User blog comment:Ytosk/Trying to define Bowers' K(n) systems/@comment-35470197-20191031094256

> Let a K(n) system be

The grammer looks broken. Is it something like "a consistent axiomatic system is said to be a K(n) system if..."?

> Let MK(n,m) be the smallest natural number greater than every natural number definable with n symbols in a K(m) system.

It does not make sense, becase a consistent axiomatic system without any specific structure related to arithmetic does not have a way to define a natural number.

Moreover, if you are just referring to proof-theoretic definability, this approach does not yield a large number significantly greater than Rayo's number, because it can be done by a first order oracle. Namely, your function is not significantly greater than busy beaver function or FGH along Kleene's O, which is much much smaller than Rayo's function.

As I commented before, it is really difficult to formalise K(n) so that the resulting function is significantly greater than Rayo's function, because you need to understand truth predicate.