User blog comment:Ikosarakt1/Uncountable function cannot exists/@comment-1605058-20130707061148/@comment-5150073-20130708192040

But isn't it true that functions works as "diagonalizators" for some structures? Ackermann function diagonalizes over primitive recursion, U(n) over current Bird's array notation, BB(n) over entire recursion, etc, and all fast-growing functions (if we speak not about ordinal collapsing ones) can have only finite arguments, and so the diagonalization of function can have only countable number of terms. Remember, w_1 have uncountable number of terms, however.