Introduction

Like with the previous chapter, we can apply the similar rules to the Primitive Expanding Array Notation right now.

Rules

• Reuse the previous rules (including similar rules).

• In this part, as we introduced \ (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.

Previous level 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

/// WIP ///

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.

OCF extension credited to: https://googology.fandom.com/wiki/User:DeepLineMadom/DeepLineMadom%27s_extension_of_the_extended_Buchholz%27s_function

• 1{1 \ 1 \ 2}2 has level ψ(I) = ψ(Φ(1,0)) = ψ(M)

• 1{2 \ 1 \ 2}2 has level ψ(I)^ω = ω^ω^(ψ(Φ(1,0))+1)

• 1{1 {1 {1 \ 1 \ 2} 2} 2}2 \ 1 \ 2}2 has level ψ(I)^ψ(I) = ω^ω^(ψ(Φ(1,0))*2)

• 1{1 {1 \ 2} 2 \ 1 \ 2}2 has level ψ(I+Ω) = ε(ψ(Φ(1,0))+1)

• 1{1 {1 \ 2} 1 {1 \ 2} 2 \ 1 \ 2}2 has level ψ(I+Ω^2) = ζ(ψ(Φ(1,0))+1)

• 1{1 {1 {1 \ 2} 2 \ 2} 2 \ 1 \ 2}2 has level ψ(I+Ω^Ω) = Γ(ψ(Φ(1,0))+1)

• 1{1 {1 {1 \ 3} 2 \ 2} 2 \ 1 \ 2}2 has level ψ(I+ψ_1(Ω_2)) = ψ(Φ(1,0)+ε(Ω+1))

• 1{1 {1 {1 \ 1,2} 2 \ 2} 2 \ 1 \ 2}2 has level ψ(I+ψ_1(Ω_ω)) = ψ(Φ(1,0)+ψ_1(Ω_ω))

• 1{1 {1 {1 \ 1 {1 \ 2} 2} 2 \ 2} 2 \ 1 \ 2}2 has level ψ(I+ψ_1(Ω_Ω)) = ψ(Φ(1,0)+ψ_1(Ω_Ω))

• 1{1 {1 {1 \ 1 \ 2} 2 \ 2} 2 \ 1 \ 2}2 has level ψ(I+ψ_1(I)) = ψ(Φ(1,0)+ψ_1(Φ(1,0)))

• 1{1 {1 \ 3} 2 \ 1 \ 2}2 has level ψ(I+Ω_2) = ψ(Φ(1,0)+Ω_2)

• 1{1 {1 {1 \ 1 \ 2} 2 \ 3} 2 \ 1 \ 2}2 has level ψ(I+ψ_2(I)) = ψ(Φ(1,0)+ψ_2(Φ(1,0)))

• 1{1 {1 \ 4} 2 \ 1 \ 2}2 has level ψ(I+Ω_2) = ψ(Φ(1,0)+Ω_3)

• 1{1 {1 \ 1,2} 2 \ 1 \ 2}2 has level ψ(I+Ω_ω) = ψ(Φ(1,0)+Ω_ω)

• 1{1 {1 \ 1 {1 \ 2} 2} 2 \ 1 \ 2}2 has level ψ(I+Ω_Ω) = ψ(Φ(1,0)+Ω_Ω)

• 1{1 {1 \ 1 {1 \ 1 \ 2} 2} 2 \ 1 \ 2}2 has level ψ(I+ψ_I(I)) = ψ(Φ(1,0)*2)

• 1{1 {1 \ 2 {1 \ 1 \ 2} 2} 2 \ 1 \ 2}2 has level ψ(I+ψ_I(I+1)) = ψ(ε(Φ(1,0)+1)) = ψ(Ω(Φ(1,0)+1))

• 1{1 {1 \ 1 {1 \ 1 {1 \ 1 \ 2} 2} 2 {1 \ 1 \ 2} 2} 2 \ 1 \ 2}2 has level ψ(I+ψ_I(I+ψ_I(I))) = ψ(Ω(Φ(1,0)2))

