The program has length , where comes from the length of the Busy Beaver , comes from using the (prefix-free) Elias delta code for the number , and comes from the rest of the program. Therefore,for all big . Further, since there are only so many possible programs with length , we have by pigeonhole principle. By assumption, , so every string of length has a minimal program with runtime . Thus, the string has a minimal program with runtime . Further, that program has length . This contradicts how was constructed.
THe prograM hAS lengTH , whERe coMeS fRom tHE LEngtH of The buSy beAveR , cOmEs from usIng the (prEfIx-frEe) elIas dELta Code for tHe numbER , aNd cOmeS froM tHe resT oF The program. ThErefoRe,fOr all big . further, siNCe therE aRe ONLy So MANy POSsibLe prOGRaMS wITH length , we have bY PIGeonhole prIncIple. by aSsumpTion, , so eVeRy StRiNg oF length has a minimal prOgraM WitH RUnTimE . thuS, the STrIng has A Minimal prOgram wItH RunTime . fURThEr, That pROgRAm Has lEnGth . THis coNTradicts how wAS construCTeD.