construct ive ▁because ▁it ▁as ser ts ▁the ▁existence ▁of ▁a ▁choice ▁set ▁but ▁says ▁nothing ▁about ▁how ▁the ▁choice ▁set ▁is ▁to ▁be ▁" construct ed ." ▁ ▁Much ▁research ▁has ▁sought ▁to ▁character ize ▁the ▁defin ability ▁( or ▁lack ▁there of ) ▁of ▁certain ▁sets ▁whose ▁existence ▁AC ▁as ser ts . ▁ ▁Mot iv ation ▁via ▁the ▁cum ulative ▁hierarchy ▁▁ ▁One ▁motiv ation ▁for ▁the ▁Z FC ▁ax i oms ▁is ▁the ▁cum ulative ▁hierarchy ▁of ▁sets ▁introduced ▁by ▁John ▁von ▁Ne umann . ▁In ▁this ▁view point , ▁the ▁universe ▁of ▁set ▁theory ▁is ▁built ▁up ▁in ▁stages , ▁with ▁one ▁stage ▁for ▁each ▁ord inal ▁number . ▁At ▁stage ▁ 0 ▁there ▁are ▁no ▁sets ▁yet . ▁At ▁each ▁following ▁stage , ▁a ▁set ▁is ▁added ▁to ▁the ▁universe ▁if ▁all ▁of ▁its ▁elements ▁have ▁been ▁added ▁at ▁previous ▁stages . ▁Thus ▁the ▁empty ▁set ▁is ▁added ▁at ▁stage ▁ 1 , ▁and ▁the ▁set ▁containing ▁the ▁empty ▁set ▁is ▁added ▁at ▁stage ▁ 2 . ▁The ▁collection ▁of ▁all ▁sets ▁that ▁are ▁obtained ▁in ▁this ▁way , ▁over ▁all ▁the ▁stages , ▁is ▁known ▁as ▁V . ▁The ▁sets ▁in ▁V ▁can ▁be ▁arranged ▁into ▁a ▁hierarchy ▁by ▁assigning ▁to ▁each ▁set ▁the ▁first ▁stage ▁at ▁which ▁that ▁set ▁was ▁added ▁to ▁V . ▁ ▁It ▁is ▁prov able ▁that ▁a ▁set ▁is ▁in ▁V ▁if ▁and ▁only ▁if ▁the ▁set ▁is ▁pure ▁and ▁well - f ounded ; ▁and ▁prov able ▁that ▁V ▁satisfies