User blog comment:Tetramur/My thoughts about functions and numbers/@comment-44079442-20191229182611

Rayo's function is probably on the level of f_{w_w^CK}. However, when we try to define f_{w_1}(n), we run into a problem. w_1 is the first uncountable ordinal, and thus is the first ordinal without a fundamental sequence of countable length. Besides, even if f_w_1(n) were possible, it would diagonalize over all functions (just as f_w diagonalizes over all primitive recursive functions and f_w_1^CK(n) diagonalizes over all computable functions). But the "ultimate function" (which is what that would be) is nonsense, just as there is no largest number.