User blog comment:JHeroJr/A computable function/@comment-35470197-20190623231332/@comment-35470197-20190624010523

You actually need a language in such a definition, and hence deleting the word "language" does not give a solution.

You have two serious problems:
 * 1) FGH is not well-defined for an ordinal. You need a well-defined system of fundamental sequences. Therefore "the fifth term is \(f_{\omega}(9))" does not make sense by the lack of a fixed system of fundamental sequences. For example, if you use the busy beaver function in the definition of the fundamental sequence of \(\omega\), the resukting value of FGH completely differs from the value associated to the Wainer hierarchy. Such ambiguity is usually ignored because we fix a suitable system of fundamental sequences following the context. However, since you diagonalise all the possible choice, the ambiguity becomes actually serious, and hence it is ill-defined.
 * 2) Since you have never defined a well-defined language, "can be represented with \(n\) symbols" does not make sense. For example, why don't you allow \(f_{\alpha}(9)\) as the fifth term, where \(\alpha\) is the countable colapse of the least Omega fixed point? Without fixing symbols, syntax, and semantics, there are infinitely many choice of a natural number described with \(n\) symbols. Therefore it is ill-defined.