User blog comment:Alemagno12/An extremely fast-growing OCF/@comment-5529393-20170730113506

How does L(ω) = I follow from the previous definition?

From the previous paragraph, ΨL_L(ω) (a) = Ω_a. Then you say that L(ω) is the smallest ordinal that cannot be made using ΨL_L(ω) and functions that are eventually overgrowed by ΨL_L(ω). It's not clear what overgrowed means here, but presumably any constant function is eventually overgrowed by ΨL_L(ω). So we can use f = a for any ordinal a, which means we can make any ordinal. So the smallest ordinal that cannot be made does not exist.

I still have a problem with the "key value" part of the definition, which seems to infer the values of a function for ordinals based merely on its values at finite ordinals. I know PsiCubed2 claims that we can always give an explicit definition because the later levels follow in a systematic way from the first level, but then Alemagno should just give that explicit definition, rather than define it by saying it is extrapolated from finite ordinals, which cannot in general be done.