User blog comment:Boboris02/Ordinal Analysis of Theories/@comment-11227630-20180213035846/@comment-30754445-20180214071749

To be fair, a computer that can run for googological timespans, can provide evidence (not proof) that a notation is well-founded. If you could systematically run all the possible combinations up to some huge index N, that would be excellent evidence that the system is well-founded. Of cource, this would require something like fLIMIT OF TON(N) steps to complete, so it can't be done on an actual computer.

At any rate, is there really no way to understand why Tranaovsky works without invoking measurable cardinals? This is... surprising.