User blog comment:Alemagno12/Making an OCF, attempt 2/@comment-5529393-20170803204913

Your definition of "eventually overgrows" is funky - are the functions in your notation functions on ordinals, or on expressions? If they are functions on ordinals, then you can't say whether an ordinal x "contains a diagonalizer", since it would depend on how x is expressed. If they are functions on expressions, then you need to define an ordering on expressions, so that x <= y makes sense.

What does y < a -> F(a) mean?

Also, I don't think anything is ever going to satisfy your definition of diagonalizer. "eventually outgrows" is a limit notion, so it is not going to depend on early values. So there's no X that is going to satisfy, for example, G(X) = sup {G(0), G(G(0)), G(G(G(0))),...} for _every_ G that eventually outgrows F, because there are going to be a lot of possible G's with varying values of G(X).