/// WIP ///
This is my array notation started in June 2022. Parts up to XAN are also an ill-definedness fixed version of Username5243's array notation (UNAN), up to the limit of the "nested array notation" in UNAN. It is also semi-homogenous with my older DeepLineMadom's array notation.
We explain parts of notations together with my expectation of the limits:
Extension 1 - Basic Array Notation (BAN) - Takes simple arrays with one entry a[c]b. FGH level ω. The definition is exactly the same as the basic array notation in UNAN, and also mirrors the up-arrow notation as well, only fixed its ambiguity of the particular definition.
Extension 2 - Extended Array Notation (XAN) - Takes linear, dimensional, and hyperdimensional arrays. FGH level ε0.
Extension 2 covers all of the first seven parts of UNAN, namely 2-entry array notation (from a[0, 1]b to a[0, 0, 1] b), linear array notation (from a[0, 0, 1]b to a[0 {1} 1] b), planar array notation (from a[0 {1} 1]b to a[0 {2} 1] b), dimensional array notation (from a[0 {2} 1]b to a[0 {0, 1} 1] b), hyperdimensional array notation (from a[0 {0 {} 1} 1]b to a[0 {0, 1} 1] b), and nested array notation (from a[0 {0 {1} 1} 1]b to a[0 {0,_1 1} 1] b where ,_1 indicates the first-order comma) in addition to the basic array notation level.
Starting from this stage, this notation have diverged from UNAN.
Extension 3 - First-order Array Notation (FoAN) - Adds the first-order separators. FGH level Bachmann-Howard ordinal (BHO, ψ0(ε(Ω+1)) = ψ0(Ω_2)).
Extension 4 - Multi-order Array Notation (MoAN) - Adds some greater nth-order separators. FGH level ψ0(Ω_ω).
Extension 5 - Hyper-order Array Notation (HoAN) - Adds some arrays into separators' subscript called "hyper-nth-order separators". FGH level BO = ψ0(Φ_1(0))]
Starting from Chapter 6, I will use Rathjen's psi for ordinals as well.
Extension 6 - Expanding Array Notation (XpaAN) - Adds the super-order separators which ranks over the subscript fixed point. FGH level ψ(M_2) = ψ(I(M+1)).
Extension 7 - Extended Expanding Array Notation (XXpaAN) - Adds separators with multiple characters and superscripts. FGH level ψ(M(1,0)) = ψ(M(2;0)) (also called Mahlo fixed point).
Extension 8 - Exploding Array Notation (XpoAN)
Extension 9 - Detonating Array Notation (DeAN)
Extension 10 - Nuclear Array Notation (NAN)
TBA