Introduction

As we introduced the Second-order Array Notation, we can apply the similar rules to the Higher-order Array Notation right now.

Rules

• Reuse the previous rules (including similar rules).

• In this part, I introduce the new symbol called \ (backslash). The \ symbol is also called "ranking separator". It works similarly to the ,, in strong array notation (SAN).

  • When you have a backslash, it works like this: First find the "diagonalization part" of the separator with the \. To do this, look for the first part separated by \'s that doesn't start with 1 (if it just starts with 1, decompose it normally). Then replace the last n by n-1, and the previous 1 becomes the "nesting point" - that's where the layers are repeated.

So 1{1 \ 2}2 becomes nests of 1{1{x \ 1}2}2, while 1{1{1 \ 2}3}2 becomes nests of 1{1{x}2{1 \ 2}2}2. 1{1 \ 1 \ 2}3 becomes nests of 1{1 \ 1{1 \ x \ 2}2}2{1 \ 1 \ 2}2, and so on.

In fact, {x \ 1} is usually equal {x}, {x \ 2} is normally <x>, {x \ 3} is normally <<x>>, {x \ y} is normally <<<...<<<x>>>...>>> with y-1 pairs of angle brackets, and so on. And {0 \ a} corresponds to Ω_(a-1) for 1 ≤ a < 1,2, but Ω_a for a ≥ 1,2. This is not always the case, however.

Examples

• 1{1 \ 2}2 = 1{1 / 2}2

• 1{2 \ 2}2 = 1{2 / 2}2

• 1{1 {1 {1 \ 2} 2} 2 \ 2} = 1{1{1 / 2}2 / 2}2

• 1{1 {1 \ 2} 2 \ 2} = 1{1 / 3}2

• 1{1 {1 {1 \ 2} 3} 1 {1 \ 2} 2 \ 2} = 1{1{1 / 3}2 / 3}2

• 1{1 {1 \ 2} 3 \ 2}2 = 1{1 / 4}2

• 1{1 {1 \ 2} 1,2 \ 2}2 = 1{1 / 1,2}2

• 1{1 {1 \ 2} 1 {1 {1 \ 2} 2} 2 \ 2}2 = 1{1 / 1{1 / 2}2}2

• 1{1 {1 \ 2} 1 {1 \ 2} 2 \ 2}2 = 1{1 / 1 / 2}2

• 1{1 {1 \ 2} 1 {1 \ 2} 1 {1 \ 2} 2 \ 2}2 = 1{1 / 1 / 1 / 2}2

• 1{1 {2 \ 2} 2 \ 2}2 = 1{1 <2> 2}2

• 1{1 {1 {1 {1 \ 2} 2} 2 \ 2} 2 \ 2}2 = 1{1 <1{1 / 2}2> 2}2

• 1{1 {1 {1 \ 2} 2 \ 2} 2 \ 2}2 = 1{1 <1 / 2> 2}2

• 1{1 {1 {1 \ 2} 1 {1 \ 2} 2 \ 2} 2 \ 2}2 = 1{1 <1 / 1 / 2> 2}2

• 1{1 {1 {1 {1 \ 2} 2 \ 2} 2 \ 2} 2 \ 2}2 = 1{1 <1 <1 / 2> 2> 2}2

• 1{1 \ 3}2 = 1{1 // 2}2

• 1{1 \ 4}2 = 1{1 /// 2}2

• 1{1 \ 5}2 = 1{1 //// 2}2

Analysis

Here're comparisons between my array notation and FGH. If separator A has recursion level α, then a[A]b has growth rate ω^ω^α.

