User blog comment:Tetramur/My thoughts about functions and numbers/@comment-35470197-20191229115044/@comment-37993808-20191229134258

1. That means that division between weak/strong uncomputability must be done without hierarchies of functions. So, my definition must be corrected.

2. I suppose that even Rayo's number is ill-defined. Why? It's value depends on the choice of some axioms of second-order set theory. It is important to clarify this 'cause another choice can change the value.

3. I read this issue many times. It is interesting 'cause some machines show value much higher than this. Can you show me a proof of upper bound in the Wainer hierarchy? I don't understand how this result is reached.

4. "The Church-Kleene ordinal, then, is the first ordinal that is not recursive."

That is, it can not be defined using \(\omega\), ordinal arithmetic and recursion.

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