I proposed the fresh, fast-growing notation that grows stronger than the extended Buchholz's function, based off the fast-growing hierarchy.
There are multiple levels of the sequence notation as follows:
Basic pipeline sequence notation (BPS)
Couple pipeline sequence notation (CPS)
Dimensional pipeline sequence notation (DPS)
/// WARNING: INCOMPLETE ///
The notation consists of natural number enclosed in parentheses, followed by the sequence of natural numbers and square brackets enclosed between the two vertical bars ("|") An example of valid notation is (16)|[[[0, 3], 2].2 + [[0, 2], 2], 3], 0|
Let x to be a natural number. The sequences are enclosed in square brackets rather than vertical bars (not including vertical bars as separators). Here are the rules for the expansion:
(x)|| = (x)|0| = (x)|0, 0| = x^^x (empty sequence).
(x)|a + [0, 0], 0| = (x)|a + 1, 0| = (((...((((x)|a, 0|)|a, 0|)|a, 0|)...)|a, 0|)|a, 0|)|a, 0| with x copies of |a, 0|.
(x)|a[b + [0, 0], c], 0| = (x)|a[b + 1, c], 0| = (x)|a[b, c].x, 0| = (x)|a[b, c]+[b, c]+...+[b, c]+[b, c], 0| with x copies of [b, c]; where a is the rest of the expression.
(x)|a[b + [0, d + 1], c], 0| = (x)|a[b + [b + [... b + [b + [0, d], d], d ...], d], c], 0| with x copies of d; where a is the rest of the expression, b, c, f can include brackets, and d >= c.
(x)|a[0, [b + [0, d + 1], c]], 0| = ???
And here are the simplifications:
[0, 0] = 1
[0, 0].n = n
[a, b].0 = 0
[a, b].n = [a, b]+[a, b]+[a, b]...[a, b]+[a, b]+[a, b] with n copies of [a, b]
"+" sign can be omitted
[a] = [a, 0]; |a, 0| = |a| (the 0 in the second argument can be omitted)
Coming soon!