• 1{1 \ 2}2 has level ε0

• 1{2 \ 2}2 has level ε0^ω = ω^ω^(ε0+1)

• 1{1 {1 {1 \ 2} 2} 2 \ 2} has level ε0^ε0 = ω^ω^(ε0*2)

• 1{1 {1 \ 2} 2 \ 2}2 has level ε1

• 1{1 {1 \ 2} 3 \ 2}2 has level ε2

• 1{1 {1 \ 2} 1,2 \ 2}2 has level εω

• 1{1 {1 \ 2} 1 {1 {1 \ 2} 2} 2 \ 2}2 has level ε(ε0)

• 1{1 {1 \ 2} 1 {1 \ 2} 2 \ 2}2 has level ζ0

• 1{1 {1 \ 2} 2 {1 \ 2} 2 \ 2}2 has level ε(ζ0+1)

• 1{1 {1 \ 2} 1 {1 \ 2} 3 \ 2}2 has level ζ1

• 1{1 {1 \ 2} 1 {1 \ 2} 1,2 \ 2}2 has level ζω

• 1{1 {1 \ 2} 1 {1 \ 2} 1 {1 {1 \ 2} 1 {1 \ 2} 2} 2 \ 2}2 has level ζ(ζ0)

• 1{1 {1 \ 2} 1 {1 \ 2} 1 {1 \ 2} 2 \ 2}2 has level η0

• 1{1 {1 \ 2} 1 {1 \ 2} 1 {1 \ 2} 1 {1 \ 2} 2 \ 2}2 has level φ(4,0)

• 1{1 {2 \ 2} 2 \ 2}2 has level φ(ω,0)

• 1{1 {1 {1 {1 \ 2} 2} 2 \ 2} 2 \ 2}2 has level φ(ε0,0)

• 1{1 {1 {1 \ 2} 2 \ 2} 2 \ 2}2 has level Γ0 = φ(1,0,0) (Feferman-Schütte ordinal)

• 1{1 {1 {1 \ 2} 2 \ 2} 3 \ 2}2 has level Γ1 = φ(1,0,1)

• 1{1 {1 {1 \ 2} 2 \ 2} 1 {1 \ 2} 2 \ 2}2 has level φ(1,1,0)

• 1{1 {1 {1 \ 2} 2 \ 2} 1 {2 \ 2} 2 \ 2}2 has level φ(1,ω,0)

• 1{1 {1 {1 \ 2} 2 \ 2} 1 {1 {1 \ 2} 2 \ 2} 2 \ 2}2 has level φ(2,0,0)

• 1{1 {2 {1 \ 2} 2 \ 2} 2 \ 2}2 has level φ(ω,0,0)

• 1{1 {1 {1 \ 2} 3 \ 2} 2 \ 2}2 has level φ(1,0,0,0) (Ackermann ordinal)

• 1{1 {1 {1 \ 2} 4 \ 2} 2 \ 2}2 has level φ(1,0,0,0,0)

• 1{1 {1 {1 \ 2} 1,2 \ 2} 2 \ 2}2 has level ψ(Ω^Ω^ω) (small Veblen ordinal)

• 1{1 {1 {1 \ 2} 1 {1 \ 2} 2 \ 2} 2 \ 2}2 has level ψ(Ω^Ω^Ω) (large Veblen ordinal)

• 1{1 {1 {1 {1 \ 2} 2 \ 2} 2 \ 2} 2 \ 2}2 has level ψ(Ω^Ω^Ω^Ω)

• 1{1 \ 3}2 has level ψ(ε(Ω+1)) = ψ(Ω_2) (Bachmann-Howard ordinal)

• 1{1 \ 4}2 has level ψ(ε(Ω_2+1)) = ψ(Ω_3)

• 1{1 \ 5}2 has level ψ(ε(Ω_3+1)) = ψ(Ω_4)

So 1{1 \ 1,2}2 has level ψ(Ω_ω), known as Buchholz's ordinal.

Moving on:

• 1{2 \ 1,2}2 has level ψ(Ω_ω)^ω = ω^ω^(ψ(Ω_ω)+1)

• 1{1 {1 {1 \ 1,2} 2} 2 \ 1,2}2 has level ψ(Ω_ω)^ψ(Ω_ω) = ω^ω^(ψ(Ω_ω)*2)

• 1{1 {1 \ 2} 2 \ 1,2}2 has level ψ(Ω_ω+Ω) = ε(ψ(Ω_ω)+1)

• 1{1 {1 \ 2} 3 \ 1,2}2 has level ψ(Ω_ω+Ω2) = ε(ψ(Ω_ω)+2)

• 1{1 {1 \ 2} 1 {1 {1 \ 2} 2} 2 \ 1,2}2 has level ψ(Ω_ω+Ωψ(Ω_ω)) = ε(ψ(Ω_ω)*2)

