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.JTOnstead20, Ordinal Array Notation - Jonathan's Googology. 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)
== Sources ==
Category:Transfinite ordinals
Category:Notations
Category:Notations by JTOnstead20
Category:Unformalised
Category:Array Notations