User blog comment:Ikosarakt1/Uncountable function cannot exists/@comment-1605058-20130707061148/@comment-5529393-20130708193239

We don't have to define a function as a diagonalization. Certainly any function can be defined as a diagonalization over some hierarchy with some fundamental sequence;  for example, if we take G_i (n) = i, then for any function f(n), we can choose the fundamental sequence for omega to be f(n), and then G_omega (n) = G_f(n) (n) = f(n). This doesn't preclude f(n) from being larger than any hierarchy f_a(n) for countable a. So f_omega_1 can exist, it just won't be defined by diagonalization over smaller f_a.