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

No. A well-order is just some way of comparing values (in this case, strings) that satisfies certain properties. A fundamental sequence system is a way of expanding ordinal( term)s to lower levels.

You don't derive one from the other, however proving a well-ordering indicates certain properties of a fundamental sequence, such as the non-existence of an infinite descending sequence.