▁generally , ▁a ▁Spe cker ▁sequence ▁is ▁called ▁a ▁recursive ▁counter example ▁to ▁the ▁least ▁upper ▁bound ▁principle , ▁i . e . ▁a ▁construction ▁that ▁shows ▁that ▁this ▁theorem ▁is ▁false ▁when ▁restricted ▁to ▁comput able ▁re als . ▁ ▁The ▁least ▁upper ▁bound ▁principle ▁has ▁also ▁been ▁analyz ed ▁in ▁the ▁program ▁of ▁reverse ▁mathematics , ▁where ▁the ▁exact ▁strength ▁of ▁this ▁principle ▁has ▁been ▁determined . ▁In ▁the ▁termin ology ▁of ▁that ▁program , ▁the ▁least ▁upper ▁bound ▁principle ▁is ▁equivalent ▁to ▁A CA 0 ▁over ▁R CA 0 . ▁ ▁In ▁fact , ▁the ▁proof ▁of ▁the ▁forward ▁imp lication , ▁i . e . ▁that ▁the ▁least ▁upper ▁bound ▁principle ▁implies ▁A CA 0 , ▁is ▁readily ▁obtained ▁from ▁the ▁text book ▁proof ▁( see ▁Sim pson , ▁ 1 9 9 9 ) ▁of ▁the ▁non - comput ability ▁of ▁the ▁supre m um ▁in ▁the ▁least ▁upper ▁bound ▁principle . ▁ ▁Const ruction ▁▁ ▁The ▁following ▁construction ▁is ▁described ▁by ▁K ush ner ▁( 1 9 8 4 ). ▁ ▁Let ▁A ▁be ▁any ▁recurs ively ▁enumer able ▁set ▁of ▁natural ▁numbers ▁that ▁is ▁not ▁decid able , ▁and ▁let ▁( ai ) ▁be ▁a ▁comput able ▁enumer ation ▁of ▁A ▁without ▁repet ition . ▁Define ▁a ▁sequence ▁( q n ) ▁of ▁rational ▁numbers ▁with ▁the ▁rule ▁ ▁It ▁is ▁clear ▁that ▁each ▁q n ▁is ▁non negative ▁and ▁rational , ▁and ▁that ▁the ▁sequence ▁q n ▁is ▁strictly ▁increasing . ▁Moreover , ▁because ▁ai