User blog comment:P進大好きbot/What is the greatest ordinal notation now?/@comment-35870936-20180623152451/@comment-1605058-20180624090627

"Is the portion of TON sufficiently large?" Well, sufficiently large for what purpose? You can find the (outline of the) proofs here. The first proof is given in ZFC, the latter ones are given under some large cardinal assumptions (measurable cardinals). Of course, this only gives an upper bound - for all we know, the same results could be proven in Z2...

Regarding psi(e(K+1)) - the way that ordinal notation is defined (by transfinite recursion), if we regard K as a Pi-3-reflecting ordinal, the map from large countable ordinals to recursive ordinals, and hence also the map from the ordinal notation to recursive ordinals, is well defined. The only thing we are unsure of is whether the ordinals we get are the same as the ones using large cardinals.