User blog comment:QuasarBooster/The Wackermann function/@comment-43798125-20200217140929/@comment-35470197-20200217152813

Every computable function is arithmetisable by Goedel-coding by the defintiion of the computability. The point is the definition will be usually too complicated if we simply apply Goedel-coding. Therefore we need several techniques, as the OP did.