Undecidable and decidable properties of constituents

V. G. Kanovei

Sbornik. Mathematics, Tome 52 (1985) no. 2, pp. 491-519

Résumé

Some assertions concerning decidability and undecidability of Luzin's problems on constituents are proved. Bibliography: 34 titles.

Bibliographie

[1] Lusin N., “Sur les voies de la theorie des ensembles”, Atti del Congresso internationale dei matematici, vol. 1 (3–10 settembre, Bologna, 1929), 295–299 | Zbl

[2] Lusin N., Lecons sur les ensembles analytiques et leurs applications, Gauthiers-Villars, Paris, 1930 ; “Лекции об аналитических множествах и их приложениях”, В кн. [9], 9–269 | Zbl

[3] Lusin N., “Sur une famille des complementaires analytiques”, Fund. math., 17 (1931), 4–7 ; “Об одном семействе аналитических дополнений”, в кн. [9], 624–626 | Zbl

[4] Lusin N., “Sur les classes des constituantes des complementaires analytiques”, Ann. r. scuola norm, super. Pisa, ser. 2, 2, fasc. 3 (1933), 269–282 ; “О классах конституант аналитических дополнений”, в кн. [9], 627–641 | MR | Zbl

[5] Luzin N. N., “O statsionarnykh posledovatelnostyakh”, Trudy fiz.-matem. in-ta, otd. matem., 5 (1934), 125–147 ; В кн. [9], 642–661 | Zbl

[6] Luzin N. N., O nekotorykh novykh rezultatakh deskriptivnoi teorii funktsii, Izd-vo AN SSSR, M.-L., 1935; В кн. [9], 552–616

[7] Lusin N., “Sur les ensembles analytiques nuls”, Fund. math., 25 (1935), 109–131 ; “О пустых аналитических множествах”, В кн. [9], 662–682 | Zbl

[8] Luzin N. N., “O chastyakh naturalnogo ryada”, Izv. AN SSSR. Seriya matem., 11:5 (1947), 403–410 ; В кн. [9], 714–722 | MR

[9] Luzin Ya. Ya., Sobr. soch., t. 2, Izd-vo AN SSSR, M., 1958

[10] Novikov P. S., “O vzaimootnoshenii vtorogo klassa proektivnykh mnozhestv i proektsii uniformnykh analiticheskikh dopolnenii”, Izv. AN SSSR. Seriya matem., 1:2 (1937), 231–252

[11] Novikov P. S., “O neprotivorechivosti nekotorykh polozhenii deskriptivnoi teorii mnozhestv”, Trudy MIAN, 38 (1951), 279–316

[12] Novikov P. S., Lyapunov A. A., “Deskriptivnaya teoriya mnozhestv”, Matematika v SSSR za 30 let, Gostekhizdat, M.-L., 1948, 243–255

[13] Novikov P. S., Keldysh L. V., “Raboty Luzina v oblasti deskriptivnoi teorii mnozhestv”, UMN, 8:2 (1953), 93–104 | MR | Zbl

[14] Keldysh L. V., “Verkhnie otsenki dlya klassov konstituant analiticheskikh dopolnenii”, Izv. AN SSSR. Seriya matem., 1 (1937), 265–284

[15] Keldysh L. V. Idei N. N., “Luzina v deskriptivnoi teorii mnozhestv”, UMN, 29:5 (1974), 183–196 | MR | Zbl

[16] Lyapunov A. A., “O nekotorykh uniformnykh analiticheskikh dopolneniyakh”, Izv. AN SSSR. Seriya matem., 1 (1937), 285–305

[17] Kuratovskii K., Topologiya, t. 1, Mir, M., 1966 | MR

[18] Kuratovskii K., Mostovskii A., Teoriya mnozhestv, Mir, M., 1970 | MR

[19] Spravochnaya kniga po matematicheskoi logike. Chast II: teoriya mnozhestv, Nauka, M., 1982 | MR

[20] Uspenskii V. A., Kanovei V. G., “Problemy Luzina o konstituantakh i ikh sudba”, Vestn. MGU. Seriya matem., mekh., 1983, no. 6, 73–87 | MR | Zbl

[21] Kanovei V. G., “O neschetnykh posledovatelnostyakh mnozhestv, davaemykh operatsiei resheta”, DAN SSSR, 257:4 (1981), 808–812 | MR | Zbl

[22] Kanovei V. G., “O strukture konstituant $\Pi_1{}^1$-mnozhestv”, Sib. matem. zhurn., 24:2 (1983), 56–76 | MR | Zbl

[23] Shenfild Dzh., Matematicheskaya logika, Nauka, M., 1975 | MR

[24] Iekh T., Teoriya mnozhestv i metod forsinga, Mir, M., 1973 | MR

[25] Lyubetskii V. A., “Nekotorye sledstviya gipotezy o neschetnosti mnozhestva konstruktivnykh deistvitelnykh chisel”, DAN SSSR, 182:4 (1968), 758–759

[26] Levy A., “Definability in axiomatic set theory, II”, Mathematical logic and foundations of set theory, North Holland, Amst., 1970, 129–145 | MR

[27] Solovay R. M., “On the cardinality of $\Sigma_2^1$ sets of reals”, Foundations of mathematics, Springer-Verlag, Berlin, 1969, 58–73 | MR

[28] Solovay R. M., “A model of set theory in which every set of reals is Lebesque measurable”, Ann. of math., 92:1 (1970), 1–56 | DOI | MR | Zbl

[29] Martin D. A., “Borel determinacy”, Ann. of math., 102:2 (1975), 363–371 | DOI | MR | Zbl

[30] Stern J., “Suites transfines d'ensembles Boreliennes”, C. r. Acad. sci. Paris, ser. A, 288:10 (1979), 527–529 | MR | Zbl

[31] Stern J., “Effective partitions of the real line into Borel sets of bounded rank”, Ann. of math. logic, 18:1 (1980), 29–60 | DOI | MR | Zbl

[32] Louveau A., “A separation theorem for $\Sigma_1^1$ sets”, Trans. Amer. Math. Soc., 260:2 (1980), 363–378 | DOI | MR

[33] Hausdorff F., “Summen von $\mathcal{N}_1$ Mengen”, Fund. math., 26 (1936), 241–255 | Zbl

[34] Selivanovski E. A., “Sur les proprietes des constituantes ensembles analytiques”, Fund. math., 21 (1933), 20–28