0 . ▁▁ ▁Int uit ively , ▁this ▁difference ▁in ▁un sol v ability ▁is ▁because ▁each ▁instance ▁of ▁the ▁" total ▁machine " ▁problem ▁represents ▁infinitely ▁many ▁instances ▁of ▁the ▁Hal ting ▁problem . ▁ ▁See ▁also : ▁Term ination ▁analysis . ▁ ▁Prov ability ▁▁ ▁One ▁may ▁be ▁interested ▁not ▁only ▁in ▁whether ▁a ▁T uring ▁machine ▁is ▁total , ▁but ▁also ▁in ▁whether ▁this ▁can ▁be ▁proven ▁in ▁a ▁certain ▁logical ▁system , ▁such ▁as ▁first ▁order ▁Pe ano ▁arithmetic . ▁ ▁In ▁a ▁sound ▁proof ▁system , ▁every ▁prov ably ▁total ▁T uring ▁machine ▁is ▁indeed ▁total , ▁but ▁the ▁con verse ▁is ▁not ▁true : ▁inform ally , ▁for ▁every ▁first - order ▁proof ▁system ▁that ▁is ▁strong ▁enough ▁( including ▁Pe ano ▁arithmetic ), ▁there ▁are ▁T uring ▁machines ▁which ▁are ▁assumed ▁to ▁be ▁total , ▁but ▁cannot ▁be ▁proven ▁as ▁such , ▁unless ▁the ▁system ▁is ▁incons istent ▁( in ▁which ▁case ▁one ▁can ▁prove ▁anything ). ▁The ▁proof ▁of ▁their ▁tot ality ▁either ▁rest s ▁on ▁some ▁assumptions ▁or ▁require ▁another ▁proof ▁system . ▁ ▁Thus , ▁as ▁one ▁can ▁enumerate ▁all ▁the ▁proofs ▁in ▁the ▁proof ▁system , ▁one ▁can ▁build ▁a ▁T uring ▁machine ▁on ▁input ▁n ▁that ▁goes ▁through ▁the ▁first ▁n ▁proofs ▁and ▁look ▁for ▁a ▁contradiction . ▁If ▁it ▁finds ▁one , ▁it ▁gets ▁into ▁an ▁infinite ▁loop ▁and ▁never ▁hal ts ; ▁otherwise , ▁it ▁hal ts . ▁If ▁the ▁system ▁is ▁consistent , ▁the ▁T uring ▁machine ▁will