User blog comment:Syst3ms/Breadth-Recursive Array Notation (BRAN)/@comment-25601061-20180424215049/@comment-30754445-20180425195215

Remember that you needed to start from a [c arrows] b to reach Graham's Number with 4 entries. It's actually the arrows the do most of the work there.

You can look at it this way:

Your own rules can generate a growth-rate level of ω. More precisely, whenever you lengthen your array by 1 entry, the growth-rate goes up by 1:

(a,b,c) is FGH level 3

(a,b,c,d) is FGH level 4

(a,b,c,d,e) is FGH level 5

and so on.

Level ω is the limit of levels 1,2,3,4,5,... And since your arrays can be of any length, your entire system (which is represented by your g(n) funciton) is level ω.

Now, if you're starting with a strong function (like arrows) then the your ω is added to the strength of that function.

So if you start with nothing (like you did in the first version), the growth rate of your g(x) would simply be ω.

But in your second version, you've artificially set {a,b,c} to have the full power of up arrows! As it turns out, up arrows also have strength ω. So you get that for free.

Now again, each additional entry increases the level by one, so in this new system:

(a,b,c) is level ω by definition of up-arrows

(a,b,c,d) is level ω+1 (which is the level of Graham's Number)

(a,b,c,d,e) is level ω+2

and so on.

And this entire system's strength would be the limit of levels ω, ω+1, ω+2... which is - of course - ω+ω. This can also be written as ωx2.

So that's also the strength of g(n) in the second version.

As for the strengths of the two versions of f(n), I'll leave that as an exercise to you. :-)