▁b q , ▁we ▁choose ▁b j ▁in ▁between ▁b p ▁and ▁b q ▁using ▁density . ▁Otherwise , ▁we ▁choose ▁a ▁suitable ▁large ▁or ▁small ▁element ▁of ▁B ▁using ▁the ▁fact ▁that ▁B ▁has ▁neither ▁a ▁maximum ▁nor ▁a ▁minimum . ▁Cho ices ▁made ▁in ▁step ▁( 2 ) ▁are ▁du ally ▁possible . ▁Finally , ▁the ▁construction ▁ends ▁after ▁count ably ▁many ▁steps ▁because ▁A ▁and ▁B ▁are ▁count ably ▁infinite . ▁Note ▁that ▁we ▁had ▁to ▁use ▁all ▁the ▁pr ere quis ites . ▁ ▁History ▁ ▁According ▁to ▁H od ges ▁( 1 9 9 3 ): ▁Back - and - for th ▁methods ▁are ▁often ▁as cribed ▁to ▁Cant or , ▁Ber tr and ▁Russell ▁and ▁C . ▁H . ▁Lang ford ▁[ … ], ▁but ▁there ▁is ▁no ▁evidence ▁to ▁support ▁any ▁of ▁these ▁att ribution s . ▁While ▁the ▁theorem ▁on ▁count able ▁dens ely ▁ordered ▁sets ▁is ▁due ▁to ▁Cant or ▁( 1 8 9 5 ), ▁the ▁back - and - for th ▁method ▁with ▁which ▁it ▁is ▁now ▁proved ▁was ▁developed ▁by ▁Hun ting ton ▁( 1 9 0 4 ) ▁and ▁Haus dor ff ▁( 1 9 1 4 ). ▁Later ▁it ▁was ▁applied ▁in ▁other ▁situations , ▁most ▁not ably ▁by ▁Roland ▁Fra ï ss é ▁in ▁model ▁theory . ▁ ▁See ▁also ▁▁ ▁Ehren fe ucht – F ra ï ss é ▁game ▁ ▁References ▁▁▁▁▁▁▁▁▁▁ ▁Category : Model ▁theory ▁Category : Math emat ical ▁proofs ▁Category : Art icles