User blog comment:P進大好きbot/List of common mistakes on formal logic appearing in googology/@comment-11227630-20181018005915/@comment-35470197-20181018051148

I think that you know well about the relation between OCFs and ordinal notation systems, but if not, please read this first.

If you do not know nothing on how we use large cardinal axioms in order to verify the well-foundedness, read this next.

After that, please read the following detail:

For our purpose, we use an OCF in order to define an ordinal notation system. So "replacing large cardinals countable ordinals" works well only when the resulting ordinal notation system has the same strength.

First of all, even when we do not use large cardinals, we need to interprete the collapsing property of an OCF \(\psi\) and the \(\in\)-relation into a recursive operator \(\alpha \mapsto \alpha[n]\) and a recursive relation \(<\) in a non-systematic (but natural case-by-case) way.

The resulting notation system \((OT,[],<)\) makes sense under \(\textrm{ZFC}\), even if we used large cardinal axioms in the definition of \(\psi\). On the other hand, the proof of the well-foundedness of \(<\) uses \(\psi\), and hence heavily depends on the large cardinal axioms. That is why the tower of PTOs of arithmetics or set theories have a rough correspondence to the tower of large cardinals.

Then we need another approach. Say, define an alternative OCF \(\psi'\) by replacing cardinals in the definition of \(\psi\) by countable ordinals. This replacement is valid for our purpose only when the resulting notation system \((OT',[]',<')\) is isomorphic to the original \((OT,[],<)\). Verifying the isomorphism is not so easy in general. If they are not isomorphic, then \((OT,[],<)\) does not admit an order-preserving natural map which sends terms which originally corresponds to cardinals to the corresponding countable ordinals.

(I am not saying that it is impossible. If you feel that you can always do this, please kindly write the proof.)

For example, I listed known published results such relations here. The explanaton by Rathjen in p. 261 in "Ordinal Notations Based on a Weakly Mahlo Cardinal" might be helpful to understand how professional feels about the difficulty.