Introduction to Almighty Array Notation

Almighty Array Notation (AAN) is a notation made by googology wiki user HaydenTheGoogologist2009. It has the format of a*(#) where # is an array. It comes in multiple parts. An example of a valid array is a*(10, 4, 1, 3, 5).

Unfortunately, I abandoned this notation on October 23th, 2023 with the statement "This notation is a chaos, and I don't think it's possible to fix the entire notation. I should probably take a break from making notations..."

Parts

We explain parts of notations together with my expectation of the limits:

• Section 1: Linear Array Notation (LAN): takes linear arrays. It's equivalent to BEAF (e. g. a*(10, 10, 10, 10)). The limit of this is approximated to ω^ω in the fast-growing hierarchy.

• Section 2: Extended Array Notation (ExAN): uses planer and dimensional arrays (e. g. a*(10, 10 {5} 2)). The limit of this is approximated to ω^ω^ω in the fast-growing hierarchy.

• Section 3: Cascading Array Notation (CAN): uses hyperdimensional arrays (e. g. a*(10, 10 {1, 2} 2)). The limit of this is approximated to ω^ω^ω^ω in the fast-growing hierarchy.

• Section 4: Nested Array Notation (NAN): uses nested arrays (e. g. a*(10, 10 {1 {2} 2} 2)). The limit of this is approximated to ε0 in the fast-growing hierarchy.

• Section 5: First-order Array Notation (FoAN): adds a separator called the first-order hyperseparators (! is the shorthand for <1> separator) (e. g. a*(10, 10 {1 ! 2} 2) = a*(10, 10 {1 <1> 2} 2)). The limit of this is approximated to ζ0 in the fast-growing hierarchy.

• Section 6: Extended First-order Array Notation (ExFoAN): uses linear expanding arrays with the first-order hyperseparators (e. g. a*(10, 10 {1 ! 1 ! 2} 2)). The limit of this is approximated to φ(ω,0) in the fast-growing hierarchy.

• Section 7: Hyper-Extended First-order Array Notation (HExFoAN): uses more than one copies of the first-order hyperseparators (e. g. a*(10, 10 {1 !! 2} 2)). The limit of this is approximated to Γ0 in the fast-growing hierarchy.

• Section 8: Mega-Extended First-order Array Notation (MExFoAN): adds inner hyperseparators to extend the first-order hyperseparators (e. g. a*(10, 10 {1 <1 ! 2> 2} 2)). The limit of this is approximated to ψ0(εΩ + 1) = ψ0(Ω2) with respect to the (extended) Buchholz's function in the fast-growing hierarchy.

For parts beyond MExFoAN, I was intended to create at least eleven other parts here and later abandoned by me:

• Section 9: Second-order Array Notation (SoAN)

• Section 10: Higher-order Array Notation (HoAN)

• Section 11: Beyond-order Array Notation (BoAN)

• Section 12: Primitive Ranking Array Notation (PRAN)

• Section 13: Recursive Ranking Array Notation (RRAN)

• Section 14: Exploding Array Notation (ExpAN)

• Section 15: Detonating Array Notation (DAN)

• Section 16: Nuclear Array Notation (NuAN)

• Section 17: Solar Array Notation (SAN)

• Section 18: Milky Array Notation (MAN)

• Section 19: Galactic Array Notation (GAN)

Etymology

The 'a' before the array in a*(#) stands for the word 'almighty'.

Comparison

For LAN, a*(#) = {#} in BEAF.