On some classical problems of descriptive set theory

Russian Mathematical Surveys

Volumn 58, Issue 5, 2003, Pages 839-927

  Kanovei, V G ^a   Lyubetskii, V A ^a  

^a INSTITUTE FOR INFORMATION TRANSMISSION PROBLEMS   (Russian Federation)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 1542291397     PISSN: 00360279     EISSN: 14684829     Source Type: Journal    
DOI: 10.1070/RM2003v058n05ABEH000666     Document Type: Review

References (117)

1. J. Addison, "Separation principles in the hierarchies of classical and effective descriptive set theory", Fund. Math. 46 (1959), 123-135.
2. J. Addison. "Some consequences of the axiom of constructibility", Fund. Math. 46 (1959), 337-357.
3. P. Alexandroff, "Sur la puissance des ensembles mesurables B", C. R. Acad. Sci. Paris 162 (1916), 323-325; Russian transl., Selected Papers of P. S. Aleksandrov. The Theory of Functions of a Real Variable and the Theory of Topological Spaces (A. N. Kolmogorov, ed.), Nauka, Moscow 1978, pp. 35-39.
4. P. Alexandroff, "Sur la puissance des ensembles mesurables B", C. R. Acad. Sci. Paris 162 (1916), 323-325; Russian transl., Selected Papers of P. S. Aleksandrov. The Theory of Functions of a Real Variable and the Theory of Topological Spaces (A. N. Kolmogorov, ed.), Nauka, Moscow 1978, pp. 35-39.
5. P. S. Aleksandrov [Aleksandroff], Introduction to set theory and to general topology, Nauka, Moscow 1977; German transl., Einführung in die Mengenlehre und in die allgemeine Topologie, (Univ. Books Math., vol. 85) VEB Deutscher Verlag der Wissenschaften, Berlin 1984.
6. P. S. Aleksandrov [Aleksandroff], Introduction to set theory and to general topology, Nauka, Moscow 1977; German transl., Einführung in die Mengenlehre und in die allgemeine Topologie, (Univ. Books Math., vol. 85) VEB Deutscher Verlag der Wissenschaften, Berlin 1984.
7. V. Ya. Arsenin and A. A. Lyapunov. "The theory of A-sets", Uspekhi Mat. Nauk 5:5 (1950), 45-108. (Russian)
8. 1-sets of reals", J. Symbolic Logic 62 (1997), 1379-1428.
9. R. Baire, "Sur les fonctions de variables reelles", Ann. Mat. Pura Appl. Ser. IIIa 3 (1899), 1-122.
10. R. Baire, É. Borel, J. Hadamard, and H. Lebesgue, "Cinq lettres sur la théorie des ensembles", Bull. Soc. Math. France 33 (1905), 261-273.
11. T. Bartoszyński. "Additivity of measure implies additivity of category", Trans. Amer. Math. Soc. 281 (1984), 209-213.
12. T. Bartoszyński and H. Judah, Set theory. On the structure of the real line, Peters, Wellesley, MA 1995.
13. K. Jon Barwise (ed.), Handbook of mathematical logic, Part B. Set theory, (Stud. Logic Found. Math., vol. 90) North-Holland, Amsterdam 1977; Russian transl., Nauka, Moscow 1982.
14. J. Baumgartner, "Iterated forcing", Surveys in Set Theory, (London Math. Soc. Lecture Note Ser., vol. 87) Cambridge Univ. Press, Cambridge 1983, pp. 1-59.
15. É. Borel, Leçons sur la théorie des fonctions, Paris 1898.
16. J. Brendle and B. Löwe, "Solovay-type characterizations for forcing-algebras", J. Symbolic Logic 64 (1999), 1307-1323.
17. P. J. Cohen, Set theory and the continuum hypothesis, Benjamin, New York 1966; Russian transl., Mir, Moscow 1969.
18. E. van Douwen, "The integers and topology", Handbook of Set-Theoretic Topology (K. Kunen and J. Vaughan, eds.), North-Holland, Amsterdam 1984, pp. 111-167.
