User blog comment:P進大好きbot/Full References of Arguments on Ordinal Notations with Large Cardinals/@comment-27513631-20180805145552/@comment-27513631-20180807212624

Oh, we're implicitly using a canonical mapping? Pretty sure that means you're right, but it wouldn't go amiss if it was clearer next time.

> So you actually know the reference of the equality KPM=\(\psi_{\Omega_1}(\varepsilon_{M+1})\), right? Please tell me.

Nah, I don't. Did I say that?

> And I am listing up what axioms googologists use in their analyses. You only use ZFC unless specified, right?

Nope. If the description of a notation uses certain axioms (for instance, a description of \(\psi_{\Omega_1}(\varepsilon_{M+1})\) usually involves a Mahlo cardinal), then I assume those axioms without stating as much explicitly. I know that's not clear-cut, but I imagine it's usually clear from context. (And if I'm defining stuff in type theory or similar, then I tend to use that sort of base axiomatic system instead of ZFC, but I don't think I've done that outside of personal notes yet.)