User blog:Eners49/New Notation Idea?

Here is a new notation I am thinking about/developing. I'm trying to make it go beyond order type w^w and maybe even epsilon-zero in the fast-growing hierarchy. It is based off of PlantStar's Notation a little bit.

Basic first-order brackets
Here are the basics. With [64], we can already surpass Graham's number. You may be thinking, this is just a shitty copy of Graham's function, but with 4s! But wait, there's more. Basically, [a, b] = [...[[[a]...]]] with b bracket sets. But what about more elements in the brackets? Let's find out. As you already notice, [a, b, c] = a, c], [b, c. So [a, b, c, d] = a, b, d], [a, c, d], [b, c, d. As you guys (and girls!) can see, w^w in the fast-growing array is already beginning to look bad. You also probably get the point for 6-entry, 7-entry, etc. brackets. But what comes next? 50 1's would be:
 * [0] = 4
 * [1] = 4^^^^4
 * [2] = 4^^^^...^^^^4 with [1] arrows
 * [3] = 4^^^^...^^^^4 with [2] arrows
 * [1, 1] = [...[[[1]...]]] with [1] bracket sets
 * [5, 10] = [...[[[5]...]]] with [10] bracket sets
 * [69, 666] = [...[[[69]...]]] with [666] bracket sets
 * [1, 1, 1] = 1, 1], [1, 1
 * [69, 666, 420] = 69, 420], [666, 420
 * [1, 1, 1, 1] = 1, 1, 1], [1, 1, 1
 * [69, 666, 420, 1337] = 69, 666, 1337], [69, 420, 1337], [666, 420, 1337
 * [1, 1, 1, 1, 1] = 1, 1, 1, 1], [1, 1, 1, 1], [1, 1, 1, 1], [1, 1, 1, 1
 * [69, 666, 420, 1337, 1000000] = 69, 666, 420, 1000000], [69, 666, 1337, 1000000], [69, 420, 1337, 1000000], [666, 420, 1337, 1000000

[1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]

Now, this is getting pretty tedious and obsolete. So we're going to switch to "second-order" brackets.

Second-order brackets and beyond
We can use second-order brackets to represent the numbers like the large string of 1's above. We can say: In general, 2[a] = [1, 1, 1, ..., 1, 1, 1] with a 1's. You could also have: Now we're going to see what 2[1, 1] would look like and why it is different from 21, 1. So 2[a, b] = 2a, a, a, ..., a, a, a with b a's. And so it goes on for 3-entry, 4-entry, etc. second-order brackets. But, eventually there arises a need for third-order brackets and beyond as well. In general, a[b] = a-1[1, 1, 1, ..., 1, 1, 1] with b 1's. And this can be used to extreme lengths. You could even have a [1]th order bracket, or a 2[5]th order bracket!! How mind-boggling is that! It would look like this: You could extend that even further, but I'm going to stop here for now. Tell me what you guys think in the comments!
 * 2[5] (not to be confused with the much smaller Mega) = [1, 1, 1, 1, 1]
 * 2[10] = [1, 1, 1, 1, 1, 1, 1, 1, 1, 1]
 * 25 = [1, 1, 1, ..., 1, 1, 1] with [5] 1's
 * 21, 1 (BE CAREFUL as 2[1, 1] and 21, 1 are two different things!!) = [1, 1, 1, ..., 1, 1, 1] with [1, 1] 1's
 * 2[1, 1] = 21, 1, 1, ..., 1, 1, 1 with [1] 1's = 2[21]
 * 2[69, 420] = 269, 69, 69, ..., 69, 69, 69 with 420 69's
 * 2[1, 1, 1] = 2[1, 1], [1, 1], [1, 1], ..., [1, 1], [1, 1], [1, 1] with [1] [1, 1]'s
 * 2[69, 666, 420] = 2[69, 666], [69, 666], [69, 666], ..., [69, 666], [69, 666], [69, 666] with [420] [69, 666]'s
 * 3[a] = 2[1, 1, 1, ..., 1, 1, 1] with a 1's
 * 4[a] = 3[1, 1, 1, ..., 1, 1, 1] with a 1's
 * [1][1] = [1] - 1[1, 1, 1, ..., 1, 1, 1] with [1] 1's
 * [1][1][1] = [1][1] - 1[1, 1, 1, ..., 1, 1, 1] with [1] 1's

