▁not ably ▁the ▁cardinal ▁number ▁ ▁and ▁the ▁set ▁, ▁where ▁ ▁is ▁any ▁infinite ▁set ▁and ▁ ▁is ▁the ▁power ▁set ▁operation . ▁Moreover , ▁one ▁of ▁Z erm elo ' s ▁ax i oms ▁invoked ▁a ▁concept , ▁that ▁of ▁a ▁" defin ite " ▁property , ▁whose ▁oper ational ▁meaning ▁was ▁not ▁clear . ▁In ▁ 1 9 2 2 , ▁Fra enk el ▁and ▁Thor alf ▁Sk ole m ▁independently ▁proposed ▁oper ational izing ▁a ▁" defin ite " ▁property ▁as ▁one ▁that ▁could ▁be ▁form ulated ▁as ▁a ▁well - formed ▁formula ▁in ▁a ▁first - order ▁logic ▁whose ▁atomic ▁formulas ▁were ▁limited ▁to ▁set ▁membership ▁and ▁identity . ▁They ▁also ▁independently ▁proposed ▁replacing ▁the ▁ax iom ▁schema ▁of ▁specification ▁with ▁the ▁ax iom ▁schema ▁of ▁replacement . ▁App ending ▁this ▁schema , ▁as ▁well ▁as ▁the ▁ax iom ▁of ▁regular ity ▁( first ▁proposed ▁by ▁John ▁von ▁Ne umann ), ▁to ▁Z erm elo ▁set ▁theory ▁yields ▁the ▁theory ▁denoted ▁by ▁Z F . ▁Adding ▁to ▁Z F ▁either ▁the ▁ax iom ▁of ▁choice ▁( AC ) ▁or ▁a ▁statement ▁that ▁is ▁equivalent ▁to ▁it ▁yields ▁Z FC . ▁ ▁A xi oms ▁▁ ▁There ▁are ▁many ▁equivalent ▁form ulations ▁of ▁the ▁Z FC ▁ax i oms ; ▁for ▁a ▁discussion ▁of ▁this ▁see ▁. ▁ ▁The ▁following ▁particular ▁ax iom ▁set ▁is ▁from ▁. ▁The ▁ax i oms ▁per ▁se ▁are ▁expressed ▁in ▁the ▁symbol ism ▁of ▁first ▁order ▁logic . ▁The ▁associated ▁English ▁pro se