User blog comment:P進大好きbot/What is the greatest ordinal notation now?/@comment-35870936-20180623152451/@comment-35470197-20180624094122

@LittlePeng9

> Well, sufficiently large for what purpose?

Ah, sorry for the ambiguity. I meant whether it presents the largest countable ordinal among completely defined ordinal notations.

> You can find the (outline of the) proofs here.

Thank you for the reference! I study more in order to understand the proof.

> Regarding psi(e(K+1))

Ah, so you mean that the ordinal notation using the real (not recursive) K is not constructed in a primitive recursive way without the large cardinal axiom, right?

Since the ordinal notation using the real (not recursive) weak Mahlo can be constructed in a primitive recursive way without the existence of weakly Mahlo and the only problem is the well-foundedness without that axiom by Section 7 of Rathjen's paper, I guessed that the same were true also for K.