User blog comment:Tetramur/My thoughts about functions and numbers/@comment-35470197-20191229115044/@comment-35470197-20191229125420

Sure.


 * 1) It is consistent (i.e. does not imply contradiction) that for any function (such as Rayo's function) f, f_ω(n) goes beyond f if you set ω[n] = Σ_{i=0}^{n} f(n). It means that there is no function such that it goes beyond any function in FGH with respect to any choice of fundamental sequences. You can see stronger and non-trivial issue on the consistency here
 * 2) The cofinality of ω_1 is Ω, and hence f_{ω_1} is ill-defined even if you fix a fundamental sequence of ω_1 (of length Ω). In general, FGH works only for countable ordinals, although you used uncountable ordinals in your criterion.
 * 3) BB is not necessarily comparable to f_{ω_1^{CK}} even if we use the system of fundamental sequences associated to Kleene's O. (You can read the precise issue here.)
 * 4) The k-th order oracle BB is not necessarily comparable to f_{ω_{k+1}^{CK}} by a similar reason. Moreover, Kleene's O is not computable by ω-th order oracle Turing machine. Also, "conjecture" usually means a statement based on some reasonable guess. Do you know the definition of ω_{k+1}^{CK}? Then why do you think that it is related to the order of oracle BBs?