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Greek notations extend chained arrow notation and themselves using the laws of recursion.[1][2]

Conway Notation is used to build powers of up arrow notation when it gets cumbersome to handle. But what occurs when Conway Notation becomes too cumbersome to handle? Greek notation can help with this by building towers in a similar way as Conway Chains build up arrow towers.

Basics[]

The first Greek function is called Alpha Function, denoted with an Α. Note that Α is the Greek letter alpha, not the Latin letter A. Α defines the instructions for building towers of Conway Notation.

The first number in front of the Α, called N, is the number repeated in the base of the Conway Tower an N amount of times. The number after the Α, called X, defines the amount of towers of Conway Chains to build.
For example, 5Α1 means the following:\(5\rightarrow5\rightarrow5\rightarrow5\rightarrow5\) with one tower (so this is the correct expansion).
Another example: 3Α100 which is equal to \(3\rightarrow3\rightarrow\cdots\rightarrow3\rightarrow3\) with a 3A99 amount of chains in between.
3ΑΑ3 is 3Α(3Α(3Α3))) using the rules of recursion. More advanced versions of Α exist such as: 4ΑΑ(100)ΑΑ4 which contains 100 Α's between the two 4s.

6th function is defined as beta. It does the same exact thing as Α does to Conway Notation: 5Β1 is 5ΑΑΑΑΑ5, 3Β100 is 3ΑΑ...ΑΑ3 with 3Β99 Α's in between. 3ΒΒ3 is 3Β(3Β(3Β)), and 4ΒΒ(100)ΒΒ4 contains 100 Α's between the two fours. 

Greek notation forms the basis of Notation Array Notation. The representation in NaN: 3Α3 is (3{5,1}3) for the two 3s in the function, 5 for the level in which Α is, and 1 is for having only one A. 3ΑΑ(100)ΑΑ3 is defined as (3{5,100}3). 3Β3 is (3{6,1}3).  
This is how the notation is used: n,qΑΑ...ΑΑx where n is the number in the base, and q is the amount of that number in the base. If the two numbers are the same, you don't need the comma. The Α's represent the level of the notation.
The 4 level in Conway Notation corresponds to one Α in the alpha notation (so by adding one A you increase the tower in a compounded way similar to that which adding a fifth chain to Conway chain notation increases the amount of arrow towers). Finally, x tells you the amount of towers (or in multi-Α numbers the expansion factor) of the particular function level. For example, a 2 in the x position would yield two towers. 

The Letters[]

Level 5: Alpha Notation which builds Conway Towers

Examples: 3ΑΑΑ3, 4ΑΑΑΑ4, 5ΑΑΑΑ5

Level 6: Beta Notation which builds Alpha Towers

Examples: 3ΒΒΒ3, 4ΒΒΒΒ4, etc. (not B)

Level 7: Gamma Notation (3ΓΓΓ3)

Level 8: Delta Notation (3ΔΔΔ3)

Level 9: Epsilon Notation (3ΕΕΕ3) (not E)

Level 10: Zeta Notation (3ΖΖΖ3) (not Z)

Level 11: Eta Notation (3ΗΗΗ3) (not H)

Level 12: Theta Notation (3ΘΘΘ3)

Level 13: Iota Notation (3ΙΙΙ3) (not I)

Level 14: Kappa Notation (3ΚΚΚ3) (not K)

Level 15: Lambda Notation (3ΛΛΛ3)

Level 16: Mu Notation (3ΜΜΜ3) (not M)

Level 17: Nu Notation (3ΝΝΝ3) (not N)

Level 18: Xi Notation (3ΞΞΞ3)

Level 19: Omicron Notation (3ΟΟΟ3) (not O)

Level 20: Pi Notation (3ΠΠΠ3)

Level 21: Rho Notation (3ΡΡΡ3) (not P)

Level 22: Sigma Notation (3ΣΣΣ3)

Level 23: Tau Notation (3ΤΤΤ3) (not T)

Level 24: Upsilon Notation (3ΥΥΥ3) (not Y)

Level 25: Phi Notation (3ΦΦΦ3)

Level 26: Chi Notation (3ΧΧΧ3) (not X)

Level 27: Psi Notation (3ΨΨΨ3)

Level 28: Omega Notation (3ΩΩΩ3)

Sources[]

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