User blog comment:Ubersketch/A question about ordinal notations/@comment-39541634-20190828194334

Yes... and no.

If you have a definition of a well-order on a notation, then you can easily use this definiton to define a system of fundamental sequences: Simply define the n-th term of the fundamental sequence of the ordinal X as "the largest ordinal smaller than X which is expressible with n symbols or less in our notation".

So yes, you can convert a well-order to a definition of a system of fundamental sequences.

Unfortunately, without an explicit comparison algorithm, the above definition is completely useless. This is because you can't tell which of the n-symbol expressions is the largest, unless you have a comparison algorithm.

And before you ask: No, there's no general way to convert a well-order to an explicit comparison algorithm. In fact, there are situation where such a feat is provably impossible.