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

M3 - ???researchoutput.researchoutputtypes.contributiontojournal.article???

AN - SCOPUS:34347404757

SN - 1064-5632

VL - 71

SP - 219

EP - 245

JO - Izvestiya Mathematics

JF - Izvestiya Mathematics

IS - 2

ER -