Vaught’s conjecture on analytic sets

Hjorth, Greg¹

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

Journal of the American Mathematical Society, Tome 14 (2001) no. 1, pp. 125-143

Résumé

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.

Détail

DOI : 10.1090/S0894-0347-00-00349-0

Bibliographie

[1] Becker, Howard The topological Vaught’s conjecture and minimal counterexamples J. Symbolic Logic 1994 757 784

[2] Becker, Howard Polish group actions: dichotomies and generalized elementary embeddings J. Amer. Math. Soc. 1998 397 449

[3] Becker, Howard, Kechris, Alexander S. Borel actions of Polish groups Bull. Amer. Math. Soc. (N.S.) 1993 334 341

[4] Becker, Howard, Kechris, Alexander S. The descriptive set theory of Polish group actions 1996

[5] Buechler, Steven, Newelski, Ludomir On the geometry of 𝑈-rank 2 types 1993 10 24

[6] Burgess, John P. Effective enumeration of classes in a Σ¹₁ equivalence relation Indiana Univ. Math. J. 1979 353 364

[7] Effros, Edward G. Transformation groups and 𝐶*-algebras Ann. of Math. (2) 1965 38 55

[8] Gao, Su On automorphism groups of countable structures J. Symbolic Logic 1998 891 896

[9] Grzegorczyk, A., Mostowski, A., Ryll-Nardzewski, C. Definability of sets in models of axiomatic theories Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 1961 163 167

[10] Hart, Bradd, Starchenko, Sergei, Valeriote, Matthew Vaught’s conjecture for varieties Trans. Amer. Math. Soc. 1994 173 196

[11] Hjorth, Greg Orbit cardinals: on the effective cardinalities arising as quotient spaces of the form 𝑋/𝐺 where 𝐺 acts on a Polish space 𝑋 Israel J. Math. 1999 221 261

[12] Hjorth, Greg, Solecki, Slawomir Vaught’s conjecture and the Glimm-Effros property for Polish transformation groups Trans. Amer. Math. Soc. 1999 2623 2641

[13] Jech, Thomas Set theory 1978

[14] Kechris, Alexander S. Classical descriptive set theory 1995

[15] Keisler, H. Jerome Model theory for infinitary logic. Logic with countable conjunctions and finite quantifiers 1971

[16] Moschovakis, Yiannis N. Descriptive set theory 1980

[17] Sami, Ramez L. Polish group actions and the Vaught conjecture Trans. Amer. Math. Soc. 1994 335 353

[18] Bouscaren, Elisabeth Martin’s conjecture for 𝜔-stable theories Israel J. Math. 1984 15 25

[19] Steel, John R. On Vaught’s conjecture 1978 193 208

[20] Stern, Jacques On Lusin’s restricted continuum problem Ann. of Math. (2) 1984 7 37

[21] Vaught, Robert Invariant sets in topology and logic Fund. Math. 1974/75 269 294