19. A. G. Dragalin and V. A. Lyubetskii, "Independence of certain problems of axiomatic set theory", Proc. All-Union Symposium on Mathematical Logic, Alma-Ata 1969, pp. 12-17. (Russian)
20. P. Du Bois Reymond, "Eine neue Theorie der Konvergenz und Divergenz von Reihen mit positiven Gliedern", J. Reine Angew. Math. 76 (1873), 61-91.
21. A. A. Fraenkel and Y. Bar-Hillel, Foundations of set theory, North-Holland, Amsterdam 1958; Russian transl., Mir, Moscow 1966.
22. K. Gödel, "The consistency of the axiom of choice and of the generalized continuum hypothesis with the axioms of set theory", Ann. Math. Studies (Princeton) 3 (1940); Russian transl., Uspekhi Mat. Nauk 3:1 (1948), 96-149.
23. K. Gödel, "The consistency of the axiom of choice and of the generalized continuum hypothesis with the axioms of set theory", Ann. Math. Studies (Princeton) 3 (1940); Russian transl., Uspekhi Mat. Nauk 3:1 (1948), 96-149.
24. L. Harrington and S. Shelah, "Some exact equiconsistency results in set theory". Notre Dame J. Formal Logic 26:2 (1985), 178-188.
25. F. Hausdorff, "Die Graduirung nach dem Endverlauf", Abh. Königlich Sächs. Ges. Wiss. Leipzig Math.-Phys. 31 (1909), 295-334.
26. F. Hausdorff, "Die Mächtigkeit der Borelschen Mengen", Math. Ann. 77 (1916), 430-437.
27. 1 Mengen". Fund. Math. 26 (1936), 241-255.
28. F. Hausdorff, Mengenlehre, de Gruyter, Berlin 1927; English transl., Set theory, 4th ed., Chelsea, New York 1991; Russian transl. of 1st ed. (P. S. Aleksandrov and A. N. Kolmogorov, eds.), ONTI, Moscow-Leningrad 1937.
29. F. Hausdorff, Mengenlehre, de Gruyter, Berlin 1927; English transl., Set theory, 4th ed., Chelsea, New York 1991; Russian transl. of 1st ed. (P. S. Aleksandrov and A. N. Kolmogorov, eds.), ONTI, Moscow-Leningrad 1937.
30. 1-sets of reals", J. Symbolic Logic 53 (1988), 636-642.
31. J. Ihoda and S. Shelah, "Martin's axioms, measurability and equiconsistency results", J. Symbolic Logic 54 (1989), 78-94.
32. 1-sets of reals", Ann. Pure Appl. Logic 42:3 (1989), 207-223.
33. T. Jech, Lectures in set theory with particular emphasis on the method of forcing, (Lecture Notes in Math., vol. 217) Springer-Verlag, Berlin 1971; Russian transl., Mir, Moscow 1973.
34. T. Jech, Set theory, Academic Press, New York 1978.
35. 1-sets of reals", Arch. Math. Logic 32:2 (1992), 101-112.
36. 1-sets of reals", Israel Math. Conf. Proc. 6 (1993), 361-384.
37. 1-sets of reals", J. Symbolic Logic 58 (1993), 72-80.
38. 1-sets of reals", Arch. Math. Logic 34:5 (1995), 301-312.
39. V. G. Kanovei, "Projective hierarchy of Luzin: Modern state of the theory", Appendix to the Russian transl, of [11], pp. 273-364.
40. V. G. Kanovei, "Undecidable and decidable properties of constituents", Mat. Sb. 124 (1984), 505-535; English transl., Math. USSR-Sb. 52 (1985), 491-519.
41. V. G. Kanovei, "Undecidable and decidable properties of constituents", Mat. Sb. 124 (1984), 505-535; English transl., Math. USSR-Sb. 52 (1985), 491-519.
