User blog comment:Fejfo/Super Fast Beaver Hierarchies and a weird OCF/@comment-1605058-20180806150910/@comment-30754445-20180806185516

I remind both of you that the usage of these ordinals in computability theory is quite different than their usage in googology.

In computability theory, Turing Machines+BB oracles correspond to the same ordinal as ordinary Turing Machines (ω1ck). In googology, BB oracles correspond to ω1ck×2 to the FGH.

LittlePeng himself is the person who tought me the former fact. The latter fact is easy to deduce from the way oracles work: If you have a computing system that reaches an ordinal x in the FGH, and give it access to an oracle that outputs the value of fy(n) in the fast growing hierarchy, then the power of the combined system would by x+y.

As an example, think of a primitive recursive computer language that has access to an oracle for n[n up-arrows] n. It isn't difficult to see that the growth-rate of any function you program in this system would be limited to fωx2(n), where "ωx2" is the sum of the strength of the language  (primitive recursive = ω) and the strength of the oracle (knuth arrows = ω).

(I actually wrote a comment about this to Fejfo a few hours ago, but it wasn't posted for some reason. Maybe I just forgot to press the "post comment" button)