User blog comment:P進大好きbot/Full References of Arguments on Ordinal Notations with Large Cardinals/@comment-27513631-20180805145552/@comment-35470197-20180805221711

So you actually know the reference of the equality \(\textrm{KPM} = \psi_{\Omega}_1(\varepsilon_{M+1})\), right? Please tell me.

And I am listing up what axioms googologists use in their analyses. You only use \(\textrm{ZFC}\) unless specified, right?

> Specifically, we can define a term εM+1 as a representation of a specific natural number, and view the defined < as a relation on A×A with A⊆N. This is how ordinal notations can generally be defined.

Right. I know how ordinal notations work by just using recursive binary relations on a subset of \(\omega\). But the point is, we are dealing with \(\psi_{\Omega_1}(\varepsilon_{M+1})\) but not with \(\varepsilon_{M+1}\) itself.

Such a correspondence from an ordinal notation to a set of countable ordinals just deals with ordinals describable by the notation, and hence sends an ordinal which might play a role of an uncountale ordinal to a countable ordinal. Then the correspondence is not commutative with \(\psi_{\Omega_1}\) at all. Regarding \(\varepsilon_{M+1}\) as a countable ordinal in that way does not yield a way to define \(\psi_{\Omega_1}(\varepsilon_{M+1})\).

> 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 κ+1 holding for a weakly Mahlo cardinal κ (if ZFC is consistent).

Oh, I guess that you are confounding the countable ordinal corresponding to \(\varepsilon_{M+1}) with the uncountable ordinal presented by \(\varepsilon_{M+1}\) through the canonical map \(o\) in the data of the ordinal notation system.

What you are talking first about (the definability in \(\textrm{ZFC}\) is the countable ordinal given as the ordinal type of \((\textrm{Im}(o |_{< \varepsilon_{M+1}}),\in)\), but \(varepsilon_{\kappa+1}\) is \(o(\varepsilon_{M+1})\). The former one is a countable ordinal, which might be possible to define in \(\textrm{ZFC}\), and the latter one is an uncountable ordinal, which is not definable in \(\textrm{ZFC}\) any more.

I recall that it is not so easy to interpret arguments on the cardinality of an uncountable ordinal \(o(\alpha)\) to arguments on the ordinal type of the countable ordinal \(\textrm{Im}(o |_{< \alpha})\), as Rathjen wrote. In the definition of \(\psi_{\Omega_1}\), we need arguments on cardinality, and this is one of the obstructions of the proof just using recursive analogues or \(\textrm{Im}(o |_{< \alpha})\) under \(\textrm{ZFC}\).