42. V. G. Kanovei, "Luzin's problems on the existence of CA-sets without perfect subsets", Mat. Zametki 41 (1987), 750-759; English transl., Math. Notes 41 (1987), 422-426.
43. V. G. Kanovei, "Luzin's problems on the existence of CA-sets without perfect subsets", Mat. Zametki 41 (1987), 750-759; English transl., Math. Notes 41 (1987), 422-426.
44. V. G. Kanovei, "The development of the descriptive theory of sets under the influence of the work of Luzin", Uspekhi Mat. Nauk 40:3 (1985), 117-153; English transl., Russian Math. Surveys 40:3 (1985), 135-180.
45. V. G. Kanovei, "The development of the descriptive theory of sets under the influence of the work of Luzin", Uspekhi Mat. Nauk 40:3 (1985), 117-153; English transl., Russian Math. Surveys 40:3 (1985), 135-180.
46. V. G. Kanovei, "Kolmogorov's ideas in the theory of operations on sets", Uspekhi Mat. Nauk 43:6 (1988), 93-128; English transl., Russian Math. Surveys 43:6 (1988), 111-155.
47. V. G. Kanovei, "Kolmogorov's ideas in the theory of operations on sets", Uspekhi Mat. Nauk 43:6 (1988), 93-128; English transl., Russian Math. Surveys 43:6 (1988), 111-155.
48. V. G. Kanovei and V. A. Uspenskii, "Luzin's problem on constituents and their fate", Vestnik Moskov. Univ. Ser. I Mat. Mekh. 1983, no. 6, 73-87; English transl., Moscow Univ. Math. Bull. 38:6 (1983), 86-102.
49. V. G. Kanovei and V. A. Uspenskii, "Luzin's problem on constituents and their fate", Vestnik Moskov. Univ. Ser. I Mat. Mekh. 1983, no. 6, 73-87; English transl., Moscow Univ. Math. Bull. 38:6 (1983), 86-102.
50. A. S. Kechris, "Measure and category in effective descriptive set theory", Ann. Math. Logic 5 (1973), 337-384.
51. A. S. Kechris, Classical descriptive set theory, Springer-Verlag, New York 1995.
52. M. Kondô, "Sur l'uniformisation des complémentaires analytiques et les ensembles projectifs de la seconde classe", Japan. J. Math. 15 (1938), 197-230.
53. K. Kunen, Set theory. An introduction to independence proofs. (Stud. Logic Found. Math., vol. 102) North-Holland, Amsterdam 1980.
54. K. Kuratowski, "Sur une généralisation de la notion de homéomorphie", Fund. Math. 22 (1934), 206-220.
55. K. Kuratowski, Topology, vol. I, rev. aug. ed., Academic Press, New York-London 1966; Russian transl., Mir, Moscow 1970.
56. H. Lebesgue, "Sur les fonctions représentables analytiquement", J. Math. Sér. 6 1 (1905), 139-216.
57. H. Lebesgue, Preface of the book [57], pp. vii-xi; Russian transl. (with comments by V. A. Uspenskii), Uspekhi Mat. Nauk 40:3 (1985), 9-14; English transl., Russian Math. Surveys 40:3 (1985), 9-14.
58. H. Lebesgue, Preface of the book [57], pp. vii-xi; Russian transl. (with comments by V. A. Uspenskii), Uspekhi Mat. Nauk 40:3 (1985), 9-14; English transl., Russian Math. Surveys 40:3 (1985), 9-14.
59. A. Levy. "Definability in axiomatic set theory. II", Mathematical Logic and Foundations of Set Theory, North-Holland, Amsterdam 1970, pp. 129-145.
60. G. G. Lorentz, "Who discovered analytic sets?", Math. Intelligencer 23:4 (2001), 28-32.
61. N. Lusin, "Sur la classification de M. Baire", C. R. Acad. Sci. Paris 164 (1917), 91-94; Russian transl., [60], pp. 270-272.
62. N. Lusin, "Sur les ensembles projectifs de M. Henri Lebesgue", C. R. Acad. Sci. Paris 180 (1925), 1572-1574; Russian transl., [60], pp. 304-306.
