User blog comment:PsiCubed2/For Newbies (and Veterans too): The Great Scale of Googology/@comment-25601061-20171219161544/@comment-30754445-20171221141907

What confused me was that I know that TM with halting oracles are stronger than ordinary TM's, at least in the googological sense.

So my reasoning was:

(1) Halting oracles fall under the general category of "orcales".

(2) TM + Halting oracles are stronger than BB's and their corresponding ordinal must therfore be strictly greater than ω1CK.

(3) Therefore, given the statement in the article, Halting oracles should have at least the strength of ω2CK.

My error, of-course, was that in #2 I've confused googological strength with computability theory (which makes me wonder whether we're confusing these two for lower ordinals as well, such as the PTO of 2nd order arithmetic).