▁used ▁in ▁, ▁, ▁and ▁in ▁the ▁Met am ath ▁implementation ▁of ▁Z FC . ▁ ▁Von ▁Ne umann – B ern ays – G ö del ▁set ▁theory ▁The ▁ax iom ▁sch em ata ▁of ▁replacement ▁and ▁separation ▁each ▁contain ▁infinitely ▁many ▁instances . ▁ ▁included ▁a ▁result ▁first ▁proved ▁in ▁his ▁ 1 9 5 7 ▁Ph . D . ▁th esis : ▁if ▁Z FC ▁is ▁consistent , ▁it ▁is ▁impossible ▁to ▁ax iom at ize ▁Z FC ▁using ▁only ▁finit ely ▁many ▁ax i oms . ▁On ▁the ▁other ▁hand , ▁von ▁Ne umann – B ern ays – G ö del ▁set ▁theory ▁( NB G ) ▁can ▁be ▁finit ely ▁ax iom at ized . ▁The ▁ont ology ▁of ▁N B G ▁includes ▁proper ▁classes ▁as ▁well ▁as ▁sets ; ▁a ▁set ▁is ▁any ▁class ▁that ▁can ▁be ▁a ▁member ▁of ▁another ▁class . ▁N B G ▁and ▁Z FC ▁are ▁equivalent ▁set ▁theories ▁in ▁the ▁sense ▁that ▁any ▁theorem ▁not ▁mention ing ▁classes ▁and ▁prov able ▁in ▁one ▁theory ▁can ▁be ▁proved ▁in ▁the ▁other . ▁ ▁Cons ist ency ▁Gö del ' s ▁second ▁in comple ten ess ▁theorem ▁says ▁that ▁a ▁recurs ively ▁ax iom at izable ▁system ▁that ▁can ▁interpret ▁Robinson ▁arithmetic ▁can ▁prove ▁its ▁own ▁consist ency ▁only ▁if ▁it ▁is ▁incons istent . ▁Moreover , ▁Robinson ▁arithmetic ▁can ▁be ▁interpreted ▁in ▁general ▁set ▁theory , ▁a ▁small ▁fragment ▁of ▁Z FC . ▁Hence ▁the ▁consist ency ▁of ▁Z FC ▁cannot ▁be