63. N. Lusin, "Les propriétés des ensembles projectifs", C. R. Acad. Sci. Paris 180 (1925), 1817-1819; Russian transl., [60], pp. 307-308.
64. N. Lusin, "Sur le problème de M. Émile Borel et la méthode des résolvantes", C. R. Acad. Sci. Paris 181 (1925), 279-281; Russian transl., [60], pp. 312-314.
65. N. Lusin, "Sur les ensembles analytiques", Fund. Math. 10 (1927), 1-95; Russian transl., [60], pp. 380-459.
66. N. Lusin, "Sur le problème de M. J. Hadamard d'uniformisation des ensembles", Mathématica (Cluj) 4 (1930), 54-66.
67. N. Lusin, Leçons sur les ensembles analytiques et leurs applications, Gauthier-Villars, Paris 1930; English transl., Lectures on analytic sets and their applications, 2nd rev. ed., Chelsea, New York 1972; Russian transl., [60], pp. 9-269; short Russian transl., Gos. Tekh. Izdat., Moscow 1953.
68. N. Lusin, Leçons sur les ensembles analytiques et leurs applications, Gauthier-Villars, Paris 1930; English transl., Lectures on analytic sets and their applications, 2nd rev. ed., Chelsea, New York 1972; Russian transl., [60], pp. 9-269; short Russian transl., Gos. Tekh. Izdat., Moscow 1953.
69. N. Lusin, Leçons sur les ensembles analytiques et leurs applications, Gauthier-Villars, Paris 1930; English transl., Lectures on analytic sets and their applications, 2nd rev. ed., Chelsea, New York 1972; Russian transl., [60], pp. 9-269; short Russian transl., Gos. Tekh. Izdat., Moscow 1953.
70. N. N. Luzin, Modern state of the theory of functions of real variables, GTTI, Moscow-Leningrad 1933 (Russian); Reprinted in [60], pp. 494-536.
71. N. N. Lusin, "Stationary sequences", Trudy Fiz.-Mat. Inst. Mat. Otdel. 5 (1934), 125-147; French transl., Sur les suites stationnaires, Hermann, Paris 1934; Russian reprint, [60], pp. 642-661.
72. N. N. Lusin, "Stationary sequences", Trudy Fiz.-Mat. Inst. Mat. Otdel. 5 (1934), 125-147; French transl., Sur les suites stationnaires, Hermann, Paris 1934; Russian reprint, [60], pp. 642-661.
73. N. N. Luzin, Collected works, vol. 2, Izdat. Akad. Nauk SSSR, Moscow 1958. (Russian)
74. N. Lusin and W. Sierpiński, "Sur quelques propriétés des ensembles (A)", Bull. Int. Acad. Sci. Cracowie 4 (1918), 35-48; Russian transl., [60], pp. 273-284.
75. N. Lusin and P. Novikoff, "Choix effectif d'un point dans un complémentaire analytique arbitraire donné par un crible", Fund. Math. 25 (1935), 559-560; independent Russian translations in [60], pp. 621-623, and [77], pp. 49-50.
76. N. Lusin and W. Sierpiński, "Sur un ensemble non mesurable B", J. Math. 2 (1923), 53-72; Russian transl., [60], pp. 285-300.
77. V. A. Lyubetskii, "Some corollaries of the hypothesis of noncountability of the set of constructible real numbers", Dokl. Akad. Nauk SSSR 182 (1968), 758-759; English transl., Soviet Math. Dokl. 9 (1968), 1195-1196.
78. V. A. Lyubetskii, "Some corollaries of the hypothesis of noncountability of the set of constructible real numbers", Dokl. Akad. Nauk SSSR 182 (1968), 758-759; English transl., Soviet Math. Dokl. 9 (1968), 1195-1196.
79. 2 there follows the existence of an uncountable set which contains no perfect subset of type CA", Dokl. Akad. Nauk SSSR 195 (1970), 548-550; English transl., Soviet Math. Dokl. 11 (1970), 1513-1515.