• 1{1 {1 \ 1 {1 \ 2 {1 \ 1 \ 2} 2} 2 {1 \ 1 \ 2} 2} 2 \ 1 \ 2}2 has level ψ(I+ψ_I(I+ψ_I(I+1))) = ψ(Ω(Ω(Φ(1,0)+1)))

• 1{1 {1 \ 1 \ 2} 2 \ 1 \ 2}2 has level ψ(I2) = ψ(Φ(1,1))

• 1{1 {1 \ 1 {1 \ 1 \ 2} 3 \ 2} 2 \ 1 \ 2}2 has level ψ(I2+ψ_I(I2)) = ψ(Φ(1,1)2)

• 1{1 {1 \ 2 {1 \ 1 \ 2} 3 \ 2} 2 \ 1 \ 2}2 has level ψ(I2+ψ_I(I2+1)) = ψ(Ω(Φ(1,1)+1))

• 1{1 {1 \ 1 {1 \ 1 {1 \ 1 \ 2} 3 \ 2} 3 \ 2} 2 \ 1 \ 2}2 has level ψ(I2+ψ_I(I2+ψ(I2))) = ψ(Ω(Φ(1,1)2))

• 1{1 {1 \ 1 \ 2} 3 \ 1 \ 2}2 has level ψ(I3) = ψ(Φ(1,2))

• 1{1 {1 \ 1 \ 2} 4 \ 1 \ 2}2 has level ψ(I4) = ψ(Φ(1,3))

• 1{1 {1 \ 1 \ 2} 1,2 \ 1 \ 2}2 has level ψ(Iω) = ψ(Φ(1,ω))

• 1{1 {1 \ 1 \ 2} 1 {1 {1 \ 1 \ 2} 2} 2 \ 1 \ 2}2 has level ψ(Iψ(I)) = ψ(Φ(1,ψ(Φ(1,0))))

• 1{1 {1 \ 1 \ 2} 1 {1 \ 2} 2 \ 1 \ 2}2 has level ψ(IΩ) = ψ(Φ(1,Ω))

• 1{1 {1 \ 1 \ 2} 2 {1 \ 2} 2 \ 1 \ 2}2 has level ψ(IΩ+I) = ψ(Φ(1,Ω+1))

• 1{1 {1 \ 1 \ 2} 1 {1 \ 2} 3 \ 1 \ 2}2 has level ψ(IΩ2) = ψ(Φ(1,Ω2))

• 1{1 {1 \ 1 \ 2} 1 {2 \ 2} 2 \ 1 \ 2}2 has level ψ(IΩ^ω) = ψ(Φ(1,Ω^ω))

• 1{1 {1 \ 1 \ 2} 1 {1 \ 3} 2 \ 1 \ 2}2 has level ψ(IΩ_2) = ψ(Φ(1,Ω_2))

• 1{1 {1 \ 1 \ 2} 1 {1 \ 1,2} 2 \ 1 \ 2}2 has level ψ(IΩ_ω) = ψ(Φ(1,Ω_ω))

• 1{1 {1 \ 1 \ 2} 1 {1 \ 1 {1 \ 2} 2} 2 \ 1 \ 2}2 has level ψ(IΩ_ω) = ψ(Φ(1,Ω_Ω))

• 1{1 {1 \ 1 \ 2} 1 {1 \ 1 {1 \ 1 \ 2} 2} 2 \ 1 \ 2}2 has level ψ(Iψ_I(I)) = ψ(Φ(1,Φ(1,0)))

• 1{1 {1 \ 1 \ 2} 1 {1 \ 2 {1 \ 1 \ 2} 2} 2 \ 1 \ 2}2 has level ψ(Iψ_I(I+1)) = ψ(Φ(1,Ω(Φ(1,0)+1)))

• 1{1 {1 \ 1 \ 2} 1 {1 \ 1 {1 \ 1 \ 2} 3} 2 \ 1 \ 2}2 has level ψ(Iψ_I(I2)) = ψ(Φ(1,Φ(1,1)))

• 1{1 {1 \ 1 \ 2} 1 {1 \ 1 {1 \ 1 \ 2} 1,2} 2 \ 1 \ 2}2 has level ψ(Iψ_I(Iω)) = ψ(Φ(1,Φ(1,ω)))

