User blog comment:Kyodaisuu/English description of Fish numbers/@comment-25418284-20131102173245/@comment-1605058-20131103193412

So, to sum everything up:

Oracle is simply a functional symbol f such that f(a)=b is true iff Rayo(a)=b.

f can be defined within FOST by a formula of finite length.

Even though we have above, by excluding all contradictions, Rayo's function is still well defined.

By using f as an oracle (distinguished symbol) we can significantly increase rate of growth of corresponding function.

By adding more oracle symbols analoguous to f we further increase rate of growth. (going up to R[φ2(0)] is defining hierarchy from this function, actually)

I'm pretty sure this is still no comparison to second order set theory, which can quantify over predicates. (this is only my opinion)