▁more ▁real ▁numbers ▁than ▁natural ▁numbers , ▁even ▁though ▁both ▁sets ▁are ▁infinite . ▁In ▁fact , ▁the ▁cardinal ity ▁of ▁the ▁re als ▁equals ▁that ▁of ▁the ▁set ▁of ▁subsets ▁( i . e . ▁the ▁power ▁set ) ▁of ▁the ▁natural ▁numbers , ▁and ▁Cant or ' s ▁diagonal ▁argument ▁states ▁that ▁the ▁latter ▁set ' s ▁cardinal ity ▁is ▁strictly ▁greater ▁than ▁the ▁cardinal ity ▁of ▁N . ▁Since ▁the ▁set ▁of ▁algebraic ▁numbers ▁is ▁count able , ▁almost ▁all ▁real ▁numbers ▁are ▁trans c endent al . ▁The ▁non - ex istence ▁of ▁a ▁subset ▁of ▁the ▁re als ▁with ▁cardinal ity ▁strictly ▁between ▁that ▁of ▁the ▁integers ▁and ▁the ▁re als ▁is ▁known ▁as ▁the ▁continu um ▁hypothesis . ▁The ▁continu um ▁hypothesis ▁can ▁neither ▁be ▁proved ▁nor ▁be ▁dis pro ved ; ▁it ▁is ▁independent ▁from ▁the ▁ax i oms ▁of ▁set ▁theory . ▁ ▁As ▁a ▁topological ▁space , ▁the ▁real ▁numbers ▁are ▁separ able . ▁This ▁is ▁because ▁the ▁set ▁of ▁rational s , ▁which ▁is ▁count able , ▁is ▁dense ▁in ▁the ▁real ▁numbers . ▁The ▁ir r ational ▁numbers ▁are ▁also ▁dense ▁in ▁the ▁real ▁numbers , ▁however ▁they ▁are ▁un count able ▁and ▁have ▁the ▁same ▁cardinal ity ▁as ▁the ▁re als . ▁ ▁The ▁real ▁numbers ▁form ▁a ▁metric ▁space : ▁the ▁distance ▁between ▁x ▁and ▁y ▁is ▁defined ▁as ▁the ▁absolute ▁value ▁. ▁By ▁virtue ▁of ▁being ▁a ▁totally ▁ordered ▁set , ▁they ▁also ▁carry ▁an ▁order ▁topology ; ▁the