User blog comment:C7X/Question About Normal Forms/@comment-35470197-20191211013048

We usually define a wider class of expressions because the recursive subset of expression of normal forms and the comparison algorithm are usually defined in a simultaneous recursion if we work with ordinal notations based on higher regularity. For example, it is good to study Rathjen's ordinal notation based on M, because the use of simultaneous recursion is quite clear. If you are interested only in ordinal notations without OCFs, then your strategy works.

Of course, if you are not interested in the computability, then you can directly define T (without specifying normal forms) in that level. For example, the set of ordinals expressed by OCFs is often directly defined in that way.