Hostname: page-component-cd9895bd7-8ctnn Total loading time: 0 Render date: 2024-12-22T11:11:02.899Z Has data issue: false hasContentIssue false
  • English
  • Français

Analytic Equivalence Relations and the Forcing Method

Published online by Cambridge University Press:  05 September 2014

Jindřich Zapletal
Jindřich Zapletal*
Affiliation:
Institute of Mathematics, Academy of Sciences ŽItná 25 11567 Praha 1, Czech Republic Department of Mathematics, University of Florida, Gainesville, FL 32611, USA, E-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

I describe several ways in which forcing arguments can be used to yield clean and conceptual proofs of nonreducibility, ergodicity and other results in the theory of analytic equivalence relations. In particular, I present simple Borel equivalence relations E, F such that a natural proof of nonreducibility of E to F uses the independence of the Singular Cardinal Hypothesis at ℵω.

Type
Research Article
Information
Bulletin of Symbolic Logic , Volume 19 , Issue 4 , September 2013 , pp. 473 - 490
Copyright
Copyright © Association for Symbolic Logic 2013

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1] Farah, Ilijas, Toms, Andrew, and Törnquist, Asger, Turbulence, orbit equivalence, and classification of C* algebras, Journal für die reine und angewandte Mathematik , to appear.Google Scholar
[2] Gao, Su, Invariant descriptive set theory, CRC Press, Boca Raton, 2009.Google Scholar
[3] Gao, Su and Kechris, Alexander S., On the classification of Polish metric spaces up to isometry, Memoirs of the American Mathematical Society, vol. 161 (2000), p. 78.Google Scholar
[4] Groszek, Marcia, Combinatorics of ideals and forcing with trees, The Journal of Symbolic Logic, vol. 52 (1987), pp. 582–593.Google Scholar
[5] Hjorth, Greg, Classification and orbit equivalence relations, Mathematical Surveys and Monographs, American Mathematical Society, Providence, 2000.Google Scholar
[6] Kanovei, Vladimir, Borel equivalence relations, University Lecture Series 44, American Mathematical Society, Providence, 2008.Google Scholar
[7] Kanovei, Vladimir, Sabok, Marcin, and Zapletal, Jindřich, Canonical Ramsey theory on Polish spaces, Cambridge University Press, 2013, to appear.Google Scholar
[8] Kechris, Alexander and Louveau, Alain, Descriptive set theory and the structure of sets of uniqueness, Cambridge University Press, Cambridge, 1989.Google Scholar
[9] Komjath, Peter and Shelah, Saharon, On uniformly antisymmetric functions, Real Analysis Exchange, vol. 19 (1993–1994), pp. 218–225.Google Scholar
[10] Komjath, Peter and Shelah, Saharon, Coloring finite subsets of uncountable sets, Proceedings of the American Mathematical Society, vol. 124 (1996), pp. 3501–3505.Google Scholar
[11] Louveau, Alain and Rosendal, Christian, Complete analytic equivalence relations, Transactions of the American Mathematical Society, vol. 357 (2005), pp. 4839–4866.Google Scholar
[12] Magidor, Menachem, On the singular cardinals problem. I, Israel Journal of Mathematics, vol. 28 (1977), pp. 1–31.Google Scholar
[13] Pestov, Vladimir, Dynamics of infinite-dimensional groups, University Lecture Series 40, American Mathematical Society, Providence, 2006.Google Scholar
[14] Pol, Roman and Zakrzewski, Piotr, On Borel mappings and σ-ideals generated by closed sets, Advances in Mathematics, vol. 231 (2012), pp. 651–663.Google Scholar
[15] Solecki, Slawomir, Covering analytic sets by families of closed sets, The Journal of Symbolic Logic, vol. 59 (1994), pp. 1022–1031.CrossRefGoogle Scholar
[16] Solecki, Slawomir and Spinas, Otmar, Dominating and unbounded free sets, The Journal of Symbolic Logic, vol. 64 (1999), pp. 75–80.Google Scholar
[17] Stern, Jacques, On Lusin's restricted continuum problem, Annals of Mathematics, vol. 120 (1984), pp. 7–37.Google Scholar
[18] Thomas, Simon, The classification problem for torsion-free abelian groups of finite rank, Journal of the American Mathematical Society, vol. 16 (2003), pp. 233–258.Google Scholar
[19] Zapletal, Jindřich, Generalizations of turbulence, preprint, 2013.Google Scholar