▁choice ▁guarante es ▁the ▁existence ▁of ▁a ▁basis ▁of ▁this ▁vector ▁space : ▁there ▁exists ▁a ▁set ▁B ▁of ▁real ▁numbers ▁such ▁that ▁every ▁real ▁number ▁can ▁be ▁written ▁uniqu ely ▁as ▁a ▁finite ▁linear ▁combination ▁of ▁elements ▁of ▁this ▁set , ▁using ▁rational ▁coefficients ▁only , ▁and ▁such ▁that ▁no ▁element ▁of ▁B ▁is ▁a ▁rational ▁linear ▁combination ▁of ▁the ▁others . ▁However , ▁this ▁existence ▁theorem ▁is ▁purely ▁theoretical , ▁as ▁such ▁a ▁base ▁has ▁never ▁been ▁explicitly ▁described . ▁ ▁The ▁well - order ing ▁theorem ▁implies ▁that ▁the ▁real ▁numbers ▁can ▁be ▁well - ordered ▁if ▁the ▁ax iom ▁of ▁choice ▁is ▁assumed : ▁there ▁exists ▁a ▁total ▁order ▁on ▁R ▁with ▁the ▁property ▁that ▁every ▁non - empty ▁subset ▁of ▁R ▁has ▁a ▁least ▁element ▁in ▁this ▁ordering . ▁( The ▁standard ▁ordering ▁ ≤ ▁of ▁the ▁real ▁numbers ▁is ▁not ▁a ▁well - order ing ▁since ▁e . g . ▁an ▁open ▁interval ▁does ▁not ▁contain ▁a ▁least ▁element ▁in ▁this ▁ordering .) ▁Again , ▁the ▁existence ▁of ▁such ▁a ▁well - order ing ▁is ▁purely ▁theoretical , ▁as ▁it ▁has ▁not ▁been ▁explicitly ▁described . ▁If ▁V = L ▁is ▁assumed ▁in ▁addition ▁to ▁the ▁ax i oms ▁of ▁Z F , ▁a ▁well ▁ordering ▁of ▁the ▁real ▁numbers ▁can ▁be ▁shown ▁to ▁be ▁explicitly ▁defin able ▁by ▁a ▁formula . ▁ ▁A ▁real ▁number ▁may ▁be ▁either ▁comput able ▁or ▁un comput able ; ▁either ▁algorithm ically ▁random ▁or ▁not ; ▁and ▁either ▁ar ithmet