User blog comment:Ubersketch/A question about ordinal notations/@comment-35392788-20190828173224/@comment-35470197-20190828221616

I note that as Plain'N'Simple explained above, if the well-ordered set is a set of finite strings for fixed letters enumerated by natural numbers, then there is a way to obtain a canonical system of fundamental sequences from the well-ordering. For a further relation, see the explanation here.

> I'm sure a well-order implies it, but I'm not sure if it goes both ways.

It is an equivalent condition fir a given total ordering in ZFC set theory. However, when we work in weaker set theories, which apear in ordinal analysis, it is not necessarily a n equivalent condition. If we work in arithmetic, since we have no way to express a subset in the theory, we often define a well-ordering as a recursive total ordering such that there is no primitive recursive infinite descending sequence.