User blog comment:MachineGunSuper/Question/@comment-30118230-20180224173316/@comment-30754445-20180224194849

The FGH is actually quite easy to use well beyond ε₀ (which is the limit of BEAF). Things do get more complex as the ordinals get larger, but at ε₀ things are still quite simple. For example, using only addition, multiplication and exponents, you can get from ε₀ to ε₁. Adding a single simple function (the ε function) gets you all the way up to ζ₀. None of this can be done with BEAF, and it really is much simpler (and far less intemidating) that it may seem at first glance.

(the biggest hurdle here is to learn the most fundamental basics: how ordinals work, how to add and multiply ordinals, how to work with "fundamental sequences" and so on. If you don't know these things, you'll have a very hard time with the FGH at any level beyond ω^ω. If you do know them, then going beyond ε₀ becomes almost trivially easy).

At any rate, the main advantage of BEAF is that it gives neat geometric interpertation to everything you're doing. You don't need do bother with ω's and stuff. You simply add more and more numbers in a row... and then add more rows to create a grid... and then add more grids to create a 3D cube... and so on.

One really cool thing you can do in BEAF is to create an entire structure with the "&" symbol. If you write 5 & 10 this means "a row of 5 tens". If you write 52 & 10, this means "a 5x5 square of 10's"... and so on.

The original idea of BEAF was to use all previously created structures to the left of the '&'. So not only you could write:

53 & 10

54 & 10

55 & 10 ...

But you could also write things like

{5,5,2} & 10 (which can also be written as 5↑↑5 & 10)

and

{5,5,5,5} & 10.

And just like "53 & 10" means "53 tens arranged in a 5x5x5 cube", you'll have "{5,5,5,5} & 10" meaning "{5,5,5,5} tens arranged in "

At least that was the original plan. Had it worked, BEAF would have been very very strong.

Unfortunately, the definitions break down beyond "tetrational arrays" (X↑↑Y & Z), and these have a mere strength of ε₀.