Polish group actions and the Vaught conjecture

Sami, Ramez L.

doi:10.1090/S0002-9947-1994-1022169-2

Polish group actions and the Vaught conjecture
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by Ramez L. Sami

Trans. Amer. Math. Soc. 341 (1994), 335-353

DOI: https://doi.org/10.1090/S0002-9947-1994-1022169-2

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Abstract:

We consider the topological Vaught conjecture: If a Polish group $G$ acts continuously on a Polish space $S$, then $S$ has either countably many or perfectly many orbits. We show 1. The conjecture is true for Abelian groups. 2. The conjecture is true whenever $G$, $S$ are recursively presented, the action of $G$ is recursive and, for $x \in S$ the orbit of $x$ is of Borel multiplicative rank $\leq \omega _1^x$. Assertion $1$ holds also for analytic $S$. Specializing $G$ to a closed subgroup of $\omega !$, we prove that nonempty invariant Borel sets, not having perfectly many orbits, have orbits of about the same Borel rank. An upper bound is derived for the Borel rank of orbits when the set of orbits is finite.

References

Jon Barwise, Admissible sets and structures, Perspectives in Mathematical Logic, Springer-Verlag, Berlin-New York, 1975. An approach to definability theory. MR 0424560
John P. Burgess, A reflection phenomenon in descriptive set theory, Fund. Math. 104 (1979), no. 2, 127–139. MR 551663, DOI 10.4064/fm-104-2-127-139
John P. Burgess, Equivalences generated by families of Borel sets, Proc. Amer. Math. Soc. 69 (1978), no. 2, 323–326. MR 476524, DOI 10.1090/S0002-9939-1978-0476524-6
John P. Burgess, Two selection theorems, Bull. Soc. Math. Grèce (N.S.) 18 (1977), no. 1, 121–136. MR 528425
John Burgess and Douglas Miller, Remarks on invariant descriptive set theory, Fund. Math. 90 (1975), no. 1, 53–75. MR 403975, DOI 10.4064/fm-90-1-53-75

