▁a ▁" core " ▁inner ▁model . ▁ ▁Crit ic isms ▁▁▁ ▁For ▁criticism ▁of ▁set ▁theory ▁in ▁general , ▁see ▁Ob jections ▁to ▁set ▁theory ▁ ▁Z FC ▁has ▁been ▁critic ized ▁both ▁for ▁being ▁excess ively ▁strong ▁and ▁for ▁being ▁excess ively ▁weak , ▁as ▁well ▁as ▁for ▁its ▁failure ▁to ▁capture ▁objects ▁such ▁as ▁proper ▁classes ▁and ▁the ▁universal ▁set . ▁ ▁Many ▁mathematical ▁the or ems ▁can ▁be ▁proven ▁in ▁much ▁we aker ▁systems ▁than ▁Z FC , ▁such ▁as ▁Pe ano ▁arithmetic ▁and ▁second - order ▁arithmetic ▁( as ▁expl ored ▁by ▁the ▁program ▁of ▁reverse ▁mathematics ). ▁Sau nd ers ▁Mac ▁Lane ▁and ▁Sol omon ▁Fe fer man ▁have ▁both ▁made ▁this ▁point . ▁Some ▁of ▁" main stream ▁mathematics " ▁( math emat ics ▁not ▁directly ▁connected ▁with ▁ax iom atic ▁set ▁theory ) ▁is ▁beyond ▁Pe ano ▁arithmetic ▁and ▁second - order ▁arithmetic , ▁but ▁still , ▁all ▁such ▁mathematics ▁can ▁be ▁carried ▁out ▁in ▁Z C ▁( Z erm elo ▁set ▁theory ▁with ▁choice ), ▁another ▁theory ▁we aker ▁than ▁Z FC . ▁Much ▁of ▁the ▁power ▁of ▁Z FC , ▁including ▁the ▁ax iom ▁of ▁regular ity ▁and ▁the ▁ax iom ▁schema ▁of ▁replacement , ▁is ▁included ▁primarily ▁to ▁facil itate ▁the ▁study ▁of ▁the ▁set ▁theory ▁itself . ▁ ▁On ▁the ▁other ▁hand , ▁among ▁ax iom atic ▁set ▁theories , ▁Z FC ▁is ▁compar atively ▁weak . ▁Un like ▁New ▁Found ations , ▁Z FC ▁does ▁not ▁admit ▁the ▁existence ▁of