80. 2 there follows the existence of an uncountable set which contains no perfect subset of type CA", Dokl. Akad. Nauk SSSR 195 (1970), 548-550; English transl., Soviet Math. Dokl. 11 (1970), 1513-1515.
81. 2 implies the existence of an uncountable set of type CA which does not contain a perfect subset", Algebra, Mathematical Logic, and Computer Mathematics, Proc. Conf. of Teacher's Training Institutes, Ivanovo 1970, pp. 22-24. (Russian)
82. V. A. Lyubetskii, "Independence of certain propositions of descriptive set theory from the Zermelo-Fraenkel set theory", Vestnik Moskov. Univ. Ser. I Mat. Mekh. 1971, no. 2, 78-82, (this paper was received by the Editorial Board in January, 1970 - V. A. Lyubetskii); English transl., Moscow Univ. Math. Bull. 26:1/2 (1972), 116-119.
83. V. A. Lyubetskii, "Independence of certain propositions of descriptive set theory from the Zermelo-Fraenkel set theory", Vestnik Moskov. Univ. Ser. I Mat. Mekh. 1971, no. 2, 78-82, (this paper was received by the Editorial Board in January, 1970 - V. A. Lyubetskii); English transl., Moscow Univ. Math. Bull. 26:1/2 (1972), 116-119.
84. V. A. Lyubetskii, Measurability and existence of a perfect kernel for protective sets, Candidate's (PhD) dissertation, Moscow State University, Moscow 1971. (Russian)
85. 2-sets", Studies in Set Theory and Non-Classical Logics, Nauka, Moscow 1976, pp. 96-122.
86. V. A. Lyubetskii, "Valuations and sheaves. On some questions of non-standard analysis", Uspekhi Mat. Nauk 44:4 (1988), 99-153; English transl., Russian Math. Surveys 44:4 (1989), 37-112.
87. V. A. Lyubetskii, "Valuations and sheaves. On some questions of non-standard analysis", Uspekhi Mat. Nauk 44:4 (1988), 99-153; English transl., Russian Math. Surveys 44:4 (1989), 37-112.
88. V. I. Malykhin, "Topology and forcing", Uspekhi Mat. Nauk 38:1 (1983), 69-118; English transl., Russian Math. Surveys 38:1 (1983), 77-136.
89. V. I. Malykhin, "Topology and forcing", Uspekhi Mat. Nauk 38:1 (1983), 69-118; English transl., Russian Math. Surveys 38:1 (1983), 77-136.
90. R. Mansfield, "Perfect subsets of definable sets of real numbers", Pacific J. Math. 35 (1970), 451-457.
91. D. Martin and R. M. Solovay, "Internal Cohen extensions", Ann. Math. Logic 2 (1970), 143-178.
92. A. R. D. Mathias, "Surrealist landscape with figures (a survey of recent results in set theory)", Period. Math. Hungar. 10:2-3 (1979), 109-175 (All our references to this survey refer to its original version known under the title: A survey of recent results in set theory, Stanford University, July 1968. This preprint contains preliminary announcements of early results in set theory. With rare exceptions, they are given without references to publications and without (even sketches of) proofs. Some of the results in the survey are cited in our paper.).
93. J. Myhill and D. Scott, "Ordinal definability", 1971 Axiomatic Set Theory, Proc. Sympos. Pure Math., vol. XIII. Part. I (Univ. California, Los Angeles, 1967), Amer. Math. Soc., Providence, RI 1971, pp. 271-278.
94. P. Novikoff, "Sur la séparabilité des ensembles projectifs de seconde classe", Fund. Math. 25 (1935), 459-466; Russian transl., [77], pp. 43-48.
95. P. S. Novikov, Selected works. Theory of sets and functions. Mathematical logic and algebra, Nauka, Moscow 1979.
96. P. S. Novikov, "On the consistency of certain propositions of the descriptive theory of sets", Trudy Mat. Inst. Steklov. 38 (1951), 279-316; reprinted in [77], pp. 170-202; English transl., Amer. Math. Soc. Transl. (2) 29 (1963), 51-89.
