▁Russell ' s ▁par ado x . ▁Von ▁Ne umann – B ern ays – G ö del ▁set ▁theory ▁( NB G ) ▁is ▁a ▁commonly ▁used ▁conserv ative ▁extension ▁of ▁Z erm elo – F ra enk el ▁set ▁theory ▁that ▁does ▁allow ▁explicit ▁treatment ▁of ▁proper ▁classes . ▁ ▁There ▁are ▁many ▁equivalent ▁form ulations ▁of ▁the ▁ax i oms ▁of ▁Z erm elo – F ra enk el ▁set ▁theory . ▁ ▁Most ▁of ▁the ▁ax i oms ▁state ▁the ▁existence ▁of ▁particular ▁sets ▁defined ▁from ▁other ▁sets . ▁For ▁example , ▁the ▁ax iom ▁of ▁pair ing ▁says ▁that ▁given ▁any ▁two ▁sets ▁ ▁and ▁ ▁there ▁is ▁a ▁new ▁set ▁ ▁containing ▁exactly ▁ ▁and ▁. ▁Other ▁ax i oms ▁describe ▁properties ▁of ▁set ▁membership . ▁A ▁goal ▁of ▁the ▁ax i oms ▁is ▁that ▁each ▁ax iom ▁should ▁be ▁true ▁if ▁interpreted ▁as ▁a ▁statement ▁about ▁the ▁collection ▁of ▁all ▁sets ▁in ▁the ▁von ▁Ne umann ▁universe ▁( also ▁known ▁as ▁the ▁cum ulative ▁hierarchy ). ▁ ▁The ▁met am ath emat ics ▁of ▁Z erm elo – F ra enk el ▁set ▁theory ▁has ▁been ▁extens ively ▁studied . ▁Land mark ▁results ▁in ▁this ▁area ▁established ▁the ▁logical ▁independence ▁of ▁the ▁ax iom ▁of ▁choice ▁from ▁the ▁remaining ▁Z FC ▁ax i oms ▁( see ▁A xi om ▁of ▁choice # Ind ep end ence ) ▁and ▁of ▁the ▁continu um ▁hypothesis ▁from ▁Z FC . ▁The ▁consist ency ▁of ▁a ▁theory ▁such ▁as ▁Z