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

This article is cited in 16 scientific papers (total in 16 papers)

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.


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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
Received: 27.05.2003

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
\by V.~G.~Kanovei, V.~A.~Lyubetskii
\paper On some classical problems of descriptive set theory
\jour Uspekhi Mat. Nauk
\yr 2003
\vol 58
\issue 5(353)
\pages 3--88
\jour Russian Math. Surveys
\yr 2003
\vol 58
\issue 5
\pages 839--927

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    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  mathnet  mathscinet  zmath
    2. V. G. Kanovei, V. A. Lyubetskii, “Perfect subsets of invariant CA-sets”, Math. Notes, 77:3 (2005), 307–310  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    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  mathnet  crossref  mathscinet
    4. V. A. Lyubetskii, S. A. Pirogov, “Nonstandard Representations of Locally Compact Groups”, Math. Notes, 82:3 (2007), 341–346  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    5. V. G. Kanovei, V. A. Lyubetskii, “Problems in set-theoretic nonstandard analysis”, Russian Math. Surveys, 62:1 (2007), 45–111  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    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  mathnet  crossref  elib  elib
    7. V. G. Kanovei, V. A. Lyubetsky, “An effective minimal encoding of uncountable sets”, Siberian Math. J., 52:5 (2011), 854–863  mathnet  crossref  mathscinet  isi
    8. V. G. Kanovei, V. A. Lyubetskii, “Effective Compactness and Sigma-Compactness”, Math. Notes, 91:6 (2012), 789–799  mathnet  crossref  crossref  mathscinet  isi  elib  elib
    9. Vladimir Kanovei, Vassily Lyubetsky, “On effective σ-boundedness and σ-compactness”, Mathematical Logic Quarterly, 2013, n/a  crossref  mathscinet  isi  scopus  scopus
    10. Marcel Nutz, Ramon van Handel, “Constructing sublinear expectations on path space”, Stochastic Processes and their Applications, 2013  crossref  mathscinet  isi  scopus  scopus
    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  crossref  mathscinet  zmath  isi  scopus  scopus
    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  mathnet  crossref  crossref  mathscinet  isi  elib
    13. Dean W., Walsh S., “The Prehistory of the Subsystems of Second-Order Arithmetic”, Rev. Symb. Log., 10:2 (2017), 357–396  crossref  mathscinet  zmath  isi  scopus  scopus
    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  mathnet  crossref  crossref  adsnasa  isi  elib
    15. V. G. Kanovei, V. A. Lyubetskii, “Opredelimye elementy opredelimykh borelevskikh mnozhestv”, Matem. zametki, 105:5 (2019), 696–707  mathnet  crossref  elib
    16. Kanovei V., Lyubetsky V., “Non-Uniformizable Sets With Countable Cross-Sections on a Given Level of the Projective Hierarchy”, Fundam. Math., 245:2 (2019), 175–215  crossref  mathscinet  zmath  isi  scopus
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