▁proved ▁within ▁Z FC ▁itself ▁( unless ▁it ▁is ▁actually ▁incons istent ). ▁Thus , ▁to ▁the ▁extent ▁that ▁Z FC ▁is ▁identified ▁with ▁ordinary ▁mathematics , ▁the ▁consist ency ▁of ▁Z FC ▁cannot ▁be ▁demonstrated ▁in ▁ordinary ▁mathematics . ▁The ▁consist ency ▁of ▁Z FC ▁does ▁follow ▁from ▁the ▁existence ▁of ▁a ▁weak ly ▁in access ible ▁cardinal , ▁which ▁is ▁un prov able ▁in ▁Z FC ▁if ▁Z FC ▁is ▁consistent . ▁Nevertheless , ▁it ▁is ▁de emed ▁unlikely ▁that ▁Z FC ▁har b ors ▁an ▁uns us pected ▁contradiction ; ▁it ▁is ▁widely ▁believed ▁that ▁if ▁Z FC ▁were ▁incons istent , ▁that ▁fact ▁would ▁have ▁been ▁un cover ed ▁by ▁now . ▁This ▁much ▁is ▁certain ▁— ▁Z FC ▁is ▁imm une ▁to ▁the ▁classic ▁par ado xes ▁of ▁na ive ▁set ▁theory : ▁Russell ' s ▁par ado x , ▁the ▁Bur ali - F ort i ▁par ado x , ▁and ▁Cant or ' s ▁par ado x . ▁▁ ▁studied ▁a ▁sub theory ▁of ▁Z FC ▁consisting ▁of ▁the ▁ax i oms ▁of ▁ext ensional ity , ▁union , ▁powers et , ▁replacement , ▁and ▁choice . ▁Using ▁models , ▁they ▁proved ▁this ▁sub theory ▁consistent , ▁and ▁proved ▁that ▁each ▁of ▁the ▁ax i oms ▁of ▁ext ensional ity , ▁replacement , ▁and ▁power ▁set ▁is ▁independent ▁of ▁the ▁four ▁remaining ▁ax i oms ▁of ▁this ▁sub theory . ▁If ▁this ▁sub theory ▁is ▁augment ed ▁with ▁the ▁ax iom ▁of ▁infinity , ▁each ▁of