User blog comment:Tetramur/My thoughts about functions and numbers/@comment-35470197-20191229115044/@comment-39541634-20200101074849

Just as I thought. It was just a semantic confusion.

Though for sake of clarity, I would like to point out that the Sacks connection also holds for Finite Oracle TM's. It's just that all of them are conencted to the same admissible (ω1CK), so this isn't useful for ranking them.

This is an interesting difference between the way ordinals are used in computability theory, and the way they are used in googology. Ordinary TM's and first-order Oracle TM's are equally powerful when it comes to calculate ordinals, but the latter is more powerful when it comes to computing large numbers (i.e. fast-growing functions). In the FGH, a simple halting Orcale seems to be intuitively equivalent to ω1CKx2.

(this could be extended by defining a sequence of oracles, to reach ω1CKxa for any recursive ordinal a... and then I'm stuck. At this point we'll need a "oracle for oracles" and I have absolutely no idea how to formally define such a thing. The FGH strength of such a super-oracle would, of-course, be (ω1CK)2).