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

> Since \(\varepsilon_{M+1}\) does not make sense under \(ZFC\), the equality \(PTO(KPM)=\psi_{\Omega_1}(\varepsilon_{M+1})\) is not provable under \(ZFC\).

I would like to contest this, as it doesn't seem to appreciate the distinction between an ordinal notation and the ordinals which it is (informally speaking) seen to represent. Specifically, we can define a term \(\varepsilon_{M+1}\) as a representation of a specific natural number, and view the defined \(<\) as a relation on \(A \times A\) with \(A \subseteq \mathbb N\). This is how ordinal notations can generally be defined.

We then often (but don't need to) define an isomorphism \(f:(A,<)\to (S,<)\), where \(S\subseteq \alpha\) for some ordinal \(\alpha\), and often neglect to mention that we're using the isomorphism (or even skip the ordinal notation system altogether). Regardless, \(\varepsilion_{M+1}\) is well-defined in \(ZFC\) - we just can't prove the existence of an isomorphism \(f\) with \(f(\varepsilion_{M+1})=\varepsilion_{\kappa+1}\) holding for a weakly Mahlo cardinal \(\kappa\) (if ZFC is consistent). However, we may still be able to prove the existence of an isomorphism for some \(S\), which corresponds to the well-foundedness of \((A,<)\). This can likely be done without extreme difficulty with \(S\subseteq \omega_1\) (possibly by excessively using elementary substructures) but there seems to be relatively academic interest in this.

(We can use to substantiate this argument in general, however the condition \(\alpha,\beta<M\) in part (v) of Definition 2.1 prevents this from being fully realised for \(\varepsilon_{M+1}\) specifically (in that notation system).)