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

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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. 
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     title = {Formula-inaccessible cardinals and a characterization of all natural models {of~Zermelo--Fraenkel} set theory},
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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/