User blog comment:Ynought/Attempt at an ordinal catching type function/@comment-35470197-20190413120241/@comment-35470197-20190413231054

I do not need to define a system of fundamental sequence in order to explain the ill-definedness of \(\gamma\). The point is that YOU have not defined a system of fundamental sequences so that \(gamma\) is well-defined.

Further, even if you define a system of fundandamental seuqences below a fixed countable ordinal \(\alpha + 1\), the enumeration of catching ordinals only makes sense below \(\alpha + 1\). Therefore \(\gamma\) never gives you a "new" ordinal, i.e. an ordinal greater than \(\alpha\). It implies the definition \(\gamma(\alpha)[n] = \cdots\) never works.

Also, the occurrence of a function such as \(\gamma\) in an expression of an ordinal in the left hand side of a definition of a fundamental sequence is invalid as I explained here.