ically ▁random ▁or ▁not . ▁ ▁App lications ▁and ▁connections ▁to ▁other ▁areas ▁ ▁Real ▁numbers ▁and ▁logic ▁▁ ▁The ▁real ▁numbers ▁are ▁most ▁often ▁formal ized ▁using ▁the ▁Z erm elo – F ra enk el ▁ax iom at ization ▁of ▁set ▁theory , ▁but ▁some ▁mathemat icians ▁study ▁the ▁real ▁numbers ▁with ▁other ▁logical ▁found ations ▁of ▁mathematics . ▁In ▁particular , ▁the ▁real ▁numbers ▁are ▁also ▁studied ▁in ▁reverse ▁mathematics ▁and ▁in ▁construct ive ▁mathematics . ▁ ▁The ▁hyper real ▁numbers ▁as ▁developed ▁by ▁Ed win ▁H ew itt , ▁Abraham ▁Robinson ▁and ▁others ▁extend ▁the ▁set ▁of ▁the ▁real ▁numbers ▁by ▁introdu cing ▁inf init es imal ▁and ▁infinite ▁numbers , ▁allowing ▁for ▁building ▁inf init es imal ▁calculus ▁in ▁a ▁way ▁closer ▁to ▁the ▁original ▁intu itions ▁of ▁Le ib n iz , ▁E uler , ▁C auch y ▁and ▁others . ▁ ▁Edward ▁Nelson ' s ▁internal ▁set ▁theory ▁en rich es ▁the ▁Z erm elo – F ra enk el ▁set ▁theory ▁s ynt act ically ▁by ▁introdu cing ▁a ▁un ary ▁predicate ▁" standard ". ▁In ▁this ▁approach , ▁inf init es im als ▁are ▁( non - " standard ") ▁elements ▁of ▁the ▁set ▁of ▁the ▁real ▁numbers ▁( r ather ▁than ▁being ▁elements ▁of ▁an ▁extension ▁there of , ▁as ▁in ▁Robinson ' s ▁theory ). ▁ ▁The ▁continu um ▁hypothesis ▁pos its ▁that ▁the ▁cardinal ity ▁of ▁the ▁set ▁of ▁the ▁real ▁numbers ▁is ▁; ▁i . e . ▁the ▁smallest ▁infinite