ed ek ind - complete ▁( and ▁vice ▁vers a ), ▁just ifying ▁using ▁" the " ▁in ▁the ▁phrase ▁" the ▁complete ▁Arch im ede an ▁field ". ▁This ▁sense ▁of ▁comple ten ess ▁is ▁most ▁closely ▁related ▁to ▁the ▁construction ▁of ▁the ▁re als ▁from ▁C auch y ▁sequences ▁( the ▁construction ▁carried ▁out ▁in ▁full ▁in ▁this ▁article ), ▁since ▁it ▁starts ▁with ▁an ▁Arch im ede an ▁field ▁( the ▁rational s ) ▁and ▁forms ▁the ▁uniform ▁completion ▁of ▁it ▁in ▁a ▁standard ▁way . ▁ ▁But ▁the ▁original ▁use ▁of ▁the ▁phrase ▁" complete ▁Arch im ede an ▁field " ▁was ▁by ▁David ▁Hil bert , ▁who ▁meant ▁still ▁something ▁else ▁by ▁it . ▁He ▁meant ▁that ▁the ▁real ▁numbers ▁form ▁the ▁largest ▁Arch im ede an ▁field ▁in ▁the ▁sense ▁that ▁every ▁other ▁Arch im ede an ▁field ▁is ▁a ▁sub field ▁of ▁R . ▁Thus ▁R ▁is ▁" complete " ▁in ▁the ▁sense ▁that ▁nothing ▁further ▁can ▁be ▁added ▁to ▁it ▁without ▁making ▁it ▁no ▁longer ▁an ▁Arch im ede an ▁field . ▁This ▁sense ▁of ▁comple ten ess ▁is ▁most ▁closely ▁related ▁to ▁the ▁construction ▁of ▁the ▁re als ▁from ▁sur real ▁numbers , ▁since ▁that ▁construction ▁starts ▁with ▁a ▁proper ▁class ▁that ▁contains ▁every ▁ordered ▁field ▁( the ▁sur re als ) ▁and ▁then ▁selects ▁from ▁it ▁the ▁largest ▁Arch im ede an ▁sub field . ▁ ▁Advanced ▁properties ▁▁ ▁The ▁re als ▁are ▁un count able ; ▁that ▁is : ▁there ▁are ▁strictly