User blog comment:P進大好きbot/Ordinal Notation Associated to a Proper Class of Ordinals/@comment-27513631-20180626200006/@comment-35470197-20180629235822

> NBG

The provability is ok for many googologists, I think. But they might concern the presentability. Given a formula on sets described in NBG with occurrence of classes, it is not so trivially done to restate it in ZFC. ...Anyway, I did not write about the presentability in my blog post either. Therefore it might be better for me to write everything in NBG without mentioning.

> The large cardinals aren't relevant to the recursive definitions

Yes. Maybe you are confounding two recursions. I explain Rathjen's method.

First, he defined an ordinal notation in a recursive way under the large cardinal axiom. Secondly, he verified that the relations used in the definition is reduced to a recursive one under ZFC. The translation is not trivial.

Also, remember that if the well-foundedness is not verified under ZFC, then it does not give a notation presenting ordinal numbers. In order to avoid the problem, he used the recursive analogue of weakly Mahlo, which gives the termination under ZFC.

Generally, one defines an ordinal notation under large cardinal axioms, and replacing them by recursive analogues under ZFC. Then the resulting ordinal notation is not necessarily the same one as the original one. It is not trivial when the recursive analogues give precise interpretations under ZFC.

Or, do you state that you can do the same for any other ordinal notations defined under large cardinal axioms...? Could you give me references?