User blog comment:Hyp cos/Fundamental Sequences in Taranovsky's Notation/@comment-40154718-20190917091450/@comment-35470197-20190920121528

FGH makes sense for any notation with fundamental sequences, but the function is not necessarily total. If you require the function to be total, then you need to require the well-foundedness of the fundamental sequences.

I note that such a notation does not necessarily forms an ordinal notation with respect to an appropriate ordering. For example, every well-founded partial ordered countable set generates a total function through FGH if you define a system of fundamental sequences in a way similar to that for TON.