User blog comment:Mr. Junky/Question about BEAF/@comment-30754445-20170719015101

Short answer: it makes the numbers so much bigger, that a direct answer "in arithmetic terms" is impossible.

Really, the only way to properly understand how the notation works, is to go about it step-by-step.

First of all, {a,b,c} is not "a [b arrows] c", but rather "a [c arrows] b".

Secondly, before you try understanding 4-number arrays, you need to really understand 3-number arrays. Knowing that "{a,b,c} is a [c arrows] b" is not enough. You need to know how the array expansion process works, because the up-arrow analogy breaks down with arrays of 4 numbers and more.

Here are the expansion rules for 3 arrays:

1. {a,1,c} = a

2. {a,b,1} = ab

3. For c>1: {a,b,c} = {a,{a,b-1,c},c-1}

Try to expand {2,3,2} following the above rules. Repeat as necessary, until you reach a single ordinary number (which you already know should be equal to 2↑↑3).

If you do this successfully, we'll be ready for the next level. And before you know it, you'll be expanding 4-number arrays on your own in no time :-)