; ▁that ▁is : ▁while ▁both ▁the ▁set ▁of ▁all ▁natural ▁numbers ▁and ▁the ▁set ▁of ▁all ▁real ▁numbers ▁are ▁infinite ▁sets , ▁there ▁can ▁be ▁no ▁one - to - one ▁function ▁from ▁the ▁real ▁numbers ▁to ▁the ▁natural ▁numbers : ▁the ▁cardinal ity ▁of ▁the ▁set ▁of ▁all ▁real ▁numbers ▁( den oted ▁ ▁and ▁called ▁cardinal ity ▁of ▁the ▁continu um ) ▁is ▁strictly ▁greater ▁than ▁the ▁cardinal ity ▁of ▁the ▁set ▁of ▁all ▁natural ▁numbers ▁( den oted ▁ ▁' ale ph - n aught '). ▁The ▁statement ▁that ▁there ▁is ▁no ▁subset ▁of ▁the ▁re als ▁with ▁cardinal ity ▁strictly ▁greater ▁than ▁ ▁and ▁strictly ▁smaller ▁than ▁ ▁is ▁known ▁as ▁the ▁continu um ▁hypothesis ▁( CH ). ▁It ▁is ▁known ▁to ▁be ▁neither ▁prov able ▁nor ▁ref utable ▁using ▁the ▁ax i oms ▁of ▁Z erm elo – F ra enk el ▁set ▁theory ▁including ▁the ▁ax iom ▁of ▁choice ▁( Z FC ), ▁the ▁standard ▁foundation ▁of ▁modern ▁mathematics , ▁in ▁the ▁sense ▁that ▁some ▁models ▁of ▁Z FC ▁satisfy ▁CH , ▁while ▁others ▁viol ate ▁it . ▁ ▁History ▁▁ ▁Simple ▁fra ctions ▁were ▁used ▁by ▁the ▁Egypt ians ▁around ▁ 1 0 0 0   BC ; ▁the ▁V edic ▁" Sh ul ba ▁S ut ras " ▁(" The ▁rules ▁of ▁ch ords ") ▁in , ▁, ▁include ▁what ▁may ▁be ▁the ▁first ▁" use " ▁of ▁ir r ational ▁numbers . ▁The ▁concept ▁of ▁ir r ational ity ▁was ▁implicitly ▁accepted ▁by