Available in electronic format
Available in print format		 Journal of the American Mathematical Society
Journal of the American Mathematical Society
ISSN: 1088-6834(e) ISSN: 0894-0347(p)
Previous issue | Table of contents | Next issue
Articles in press | Recently posted articles | Most recent issue | All issues
Previous Article | Next Article
     

Vaught's conjecture on analytic sets

Author(s): Greg Hjorth
Journal: J. Amer. Math. Soc. 14 (2001), 125-143.
MSC (2000): Primary 03E15
Posted: September 18, 2000
Retrieve article in: PDF DVI PostScript
This article is available free of charge

Abstract | References | Similar articles | Additional information

Abstract:

Let $G$ be a Polish group. We characterize when there is a Polish space $X$ with a continuous $G$-action and an analytic set (that is, the Borel image of some Borel set in some Polish space) $A\subset X$ having uncountably many orbits but no perfect set of orbit inequivalent points.

Such a Polish $G$-space $X$ and analytic $A$ exist exactly when there is a continuous, surjective homomorphism from a closed subgroup of $G$ onto the infinite symmetric group, $S_\infty$, consisting of all permutations of $\mathbb{N} $ equipped with the topology of pointwise convergence.

References:

[1]
H. Becker, The topological Vaught's conjecture and minimal counterexamples, Journal of Symbolic Logic, vol. 59(1994), pp. 757-784. MR 95k:03077

[2]
H. Becker, Polish group actions: dichotomies and generalized elementary embeddings, Journal of the American Mathematical Society, vol. 11 (1998), pp. 397-449. MR 99g:03051

[3]
H. Becker, A.S. Kechris, Borel actions of Polish groups, Bulletin American Mathematical Society, vol. 28(1993), pp. 334-341. MR 93m:03083

[4]
H. Becker, A.S. Kechris, The descriptive set theory of Polish group actions, London Mathematical Society Lecture Notes Series, 232, Cambridge University Press, Cambridge, 1996. MR 98d:54068

[5]
S. Buechler, L. Newelski, On the geometry of $U$-rank $2$ types, Logic Colloquium '90 (Helsinki, 1990), pp. 10-24, Lecture Notes Logic, 2, Springer, Berlin, 1993. MR 95e:03104

[6]
J. Burgess, Effective enumeration of $\Sigma^1_1$ equivalence relations, Indiana University Mathematics Journal, vol. 28(1979), pp. 353-364. MR 80f:03053

[7]
E. Effros, Transformation groups and $C^*$-algebras, Annals of Mathematics, ser. 2, vol. 81(1975), pp. 38-55. MR 30:5175

[8]
S. Gao, Automorphism groups of countable structures, Journal of Symbolic Logic, vol. 63(1998), pp. 891-896. MR 2000b:03118

[9]
A. Gregorczyk, A. Mostowski, C. Ryall-Nardzewski, Definability of sets of models of axiomatic theories, Bulletin of the Polish Academy of Sciences (series Mathematics, Astronomy, Physics), vol. 9(1961), pp. 163-7. MR 29:1138

[10]
B. Hart, S. Starchenko, M. Valeriote, Vaught's conjecture for varieties, Transactions of the American Mathematical Society, vol. 342(1994), pp. 173-196. MR 94e:03036

[11]
G. Hjorth, Orbit cardinals: On the effective cardinalities of quotients of the form $X/G$ for $X$ a Polish $G$-space, Israel Journal of Mathematics, vol. 111(1999), pp. 221-261. MR 2000h:03091

[12]
G. Hjorth, Classification and orbit equivalence relations, Mathematical Surveys and Monographs, 75, American Mathematical Society, Providence, RI, 2000.

[13]
G. Hjorth, S. Solecki, Vaught's conjecture and the Glimm-Effros property for Polish transformation groups, Transactions of the American Mathematical Society, vol. 351(1999), pp. 2623-2641. MR 99j:54037

[14]
T. Jech, Set theory, Academic Press, New York, 1978. MR 80a:03062

[15]
A.S. Kechris, Classical descriptive set theory, Graduate Texts in Mathematics, Springer-Verlag, Berlin, 1995. MR 96e:03057

[16]
H.J. Keisler, Model theory for infinitary logic, North-Holland, Amsterdam, 1971. MR 49:8855

[17]
Y.N. Moschovakis, Descriptive set theory, North-Holland, Amsterdam, 1980. MR 82e:03002

[18]
R. Sami, Polish group actions and the Vaught conjecture, Transactions of the American Mathematical Society, vol. 341(1994), pp. 335-353. MR 94c:03068

[19]
S. Shelah, L. Harrington, M. Makkai, A proof of Vaught's conjecture for $\omega$-stable theories, Israel Journal of Mathematics, vol. 49(1984), pp. 259-280. MR 86j:03029b

[20]
J.R. Steel, On Vaught's conjecture, Cabal Seminar 76-77 (Proceedings Caltech-UCLA Logic Seminar, 1976-77), pp. 193-208, Lecture Notes in Mathematics, 689, Springer, Berlin, 1978. MR 81b:03036

[21]
J. Stern, Lusin's restricted continuum problem, Annals of Mathematics, ser. 2, vol. 120(1984), pp. 7-37. MR 85h:03051

[22]
R. Vaught, Invariant sets in topology and logic, Fundamenta Mathematicae, vol. 82(1974), pp. 269-94. MR 51:167

Similar Articles:

Retrieve articles in Journal of the American Mathematical Society with MSC (2000): 03E15

Retrieve articles in all Journals with MSC (2000): 03E15

Additional Information:

Greg Hjorth
Affiliation: Department of Mathematics, University of California, Los Angeles, California 90095-1555
Email: greg@math.ucla.edu

DOI: 10.1090/S0894-0347-00-00349-0
PII: S 0894-0347(00)00349-0
Keywords: Polish group, group actions, topological Vaught conjecture
Received by editor(s): June 8, 1998
Received by editor(s) in revised form: June 22, 2000
Posted: September 18, 2000
Additional Notes: The author's research was partially supported by NSF grant DMS 96-22977.
Copyright of article: Copyright 2000, American Mathematical Society

  AMS Website Logo Small Comments: webmaster@ams.org
© Copyright 2008, American Mathematical Society
Privacy Statement 		Search the AMSPowered by Google