User blog comment:Hyp cos/Analysis - BEAF, FGH and SGH (part 3)/@comment-2033667-20141013055709/@comment-1605058-20141013200420

By the way, Hyp cos, our main concern wasn't really the problem with BEAF, but with Catching hierarchy. How are you so sure that it does fall below the level of Taranovsky's notation?

Here is your way of thinking: these fundamental sequences are undefined, but it certainly won't go beyond Taranavsky's notation, so we can use Taranovsky's notation to define it. This is a vicious circle - we cannot claim anything about the functions before they are defined. What if the fundamental sequences based on Taranovsky's function turn out to be so pathological that, say, the w^2-th catching point won't fall beyond it's reach? Even worse, what is there is no such catching point using these fundamental sequences?

Note that I'm not claiming that this would indeed happen - I do believe that the notation isn't that pathological, but note that we have pretty much nothing to support this.