Polish group actions: Dichotomies and generalized elementary embeddings

Becker, Howard¹

¹ Department of Mathematics, The University of South Carolina, Columbia, South Carolina 29208

Journal of the American Mathematical Society, Tome 11 (1998) no. 2, pp. 397-449

Résumé

We prove that any Polish group which admits a complete left-invariant metric satisfies the Topological Vaught Conjecture. We also generalize some theorems of model theory from the logic actions to other Polish group actions.

Détail

DOI : 10.1090/S0894-0347-98-00258-6

Bibliographie

[1] Barwise, Jon Admissible sets and structures 1975

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

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

[4] Burgess, John P. Equivalences generated by families of Borel sets Proc. Amer. Math. Soc. 1978 323 326

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

[6] Dixmier, J. Dual et quasi-dual d’une algèbre de Banach involutive Trans. Amer. Math. Soc. 1962 278 283

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

[8] Effros, Edward G. Polish transformation groups and classification problems 1981 217 227

[9] Friedman, Harvey Countable models of set theories 1973 539 573

[10] Friedman, Harvey, Stanley, Lee A Borel reducibility theory for classes of countable structures J. Symbolic Logic 1989 894 914

[11] Glimm, James Locally compact transformation groups Trans. Amer. Math. Soc. 1961 124 138

[12] Harrington, Leo Analytic determinacy and 0^{♯} J. Symbolic Logic 1978 685 693

[13] Harrington, L. A., Kechris, A. S., Louveau, A. A Glimm-Effros dichotomy for Borel equivalence relations J. Amer. Math. Soc. 1990 903 928

[14] Harrington, L., Sami, R. Equivalence relations, projective and beyond 1979 247 264

[15] Harrington, Leo, Shelah, Saharon Counting equivalence classes for co-𝜅-Souslin equivalence relations 1982 147 152

[16] Hodges, Wilfrid Model theory 1993

[17] Jech, Thomas Set theory 1978

[18] Kanamori, Akihiro The higher infinite 1994

[19] Kechris, Alexander S. The structure of Borel equivalence relations in Polish spaces 1992 89 102

[20] Kechris, Alexander S. Topology and descriptive set theory Topology Appl. 1994 195 222

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

[22] Kechris, Alexander S., Louveau, Alain The classification of hypersmooth Borel equivalence relations J. Amer. Math. Soc. 1997 215 242

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

[24] Kunen, Kenneth Set theory 1980

[25] Miller, Douglas E. The invariant Π⁰_{𝛼} separation principle Trans. Amer. Math. Soc. 1978 185 204

[26] Maclane, Saunders Steinitz field towers for modular fields Trans. Amer. Math. Soc. 1939 23 45

[27] Morel, Anne C. Structure and order structure in Abelian groups Colloq. Math. 1968 199 209

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

[29] Rotman, Joseph J. The theory of groups. An introduction 1965

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

[31] Silver, Jack H. Counting the number of equivalence classes of Borel and coanalytic equivalence relations Ann. Math. Logic 1980 1 28

[32] Solecki, Så‚Awomir Equivalence relations induced by actions of Polish groups Trans. Amer. Math. Soc. 1995 4765 4777

[33] Steel, John R. Forcing with tagged trees Ann. Math. Logic 1978 55 74

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

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

[36] Suzuki, Y. Orbits of denumerable models of complete theories Fund. Math. 1970 89 95

[37] Vaught, R. L. Denumerable models of complete theories 1961 303 321

[38] Wagner, C. M. On Martin’s conjecture Ann. Math. Logic 1982 47 67