User blog comment:P進大好きbot/Ordinal Notation Associated to a Proper Class of Ordinals/@comment-27513631-20180626200006/@comment-35470197-20180627010251

I prefer NBG, and hence if almost all googologists here prefer it too, then I would like to use it. So they are familiar with NBG, right?

> If the large numbers you're defining have to be comutable, then googologists generally don't care about what foundation you use.

Maybe right. But I care about it :D

As Rathjen did in his paper, it is interesting (at least for me) when an ordinal notation defined in ZFC plus large cardinal axioms is precisely translated in one in ZFC.

You may replace large cardinals by their recursive analogues, but it is not so trivial in general that the corresponding ordinal notation is actually equivalent to the original one.

You may replace the order on the ordinal notation with large cardinals by a premitive recursive relation defined in ZFC, but it is not so trivial in general that the corresponding order is actually well-founded.

Such arguments are necessary if you want to construct not only a well-defined notation but also a well-defined large number in ZFC. Of course, you do not have to care about it if you work in any axiom such as \(0=1\). But if you want to compare your large number with another one, it is fair to fix a reasonable axiom.

> It only becomes relevant when we get sufficiently far into the non-computables.

Nope. It neglects the great effort of Rathjen and others to interpret the ordinal notations with large cardinals to ones well-defined in ZFC...