Ordinal Array Notation

Ordinal Array Notation was a notation created by JTOnstead20 under the name of Jonathan, and was supposed to be able to be used to express ordinals or be used in a hierarchy. It was based on vague and inconsistent rules, and was ill-defined. It was based off of Veblen's function. It was expressed as follows:

[A,B,C,D,...]

Where the numbers inside the bracket can determine the ordinal under a set of "rules" explained below.

Later, Googology Wiki user DeepLineMadom clarified in the blog post that the notation no longer meaningfully exist.

Rules

Rule 1: All rules that form the basis of Veblen's Phi notation form the basis of this array notation

Rule 2: Phi notation can be directly converted to OAN by the following property: phi(x,0) = [x+1]. Set omega to 1 in OAN.

Example: phi(3,0) = [4]

Rule 3: Extended phi notation can be directly converted to OAN by the following property: phi(x,0,0,...0) (with n 0) = [x,n]

Example: phi(1,0,0,0) = [1,3]

Rule 4: As n approaches omega, make symbol Q that can be used as a base as an extension to extended phi notation. Q^n can be converted into [x,n] and Q^w is directly convertable to [x,n,1]. Q is based off of Weiermann's theta function where the Qs in the tower are equal to the uppercase Omega in the theta function.

Example: phi(1,0,0,0...0) (with w 0) = Q^w = [1,1,1]. 

Rule 5: Adding more Qs to the power tree of Q will increase the third number in OAN by 1 every time.

Example: Q^Q^Q^Q = [1,1,4]

Rule 6: Once a power tower of Q reaches a height of omega or over, add a fourth number to the OAN bracket

Example: Q^Q^Q^Q^...Q (with w Q) = [1,1,1,1]

Rule 7: The final rule is one of truncation. A series of 1's equal to [1,1,...,1] with n 1s is represented as [1^n] in the notation.

Example: [1^5] = [1,1,1,1,1]

Ordinals originally given as examples

• Omega = [0]
• Epsilon = [1]
• Cantor = [2]
• Feferman = [1,1]
• Ackermann = [1,3]
• Small Veblen = [1,1,1]
• Large Veblen = [1,1,2]
• Subcubic = [1,1,1,1]
• Rathjen = [1,1,1,1,1] = [1^5]

WARNING: Although JTOnstead20 called ψ_0(Ω_ω) and the countable limit of Extended Buchholz's function "Subcubic" and "Rathjen" respectively, there are not such common names. In particular, the latter one is irrelevant to Rathjen.

Extension

Despite being ill-defined beyond [1,1,1,1], JTOnstead20 gives another set of rules on how to extend arrays to transfinite numbers of entries:

[1^[n]] = [1^1,n]

[1^ω,1^ω,1^ω,...] with ω "1^ω"s = [1^(1+1)]

[1^(1+n),1^(1+n),1^(1+n),...] with ω "1^(1+n)"s = [1^(1+(n+1))]

[1^(1+ω)] = [1^2]

[1^(ω+ω)] = [1^2ω], and so on through the main hierarchy

[1^(1^1)] = [1^1^1] = [1^[1^1]] = [1^^3]

"If 1^1^...1 is repeated ω amount of times, then the notation is [1^^1]."

"Once you get to [1^^(1^1^1...1)], use three up arrows"

[1{ω}a] is the limit of this notation.

Hyper-Extension

JTOnstead20 later gave even more vague "Hyper-Extended rules":

Represent the recursion path used in extended notation as A, which is equal to [1{ω}a].

"B is equal to additional recursion using the extended notation rules with a ω amount of using a new way of expressing the notation once a number reaches ω" (what does this mean?)

[1->1] is "the supremum of all recursion" (probably referring to the Church-Kleene ordinal)