User blog comment:Tetramur/My thoughts about functions and numbers/@comment-35470197-20191229115044/@comment-35470197-20200101083325

> EDIT

Uh, sorry. My statement "the oracle for Kleene's O requires ω_1^{CK}-th order oracle Turing machine" just means "it is not computable by an α-th order oracle Turing machine for some α<ω_1^{CK}". Sorry for the ambiguity. It does not mean that Kleene's O can be computed by an ω_1^{CK}-th order oracle Turing machine. Moreover, I do not have an idea how to reasonably formulate the notion of an ω_1^{CK}-th order oracle Turing machine, because the construction itself seems to use an enumeration of ω_1^{CK}. For example, when we formalise ω-th order oracle Turing machine, then we need to enumerate n-th order oracle Turing machine for each n<ω.