▁trans c endent al ▁numbers ; ▁Georg ▁Cant or ▁( 1 8 7 3 ) ▁extended ▁and ▁greatly ▁simplified ▁this ▁proof . ▁Charles ▁Herm ite ▁( 1 8 7 3 ) ▁first ▁proved ▁that ▁e ▁is ▁trans c endent al , ▁and ▁Ferdinand ▁von ▁Lin dem ann ▁( 1 8 8 2 ), ▁showed ▁that ▁ ▁is ▁trans c endent al . ▁Lin dem ann ' s ▁proof ▁was ▁much ▁simplified ▁by ▁We ier str ass ▁( 1 8 8 5 ), ▁still ▁further ▁by ▁David ▁Hil bert ▁( 1 8 9 3 ), ▁and ▁has ▁finally ▁been ▁made ▁elementary ▁by ▁Adolf ▁Hur witz ▁and ▁Paul ▁G ord an . ▁ ▁The ▁development ▁of ▁calculus ▁in ▁the ▁ 1 8 th ▁century ▁used ▁the ▁entire ▁set ▁of ▁real ▁numbers ▁without ▁having ▁defined ▁them ▁clean ly . ▁The ▁first ▁rig orous ▁definition ▁was ▁published ▁by ▁Georg ▁Cant or ▁in ▁ 1 8 7 1 . ▁In ▁ 1 8 7 4 , ▁he ▁showed ▁that ▁the ▁set ▁of ▁all ▁real ▁numbers ▁is ▁un count ably ▁infinite ▁but ▁the ▁set ▁of ▁all ▁algebraic ▁numbers ▁is ▁count ably ▁infinite . ▁Con tr ary ▁to ▁widely ▁held ▁belief s , ▁his ▁first ▁method ▁was ▁not ▁his ▁famous ▁diagonal ▁argument , ▁which ▁he ▁published ▁in ▁ 1 8 9 1 . ▁See ▁Cant or ' s ▁first ▁un count ability ▁proof . ▁ ▁Definition ▁▁ ▁The ▁real ▁number ▁system ▁ ▁can ▁be ▁defined ▁ax iom atically ▁up ▁to ▁an ▁isomorphism , ▁which ▁is ▁described ▁here after . ▁There