ized ▁such ▁that ▁the ▁unit ▁interval ▁[ 0 ; 1 ] ▁has ▁measure ▁ 1 . ▁There ▁exist ▁sets ▁of ▁real ▁numbers ▁that ▁are ▁not ▁Leb es gue ▁meas urable , ▁e . g . ▁V ital i ▁sets . ▁ ▁The ▁supre m um ▁ax iom ▁of ▁the ▁re als ▁refers ▁to ▁subsets ▁of ▁the ▁re als ▁and ▁is ▁therefore ▁a ▁second - order ▁logical ▁statement . ▁It ▁is ▁not ▁possible ▁to ▁character ize ▁the ▁re als ▁with ▁first - order ▁logic ▁alone : ▁the ▁Lö wen heim – Sk ole m ▁theorem ▁implies ▁that ▁there ▁exists ▁a ▁count able ▁dense ▁subset ▁of ▁the ▁real ▁numbers ▁satisfying ▁exactly ▁the ▁same ▁sentences ▁in ▁first - order ▁logic ▁as ▁the ▁real ▁numbers ▁themselves . ▁The ▁set ▁of ▁hyper real ▁numbers ▁satisfies ▁the ▁same ▁first ▁order ▁sentences ▁as ▁R . ▁Order ed ▁fields ▁that ▁satisfy ▁the ▁same ▁first - order ▁sentences ▁as ▁R ▁are ▁called ▁non standard ▁models ▁of ▁R . ▁This ▁is ▁what ▁makes ▁non standard ▁analysis ▁work ; ▁by ▁prov ing ▁a ▁first - order ▁statement ▁in ▁some ▁non standard ▁model ▁( which ▁may ▁be ▁easier ▁than ▁prov ing ▁it ▁in ▁R ), ▁we ▁know ▁that ▁the ▁same ▁statement ▁must ▁also ▁be ▁true ▁of ▁R . ▁ ▁The ▁field ▁R ▁of ▁real ▁numbers ▁is ▁an ▁extension ▁field ▁of ▁the ▁field ▁Q ▁of ▁rational ▁numbers , ▁and ▁R ▁can ▁therefore ▁be ▁seen ▁as ▁a ▁vector ▁space ▁over ▁Q . ▁Z erm elo – F ra enk el ▁set ▁theory ▁with ▁the ▁ax iom ▁of