▁cardinal ▁number ▁after ▁, ▁the ▁cardinal ity ▁of ▁the ▁integers . ▁Paul ▁C ohen ▁proved ▁in ▁ 1 9 6 3 ▁that ▁it ▁is ▁an ▁ax iom ▁independent ▁of ▁the ▁other ▁ax i oms ▁of ▁set ▁theory ; ▁that ▁is : ▁one ▁may ▁choose ▁either ▁the ▁continu um ▁hypothesis ▁or ▁its ▁neg ation ▁as ▁an ▁ax iom ▁of ▁set ▁theory , ▁without ▁contradiction . ▁ ▁In ▁physics ▁▁ ▁In ▁the ▁physical ▁sciences , ▁most ▁physical ▁constants ▁such ▁as ▁the ▁universal ▁gravit ational ▁constant , ▁and ▁physical ▁variables , ▁such ▁as ▁position , ▁mass , ▁speed , ▁and ▁electric ▁charge , ▁are ▁mode led ▁using ▁real ▁numbers . ▁In ▁fact , ▁the ▁fundamental ▁physical ▁theories ▁such ▁as ▁classical ▁mechan ics , ▁elect romagnet ism , ▁quantum ▁mechan ics , ▁general ▁relativ ity ▁and ▁the ▁standard ▁model ▁are ▁described ▁using ▁mathematical ▁structures , ▁typically ▁smooth ▁manif olds ▁or ▁Hil bert ▁spaces , ▁that ▁are ▁based ▁on ▁the ▁real ▁numbers , ▁although ▁actual ▁measurements ▁of ▁physical ▁quantities ▁are ▁of ▁finite ▁accuracy ▁and ▁precision . ▁ ▁Phys ic ists ▁have ▁occasionally ▁suggested ▁that ▁a ▁more ▁fundamental ▁theory ▁would ▁replace ▁the ▁real ▁numbers ▁with ▁quantities ▁that ▁do ▁not ▁form ▁a ▁continu um , ▁but ▁such ▁propos als ▁remain ▁spec ulative . ▁ ▁In ▁computation ▁▁ ▁With ▁some ▁exceptions , ▁most ▁calcul ators ▁do ▁not ▁operate ▁on ▁real ▁numbers . ▁Instead , ▁they ▁work ▁with ▁finite - prec ision ▁approxim ations ▁called ▁floating - point ▁numbers . ▁In ▁fact , ▁most ▁scientific ▁computation ▁uses ▁floating - point ▁arithmetic .