User blog comment:P進大好きbot/Full References of Arguments on Ordinal Notations with Large Cardinals/@comment-27513631-20180805145552/@comment-35470197-20180807230238

> Oh, we're implicitly using a canonical mapping?

Actually I was assuming the canonical map \(o\) associated to the ordinal notation system.

In addtion, even if you use a non-canonical map, the equality \(f(\varepsilon_{M+1}\) = \varepsilon_{\kappa+1}\) never holds as long as \(\varepsilon_{M+1}\) is regarded as an ordinal type of a countable well-ordered set. Or I might be misunderstanding what you wrote.

> Pretty sure that means you're right, but it wouldn't go amiss if it was clearer next time.

OK. Sorry for it.

>> So you actually know the reference of the equality KPM=ψ Ω 1 (ε M+1 ) ψΩ1(εM+1), right? Please tell me. > > Nah, I don't. Did I say that?

No, you did not. I just guessed so, because you contested my assertion "Since \(\varepsilon_{M+1}\) does not make sense under \(\textrm{ZFC}\), the equality \(\textrm{PTO}(\textrm{KPM}) = \psi_{\Omega_1}(\varepsilon_{M+1})\) is not provable under \(\textrm{ZFC}\)". Also, since you are a reliable specialist for me, I asked you to tell me if you had a reference.

> I know that's not clear-cut, but I imagine it's usually clear from context.

Exactly. However, many googologists here often use symbols like \(I,M,K\), and so on just as placeholders stating that they are irrerevant to lardi cardinal axioms, and simultaneously use the expressions like \(\varpsilon_{M+1}\) stating that the analysis holds without the assumption of large cardinal axioms. Then it will be actually unclear. That is why I am listing up references.