User blog:Alemagno12/An extremely fast-growing OCF

Sometimes, there are notations that are so strong that norrmal ordinal notation can't handle it. So I will make a new ordinal OCF, Ψa(b), and two functions to generate the uncountable ordinals used in the OCF, ΨLa(b) and L(x).

WIP!

What's an omega-limit ordinal?
We all know what sequence and limit ordinals are, but what's an omega-limit ordinal?

An omega-limit ordinal is the limit of a set of ordinals of the form ΨLL(a)(b), for a specific a. But it's not like a limit ordinal, like the relation ω has to the set of non-negative integers, it's more of like the relation that the first uncountable inaccesible cardinal has to the set of alephx for all x.

If an ordinal a is the omega-limit of a set of ordinals of the form ΨLa(b) for all b, then that ordinal is the smallest ordinal that does not belong to that set.

For example, L(ω) is the omega-limit of the sequence of ordinals ΨLL(ω)(x) for all x.

Defining L(x) and ΨL_a(b)
For sequence ordinals, L(x+1) is the smallest ordinal with greater cardinality than L(x). Else, L(x) is the omega-limit of the sequence of ordinals ΨLL(x)(y) for all y.

If x is of the form L(y) or ΨLz(y), where y has a fundamental sequence:
 * If n < ω, then ΨLx(n) = x', where x' is the same as x, but y is replaced with y[n].
 * If n+1 > ω, find the way to get from one element of the fundamental sequence of y to another for all of the elements, then do that to ΨLx(n) to get ΨLx(n+1).