User blog comment:Tetramur/My thoughts about functions and numbers/@comment-35470197-20191229115044/@comment-35470197-20200101224133

Thank you for the detail. It is helpful for me.

> 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)

I tried to construct an example, but I could not. At least if there are first order oracles which compute whether given n is a code of a recursive well-ordering and whether m is a code of recursive system of fundamental seuqueces for the recursive well-ordering encoded by n, then it is the case.

But I got confused here. If we consider intuitive estimation "ω_1^{CK} = diagonalisation of TMs = BB" and "ω_1^{CK}×2 = diagonalisation of first order oracle TMs = first order oracle BB" in that way, then don't we have "ω_1^{CK}×2^2 ≦ diagonalisation of second order oracle TMs = second order oracle BB" instead of ω_1^{CK}×3, because each first order oracle recursive extension of first order oracle BB is estimated by (ω_1^{CK}×2)+α for some α < ω_1^{CK}×2?