▁all ▁the ▁ax i oms ▁of ▁Z FC , ▁if ▁the ▁class ▁of ▁ord inals ▁has ▁appropriate ▁reflection ▁properties . ▁For ▁example , ▁suppose ▁that ▁a ▁set ▁x ▁is ▁added ▁at ▁stage ▁α , ▁which ▁means ▁that ▁every ▁element ▁of ▁x ▁was ▁added ▁at ▁a ▁stage ▁earlier ▁than ▁α . ▁Then ▁every ▁subset ▁of ▁x ▁is ▁also ▁added ▁at ▁stage ▁α , ▁because ▁all ▁elements ▁of ▁any ▁subset ▁of ▁x ▁were ▁also ▁added ▁before ▁stage ▁α . ▁This ▁means ▁that ▁any ▁subset ▁of ▁x ▁which ▁the ▁ax iom ▁of ▁separation ▁can ▁construct ▁is ▁added ▁at ▁stage ▁α , ▁and ▁that ▁the ▁powers et ▁of ▁x ▁will ▁be ▁added ▁at ▁the ▁next ▁stage ▁after ▁α . ▁For ▁a ▁complete ▁argument ▁that ▁V ▁satisfies ▁Z FC ▁see ▁. ▁ ▁The ▁picture ▁of ▁the ▁universe ▁of ▁sets ▁strat ified ▁into ▁the ▁cum ulative ▁hierarchy ▁is ▁characteristic ▁of ▁Z FC ▁and ▁related ▁ax iom atic ▁set ▁theories ▁such ▁as ▁Von ▁Ne umann – B ern ays – G ö del ▁set ▁theory ▁( o ften ▁called ▁N B G ) ▁and ▁Mor se – K elle y ▁set ▁theory . ▁The ▁cum ulative ▁hierarchy ▁is ▁not ▁compatible ▁with ▁other ▁set ▁theories ▁such ▁as ▁New ▁Found ations . ▁ ▁It ▁is ▁possible ▁to ▁change ▁the ▁definition ▁of ▁V ▁so ▁that ▁at ▁each ▁stage , ▁instead ▁of ▁adding ▁all ▁the ▁subsets ▁of ▁the ▁union ▁of ▁the ▁previous ▁stages , ▁subsets ▁are ▁only ▁added ▁if ▁they ▁are ▁defin able ▁in ▁a ▁certain ▁sense . ▁This ▁results ▁in ▁a ▁more ▁"