▁the ▁ax i oms ▁of ▁union , ▁choice , ▁and ▁infinity ▁is ▁independent ▁of ▁the ▁five ▁remaining ▁ax i oms . ▁Because ▁there ▁are ▁non - well - f ounded ▁models ▁that ▁satisfy ▁each ▁ax iom ▁of ▁Z FC ▁except ▁the ▁ax iom ▁of ▁regular ity , ▁that ▁ax iom ▁is ▁independent ▁of ▁the ▁other ▁Z FC ▁ax i oms . ▁ ▁If ▁consistent , ▁Z FC ▁cannot ▁prove ▁the ▁existence ▁of ▁the ▁in access ible ▁card inals ▁that ▁category ▁theory ▁requires . ▁H uge ▁sets ▁of ▁this ▁nature ▁are ▁possible ▁if ▁Z F ▁is ▁augment ed ▁with ▁Tar ski ' s ▁ax iom . ▁Assuming ▁that ▁ax iom ▁turns ▁the ▁ax i oms ▁of ▁infinity , ▁power ▁set , ▁and ▁choice ▁( 7 ▁− ▁ 9 ▁above ) ▁into ▁the or ems . ▁ ▁Independ ence ▁▁ ▁Many ▁important ▁statements ▁are ▁independent ▁of ▁Z FC ▁( see ▁list ▁of ▁statements ▁und ec id able ▁in ▁Z FC ). ▁The ▁independence ▁is ▁usually ▁proved ▁by ▁forcing , ▁where by ▁it ▁is ▁shown ▁that ▁every ▁count able ▁trans itive ▁model ▁of ▁Z FC ▁( s ometimes ▁augment ed ▁with ▁large ▁cardinal ▁ax i oms ) ▁can ▁be ▁expanded ▁to ▁satisfy ▁the ▁statement ▁in ▁question . ▁A ▁different ▁expansion ▁is ▁then ▁shown ▁to ▁satisfy ▁the ▁neg ation ▁of ▁the ▁statement . ▁An ▁independence ▁proof ▁by ▁forcing ▁automatically ▁proves ▁independence ▁from ▁ar ithmet ical ▁statements , ▁other ▁concrete ▁statements , ▁and ▁large ▁cardinal ▁ax i oms . ▁Some ▁statements ▁independent ▁of ▁Z FC ▁can ▁be ▁proven