• 1{1 {1 \ 2} 1 {1 \ 2} 2 \ 1,2}2 has level ψ(Ω_ω+Ω^2) = ζ(ψ(Ω_ω)+1)

• 1{1 {2 \ 2} 2 \ 1,2}2 has level ψ(Ω_ω+Ω^ω) = φ(ω,ψ(Ω_ω)+1)

• 1{1 {1 {1 {1 \ 1,2} 2} 2 \ 2} 2 \ 1,2}2 has level ψ(Ω_ω+Ω^ψ(Ω_ω)) = φ(ψ(Ω_ω),1)

• 1{1 {1 {1 \ 2} 2 \ 2} 2 \ 1,2}2 has level ψ(Ω_ω+Ω^Ω) = Γ(ψ(Ω_ω)+1)

• 1{1 {1 {1 \ 2} 1 {1 \ 2} 2 \ 2} 2 \ 1,2}2 has level ψ(Ω_ω+Ω^Ω^Ω)

• 1{1 {1 {1 {1 \ 2} 2 \ 2} 2 \ 2} 2 \ 1,2}2 has level ψ(Ω_ω+Ω^Ω^Ω^Ω)

• 1{1 {1 {1 \ 3} 2 \ 2} 2 \ 1,2}2 has level ψ(Ω_ω+ψ_1(Ω_2)) = ψ(Ω_ω+ε(Ω+1))

• 1{1 {1 {1 \ 3} 3 \ 2} 2 \ 1,2}2 has level ψ(Ω_ω+ψ_1(Ω_2*2))

• 1{1 {1 {1 \ 3} 1,2 \ 2} 2 \ 1,2}2 has level ψ(Ω_ω+ψ_1(Ω_2*ω))

• 1{1 {1 {1 \ 3} 1 {1 \ 3} 2 \ 2} 2 \ 1,2}2 has level ψ(Ω_ω+ψ_1(Ω_2^2))

• 1{1 {1 {1 {1 \ 3} 2 \ 3} 2 \ 2} 2 \ 1,2}2 has level ψ(Ω_ω+ψ_1(Ω_2^Ω_2))

• 1{1 {1 \ 3} 2 \ 1,2}2 has level ψ(Ω_ω+Ω_2)

• 1{1 {1 \ 4} 2 \ 1,2}2 has level ψ(Ω_ω+Ω_3)

• 1{1 {1 \ 5} 2 \ 1,2}2 has level ψ(Ω_ω+Ω_4)

• 1{1 {1 \ 1,2} 2 \ 1,2}2 has level ψ(Ω_ω*2)

• 1{1 {1 \ 2} 2 {1 \ 1,2} 2 \ 1,2}2 has level ψ(Ω_ω*2+Ω)

• 1{1 {1 \ 3} 2 {1 \ 1,2} 2 \ 1,2}2 has level ψ(Ω_ω*2+Ω_2)

• 1{1 {1 \ 1,2} 3 \ 1,2}2 has level ψ(Ω_ω*3)

• 1{1 {1 \ 1,2} 1,2 \ 1,2}2 has level ψ(Ω_ω*ω)

• 1{1 {1 \ 1,2} 1 {1 \ 2} 2 \ 1,2}2 has level ψ(Ω_ω*Ω)

• 1{1 {1 \ 1,2} 1 {1 \ 3} 2 \ 1,2}2 has level ψ(Ω_ω*Ω_2)

• 1{1 {1 \ 1,2} 1 {1 \ 1,2} 2 \ 1,2}2 has level ψ(Ω_ω^2)

• 1{1 {1 \ 1,2} 1 {1 \ 1,2} 1 {1 \ 1,2} 2 \ 1,2}2 has level ψ(Ω_ω^3)

• 1{1 {2 \ 1,2} 2 \ 1,2}2 has level ψ(Ω_ω^ω)

• 1{1 {1 {1 \ 2} 2 \ 1,2} 2 \ 1,2}2 has level ψ(Ω_ω^Ω)

• 1{1 {1 {1 \ 1,2} 2 \ 1,2} 2 \ 1,2}2 has level ψ(Ω_ω^Ω_ω)

