User blog comment:Wythagoras/All my stuff/@comment-10429372-20130715145943/@comment-5529393-20130715164916

The natural ordinal to correspond to the ¥ function will be the smallest undefinable ordinal, which would be a countable ordinal.

While functional hierarchies certainly exist that go all the way to the ordinal omega_1, by the axiom of choice, the problem is we have to pick a specific functional hierarchy by which to measure our functions. I don't believe we can do this up to omega_1, due to its uncountability, nor do I think we can do it up to the smallest undefinable ordinal. The question is whether there "exists" a notion of canoical rate of growth for a given definable ordinal, even if we can't define it. But since ¥ exceeds all definable functions, it will exceed F_a for all definable ordinals a, so the smallest undefinable ordinal is the natural ordinal to associate it with.