[xamuel]

Not sure what you mean exactly. Note, the set P of intuitive ordinal notations is not computably enumerable. Proof: if X were a computer program which enumerated all the intuitive ordinal notations and nothing else, then X itself would (by definition) be an intuitive ordinal notation. It would notate an ordinal number bigger than all ordinal numbers that can be notated by intuitive ordinal notations. Absurd. (Of course, to make this proof fully rigorous I'd have to give the formal definition of which ordinal an ION notates, which I left out above. But you can probably fill in those details yourself.)