User blog comment:DontDrinkH20/Some Explanations of Certain Large Cardinals/@comment-35870936-20181017021650/@comment-35470197-20181018005531

> it does not necessarily mean that it is impossible

Right. That is why I wrote that "there is no systematic way".

> then we just need to use a stronger set theory, like ZFC + [large cardinal axioms].

Exactly. Since the OP mentions the difference of the consistency, I did not assume more than \(\textrm{ZFC}\).

Trivially, we can replace large cardinals appearing in an OCF \(\psi\) by countable ordinals corresponding to them through the associated ordinal notation system.

If such countable ordinals admit other description without using \(\psi\), then there is no problem.

On the other hand, if such countable ordinals are just defined as "the countable ordinals corresponding to large cardinals through the ordinal notation system associated to \(\psi\)", then you need to use \(\psi\) and collapse the large cardinals again in order to define the resulting OCF.

But Ecl1psed276 might not mind what I minded. Thank you for the complement.