FC ▁cannot ▁be ▁proved ▁within ▁the ▁theory ▁itself , ▁as ▁shown ▁by ▁Gö del ' s ▁second ▁in comple ten ess ▁theorem . ▁ ▁Form ally , ▁Z FC ▁is ▁a ▁one - sorted ▁theory ▁in ▁first - order ▁logic . ▁The ▁signature ▁has ▁equality ▁and ▁a ▁single ▁primitive ▁binary ▁relation , ▁set ▁membership , ▁which ▁is ▁usually ▁denoted ▁. ▁The ▁formula ▁ ▁means ▁that ▁the ▁set ▁ ▁is ▁a ▁member ▁of ▁the ▁set ▁ ▁( which ▁is ▁also ▁read , ▁" ▁is ▁an ▁element ▁of ▁" ▁or ▁" ▁is ▁in ▁" ). ▁ ▁History ▁▁ ▁The ▁modern ▁study ▁of ▁set ▁theory ▁was ▁initi ated ▁by ▁Georg ▁Cant or ▁and ▁Richard ▁D ed ek ind ▁in ▁the ▁ 1 8 7 0 s . ▁However , ▁the ▁discovery ▁of ▁par ado xes ▁in ▁na ive ▁set ▁theory , ▁such ▁as ▁Russell ' s ▁par ado x , ▁led ▁to ▁the ▁desire ▁for ▁a ▁more ▁rig orous ▁form ▁of ▁set ▁theory ▁that ▁was ▁free ▁of ▁these ▁par ado xes . ▁ ▁In ▁ 1 9 0 8 , ▁Ernst ▁Z erm elo ▁proposed ▁the ▁first ▁ax iom atic ▁set ▁theory , ▁Z erm elo ▁set ▁theory . ▁However , ▁as ▁first ▁pointed ▁out ▁by ▁Abraham ▁Fra enk el ▁in ▁a ▁ 1 9 2 1 ▁letter ▁to ▁Z erm elo , ▁this ▁theory ▁was ▁in cap able ▁of ▁prov ing ▁the ▁existence ▁of ▁certain ▁sets ▁and ▁cardinal ▁numbers ▁whose ▁existence ▁was ▁taken ▁for ▁granted ▁by ▁most ▁set ▁the or ists ▁of ▁the ▁time ,