User blog comment:PsiCubed2/For Newbies (and Veterans too): The Great Scale of Googology/@comment-25601061-20171219161544/@comment-30754445-20171221130618

The article in question is the one titled "admissable ordinals".

And it doesn't actually make the claim that oracle TMs reach ω2CK. It just says that "the countable admissible ordinals are exactly those constructed in a manner similar to the Church-Kleene ordinal, but for Turing machines with oracles"

Which seems technically true given what you've said already. It's just a terribly confusing way to put it (i dislike it when articles give me the impression that I understand something, when in the end it turns out that I don't).

At any rate, it seems that the needs of googolgists and the needs of mainstream mathematicians diverge here. The statement "the countable admissible ordinals are precisely the ones which are limits for TMs with some fixed oracle" doesn't make for useful googological definitions.

For example, if the ω1CK-th function in our hierarchy is BB(n), we would want ω1CK+1 to represent BB(BB(...(n)...)) and so on. And in general, for any recursive ordinal x, we would want ω1CK+x to represent an x-level recursion starting with BB's.

Now, it is trivial to build a Turing machine with a simple halting oracle that emulates the above process for any given recursive ordinal x. So we would want the ordinal corresponding to "the largest number outputed by an n-state TM with a halting oracle" to be - at the very least - to the ordinal ω1CKx2.

In short, the fact that such Oracle TM's can't actually represent ordinals beyond ω1CK isn't very helpful to us as googologists. We want to know how large a number they can produce, and for this we need an actual scale (either ordinal or otherwise) between "ordinary TM's" (ω1CK) and "TM's with hyperjumps" (ω2CK).