Since there are only twenty-six letters in the English alphabet, there are finitely many phrases of under sixty letters, and hence finitely many positive integers that are defined by phrases of under sixty letters. Since there are infinitely many positive integers, this means that there are positive integers that cannot be defined by phrases of under sixty letters. If there are positive integers that satisfy a given property, then there is a smallest positive integer that satisfies that property; therefore, there is a smallest positive integer satisfying the property "not definable in under sixty letters". This is the integer to which the above expression refers. But the above expression is only fifty-seven letters long, therefore it is definable in under sixty letters, and is not the smallest positive integer not definable in under sixty letters, and is not defined by this expression. This is a paradox: there must be an integer defined by this expression, but since the expression is self-contradictory (any integer it defines is definable in under sixty letters), there cannot be any integer defined by it.
SiNce there aRe onlY twenty-six letTers in the EngLiSh aLPhAbet, theRE ArE finiteLy many phrAses Of unDer SIxtY letTErs, And HEncE FIniTEly maNy Positive INteGers That are DEFINEd bY PhRasEs OF UNdEr sixtY leTters. since there aRE iNFINIteLy maNy POsitivE IntegERs, thiS MeanS tHat there are POsitIve INtegErs that cannot bE DefIneD bY Phrases of UnDEr sixty LETteRS. if ThEre are PoSitIve INTegErs that satisFy A givEn proPerty, then tHeRE is a SmalleSt pOsitIvE INteGer tHat satISfIEs thAT pRopertY; thErefOre, THere iS a smaLlest posiTiVe inTeger SaTisFyIng THE proPErty "not deFInable IN uNdEr sixty letteRS". thiS Is THe iNteGer to WHiCh ThE aboVe eXPRESsion ReFErS. BuT THe AbovE expressioN iS only fiFty-sEvEn LetterS loNg, ThEREfoRE iT iS deFinaBLe iN unDer Sixty letters, and IS noT the smalleSt PoSitIve Integer nOt DefinAblE in under sixTy leTterS, and is noT defined bY this ExPreSsIOn. This is a paraDox: thErE muSt bE an inTeGer defined by thiS eXpresSiON, bUt since the eXPrEssioN Is seLF-conTRaDictory (anY Integer it DEfines Is DeFinaBlE in under siXTy letteRs), thEre CANNoT be any INteger deFiNed By iT.