User blog comment:Tetramur/My thoughts about functions and numbers/@comment-35470197-20191229115044/@comment-35470197-20200101082844

> Ordinary TM's and first-order Oracle TM's are equally powerful when it comes to calculate ordinals

I roughly recognise this phenomenon by understanding that Kleene's O (or any other notation up to ω_1^{CK}) is so uncomputable that its segments are also sufficiently complicated. Now only the fundamental sequence of the limit ω_1^{CK}, but also notations given by initial segments such as {n∈K|O(n)<α} for a sufficiently large recursive ordinal α are far beyond the computable realm.

(I do not know what occurs when we consider a weaker computation model and oracles, though.)

> In the FGH, a simple halting Orcale seems to be intuitively equivalent to ω_1^{CK}x2.

I have heard such a correspondence several times in googology, but I honestly do not understand the philosophy of the correspondence. Is there an intuitive explanation how the contribution of a first order oracle to usual TMs is approximately the same as the contribution of a second order oracle to first order oracle TMs when we consider ordinals in FGH?