TY - JOUR
T1 - Formula-inaccessible cardinals and a characterization of all natural models of Zermelo-Fraenkel set theory
AU - Bunina, E. I.
AU - Zakharov, Valeriy K.
PY - 2007/4
Y1 - 2007/4
N2 - E. Zermelo (1930) and J. C. Sheperdson (1952) proved that a cumulative set Vα is a standard model of von Neumann-Bernays-Gödel set theory if and only if α = א + 1 for some inaccessible cardinal number א. 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 θ using a schema of formulae and its relativization on the set Vθ, and prove a formula-analogue of the Zermelo-Sheperdson theorem giving a canonical form for all natural models of ZF theory.
AB - E. Zermelo (1930) and J. C. Sheperdson (1952) proved that a cumulative set Vα is a standard model of von Neumann-Bernays-Gödel set theory if and only if α = א + 1 for some inaccessible cardinal number א. 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 θ using a schema of formulae and its relativization on the set Vθ, and prove a formula-analogue of the Zermelo-Sheperdson theorem giving a canonical form for all natural models of ZF theory.
UR - http://www.scopus.com/inward/record.url?scp=34347404757&partnerID=8YFLogxK
U2 - 10.1070/im2007v071n02abeh002356
DO - 10.1070/im2007v071n02abeh002356
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AN - SCOPUS:34347404757
SN - 1064-5632
VL - 71
SP - 219
EP - 245
JO - Izvestiya Mathematics
JF - Izvestiya Mathematics
IS - 2
ER -