F ▁ax iom at izations ▁include ▁an ▁ax iom ▁as ser ting ▁that ▁the ▁empty ▁set ▁exists . ▁The ▁ax i oms ▁of ▁pair ing , ▁union , ▁replacement , ▁and ▁power ▁set ▁are ▁often ▁stated ▁so ▁that ▁the ▁members ▁of ▁the ▁set ▁ ▁whose ▁existence ▁is ▁being ▁assert ed ▁are ▁just ▁those ▁sets ▁which ▁the ▁ax iom ▁as ser ts ▁ ▁must ▁contain . ▁ ▁The ▁following ▁ax iom ▁is ▁added ▁to ▁turn ▁Z F ▁into ▁Z FC : ▁▁ 9 . ▁Well - order ing ▁theorem ▁▁ ▁For ▁any ▁set ▁, ▁there ▁is ▁a ▁binary ▁relation ▁ ▁which ▁well - orders ▁. ▁ ▁This ▁means ▁ ▁is ▁a ▁linear ▁order ▁on ▁ ▁such ▁that ▁every ▁non empty ▁subset ▁of ▁ ▁has ▁a ▁member ▁which ▁is ▁minimal ▁under ▁. ▁▁▁▁ ▁Given ▁ax i oms ▁ 1 – 8 , ▁there ▁are ▁many ▁statements ▁ ▁equivalent ▁to ▁ax iom ▁ 9 , ▁the ▁best ▁known ▁of ▁which ▁is ▁the ▁ax iom ▁of ▁choice ▁( AC ), ▁which ▁goes ▁as ▁follows . ▁Let ▁ ▁be ▁a ▁set ▁whose ▁members ▁are ▁all ▁non - empty . ▁Then ▁there ▁exists ▁a ▁function ▁ ▁from ▁ ▁to ▁the ▁union ▁of ▁the ▁members ▁of ▁, ▁called ▁a ▁" choice ▁function ", ▁such ▁that ▁for ▁all ▁ ▁one ▁has ▁. ▁Since ▁the ▁existence ▁of ▁a ▁choice ▁function ▁when ▁ ▁is ▁a ▁finite ▁set ▁is ▁easily ▁proved ▁from ▁ax i oms ▁ 1 – 8 , ▁AC ▁only ▁matters ▁for ▁certain ▁infinite ▁sets . ▁AC ▁is ▁character ized ▁as ▁non