▁ax iom ▁of ▁choice ▁( AC ) ▁included , ▁is ▁the ▁standard ▁form ▁of ▁ax iom atic ▁set ▁theory ▁and ▁as ▁such ▁is ▁the ▁most ▁common ▁foundation ▁of ▁mathematics . ▁Z erm elo – F ra enk el ▁set ▁theory ▁with ▁the ▁ax iom ▁of ▁choice ▁included ▁is ▁abbre vi ated ▁Z FC , ▁where ▁C ▁stands ▁for ▁" choice ", ▁and ▁Z F ▁refers ▁to ▁the ▁ax i oms ▁of ▁Z erm elo – F ra enk el ▁set ▁theory ▁with ▁the ▁ax iom ▁of ▁choice ▁ex cluded . ▁▁ ▁Z erm elo – F ra enk el ▁set ▁theory ▁is ▁intended ▁to ▁formal ize ▁a ▁single ▁primitive ▁notion , ▁that ▁of ▁a ▁her edit ary ▁well - f ounded ▁set , ▁so ▁that ▁all ▁entities ▁in ▁the ▁universe ▁of ▁disc ourse ▁are ▁such ▁sets . ▁Thus ▁the ▁ax i oms ▁of ▁Z erm elo – F ra enk el ▁set ▁theory ▁refer ▁only ▁to ▁pure ▁sets ▁and ▁prevent ▁its ▁models ▁from ▁containing ▁u re lement s ▁( elements ▁of ▁sets ▁that ▁are ▁not ▁themselves ▁sets ). ▁Furthermore , ▁proper ▁classes ▁( collections ▁of ▁mathematical ▁objects ▁defined ▁by ▁a ▁property ▁shared ▁by ▁their ▁members ▁where ▁the ▁collections ▁are ▁too ▁big ▁to ▁be ▁sets ) ▁can ▁only ▁be ▁treated ▁indirect ly . ▁Specifically , ▁Z erm elo – F ra enk el ▁set ▁theory ▁does ▁not ▁allow ▁for ▁the ▁existence ▁of ▁a ▁universal ▁set ▁( a ▁set ▁containing ▁all ▁sets ) ▁nor ▁for ▁un rest rict ed ▁compreh ension , ▁thereby ▁avoid ing