User blog comment:P進大好きbot/What is the greatest ordinal notation now?/@comment-1605058-20180623095923/@comment-1605058-20180623131207

The only definition which is actually ever used in practice is as follows: PTO of a theory T is the supremum of the ordinals \(\alpha\) such that there is a recursive ordering \(\prec\) of length \(\alpha\) such that T proves \(\prec\) is well-founded (which means that T can prove transfinite induction along \(\prec\)).

The other definition you mention, the least ordinal \(\alpha\) such that we can deduce consistency of T from some ordering of length \(\alpha\) being well-founded, is sometimes quoted, but is never used in practice, since it's not hard to show that it's equal to \(\omega\) for every consistent T.