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

The original catching function is ill-defined because it heavily depends on a choice of fundamental sequences. By the same reason, \(\gamma\) is ill-defined.

I note that Hyp cos tried to fix the original catching function by using TON, but as others pointed out, it does not work as desired, either by the lack of the closedness of TON by the catching function itself.