▁is ▁only ▁intended ▁to ▁aid ▁the ▁intu ition . ▁ ▁All ▁form ulations ▁of ▁Z FC ▁imply ▁that ▁at ▁least ▁one ▁set ▁exists . ▁Kun en ▁includes ▁an ▁ax iom ▁that ▁directly ▁as ser ts ▁the ▁existence ▁of ▁a ▁set , ▁in ▁addition ▁to ▁the ▁ax i oms ▁given ▁below ▁( although ▁he ▁notes ▁that ▁he ▁does ▁so ▁only ▁“ for ▁emphas is ” ). ▁Its ▁om ission ▁here ▁can ▁be ▁just ified ▁in ▁two ▁ways . ▁First , ▁in ▁the ▁standard ▁semantics ▁of ▁first - order ▁logic ▁in ▁which ▁Z FC ▁is ▁typically ▁formal ized , ▁the ▁domain ▁of ▁disc ourse ▁must ▁be ▁non empty . ▁Hence , ▁it ▁is ▁a ▁logical ▁theorem ▁of ▁first - order ▁logic ▁that ▁something ▁exists ▁— ▁usually ▁expressed ▁as ▁the ▁assertion ▁that ▁something ▁is ▁identical ▁to ▁itself , ▁. ▁Con sequently , ▁it ▁is ▁a ▁theorem ▁of ▁every ▁first - order ▁theory ▁that ▁something ▁exists . ▁However , ▁as ▁noted ▁above , ▁because ▁in ▁the ▁intended ▁semantics ▁of ▁Z FC ▁there ▁are ▁only ▁sets , ▁the ▁interpretation ▁of ▁this ▁logical ▁theorem ▁in ▁the ▁context ▁of ▁Z FC ▁is ▁that ▁some ▁set ▁exists . ▁Hence , ▁there ▁is ▁no ▁need ▁for ▁a ▁separate ▁ax iom ▁as ser ting ▁that ▁a ▁set ▁exists . ▁Second , ▁however , ▁even ▁if ▁Z FC ▁is ▁form ulated ▁in ▁so - called ▁free ▁logic , ▁in ▁which ▁it ▁is ▁not ▁prov able ▁from ▁logic ▁alone ▁that ▁something ▁exists , ▁the ▁ax iom ▁of ▁infinity ▁( below ) ▁as ser ts ▁that