/// 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.
It was attempted to fix this notation five times before the notation has been eventually created.
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