User blog comment:Alemagno12/Huge Ordinal Analysis I: Z 2/@comment-1605058-20180204192308

"This allows us to skip the proving process" No, it does not. We can give formulas for orders of any order type up to \(\omega_1^\mathrm{CK}\). The very point of ordinal analysis is that not all of them can be proven to be well-orders in a given theory, and the PTO is precisely the limit of which ones can be proven.

Long story short, there is a reason for "proof" in the name "proof-theoretic ordinal".