▁a ▁set ▁ ▁such ▁that ▁the ▁empty ▁set ▁ ▁is ▁a ▁member ▁of ▁ ▁and , ▁whenever ▁a ▁set ▁ ▁is ▁a ▁member ▁of ▁, ▁then ▁ ▁is ▁also ▁a ▁member ▁of ▁. ▁▁▁▁ ▁More ▁col lo qu ial ly , ▁there ▁exists ▁a ▁set ▁ ▁having ▁infinitely ▁many ▁members . ▁( It ▁must ▁be ▁established , ▁however , ▁that ▁these ▁members ▁are ▁all ▁different , ▁because ▁if ▁two ▁elements ▁are ▁the ▁same , ▁the ▁sequence ▁will ▁loop ▁around ▁in ▁a ▁finite ▁cycle ▁of ▁sets . ▁The ▁ax iom ▁of ▁regular ity ▁prevents ▁this ▁from ▁happening .) ▁The ▁minimal ▁set ▁ ▁satisfying ▁the ▁ax iom ▁of ▁infinity ▁is ▁the ▁von ▁Ne umann ▁ord inal ▁, ▁which ▁can ▁also ▁be ▁thought ▁of ▁as ▁the ▁set ▁of ▁natural ▁numbers ▁. ▁▁ 8 . ▁A xi om ▁of ▁power ▁set ▁▁ ▁By ▁definition ▁a ▁set ▁ ▁is ▁a ▁subset ▁of ▁a ▁set ▁ ▁if ▁and ▁only ▁if ▁every ▁element ▁of ▁ ▁is ▁also ▁an ▁element ▁of ▁: ▁▁▁▁ ▁The ▁A xi om ▁of ▁Power ▁Set ▁states ▁that ▁for ▁any ▁set ▁, ▁there ▁is ▁a ▁set ▁ ▁that ▁ ▁contains ▁every ▁subset ▁of ▁: ▁▁▁▁ ▁The ▁ax iom ▁schema ▁of ▁specification ▁is ▁then ▁used ▁to ▁define ▁the ▁power ▁set ▁ ▁as ▁the ▁subset ▁of ▁such ▁a ▁ ▁containing ▁the ▁subsets ▁of ▁ ▁exactly : ▁▁▁▁ ▁A xi oms ▁ 1 – 8 ▁define ▁Z F . ▁Altern ative ▁forms ▁of ▁these ▁ax i oms ▁are ▁often ▁encountered , ▁some ▁of ▁which ▁are ▁listed ▁in ▁. ▁Some ▁Z