User blog comment:Tetramur/My thoughts about functions and numbers/@comment-35470197-20191229115044/@comment-37993808-20191229175049

> 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?

Yes, there is such a theorem.

> 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.

I'm surprised. You got my idea. Yes, I had in mind some form of "hyper-recursion". I dunno what is it going to be, but it sounds... interesting.

> 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).

I agree with you also in this part. I myself imagined such a well-behaved version of the FGH that had 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?

It seems to me that we have the same thoughts... I think that Kleene's O system is pathological at the moment. How is it even possible to compare fω 1CK (n) with fω+3(n) when they aren't even in the same realm? It seems plausible to me to say that Kleene's O system is incompatible with "standard" systems - Wainer's hierarchy, Veblen's function, "collapsing" functions etc. fω 1CK (n) in the Kleene's O hierarchy with specific enumeration of TMs is equivalent to fω+3(n) in Wainer's hierarchy, but these two ordinals are very different, and fω 1CK (n) in some "standard" system should eventually dominate fα(n) for every recursive ordinal α and approximate BB(n). Unfortunately, I know no research about systems of FS above recursive ordinals.

> Is Rayo's number well-defined?

> Partially yes, but partially no.

This is impossible. This number is unformalized in such case, and it follows that Rayo's number is ill-defined.