User blog comment:Hyp cos/Analysis - BEAF, FGH and SGH (part 3)/@comment-2033667-20141013055709/@comment-1605058-20141013135556

We won't have to use a fixed notation, if we make following modification: $$\alpha[n]$$ is the largest ordinal $$<\alpha$$ which can be expressed in n symbols by using n-th notation. Of course, it makes the definition look a bit more ad hoc, but it would work.

The problem with suggested definition is that it can be quite hard to work with, and it would probably require quite lengthly proof that these fundamental sequences will make it work as expected, and not like Weiermann's systems which make SGH catch FGH at $$\varepsilon_0$$ or lower.

If you are interested in a notation which "just works" for a long time we can simply use Turing machines or a stronger system.