▁model ▁existed ▁then ▁each ▁of ▁its ▁computers ▁could ▁be ▁sim ulated ▁by ▁a ▁T uring ▁machine . ▁ ▁Thus ▁if ▁this ▁new ▁model ▁of ▁computation ▁consisted ▁of ▁a ▁sequence ▁ ▁of ▁machines , ▁there ▁would ▁be ▁a ▁recurs ively ▁enumer able ▁sequence ▁ ▁of ▁T uring ▁machines ▁that ▁compute ▁total ▁functions ▁and ▁so ▁that ▁every ▁total ▁comput able ▁function ▁is ▁comput able ▁by ▁one ▁of ▁the ▁machines ▁Ti . ▁ ▁This ▁is ▁impossible , ▁because ▁a ▁machine ▁ ▁could ▁be ▁constructed ▁such ▁that ▁on ▁input ▁i ▁the ▁machine ▁T ▁returns ▁. ▁ ▁This ▁machine ▁cannot ▁be ▁equivalent ▁to ▁any ▁machine ▁T ▁on ▁the ▁list : ▁suppose ▁it ▁were ▁on ▁the ▁list ▁at ▁index ▁j . ▁ ▁Then ▁, ▁which ▁does ▁not ▁return ▁an ▁integer ▁result . ▁ ▁Therefore , ▁it ▁cannot ▁be ▁total , ▁but ▁the ▁function ▁by ▁construction ▁must ▁be ▁total ▁( if ▁total ▁functions ▁are ▁recurs ively ▁enumer able , ▁then ▁this ▁function ▁can ▁be ▁constructed ), ▁which ▁is ▁a ▁contradiction . ▁ ▁This ▁shows ▁that ▁the ▁second ▁question ▁has ▁a ▁negative ▁answer . ▁ ▁The ▁set ▁of ▁indices ▁of ▁total ▁T uring ▁machines ▁▁ ▁The ▁decision ▁problem ▁of ▁whether ▁the ▁T uring ▁machine ▁with ▁index ▁e ▁will ▁halt ▁on ▁every ▁input ▁is ▁not ▁decid able . ▁ ▁In ▁fact , ▁this ▁problem ▁is ▁at ▁level ▁ ▁of ▁the ▁ar ithmet ical ▁hierarchy . ▁ ▁Thus ▁this ▁problem ▁is ▁strictly ▁more ▁difficult ▁than ▁the ▁Hal ting ▁problem , ▁which ▁asks ▁whether ▁the ▁machine ▁with ▁index ▁e ▁hal ts ▁on ▁input ▁