User blog comment:Ynought/Attempt at an ordinal catching type function/@comment-35470197-20190413120241/@comment-35470197-20190414220141

> No,i meant how do you define even define some fundamental sequences without defining them individualy

Usually we define a system of fundamental sequences of ordinals (or notations of ordinals) in a recursive way. For example, you can see the recursive definition of Wainer hierarchy here. Compare it with the ill-defined example of a system of fundamental sequences in my blog post to which I referred above.

> would that work?

No. It just gives a fundamental sequence of the limit of \(\gamma\) if \(\gamma\) is well-defined. On the other hand, you need a system of fundamental sequences for all countable (or sufficiently large) ordinals so that \(\gamma\) becomes well-defined.