▁halt ▁on ▁every ▁input , ▁but ▁one ▁cannot ▁prove ▁this ▁in ▁a ▁strong ▁enough ▁proof ▁system ▁due ▁to ▁Gö del ' s ▁in comple ten ess ▁the or ems . ▁ ▁One ▁can ▁also ▁create ▁a ▁T uring ▁machine ▁that ▁will ▁halt ▁if ▁and ▁only ▁if ▁the ▁proof ▁system ▁is ▁incons istent , ▁and ▁is ▁thus ▁non - total ▁for ▁a ▁consistent ▁system ▁but ▁cannot ▁be ▁proven ▁such : ▁This ▁is ▁a ▁T uring ▁machine ▁that , ▁regardless ▁of ▁input , ▁enumer ates ▁all ▁proofs ▁and ▁hal ts ▁on ▁a ▁contradiction . ▁ ▁A ▁T uring ▁machine ▁that ▁goes ▁through ▁Good stein ▁sequences ▁and ▁hal ts ▁at ▁zero ▁is ▁total ▁but ▁cannot ▁be ▁proven ▁as ▁such ▁in ▁Pe ano ▁arithmetic . ▁ ▁See ▁also ▁▁ ▁Blo o P ▁and ▁Flo o P ▁ ▁Total ▁functional ▁programming ▁ ▁Term ination ▁analysis ▁ ▁References ▁▁▁ ▁Bra in erd , ▁W . S ., ▁Land we ber , ▁L . H . ▁( 1 9 7 4 ), ▁Theory ▁of ▁Comput ation , ▁W iley . ▁ ▁Meyer , ▁A . R ., ▁R itch ie , ▁D . M . ▁( 1 9 6 7 ), ▁The ▁complexity ▁of ▁loop ▁programs , ▁Pro c . ▁of ▁the ▁AC M ▁National ▁Me et ings , ▁ 4 6 5 . ▁ ▁S ips er , ▁M . ▁( 2 0 0 6 ), ▁Introduction ▁to ▁the ▁Theory ▁of ▁Comput ation , ▁P WS ▁Publishing ▁Co . ▁ ▁K oz en , ▁D . C . ▁(