User blog:ArtismScrub/Generalization of chained arrow notation

This is a generalization from the relations between up-arrow notation and Peter Hurford's extended chained arrow notation.

Here, we can think of chained arrows as second-order up arrows, or up-arrows as 0th order chained arrows.

a↑b↑c = ab c

a→b→c = a ↑c b

Where am I going with this? Well, consider the following:

a →→ b = a → (a →→ b-1)

This is equivalent to a →2 b in Peter Hurford's extension, and this can continue towards higher →n, analogous to the rules of ↑n.

However, for the purpose of making this extra strong, we will use Cookiefonster's rules that allow mixed-order chained arrows, allowing for growth rate ωω instead of ω3.

NOW, the second order chained arrows:

a →2 b = a →b a

All other rules remain unchanged, so a →2 b → c is comparable to {a,b,c(1)2} in BEAF.

But, we can still mix the arrows further.

a →2→ b = a →2 (a →2→ b-1)

etc.

a →2→2 b = a →2→b a

etc.

a →3 b = a (→2)b a

a →n b = a (→n-1)b a

This essentially allows us to create arrays of arrows.

How strong is this at its limits? I suspect either ωω ω or ε0.