User blog comment:Rgetar/Square brackets and I ordinal notation/@comment-35470197-20190107041339

> For other strings this correspondence is defined using fundamental sequences (see next section).

No. In order to define such a correspondence compatible with fundamental seuqences, you need transcendental induction. Therefore the whole correspondence has not been defined yet, because you have not completed a definition of a well-fonded order on the set of strings. Or you need an OCF which actually yields your notation system.