Notation Array Notation

Notation Array Notation, although it sounds redundant, is a very quickly growing function written by JTOnstead20.Notation Array Notation - Jonathan's Googology It catalogues all the levels of operation. It is also known as NaN (Not to be confused with "Not A Number"). The notation is written as the following: \((a\{X,Y\}b)\). The rules for the notation are as follows.
== Informal Rules ==
# follows a similar rule to that of the Hyper-L notation parenthesis
# Placed in a parenthesis \(()\)
# The first and last number \((3,3)\) tells you which numbers are on each side of the operation
# the brackets \({}\) tell you the notation level (addition is 1, mult and exp is 2, etc.)
# They also tell you which level of operation in the notation you are describing
An example:
\((4\{2,2\}2)\)
In this example, 4 and 2 are on both sides of the operation
2 tells you it is Up Arrow Notation
2 tells you there are 2 up arrows separating the 4 and 2
For beyond the 4th level, these are the new function rules that do not depend on any other function:
Each level of the first bracket number is an operational recursion of the previous function. For example, if you want to make large towers of exponents, you wouldn't write them out. Instead, you would use up arrow notation to make the towers. Likewise, eventually you will get to a point where you will need a function to build towers of up arrows in the same way you needed the up arrows to build the exponentiation towers. This function, also known as Conway Chains, already exists so it becomes level 3. Level 4 is defined as the following: Every 4th function is called an Alpha Function, denoted with an \(A\). \(A\) defines the instructions for building towers of Conway Notation. The first number in front of the \(A\), called \(N\), is the number repeated in the base of the Conway Tower an \(N\) amount of times. The number after the \(A\), called \(X\), defines the amount of towers of Conway Chains to build. For example, \(5A1\) means the following: \(5\rightarrow 5\rightarrow 5\rightarrow 5\rightarrow 5\) with one tower (so this is the correct expansion). Another example: 3A100 which is equal to \(3\rightarrow 3\rightarrow 3\rightarrow ...\rightarrow 3\) with a \(3A99\) amount of chains in between. \(3AA3\) is \(3A(3A(3A3)))\) using the rules of recursion. More advanced versions of A exist such as: \(4AA(100)AA4\) which contains \(100 A's\) between the two \(4's\). 
5th function is defined as beta. It does the same exact thing as A does to Conway Notation: \(5B1\) is \(5AAAAA5\), \(3B100\) is \(3AA...AA3\) with \(3B99\) \(A's\) in between. \(3BB3\) is \(3B(3B(3B))\), and \(4BB(100)BB4\) contains 100 \(A's\) between the two fours. 
For more info on this notation, see Greek Notation please.
== Formal Definition (FGH) ==
FGH follows like this: \(1,2,3,4...\omega,\omega+1,\omega+2,\omega+3...\omega2,\omega3,\omega4...\) So, every time \(\omega\) upgrades, a new notation has been achieved. The \(0,1\) set is the first level of NaN as it includes addition, multiplication, and exponentiation. \(2,3,4,5...\) is the next step as it includes all of up arrow notation. \(\omega+1,\omega+2,\omega+3\) includes chained arrow notation, so that's level 3. Chained arrows are also included in \(\omega2\), so \(\omega+1,\omega+2,\omega+3...\omega2\) are all in level 3. Alpha notation would start at w2 and end at w^2. Beta is at w^2...(w^w), and so forth. When the ordinal of FGH changes, the \(Z\) (final bracket number) changes as well by one (see Three Bracket Notation).
Definition of Y: Y is defined as the set 1,2,3,... after the ordinal value at that particular X level. For example, in (a{4,Y}b), 4 represents the X level f_w2 ... f_w^2. The base value is always a y of one, so f_w2 is has a y of 1. If the function is f_w2+1, then the value of y is 2. The same goes for w^2...w^w. w^2 is equal to a y of 1, while w^3 is 2 and w^4 is 3. The significant operation between the base level of X and its limit iterated by n times will yield a result of Y equal to n with the manipulation of one in either direction based off of the base value. 
Full List: In terms of FGH.
Z=1 (Omega)
# 0,1,2 (addition, multiplication, exponentiation)
# 3,4,5...w (Up-arrow notation)
# w+1,w+2,w+3...w2...w^2 (Chained arrow notation)
# w^2...w^3 (Alpha)
# w^3... w^w (Beta)
# w^w, w^w+1...w2^w (Gamma)
# w2^w...w^w2
# w^w2+1...w^w^2
# w^w^2...w^w^3
# w^w^3... w^w^w
# w^w^w, w^w^w+1...w2^w^w
# w2^w^w...w^w2^w
# w^w2^w...w^w^w2
# w^w^w2...w^w^w^2
# w^w^w^2...w^w^w^3
# w^w^w^3...w^w^w^w
...
Z=2 (Epsilon)
# E0
# E1, E2, E3, E4, ... E(w)
# E(w)...E(w)
# E(w)...E(w2)
# E(w2)...E(w^2)
# E(w^2)...E(w2^w)
# E(w^w)...E(E0)
Veblen Hierarchy:
If psi(n,0) then Z= n+1. For example, if phi(100,0) then the expression in NaN would be Z= 101. This process goes on until psi(omega,1), which is the limit of three bracket NaN.
== Levels ==
The levels are:
Level One (addition,multiplication and exponentiation ) = (x{1,1}x)
Ex: (3{1,1}1) = 3+1 = 4
Level Two: (arrow ) = (x{2,n}x) 
Ex: (10{2,5}6) = 10^^^^^6 (^ is an up arrow)
Level Three: (Conway ) = (x{3,n}x)
Ex: (4{3,4}4) = 4->4->4->4 = Conway's Tetratet
Level 5: (Alpha) = (x{4,n}x)
Ex: (3{4,6}4) = \(3AAAAAA4\)
For those beyond notation:
Level 29: Not29 \((3\{29,3\}3)\)
Level 30: Not30 \((3\{30,3\}3)\)
== NaN Notation Extender ==
Comma NaN:
Comma NaN adds commas to the first number in NaN notation to define more numbers. The comma is unnecessary for notations up to chained arrow notation. In chained arrow notation, a good approximation of 4->5->6->7 is (4{3,3}7). However, adding commas will define all the numbers in the chain up until the last number (which always has to be B). Therefore, the example from earlier becomes (4,5,6{3,3}7). This comma notation also works for the Greek ordinal commas.
X NaN:
X NaN is a notation that adds another bracket to NaN to define a certain large number input in one of the places. An example of X NaN is (3{2,X}4). Let's say you have 3 and 4 separated by up arrows, and there are 4^^4 up arrows in between them. You can approximate using the next level up (in this case Chained Arrow Notation), but if you want a more exact answer, then you can use X NaN. The example would then become (3{2,X}4) (4{2,2}4).
== Sources ==
Category:Notations
Category:Notations by JTOnstead20
Category:Functions
Category:Array Notations