On the Set of Constructible Reals

V. G. Kanovei ; V. A. Lyubetskii

Informatics and Automation, Geometric topology and set theory, Tome 247 (2004), pp. 95-128.

Résumé

The most important results concerning the set of Gödel constructible reals are presented together with full proofs.

Bibliographie

[1] Bartoszyński T., Judah H., Set theory, on the structure of the real line, A. K. Peters, Wellesley, 1995 | MR | Zbl

[2] Koen P. Dzh., Teoriya mnozhestv i kontinuum-gipoteza, Mir, M., 1969 | MR

[3] Dragalin A. G., Lyubetskii V. A., “Nezavisimost nekotorykh problem aksiomaticheskoi teorii mnozhestv”, Tezisy dokladov na Vsesoyuznom simpoziume po matematicheskoi logike, Alma-Ata, 1969, 12–17

[4] Gedel K., “Sovmestimost aksiomy vybora i obobschennoi kontinuum-gipotezy s aksiomami teorii mnozhestv”, UMN, 3:1 (1948), 96–149 | MR

[5] Groszek M., Slaman T., “A basis theorem for perfect sets”, Bull. Symbol. Log., 4:2 (1998), 204–209 | DOI | MR | Zbl

[6] Spravochnaya kniga po matematicheskoi logike. Ch. 2. Teoriya mnozhestv, Nauka, M., 1983

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

[8] Jech T., Set theory, Acad. Press, New York, 1978 | MR | Zbl

[9] Kanovei V. G., “O stepenyakh konstruktivnosti i deskriptivnykh svoistvakh mnozhestva deistvitelnykh chisel v iskhodnoi modeli i v ee rasshireniyakh”, DAN SSSR, 216:4 (1974), 728–729 | MR | Zbl

[10] Kanovei V. G., “Proektivnaya ierarkhiya N. N. Luzina: Sovremennoe sostoyanie teorii”, Spravochnaya kniga po matematicheskoi logike, Ch. 2, Nauka, M., 1983, 273–364 | MR

[11] Kanovei V. G., “Razvitie deskriptivnoi teorii mnozhestv pod vliyaniem trudov N. N. Luzina”, UMN, 40:3 (1985), 117–153 | MR

[12] Kanovei V. G., Lyubetskii V. A., “O nekotorykh klassicheskikh problemakh deskriptivnoi teorii mnozhestv”, UMN, 58:5 (2003), 3–88 | MR | Zbl

[13] Kechris A. S., “Measure and category in effective descriptive set theory”, Ann. Math. Log., 5 (1973), 337–384 | DOI | MR | Zbl

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

[15] Lyubetskii V. A., “Iz suschestvovaniya neizmerimogo mnozhestva tipa $A_2$ vytekaet suschestvovanie neschetnogo mnozhestva, ne soderzhaschego sovershennogo podmnozhestva, tipa $CA$”, DAN SSSR, 195:3 (1970), 548–550 | Zbl

[16] Lyubetskii V. A., “Iz suschestvovaniya neizmerimogo mnozhestva tipa $A_2$ sleduet suschestvovanie neschetnogo mnozhestva tipa $CA$, ne soderzhaschego sovershennogo podmnozhestva”, Tezisy dokladov po algebre, matematicheskoi logike i vychislitelnoi matematike konferentsii pedagogicheskikh vuzov tsentralnoi zony RSFSR, Ivanovo, 1970, 22–24

[17] Lyubetskii V. A., “Nezavisimost nekotorykh predlozhenii deskriptivnoi teorii mnozhestv ot teorii mnozhestv Tsermelo–Frenkelya”, Vestn. MGU. Matematika. Mekhanika, 1971, no. 2, 78–82

[18] Lyubetskii V. A., Izmerimost i nalichie sovershennogo yadra u proektivnykh mnozhestv, Dis. $\dots$ kand. fiz.-mat. nauk, MGU, M., 1971

[19] Lyubetskii V. A., “Sluchainye posledovatelnosti chisel i $\mathrm{A}_2$-mnozhestva”, Issledovaniya po teorii mnozhestv i neklassicheskim logikam, Nauka, M., 1976, 96–122 | MR

[20] Martin D., Solovay R. M., “Internal Cohen extensions”, Ann. Math. Log., 2 (1970), 143–178 | DOI | MR | Zbl

[21] Mathias A. R. D., “Surrealist landscape with figures (a survey of recent results in set theory)”, Periodica Math. Hung., 10 (1979), 109–175 | DOI | MR | Zbl

[22] Novikov P. S., “O neprotivorechivosti nekotorykh polozhenii deskriptivnoi teorii mnozhestv”, Tr. MIAN, 38, 1951, 279–316 | MR

[23] Sacks G. E., “Forcing with perfect closed sets”, Proc. Symp. Pure Math., 13:1 (1971), 331–355 | MR | Zbl

[24] Shoenfield J., “The problem of predicativity”, Essays on the foundations of mathematics, Magnes Press, Hebrew Univ., Jerusalem, 1961, 132–139 | MR

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

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

[27] Solovay R. M., Tennenbaum S., “Iterated Cohen extensions and Suslin's problem”, Ann. Math., 94 (1971), 201–245 | DOI | MR | Zbl

[28] Uspenskii V. A., “Vklad N. N. Luzina v deskriptivnuyu teoriyu mnozhestv i funktsii: ponyatiya, problemy, predskazaniya”, UMN, 40:3 (1985), 85–116 | MR

[29] Velickovic B., Woodin W. H., “Complexity of the reals of inner models of set theory”, Ann. Pure and Appl. Logic., 92:3 (1998), 283–295 | DOI | MR | Zbl