User blog comment:Rgetar/Notation without converting into standard form. Program and lists of ordinals/@comment-35470197-20190316121157

> an ordinal notation (that is correspondence "string - ordinal")

It is not an ordinal notation in the usual sense in mathematics, but is just a notation with a correspondence to ordinals. See the difference between these notions here.

> If we have this notation and this algorythm, is a system of fundamental sequences defined? Maybe not.

It depends on what "a system of fundamental sequences" means. It is not a system of fundamental sequences for ordinals, but can be a system of fundamental sequences for strings. For example, the bracket in Buchholz's notation system \(T\) (containing the subset \(OT\) of his ordinal notation system) does not yield a system of fundamental sequences because two distinct strings correspond to the same ordinal, but it is harmless to call it a system of fundamental sequences.