User blog comment:Tetramur/My thoughts about functions and numbers/@comment-35470197-20191229115044/@comment-35470197-20191229144846

> 1

One of known classification is the hypercomputability.

> 2

The value itself does not depend on the choice of a second order axiom which makes Rayo's number well-defined. It is true that the full collection of aixoms should be declared, but I note that Rayo's function is kind of the first example which helps us to recognise the importance of the declaration of axioms. Namely, I guess that googologists were not required to declare axioms before Rayo defined that function. Anyway, you can see the issue here.

> 3

I added a link to the proof to the article on Church-Kleene ordinal.

> 4

> I suppose that CK ordinals with higher indices are such ordinals that they can not be defined using ω, all previous CK ordinals, ordinal arithmetic and recursion in the "body" of ordinal (but not in indices).

It has nothing to do with the definition of ω_α^{CK}. It is defined by the admissibility, which is a notion related to KP set theory.