User blog comment:Hyp cos/Catching hierarchy fails/@comment-31580368-20180116074332/@comment-1605058-20180116094647

Unless we start off with having fundamental sequences for all the ordinals we are defining, then already C(0) is not well-defined. This is because (as Deedlit mentions) for different choices of FSes, the first catching point can vary. It can be as large as Bachmann-Howard ordinal (using fundamental sequences coming from OCF) or as small as \(\omega^2\) (using certain fundamental sequences coming from orderings on trees).