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This article is cited in 6 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.
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
Citation in format AMSBIB:
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\paper On some classical problems of descriptive set theory
\jour Uspekhi Mat. Nauk
\yr 2003
\vol 58
\issue 5(353)
\pages 3--88
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\transl
\jour Russian Math. Surveys
\yr 2003
\vol 58
\issue 5
\pages 839--927
\crossref{http://dx.doi.org/10.1070/RM2003v058n05ABEH000666}
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English version:
Russian Mathematical Surveys, 2003, 58:5, 839–927
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000189179400001
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 ; V. G. Kanovei, V. A. Lyubetskii, “On the Set of Constructible Reals”, Proc. Steklov Inst. Math., 247 (2004), 83–114 -
В. Г. Кановей, В. А. Любецкий, “О совершенных подмножествах инвариантных CA-множеств”, Матем. заметки, 77:3 (2005), 334–338 ; V. G. Kanovei, V. A. Lyubetskii, “Perfect subsets of invariant CA-sets”, Math. Notes, 77:3 (2005), 307–310
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В. Г. Кановей, В. А. Любецкий, “Конфинальное семейство отношений эквивалентности и порождающих их борелевских идеалов”, Геометрическая топология, дискретная геометрия и теория множеств, Сборник статей, Тр. МИАН, 252, Наука, М., 2006, 94–113 ; 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
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В. А. Любецкий, С. А. Пирогов, “Нестандартные представления локально компактных групп”, Матем. заметки, 82:3 (2007), 383–389 ; V. A. Lyubetskii, S. A. Pirogov, “Nonstandard Representations of Locally Compact Groups”, Math. Notes, 82:3 (2007), 341–346
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В. Г. Кановей, В. А. Любецкий, “Проблемы теоретико-множественного нестандартного анализа”, УМН, 62:1(373) (2007), 51–122 ; V. G. Kanovei, V. A. Lyubetskii, “Problems in set-theoretic nonstandard analysis”, Russian Math. Surveys, 62:1 (2007), 45–111
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В. Г. Кановей, В. А. Любецкий, “Борелевская сводимость как аддитивное свойство областей”, Исследования по конструктивной математике и математической логике. XI, Зап. научн. сем. ПОМИ, 358, ПОМИ, СПб., 2008, 189–198
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