User blog comment:Naruyoko/My Tottaly Not Broken "Ordinal Notation", Which Totally is Well-Defined and Covers All Ordinals To the Limit Ok Why Is the Title so Long/@comment-35470197-20190318222152/@comment-35470197-20190319013220

In order to define a correspondence to ordinals through \(<\), you need to define it as a (primitive) recursive well-ordering. Explanations on the notion of ordinal notations is available here.

Please keep doing it, which is not non-sense after you fully understand the definition of an ordinal notation. Studying ordinal notations is very good direction in googology, and hence I recommend you to continue your works.