User blog comment:Tetramur/My thoughts about functions and numbers/@comment-35470197-20191229115044/@comment-35470197-20191230105157

@Tetramur

> It is not clear for me how it is connected, but this is a result of Sacks, not me.

The point is that you expect that k-th order oracle Turing machines are related to ω_{k+1}^{CK}, but the statement which you quoted is perhaps referring to the connection between Turing machines with oracles for k-th iterated relativisation of Kleene's O and ω_{k+1}^{CK}. They are not computable by a k-th order oracle Turing machine.

> So, I want to divide all uncomputable functions on two classes:

It sounds interesting, but we have a trouble when we consider set thoeries on constructible universes. Since they are set thoeries, functions deeply related to them should be the second class, right? On the other hand, there are many known connection between constructible universes and hyperarithmetic, in which oracle Turing machines appear. Say, if we consider f_{ω_ω^{CK}} with respect to a fixed system of fundamental sequences, then which class should it belong to? It has a connection with KP set theory by the definition of ω_ω^{CK}, while it is related to relativised Kleene's O, which has of course connection to TMs.

@Plain'N'Simple

> Doesn't your statement imply that

The statement in the question is correct. The point is that the oracle for Kleene's O is not computable by an oracle Turing machine. In other words, it is not a connection between admissible ordinals and oracle Turing machines, but is a connection between admissible ordinals and higher computation models, which are not oracle Turing machines any more.

More storngly, the oracle for Kleene's O requires ω_1^{CK}-th order oracle Turing machine in the sense that it is not computable by an α-th order oracle Turing machine for some α<ω_1^{CK}. Therefore it is not recursively constructed from higher order oracle Turing machines without circular logic.