▁existence ▁of ▁a ▁strongly ▁in access ible ▁cardinal . ▁ ▁A ▁variation ▁on ▁the ▁method ▁of ▁forcing ▁can ▁also ▁be ▁used ▁to ▁demonstrate ▁the ▁consist ency ▁and ▁un prov ability ▁of ▁the ▁ax iom ▁of ▁choice , ▁i . e ., ▁that ▁the ▁ax iom ▁of ▁choice ▁is ▁independent ▁of ▁Z F . ▁The ▁consist ency ▁of ▁choice ▁can ▁be ▁( rel atively ) ▁easily ▁verified ▁by ▁prov ing ▁that ▁the ▁inner ▁model ▁L ▁satisfies ▁choice . ▁( Th us ▁every ▁model ▁of ▁Z F ▁contains ▁a ▁sub model ▁of ▁Z FC , ▁so ▁that ▁Con ( Z F ) ▁implies ▁Con ( Z FC ). ) ▁Since ▁forcing ▁pres erves ▁choice , ▁we ▁cannot ▁directly ▁produce ▁a ▁model ▁contradict ing ▁choice ▁from ▁a ▁model ▁satisfying ▁choice . ▁However , ▁we ▁can ▁use ▁forcing ▁to ▁create ▁a ▁model ▁which ▁contains ▁a ▁suitable ▁sub model , ▁namely ▁one ▁satisfying ▁Z F ▁but ▁not ▁C . ▁ ▁Another ▁method ▁of ▁prov ing ▁independence ▁results , ▁one ▁ow ing ▁nothing ▁to ▁forcing , ▁is ▁based ▁on ▁Gö del ' s ▁second ▁in comple ten ess ▁theorem . ▁This ▁approach ▁emp lo ys ▁the ▁statement ▁whose ▁independence ▁is ▁being ▁exam ined , ▁to ▁prove ▁the ▁existence ▁of ▁a ▁set ▁model ▁of ▁Z FC , ▁in ▁which ▁case ▁Con ( Z FC ) ▁is ▁true . ▁Since ▁Z FC ▁satisfies ▁the ▁conditions ▁of ▁Gö del ' s ▁second ▁theorem , ▁the ▁consist ency ▁of ▁Z FC ▁is ▁un prov able ▁in ▁Z FC ▁( prov ided ▁that ▁Z