Applications séparément continues

288

Model theory

J. Dixmier, Dual et quasi-dual d’une algèbre de Banach involutive, Trans. Amer. Math. Soc. 104 (1962), 278–283 (French). MR 139960, DOI 10.1090/S0002-9947-1962-0139960-6
R. O. Gandy, On a problem of Kleene’s, Bull. Amer. Math. Soc. 66 (1960), 501–502. MR 122724, DOI 10.1090/S0002-9904-1960-10518-6
A. Grzegorczyk, A. Mostowski, and C. Ryll-Nardzewski, Definability of sets in models of axiomatic theories, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 9 (1961), 163–167. MR 163839
Peter G. Hinman, Some applications of forcing to hierarchy problems in arithmetic, Z. Math. Logik Grundlagen Math. 15 (1969), 341–352. MR 280376, DOI 10.1002/malq.19690152004
Alexander S. Kechris, Measure and category in effective descriptive set theory, Ann. Math. Logic 5 (1972/73), 337–384. MR 369072, DOI 10.1016/0003-4843(73)90012-0
K. Kuratowski, Topology. Vol. I, Academic Press, New York-London; Państwowe Wydawnictwo Naukowe [Polish Scientific Publishers], Warsaw, 1966. New edition, revised and augmented; Translated from the French by J. Jaworowski. MR 0217751
Alain Louveau, A separation theorem for $\Sigma ^{1}_{1}$ sets, Trans. Amer. Math. Soc. 260 (1980), no. 2, 363–378. MR 574785, DOI 10.1090/S0002-9947-1980-0574785-X
M. Makkai, An example concerning Scott heights, J. Symbolic Logic 46 (1981), no. 2, 301–318. MR 613284, DOI 10.2307/2273623
Douglas E. Miller, On the measurability of orbits in Borel actions, Proc. Amer. Math. Soc. 63 (1977), no. 1, 165–170. MR 440519, DOI 10.1090/S0002-9939-1977-0440519-8
Douglas E. Miller, The invariant $\Pi ^{0}_{\alpha }$ separation principle, Trans. Amer. Math. Soc. 242 (1978), 185–204. MR 496802, DOI 10.1090/S0002-9947-1978-0496802-9
Douglas E. Miller, Borel selectors for separated quotients, Pacific J. Math. 91 (1980), no. 1, 187–198. MR 612898
Yiannis N. Moschovakis, Descriptive set theory, Studies in Logic and the Foundations of Mathematics, vol. 100, North-Holland Publishing Co., Amsterdam-New York, 1980. MR 561709
Jan Mycielski, Independent sets in topological algebras, Fund. Math. 55 (1964), 139–147. MR 173645, DOI 10.4064/fm-55-2-139-147
Mark Nadel, Scott sentences and admissible sets, Ann. Math. Logic 7 (1974), 267–294. MR 384471, DOI 10.1016/0003-4843(74)90017-5
John C. Oxtoby, Measure and category, 2nd ed., Graduate Texts in Mathematics, vol. 2, Springer-Verlag, New York-Berlin, 1980. A survey of the analogies between topological and measure spaces. MR 584443
Michel Parigot, Omission des types dans les théories $\Pi _2^0(\Delta )$, C. R. Acad. Sci. Paris Sér. I Math. 299 (1984), no. 7, 195–198 (French, with English summary). MR 762719
J. P. Ressayre, Models with compactness properties relative to an admissible language, Ann. Math. Logic 11 (1977), no. 1, 31–55. MR 465849, DOI 10.1016/0003-4843(77)90009-2
Analytic sets, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London-New York, 1980. Lectures delivered at a Conference held at University College, University of London, London, July 16–29, 1978. MR 608794
C. Ryll-Nardzewski, On Borel measurabilty of orbits, Fund. Math. 56 (1964), 129–130. MR 172810, DOI 10.4064/fm-56-1-129-130
Gerald E. Sacks, Countable admissible ordinals and hyperdegrees, Advances in Math. 20 (1976), no. 2, 213–262. MR 429523, DOI 10.1016/0001-8708(76)90187-0
Ramez L. Sami, Sur la conjecture de Vaught en théorie descriptive, C. R. Acad. Sci. Paris Sér. I Math. 295 (1982), no. 6, 413–416 (French, with English summary). MR 683391
Ramez L. Sami, On $\Sigma _1^1$ equivalence relations with Borel classes of bounded rank, J. Symbolic Logic 49 (1984), no. 4, 1273–1283. MR 771794, DOI 10.2307/2274278
Jack H. Silver, Counting the number of equivalence classes of Borel and coanalytic equivalence relations, Ann. Math. Logic 18 (1980), no. 1, 1–28. MR 568914, DOI 10.1016/0003-4843(80)90002-9
John R. Steel, On Vaught’s conjecture, Cabal Seminar 76–77 (Proc. Caltech-UCLA Logic Sem., 1976–77) Lecture Notes in Math., vol. 689, Springer, Berlin, 1978, pp. 193–208. MR 526920
Jacques Stern, Effective partitions of the real line into Borel sets of bounded rank, Ann. Math. Logic 18 (1980), no. 1, 29–60. MR 568915, DOI 10.1016/0003-4843(80)90003-0
Jacques Stern, Évaluation du rang de Borel de certains ensembles, C. R. Acad. Sci. Paris Sér. A-B 286 (1978), no. 20, A855–A857 (French, with English summary). MR 498855
Y. Suzuki, Orbits of denumerable models of complete theories, Fund. Math. 67 (1970), 89–95. MR 266748, DOI 10.4064/fm-67-1-89-95
Robert Vaught, Invariant sets in topology and logic, Fund. Math. 82 (1974/75), 269–294. MR 363912, DOI 10.4064/fm-82-3-269-294