User blog:Ytosk/Trying to define Bowers' K(n) systems

So, apparently K(n) systems are ill-defined, and i'm still not quite sure what exactly ill-defined means, but i know that Jonathan Bowers really didn't make the definition formal. I think i understood a bit what he meant, so here i am, trying to formalize it. This could be wrong in all possible ways, so i'm sorry if that's the case.

Let a K(n) system be a consistent axiomatic system with at most n symbols in the axioms in total.

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

Let the definability of x in an axiomatic system s be the smallest natural number n, such that x is definable with n symbols in s.

x is definable with n symbols in an axiomatic system s iff there exists a formula φ in s, such that φ has one free variable and φ is true iff that variable is x, and (the sum of the definabilities of functions and variables used in φ)+(the amount of symbols in φ)≤n.

This definition doesn't work perfectly yet, because if a function f used a function g in its definition, and g used f in its definition, then they wouldn't be definable with a finite amount of symbols. The same would happen if g, instead of using f, used another function, h, in its definition, which uses f in its definition, and it would happen with any loop of functions, each using the next function in its definition, but the last function uses the first function. But when that gets fixed, then Oblivion=MK(kungulus,gongulus)=MK(10^^^100&10,10^100&10)