scholar 

  26. R. Sami,The topological Vaught conjecture, Transactions of the American Mathematical Society341 (1994), 335–353.

    Article  MATH  MathSciNet  Google Scholar 

  27. E. Schimmerling,Notes on some lectures by Woodin, handwritten notes, UC Berkeley, Spring, 1990.

  28. J. Stern,On Lusin’s restricted continuum hypothesis, Annals of Mathematics (2)120 (1984), 7–37.

    Article  MathSciNet  Google Scholar 

  29. R. Vaught,Invariant sets in topology and logic, Fundamenta Mathematica82 (1974), 269–294.

    MATH  MathSciNet  Google Scholar 

  30. W. H. Woodin,Supercompact cardinals, sets of reals, and weakly homogeneous trees, Proceedings of the National Academy of Sciences of the United States of America85 (1988), 6587–6591.

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of California, 90095-1555, Los Angeles, CA, USA

    Greg Hjorth

Authors
  1. Greg Hjorth
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Greg Hjorth.

Additional information

Research partially supported by NSF grant DMS 96-22977.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Hjorth, G. Orbit cardinals: On the effective cardinalities arising as quotient spaces of the formX/G whereG acts on a Polish spaceX . Isr. J. Math. 111, 221–261 (1999). https://doi.org/10.1007/BF02810686

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02810686

Keywords

Search

Navigation