User blog comment:Hyp cos/Googolisms hard to extend?/@comment-35470197-20190419220952/@comment-35470197-20190420021922

> One way to generally extend FGH + OCF is: let λ be the computable limit of that OCF, and λ[n] be the maximal computable ordinal expressible by the OCF within n symbols

No. The "computable" limit does not sense without intrepreting OCFs into arithmetic. Also, you need such interpretation in order to ensure the computability of the sequence given by λ[n]'s.

> (using base-λ Wainer hierarchy for ordinals >λ)

It does not work in general because such an expression for a large ordinal is not unique unlike Wainer heirarchy. Since you need to introduce a notion of normal forms using OCFs with carefu study of the uniqueness of expressions given by observing all fixed points, it is not a naive extension. In order to go beyond the computable limit of a given OCF. you actually need a non-naive argument on expressions unless you create another new OCF extending the original one.