Geodesic
Formula-inaccessible cardinals and a characterization of all natural models of Zermelo–Fraenkel set theory
Izvestiya. Mathematics, Tome 71 (2007) no. 2, pp. 219-245 
E. I. Bunina; V. K. Zakharov. Formula-inaccessible cardinals and a characterization of all natural models of Zermelo–Fraenkel set theory. Izvestiya. Mathematics, Tome 71 (2007) no. 2, pp. 219-245. http://geodesic.mathdoc.fr/item/IM2_2007_71_2_a0/
@article{IM2_2007_71_2_a0,
     author = {E. I. Bunina and V. K. Zakharov},
     title = {Formula-inaccessible cardinals and a characterization of all natural models {of~Zermelo{\textendash}Fraenkel} set theory},
     journal = {Izvestiya. Mathematics},
     pages = {219--245},
     year = {2007},
     volume = {71},
     number = {2},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_2007_71_2_a0/}
}
TY  - JOUR
AU  - E. I. Bunina
AU  - V. K. Zakharov
TI  - Formula-inaccessible cardinals and a characterization of all natural models of Zermelo–Fraenkel set theory
JO  - Izvestiya. Mathematics
PY  - 2007
SP  - 219
EP  - 245
VL  - 71
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/IM2_2007_71_2_a0/
LA  - en
ID  - IM2_2007_71_2_a0
ER  - 
%0 Journal Article
%A E. I. Bunina
%A V. K. Zakharov
%T Formula-inaccessible cardinals and a characterization of all natural models of Zermelo–Fraenkel set theory
%J Izvestiya. Mathematics
%D 2007
%P 219-245
%V 71
%N 2
%U http://geodesic.mathdoc.fr/item/IM2_2007_71_2_a0/
%G en
%F IM2_2007_71_2_a0

Voir la notice de l'article provenant de la source Math-Net.Ru

E. Zermelo (1930) and J. C. Sheperdson (1952) proved that a cumulative set $V_\alpha$ is a standard model of von Neumann–Bernays–Gödel set theory if and only if $\alpha=\varkappa+1$ for some inaccessible cardinal number $\varkappa$. The problem of a canonical form for all natural models of ZF theory turned out to be more complicated. Since the notion of a model of ZF theory cannot be defined by a finite set of formulae, we introduce a new notion of (strongly) formula-inaccessible cardinal number $\theta$ using a schema of formulae and its relativization on the set $V_\theta$, and prove a formula-analogue of the Zermelo–Sheperdson theorem giving a canonical form for all natural models of ZF theory.

[1] A. Fraenkel, “Zu den Grundlagen der Cantor–Zermeloschen Mengenlehre”, Math. Ann., 86:3–4 (1922), 230–237 | DOI | MR | Zbl

[2] K. Gedel, “Sovmestimye aksiomy vybora i obobschennoi kontinuum-gipotezy s aksiomami teorii mnozhestv”, UMN, 3:1 (1948), 96–149 ; K. Gödel, The consistency of the continuum hypothesis, Ann. of Math. Stud., 3, Princeton Univ. Press, Princeton, NJ, 1940 | MR | MR | Zbl

[3] D. Mirimanoff, “Les antinomies de Russel et de Burali–Forti et le probléme fondamental de la théorie des ensembles”, Enseign. Math., 19 (1917), 37–52 | Zbl

[4] J. von Neumann, “Über eine Widerspruchfreiheitsfrage in der axiomatischen Mengenlehre”, J. Reine Angew. Math., 160 (1929), 227–241 | Zbl

[5] A. Tarski, “Notions of proper models for set theories”, Bull. Amer. Math. Soc., 62 (1956), 601

[6] R. Montague, L. R. Vaught, “Natural models of set theories”, Fund. Math., 47 (1959), 219–242 | MR | Zbl

[7] E. Zermelo, “Über Grenzzahlen und Mengenbereiche. Neue Untersuchungen uber die Grundlagen der Mengenlehre”, Fund. Math., 16 (1930), 29–47 | Zbl

