User blog comment:Mh314159/FOX notation/@comment-35470197-20191204015153

The function f(x) roughly corresponds to ω^n×a_n+ω^{n-1}×(a_{n-1}-1)+…+ω×(a_1-1)+a_0 if I correctly understand the definition. Therefore the limit is ω^ω.

I note that there are a little weird correspondences when the array includes 0. For example, f(x) roughly corresponds to ω^2×c, which is independent of a. Of course, f(x) is much stronger than f(x), but the difference is much smaller than the difference between ω^2×c and ω^2×c + 1, If it is unintensional, then it is good for you to study the behaviour of  in order to understand FGH more.

Although this weird phenomenon does not effect the limit, studying it will help you to create a stronger recursion, which requires the understanding of "how to diagonalise the strongest functions". The point is that f(a+1,0,c+1>(x) diagonalises (f(x))_n, while f(x) is the strongest below f(x). Therefore it prefered to define f(x) as f_x(x) or something like that. Anyway, as I emphasised, this replacement does not change the limit. Therefore you do not have to care about my alternative choice if it is intensional.