Ordinal Array Notation

Ordinal Array Notation is a notation that can be used to express ordinals or be used in a hierarchy. It does this by grouping sets of infinities into brackets to define different notations. The way it looks is as follows:

[k(A,B,C,D,...)]

Where the numbers inside the bracket, as well as K, can determine the ordinal under a set of rules explained below.

Rules:
Rule 0: Infinitize is to increase an object or number, typically a notation, to omega

Rule #1: [0] = w

Rule #2: phi(n,0) = [n]

Rule #3: Placing a number in front of N, called K, will yield a magnitude of infinity equal to [n], in other words, will infinitize n

Ex: [2(0)] - has one input meaning to refer to first rule, 0 means Omega, and 2 means to infinitize omega once. This translates to phi(w,0). [3(0)] means phi(phi(phi...0)...0) an omega times. [4(0)] means to infinitize [2(0)], and so on.

Rule #4: Once the number in front of n reaches omega, add a second bracket number as 1

Rule #5: the second bracket number increases dependent on how many times the number in front of n is infinitized

Common Ordinals and their Notation form:

Omega = [0]

Epsilon = [1]

Cantor = [2]

Feferman = [3(0)]

Ackermann = [5(0)]

Small Veblen = [0,1]

Large Veblen = [0,3]

Psi(Omega_omega) = [1(0,3)]

Usage in NaN
(called Notational Exploding Ordinal Notation, or NEON for short)

Ordinal array notation can be placed in the Z position of NaN. For the first two rules, add one to the bracket and you will get Z. Then remove the brackets. However, for the rest of the rules, adding one will not matter that much and thus you may keep the full notation in place of Z. If you want to be really specific, you may place a +1 after the bracket in the Z position.