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-35470197-20190319004900

Your explanation of the transivity is completely wrong.

Also, you seem to be comfounding ordinals and their expressions. For example, "\(t(r)\) is the smallest ordinal satisfying \(r < t(r)\)" does not make sense, because \(t(r)\) is just an expression, which is not an ordinal.

Moreover, you have not defined \(<\) yet, either. I guess that you are "defining" it as the usual order of "the corresponding ordinals", but it is a circular logic derived from your comfusion of ordinals and expressions. Remember why you need \(<\) to be primitive recursive.

Or I guess that you do not understand the definition of the notion of an ordinal notation. It is not a notation of ordinals.