97. P. S. Novikov, "On the consistency of certain propositions of the descriptive theory of sets", Trudy Mat. Inst. Steklov. 38 (1951), 279-316; reprinted in [77], pp. 170-202; English transl., Amer. Math. Soc. Transl. (2) 29 (1963), 51-89.
98. J. Raisonnier, "A mathematical proof of S. Shelah's theorem on the measure problem and related results", Israel J. Math. 48 (1984), 48-56.
99. J. Raisonnier and J. Stern, "The strength of the measurability hypotheses", Israel J. Math. 50 (1985), 337-349.
100. ω", J. Appl. Anal. 3:1 (1997), 103-127.
101. G. E. Sacks, "Forcing with perfect closed sets", Proc. Sympos. Pure Math. 13, Part 1 (1971), 331-355.
102. E. Selivanowski, "Sur les propriétés des constituantes des ensembles analytiques", Fund. Math. 21 (1933), 20-28.
103. S. Shelah, "Can you take Solovay's inaccessible away?", Israel J. Math. 48 (1984), 1-47.
104. J. Shoenfield, "The problem of predicativity", Essays on the Foundations of Mathematics, Magnes Press, Hebrew University, Jerusalem 1961, pp. 132-139.
105. J. R. Shoenfield, Mathematical logic, Addison-Wesley, Reading, Mass.-London-Don Mills, Ont. 1967; Russian transl., Nauka, Moscow 1975.
106. J. R. Shoenfield. "Unramified forcing", 1971 Axiomatic Set Theory, Proc. Sympos. Pure Math., vol. XIII, Part. I (Univ. California, Los Angeles, 1967), Amer. Math. Soc., Providence, RI 1971, pp. 357-381; published in Russian as an appendix to the Russian translation of [86], pp. 482-519.
107. 1 sets of reals", Foundations of Mathematics. Symposium papers commemorating the 60th birthday of Kurt Gödel (Columbus, 1966), Springer-Verlag, New York 1969, pp. 58-73.
108. R. M. Solovay, "A model of set theory in which every set of reals is Lebesgue measurable", Ann. of Math. (2) 92:1 (1970), 1-56.
109. R. M. Solovay and S. Tennenbaum, "Iterated Cohen extensions and Suslin's problem", Ann. of Math. (2) 94 (1971), 201-245.
110. M. Souslin, "Sur une définition des ensembles mesurables B sans nombres transfinis", C. R. Acad. Sci. Paris 164 (1917), 88-90.
111. J. Stern, "Generic extensions which do not add random reals", Methods in Mathematical Logic (Caracas, 1983), (Lecture Notes in Math., vol. 1130) Springer-Verlag, Berlin 1985, pp. 395-407.
112. J. Stern, "On Lusin's restricted continuum problem", Ann. of Math. (2) 120:1 (1984), 7-37.
113. J. Stern, "Regularity properties of definable sets of reals", Ann. Pure Appl. Logic 29:3 (1985), 289-324.
114. V. M. Tikhomirov, "The discovery of A-sets", Istor.-Mat. Issled. 1993, no. 34, 129-139. (Russian)
115. 1", Logic Colloquium 76 (Oxford, 1976), (Stud. Logic Found. Math., vol. 87) North-Holland, Amsterdam 1977, pp. 595-612.
116. V. A. Uspenskii, "Luzin's contribution to the descriptive theory of sets and functions: concepts, problems, predictions", Uspekhi Mat. Nauk 40:3 (1985), 85-116; English transl., Russian Math. Surveys 40:3 (1985), 97-134.
117. V. A. Uspenskii, "Luzin's contribution to the descriptive theory of sets and functions: concepts, problems, predictions", Uspekhi Mat. Nauk 40:3 (1985), 85-116; English transl., Russian Math. Surveys 40:3 (1985), 97-134.