▁an ▁infinite ▁set ▁exists . ▁This ▁implies ▁that ▁a ▁set ▁exists ▁and ▁so , ▁once ▁again , ▁it ▁is ▁super fl uous ▁to ▁include ▁an ▁ax iom ▁as ser ting ▁as ▁much . ▁▁ 1 . ▁A xi om ▁of ▁ext ensional ity ▁▁ ▁Two ▁sets ▁are ▁equal ▁( are ▁the ▁same ▁set ) ▁if ▁they ▁have ▁the ▁same ▁elements . ▁▁▁▁ ▁The ▁con verse ▁of ▁this ▁ax iom ▁follows ▁from ▁the ▁substitution ▁property ▁of ▁equality . ▁If ▁the ▁background ▁logic ▁does ▁not ▁include ▁equality ▁"", ▁ ▁may ▁be ▁defined ▁as ▁an ▁abbre viation ▁for ▁the ▁following ▁formula : ▁▁▁▁ ▁In ▁this ▁case , ▁the ▁ax iom ▁of ▁ext ensional ity ▁can ▁be ▁reform ulated ▁as ▁▁▁▁ ▁which ▁says ▁that ▁if ▁▁ ▁and ▁ ▁have ▁the ▁same ▁elements , ▁then ▁they ▁belong ▁to ▁the ▁same ▁sets . ▁▁ 2 . ▁A xi om ▁of ▁regular ity ▁( also ▁called ▁the ▁ax iom ▁of ▁foundation ) ▁▁ ▁Every ▁non - empty ▁set ▁ ▁contains ▁a ▁member ▁ ▁such ▁that ▁ ▁and ▁ ▁are ▁dis joint ▁sets . ▁▁▁ ▁or ▁in ▁modern ▁notation : ▁▁▁ ▁This ▁( al ong ▁with ▁the ▁A xi om ▁of ▁P air ing ) ▁implies , ▁for ▁example , ▁that ▁no ▁set ▁is ▁an ▁element ▁of ▁itself ▁and ▁that ▁every ▁set ▁has ▁an ▁ord inal ▁rank . ▁▁ 3 . ▁A xi om ▁schema ▁of ▁specification ▁( also ▁called ▁the ▁ax iom ▁schema ▁of ▁separation ▁or ▁of ▁restricted ▁compreh ension ) ▁▁ ▁Sub sets ▁are ▁commonly ▁constructed ▁using ▁set ▁builder