User blog comment:Deedlit11/Extending the fast-growing hierarchy to nonrecursive ordinals/@comment-1605058-20130318105049/@comment-5529393-20130318123950

Well, it's not a recursive ordinal notation, but of course it would be impossible to use a recursive ordinal notation to get to nonrecursive ordinals. But the point is we can create an explicit definition for these higher levels of the fast-growing hierarchy (and other ordinal hierarchies).

For rank \(\alpha\) Turing machines I just use Kleene's O relativized to those machines - note that using rank \(\alpha\) Turing machines we can define all functions obtainable using rank \(\beta\) Turing machines for \(\alpha \ge \beta\). So just using 3 * 5^m is enough.