User blog:JHeroJr/Why I need a system that inductively defines set theories for each ordinal

I want to come up with a gigantic Googological function. Here's my basic plan:

So, we start off with functions that are variations of "the smallest number bigger than any finite number named by an expression in the language of first-order set theory with a googol symbols or less."

Step A: We approximate the function in a certain Fast-Growing Hierarchy which is defined as follows:

 Approximated to the closest ordinal.
 * 1) \(f_0(n) = f(n) = n+1\)
 * 2) \(f_\alpha^{m+1}(n) = f_\alpha(f_\alpha^m(n))\)
 * 3) \(f_\alpha^0(n) = f_\alpha(n)\)
 * 4) \(f_{\alpha+1}(n) = f_\alpha^n(n)\)
 * 5) \(f_\alpha(n) = f_{\alpha[n]}(n)\)
 * 6) where {\alpha[n]} is {\alpha} 's fundamental sequence. Some ordinals have more than one fundamental sequence, so in this case {\alpha[n]} is {\alpha} 's fundamental sequence with the smallest terms.

Step B: We feed that ordinal into that variation of "the smallest number bigger than any finite number named by an expression in the language of [Put ordinal here]th-order set theory with a googol symbols or less." to create another function.

Repeating Step A and Step B n times, we get a function, which I call WATERPOWER(n)(x), where x is the input of that function that we get from repeating it n times.

But I need that system that defines set theories corresponding to certain ordinals. It can't be a proof-theoretic variation as Pbot explained to me. It might not exist. But someone please try to come up with it!