User blog comment:Tetramur/My thoughts about functions and numbers/@comment-35470197-20191229115044/@comment-39541634-20200101093750

"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?"

Hmmm...

I've thought about this a long time ago, but when I try to reconstruct the argument I can only get half-way.

The idea, basically, is to notice that appending two FGH-like systems to one another is equivalent to ordinal addition (due to the rule (α+β)[n] =α+(β[n])). And since an (n+1)-th order OTM can compute any recursive extension of an n-th order OTM, it follows that the FGH limit of 1st order OTMs should be at least:

ω1CK+[the limit of all recursive ordinals] = ω1CK+ω1CK = ω1CKx2.

The problem is that this is just a lower bound. I can't - for the life of me - reconstruct the other half of the reasoning, and I now suspect that it was probably faulty.

So I take back my claim. ω1CKx2 is just a lower bound.

(I now wonder - assuming it is possible - how difficult would it be to provide an example of a 1st-order OTM which outputs a function that outgrows any recursive extension of Σ(n))