User blog comment:P進大好きbot/What is the greatest ordinal notation now?/@comment-35870936-20180623152451/@comment-35470197-20180624001443

Thank you very much for the detailed information!

> BMS

I heard that BMS is not total, and it is not known when the computation actually halts. So it is not completely defined as an ordinal notation (or a wider notation for trees), is it? So I wonder whether TON is completely defined or not.

> psi(e(K+1))

Is the well-foundedness also verified in ZFC? For example, in Rathjen's paper wehre he constructed an ordinal notation with weak Mahlo's, he verified that the ordinal notation and its ordering are primitive recursively well-defined without the weak Mahlo axiom by the technique using the translation of the ordering into a relation between recursively defined sets, but does not the well-foundedness. He wrote that the well-foundedness of his original construction needs weak Mahlo axiom, but that of another ordinal notation obtained by replacing weak Mahlo's by recursive analogues can be verified without weak Mahlo axiom.

So for "standard" notation in your sense, I would like to know whether the well-foundedness up to psi(e(K+1)) has already been verified without any axioms beyond ZFC.

> UNOCF

I read the definition before, but found that the definition of \(\psi(\alpha)\) even for \(\alpha\) with \(\textrm{Cof}(\alpha) = \Omega\) is not completely written, because one needs \(\psi(\Omega^{\Omega+n})\) in the "definition" of \(\psi(\Omega)\), which should be computed to be \(\epsilon_0\). To what extent it has a well-defined ruleset? Just for countable ordinals with fixed fundamental systems?

> Aarex's OCF

Sounds great! I am looking forward seeing a completed definition.