• 1{1 {1 {1 \ 1,2} 1 {1 \ 1,2} 2 \ 1,2} 2 \ 1,2}2 has level ψ(Ω_ω^Ω_ω^Ω_ω)

• 1{1 {1 {1 {1 \ 1,2} 2 \ 1,2} 2 \ 1,2} 2 \ 1,2}2 has level ψ(Ω_ω^Ω_ω^Ω_ω^Ω_ω)

So 1{1 \ 2,2}2 separator has level ψ(ε(Ω_ω+1)), which is the Takeuti-Feferman-Buchholz ordinal. It eventually dominates all functions probably recursive in Π1^1-CA0+BI, such as the limit of the Buchholz hydra (BH(n)), the upper bounds of subcubic graph number (SCG(n)) and simple subcubic graph number (SSCG(n)). It is also equal to ψ(Ω_(ω+1)) using extended Buchholz's function.

Let's speed up!

• 1{1 \ 3,2}2 has level ψ(Ω_(ω+2))

• 1{1 \ 4,2}2 has level ψ(Ω_(ω+3))

• 1{1 \ 1,3}2 has level ψ(Ω_(ω2))

• 1{1 \ 1,4}2 has level ψ(Ω_(ω3))

• 1{1 \ 1,1,2}2 has level ψ(Ω_(ω^2))

• 1{1 \ 1{2}2}2 has level ψ(Ω_(ω^ω))

• 1{1 \ 1 {1 {1 \ 2} 2} 2}2 has level ψ(Ω_(ε0))

• 1{1 \ 1 {1 {1 \ 1,2} 2} 2}2 has level ψ(Ω_(ψ(Ω_ω)))

• 1{1 \ 1 {1 {1 \ 1 {1 {1 \ 1,2} 2} 2} 2} 2}2 has level ψ(Ω_(ψ(Ω_ψ(Ω_ω))))

• 1{1 \ 1 {1 {1 \ 1 {1 {1 \ 1 {1 {1 \ 1,2} 2} 2} 2} 2} 2} 2}2 has level ψ(Ω_(ψ(Ω_ψ(Ω_ψ(Ω_ω)))))

• 1{1 \ 1 {1 \ 2} 2}2 has level ψ(Ω_Ω)

• 1{1 \ 2 {1 \ 2} 2}2 has level ψ(Ω_(Ω+1))

• 1{1 \ 1 {1 \ 2} 3}2 has level ψ(Ω_(Ω2))

• 1{1 \ 1 {1 \ 2} 1 {1 \ 2} 2}2 has level ψ(Ω_(Ω^2))

• 1{1 \ 1 {1 {1 \ 2} 2 \ 2} 2}2 has level ψ(Ω_(Ω^Ω))

• 1{1 \ 1 {1 {1 \ 3} 2 \ 2} 2}2 has level ψ(Ω_(ψ1(Ω_2)))

• 1{1 \ 1 {1 {1 \ 1,2} 2 \ 2} 2}2 has level ψ(Ω_(ψ1(Ω_ω)))

• 1{1 \ 1 {1 \ 3} 2}2 has level ψ(Ω_Ω_2)

• 1{1 \ 1 {1 \ 4} 2}2 has level ψ(Ω_Ω_3)

• 1{1 \ 1 {1 \ 1,2} 2}2 has level ψ(Ω_Ω_ω)

• 1{1 \ 1 {1 \ 1 {1 \ 2} 2} 2}2 has level ψ(Ω_Ω_Ω)

• 1{1 \ 1 {1 \ 1 {1 \ 1 {1 \ 2} 2} 2} 2}2 has level ψ(Ω_Ω_Ω_Ω)

• 1{1 \ 1 {1 \ 1 {1 \ 1 {1 \ 1 {1 \ 2} 2} 2} 2} 2}2 has level ψ(Ω_Ω_Ω_Ω_Ω)

And finally, the limit of the Primitive Expanding Array Notation is ψ(Φ(1,0)) with respect to the extended Buchholz's function, and Φ(1,0) is the least omega fixed point, known as the countable limit of extended Buchholz's function.