User blog comment:Tetramur/My thoughts about functions and numbers/@comment-35470197-20191229115044/@comment-35470197-20191230082139

Although I do not have a precise connection between admissibility and oracle TMs, it is known that the notation associated to an enumeration of functions computable by Turing machines with the oracle for Kleene's O gives a (non-recursive) ordinal notation up to ω_2^{CK}. (Since Kleene's O is not computable by an oracle Turing machine, this result does not mean that ω_2^{CK} is computable by a second-order oracle Turing machine.)

Also, my conjecture is that Tetramur will say that the statement above is the one which Tetramur mentioned in the comment "Yes, there is such a theorem", although it is not "such" a theorem because Kleene's O itself is not computable by an oracle Turing machine.