▁the ▁word ▁" the " ▁in ▁the ▁phrase ▁" complete ▁ordered ▁field " ▁when ▁this ▁is ▁the ▁sense ▁of ▁" complete " ▁that ▁is ▁meant . ▁This ▁sense ▁of ▁comple ten ess ▁is ▁most ▁closely ▁related ▁to ▁the ▁construction ▁of ▁the ▁re als ▁from ▁D ed ek ind ▁cut s , ▁since ▁that ▁construction ▁starts ▁from ▁an ▁ordered ▁field ▁( the ▁rational s ) ▁and ▁then ▁forms ▁the ▁D ed ek ind - comple tion ▁of ▁it ▁in ▁a ▁standard ▁way . ▁ ▁These ▁two ▁not ions ▁of ▁comple ten ess ▁ignore ▁the ▁field ▁structure . ▁However , ▁an ▁ordered ▁group ▁( in ▁this ▁case , ▁the ▁add itive ▁group ▁of ▁the ▁field ) ▁defines ▁a ▁uniform ▁structure , ▁and ▁uniform ▁structures ▁have ▁a ▁notion ▁of ▁comple ten ess ▁( top ology ); ▁the ▁description ▁in ▁the ▁previous ▁section ▁Comple ten ess ▁is ▁a ▁special ▁case . ▁( We ▁refer ▁to ▁the ▁notion ▁of ▁comple ten ess ▁in ▁uniform ▁spaces ▁rather ▁than ▁the ▁related ▁and ▁better ▁known ▁notion ▁for ▁metric ▁spaces , ▁since ▁the ▁definition ▁of ▁metric ▁space ▁re lies ▁on ▁already ▁having ▁a ▁character ization ▁of ▁the ▁real ▁numbers .) ▁It ▁is ▁not ▁true ▁that ▁R ▁is ▁the ▁only ▁uniformly ▁complete ▁ordered ▁field , ▁but ▁it ▁is ▁the ▁only ▁uniformly ▁complete ▁Arch im ede an ▁field , ▁and ▁indeed ▁one ▁often ▁he ars ▁the ▁phrase ▁" complete ▁Arch im ede an ▁field " ▁instead ▁of ▁" complete ▁ordered ▁field ". ▁Every ▁uniformly ▁complete ▁Arch im ede an ▁field ▁must ▁also ▁be ▁D