[8] W. Sierpinski, A. Tarski, “Sur une propriété caractéristique des nombres inaccessibles”, Fund. Math., 15 (1930), 292–300 | Zbl

[9] J. C. Shepherdson, “Inner models for set theory”, part I, J. Symbolic Logic, 16:3 (1951), 161–190 ; part II, 17:4 (1952), 225–237 ; part III, 18:2 (1953), 145–167 | DOI | MR | Zbl | DOI | MR | Zbl | DOI | MR | Zbl

[10] A. Kanamori, The higher infinite: Large cardinals in set theory from their beginnings, Perspectives in Mathematical Logic, Springer-Verlag, Berlin, 1994 | MR | Zbl

[11] C. Ehresmann, “Guttungen von lokalen Strukturen”, Jahresber. Deutsch. Math.-Verein. Verein., 60 (1957), 49–77 | MR | Zbl

[12] P. Dedecker, “Introduction aux structures locales”, Centre Belge Rech. Math., Colloque Géom. Différ. Globale (Bruxelles, 1958), 1959, 103–135 | MR | Zbl

[13] J. Sonner, “On the formal definition of categories”, Math. Z., 80:1 (1962), 163–176 | DOI | MR | Zbl

[14] P. Gabriel, “Des catégories abéliennes”, Bull. Soc. Math. France, 90 (1962), 323–448 | MR | Zbl

[15] S. Maklein, Kategorii dlya rabotayuschego matematika, Fizmatlit, M., 2004; S. MacLane, Categories for the working mathematician, Grad. Texts in Math., 5, Springer-Verlag, New York, 1971 | MR | Zbl

[16] T. E. Forster, Set theory with a universal set. Exploring an untyped universe, Oxford Logic Guides, 31, The Clarendon Press, Oxford Univ. Press, New York, 1995 | MR

[17] M. R. Holmes, Elementary set theory with a universal set, Cahiers du Centre de Logique, 10, Université Catholique de Louvain, Département de Philosophie, Louvain-la-Neuve, 1998 | MR | Zbl

[18] A. Tarski, “Über unerreichbare Kardinalzahlen”, Fund. Math., 30 (1938), 68–89 | Zbl

[19] K. Kuratovskii, A. Mostovskii, Teoriya mnozhestv, Mir, M., 1970 ; K. Kuratowski, A. Mostowski, Set theory, PWN – Polish Sci. Publ., Warszawa, 1968 | MR | Zbl | MR | Zbl

[20] E. I. Bunina, V. K. Zakharov, “Universal sets, Tarski sets, and inaccessible von Neumann sets”, Abstracts of the International Meeting “Second St. Petersburg Days of Logic and Computability”, St. Petersburg, 2003, 19–21

[21] E. I. Bunina, V. K. Zakharov, “Kanonicheskii vid supertranzitivnykh standartnykh modelei v teorii mnozhestv Tsermelo–Frenkelya”, UMN, 58:4 (2003), 143–144 | MR | Zbl

[22] T. Iekh, Teoriya mnozhestv i metod forsinga, Mir, M., 1973 ; T. J. Jech, Lectures in set theory with particular emphasis on the method of forcing, Lecture Notes in Math., 217, Springer-Verlag, Berlin, 1971 | MR | Zbl | MR | Zbl

[23] T. Jech, Set theory, The third millennium edition, revised and expanded, Springer Monogr. Math., Springer-Verlag, Berlin, 2003 | MR | Zbl

[24] Dzh. Shenfild, Matematicheskaya logika, Matematicheskaya logika i osnovaniya matematiki, Nauka, M., 1975 ; J. R. Shönfield, Mathematical logic, Addison-Wesley, Reading, Mass., 1967 | MR | MR | Zbl

[25] P. Dzh. Koen, Teoriya mnozhestv i kontinuum-gipoteza, Mir, M., 1969 ; P. J. Cohen, Set theory and the continuum hypothesis, W. A. Benjamin, Inc., New York, 1966 | MR | Zbl | MR | Zbl

[26] A. N. Kolmogorov, A. G. Dragalin, Matematicheskaya logika, URSS, M., 2004