User blog comment:P進大好きbot/List of common mistakes on formal logic appearing in googology/@comment-11227630-20181018005915/@comment-35470197-20181018111930

There is no difference at least in my blog posts. I just use "ordinal notation" as a shorthand of "ordinal notation system". It is just a recursive subset of \(\mathbb{N}\) equipped with a recursive order which is provably well-founded under specific axioms, and is often equipped with the system \([]\) of fundamental sequences and an order-preserving map \(o\) into the class of ordinals.