User blog comment:Deedlit11/Ordinal Notations V: Up to a weakly Mahlo cardinal/@comment-5029411-20130811125215/@comment-5529393-20140623205046

Remember a Turing machine can only handle a natural number input and a natural number output, so a computable function is necessarily from a natural number to a natural number. So a computable function on ordinals can only be defined with respect to some specific notation that corresponds a countable set of ordinals to the natural numbers. For a function on all ordinals, we cannot even speak of computability.