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 Uspekhi Mat. Nauk, 2003, Volume 58, Issue 5(353), Pages 3–88 (Mi umn666)

On some classical problems of descriptive set theory

V. G. Kanovei, V. A. Lyubetskii

Institute for Information Transmission Problems, Russian Academy of Sciences

Abstract: The centenary of P. S. Novikov's birth provides an inspiring motivation to present, with full proofs and from a modern standpoint, the presumably definitive solutions of some classical problems in descriptive set theory which were formulated by Luzin [Lusin] and, to some extent, even earlier by Hadamard, Borel, and Lebesgue and relate to regularity properties of point sets. The solutions of these problems began in the pioneering works of Aleksandrov [Alexandroff], Suslin [Souslin], and Luzin (1916–17) and evolved in the fundamental studies of Gödel, Novikov, Cohen, and their successors. Main features of this branch of mathematics are that, on the one hand, it is an ordinary mathematical theory studying natural properties of point sets and functions and rather distant from general set theory or intrinsic problems of mathematical logic like consistency or Gödel's theorems, and on the other hand, it has become a subject of applications of the most subtle tools of modern mathematical logic.

DOI: https://doi.org/10.4213/rm666

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English version:
Russian Mathematical Surveys, 2003, 58:5, 839–927

Bibliographic databases:

UDC: 510.225
MSC: Primary 03E15, 03E30, 03E45; Secondary 03E40, 28A05, 54H05, 03C25, 54E52

Citation: V. G. Kanovei, V. A. Lyubetskii, “On some classical problems of descriptive set theory”, Uspekhi Mat. Nauk, 58:5(353) (2003), 3–88; Russian Math. Surveys, 58:5 (2003), 839–927

Citation in format AMSBIB
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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. V. G. Kanovei, V. A. Lyubetskii, “On the Set of Constructible Reals”, Proc. Steklov Inst. Math., 247 (2004), 83–114
2. V. G. Kanovei, V. A. Lyubetskii, “Perfect subsets of invariant CA-sets”, Math. Notes, 77:3 (2005), 307–310
3. V. G. Kanovei, V. A. Lyubetskii, “A Cofinal Family of Equivalence Relations and Borel Ideals Generating Them”, Proc. Steklov Inst. Math., 252 (2006), 85–103
4. V. A. Lyubetskii, S. A. Pirogov, “Nonstandard Representations of Locally Compact Groups”, Math. Notes, 82:3 (2007), 341–346
5. V. G. Kanovei, V. A. Lyubetskii, “Problems in set-theoretic nonstandard analysis”, Russian Math. Surveys, 62:1 (2007), 45–111
6. V. G. Kanovei, V. A. Lyubetskii, “Borel reducibility as an additive property of domains”, J. Math. Sci. (N. Y.), 158:5 (2009), 708–712
7. V. G. Kanovei, V. A. Lyubetsky, “An effective minimal encoding of uncountable sets”, Siberian Math. J., 52:5 (2011), 854–863
8. V. G. Kanovei, V. A. Lyubetskii, “Effective Compactness and Sigma-Compactness”, Math. Notes, 91:6 (2012), 789–799
9. Vladimir Kanovei, Vassily Lyubetsky, “On effective σ-boundedness and σ-compactness”, Mathematical Logic Quarterly, 2013, n/a
10. Marcel Nutz, Ramon van Handel, “Constructing sublinear expectations on path space”, Stochastic Processes and their Applications, 2013
11. Fischer V., Friedman S.D., Khomskii Yu., “Cichon's Diagram, Regularity Properties and Delta(1)(3) Sets of Reals”, Arch. Math. Log., 53:5-6 (2014), 695–729
12. V. G. Kanovei, V. A. Lyubetskii, “On Effective $\sigma$-Boundedness and $\sigma$-Compactness in Solovay's Model”, Math. Notes, 98:2 (2015), 273–282
13. Dean W., Walsh S., “The Prehistory of the Subsystems of Second-Order Arithmetic”, Rev. Symb. Log., 10:2 (2017), 357–396
14. V. G. Kanovei, V. A. Lyubetsky, “Non-uniformizable sets of second projective level with countable cross-sections in the form of Vitali classes”, Izv. Math., 82:1 (2018), 61–90
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