User blog comment:Plain'N'Simple/A question for proof-theory experts/@comment-35470197-20191029224813/@comment-35470197-20191031093222

By the way, my personal position for this problem, which I consider many times, is that to aim at creating a notation with a (conjecturelly well-founded) partial order rather than createing a notation with fundamental sequences. If there will be found a new strategy to create a "desired" system of fundamental sequences for a given ordinal notation, then we can apply it.

Another solution is to diagonalise all algorithms which gives provably well-founded decreasing sequences. This method is what I did when I created a notation aiming at PTO(ZFC). We have no method to know which algorithm actually gives a real fundamental sequences of a real ordinal notation in this strategy, but at least it works.