User blog comment:Tetramur/My thoughts about functions and numbers/@comment-35470197-20191229115044/@comment-39541634-20191229154454

Isn't there a theorem that says that admissible ordinals (those of the form ωαCK) are exactly those who can be "recursively" defined with Oracle TM's?

As far as I know the concepts are closely related.

The catch is that the word "recursion" here does not refer to ordinary recursion, but to computability in a stronger computing model. So you could say that ω2CK is the smallest ordinal which is unreachable using some form of "hyper-recursion", which is similar - at least in spirit - to what Tetramur said.

At any rate, none of this is really relevant to the FGH, which needs ordinary recursion to work. And it certainly isn't true that ω2CK can be reached from ω1CK with ordinary recursion.

As for equating fω 1CK (n) with BB(n)-style growth:

That would certainly be a desirable outcome. We naturally want every computable function to be bounded by fα(n) for some recursive ordinal α. BB(n) can be seen as the limit of such functions, so any well-behaved version of the FGH should indeed have fω 1CK (n)~BB(n).

Actually guaranteeing this... well, that's the tricky part. How on earth are we expected to prove that our fundamental sequences aren't pathological, when they are not even computable?

Also, I think that simple BB oracles should be comparable to the ordinal ω1CKx2 in the FGH. The oracle itself gives you BB(n) (that's one ω1CK) and the associated (ordinary) Turing Machine gives you another ω1CK. I have no idea what we could associate with ω2CK, but any reasonable candidate will have to be insanely more powerful then simple BB oracles.