User blog comment:MachineGunSuper/Question/@comment-32213734-20180225032808

FGH depends on fundamental sequence system (fss). So, there is no single FGH, there are a lot of them - for each fss there is separate FGH. So, as long as we can describe ordinals and fss for them, we can can use corresponding FGH.

BEAF does not depend on fss, so, there is single BEAF and we do not need fss.

Some advantages to BEAF: we can easily express up-arrow notation using BEAF, and it is not so easy with FGH (even for one arrow, that is exponentiation). Also, we can easily express Graham's number using BEAF (as well as majority of Bowers' numbers), and it is not so easy using FGH.