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New Directions in Descriptive Set Theory

Published online by Cambridge University Press:  15 January 2014

Alexander S. Kechris*
Affiliation:
Department of Mathematics, California Institute of Technology, Pasadena, CA 91125, USAE-mail:kechris@caltech.edu

Extract

§1. I will start with a quick definition of descriptive set theory: It is the study of the structure of definable sets and functions in separable completely metrizable spaces. Such spaces are usually called Polish spaces. Typical examples are ℝn, ℂn, (separable) Hilbert space and more generally all separable Banach spaces, the Cantor space 2, the Baire space, the infinite symmetric group S∞, the unitary group (of the Hilbert space), the group of measure preserving transformations of the unit interval, etc.

In this theory sets are classified in hierarchies according to the complexity of their definitions and the structure of sets in each level of these hierarchies is systematically analyzed. In the beginning we have the Borel sets in Polish spaces, obtained by starting with the open sets and closing under the operations of complementation and countable unions, and the corresponding Borel hierarchy ( sets). After this come the projective sets, obtained by starting with the Borel sets and closing under the operations of complementation and projection, and the corresponding projective hierarchy ( sets).

There are also transfinite extensions of the projective hierarchy and even much more complex definable sets studied in descriptive set theory, but I will restrict myself here to Borel and projective sets, in fact just those at the first level of the projective hierarchy, i.e., the Borel (), analytic () and coanalytic () sets.

Type
Research Article
Copyright
Copyright © Association for Symbolic Logic 1999

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References

[1] Becker, H. and Kechris, A. S., The descriptive set theory of Polish group actions, London Mathematical Society Lecture Note Series, no. 232, Cambridge University Press, 1966.Google Scholar
[2] Connes, A., Noncommutative geometry, Academic Press, 1994.Google Scholar
[3] Connes, A., Feldman, J., and Weiss, B., An amenable equivalence relation is generated by a single transformation, Ergodic Theory and Dynamical Systems, vol. 1 (1981), pp. 430450.CrossRefGoogle Scholar
[4] Dougherty, R., Jackson, S., and Kechris, A. S., The structure of hyperfinite Borel equivalence relations, Transactions of the American Mathematical Society, vol. 341 (1994), no. 1, pp. 193225.CrossRefGoogle Scholar
[5] Effros, E. G., Transformation groups and C*-algebras, Annals of Mathematics, vol. 81 (1965), pp. 3855.CrossRefGoogle Scholar
[6] Feldman, J. and Moore, C. C., Ergodic equivalence relations and von Neumann algebras, I, Transactions of the American Mathematical Society, vol. 234 (1977), pp. 289324.CrossRefGoogle Scholar
[7] Friedman, H. and Stanley, L., A Borel reducibility theory for classes of countable models, this Journal, vol. 54 (1989), pp. 894914.Google Scholar
[8] Fuchs, L., Abelian groups, Academic Press, 1967.Google Scholar
[9] Glimm, J., Locally compact transformation groups, Transactions of the American Mathematical Society, vol. 101 (1961), pp. 124138.CrossRefGoogle Scholar
[10] Harrington, L., Kechris, A. S., and Louveau, A., A Glimm-Effros dichotomy for Borel equivalence relations, Journal of the American Mathematical Society, vol. 3 (1990), no. 4, pp. 903928.CrossRefGoogle Scholar
[11] Hjorth, G., Vaught's conjecture on analytic sets, preprint, 1997.Google Scholar
[12] Hjorth, G., Around nonclassifiability for countable torsion-free abelian groups, preprint, 1998.Google Scholar
[13] Hjorth, G., Classification and orbit equivalence relations, preprint, 1998.Google Scholar
[14] Hjorth, G. and Kechris, A. S., The complexity of the classification of Riemann surfaces and complex manifolds, Illinois Journal of Mathematics, to appear.Google Scholar
[15] Hjorth, G. and Kechris, A. S., Borel equivalence relations and classifications of countable models, Annals of Pure and Applied Logic, vol. 82 (1996), pp. 221272.CrossRefGoogle Scholar
[16] Hjorth, G. and Kechris, A. S., New dichotomies for Borel equivalence relations, Bulletin of Symbolic Logic, vol. 3 (1997), no. 3, pp. 329346.CrossRefGoogle Scholar
[17] Hjorth, G., Kechris, A. S., and Louveau, A., Borel equivalence relations induced by actions of the symmetric group, Annals of Pure and Applied Logic, vol. 92 (1998), pp. 63112.CrossRefGoogle Scholar
[18] Jackson, S., Kechris, A. S., and Louveau, A., Countable Borel equivalence relations, in preparation.Google Scholar
[19] Kechris, A. S., Amenable equivalence relations and Turing degrees, this Journal, vol. 56 (1991), pp. 182194.Google Scholar
[20] Kechris, A. S., Classical descriptive set theory, Graduate Texts in Mathematics, no. 156, Springer-Verlag, 1995.CrossRefGoogle Scholar
[21] Kechris, A. S. and Louveau, A., The classification of hypersmooth Borel equivalence relations, Journal of the American Mathematical Society, vol. 10 (1997), no. 1, pp. 215242.CrossRefGoogle Scholar
[22] Kechris, A. S. and Moschovakis, Y. N. (editors), Cabal seminar 76–77, Lecture Notes in Mathematics, no. 689, Springer-Verlag, 1978.CrossRefGoogle Scholar
[23] Kechris, A. S. and Sofronidis, N., A strong generic ergodicity property of unitary conjugacy, preprint, 1997.Google Scholar
[24] Louveau, A. and Velickovic, B., A note on Borel equivalence relations, Proceedings of the American Mathematical Society, vol. 120 (1994), pp. 255259.CrossRefGoogle Scholar
[25] Moschovakis, Y. N., Descriptive set theory, North-Holland, 1980.Google Scholar
[26] Ornstein, D. and Weiss, B., Ergodic theory of amenable group actions, I. The Rohlin lemma, Bulletin of the American Mathematical Society, vol. 2 (1980), pp. 161164.CrossRefGoogle Scholar
[27] Silver, J., Counting the number of equivalence classes of Borel and co-analytic equivalence relations, Annals of Mathematical Logic, vol. 18 (1980), pp. 128.CrossRefGoogle Scholar
[28] Slaman, T. and Steel, J., Definable functions on degrees, Cabal seminar 81–85, Lecture Notes in Mathematics, no. 1333, Springer-Verlag, 1988, pp. 3755.CrossRefGoogle Scholar
[29] Solecki, S., Analytic ideals, Bulletin of Symbolic Logic, vol. 2 (1996), no. 3, pp. 339348.CrossRefGoogle Scholar
[30] Thomas, S. and Velickovic, B., On the complexity of the isomorphism relation for fields of finite transcendence degree, preprint, 1998.Google Scholar
[31] Thomas, S. and Velickovic, B., On the complexity of the isomorphism relation for finitely generated groups, preprint, 1998.Google Scholar
[32] Weiss, B., Measurable dynamics, Contemporary Mathematics, vol. 26 (1984), pp. 395421.CrossRefGoogle Scholar
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