• 1{1 {1 \ 1 \ 2} 1 {1 \ 1 {1 \ 1 {1 \ 1 \ 2} 2} 2} 2 \ 1 \ 2}2 has level ψ(Iψ_I(Iψ_I(I))) = ψ(Φ(1,Φ(1,Φ(1,0))))

• 1{1 {1 \ 1 \ 2} 1 {1 \ 1 \ 2} 2 \ 1 \ 2}2 has level ψ(I^2) = ψ(Φ(2,0))

• 1{1 {1 \ 1 {1 \ 1 \ 2} 2} 2 {1 \ 1 \ 2} 1 {1 \ 1 \ 2} 2 \ 1 \ 2}2 has level ψ(I^2+ψ_I(I))

• 1{1 {1 \ 1 {1 \ 1 \ 2} 1 {1 \ 1 \ 2} 2} 2 {1 \ 1 \ 2} 1 {1 \ 1 \ 2} 2 \ 1 \ 2}2 has level ψ(I^2+ψ_I(I^2))

• 1{1 {1 \ 1 \ 2} 2 {1 \ 1 \ 2} 2 \ 1 \ 2}2 has level ψ(I^2+I) = ψ(Φ(1,Φ(2,0)+1))

• 1{1 {1 \ 1 \ 2} 1 {1 \ 1 {1 \ 1 \ 2} 1 {1 \ 1 \ 2} 2} 2 {1 \ 1 \ 2} 2 \ 1 \ 2}2 has level ψ(I^2+Iψ_I(I^2)) = ψ(Φ(1,Φ(2,0)2))

• 1{1 {1 \ 1 \ 2} 1 {1 \ 1 \ 2} 3 \ 1 \ 2}2 has level ψ(I^2*2) = ψ(Φ(2,1))

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

• 1{1 {1 \ 1 \ 2} 1 {1 \ 1 \ 2} 1 {1 \ 2} 2 \ 1 \ 2}2 has level ψ(I^2*Ω) = ψ(Φ(2,Ω))

• 1{1 {1 \ 1 \ 2} 1 {1 \ 1 \ 2} 1 {1 \ 1 {1 \ 1 \ 2} 2} 2 \ 1 \ 2}2 has level ψ(I^2*ψ_I(I^2)) = ψ(Φ(2,Φ(2,0)))

• 1{1 {1 \ 1 \ 2} 1 {1 \ 1 \ 2} 1 {1 \ 1 \ 2} 2 \ 1 \ 2}2 has level ψ(I^3) = ψ(Φ(3,0))

• 1{1 {1 \ 1 \ 2} 1 {1 \ 1 \ 2} 1 {1 \ 1 \ 2} 1 {1 \ 1 \ 2} 2 \ 1 \ 2}2 has level ψ(I^4) = ψ(Φ(4,0))

• 1{1 {2 \ 1 \ 2} 2 \ 1 \ 2}2 has level ψ(I^ω) = ψ(Φ(ω,0))

• 1{1 {1 {1 \ 2} 2 \ 1 \ 2} 2 \ 1 \ 2}2 has level ψ(I^Ω) = ψ(Φ(Ω,0))

• 1{1 {1 {1 \ 1 {1 \ 1 \ 2} 2} 2 \ 1 \ 2} 2 \ 1 \ 2}2 has level ψ(I^ψ_I(I)) = ψ(Φ(Φ(1,0),0))

• 1{1 {1 {1 \ 1 \ 2} 2 \ 1 \ 2} 2 \ 1 \ 2}2 has level ψ(I^I) = ψ(Φ(1,0,0))

• 1{1 {1 {1 \ 1 \ 2} 2 \ 1 \ 2} 3 \ 1 \ 2}2 has level ψ(I^I*2) = ψ(Φ(1,0,1))

• 1{1 {1 {1 \ 1 \ 2} 2 \ 1 \ 2} 1 {1 \ 1 \ 2} 2 \ 1 \ 2}2 has level ψ(I^(I+1)) = ψ(Φ(1,1,0))

• 1{1 {1 {1 \ 1 \ 2} 2 \ 1 \ 2} 1 {1 {1 \ 1 \ 2} 2 \ 1 \ 2} 2 \ 1 \ 2}2 has level ψ(I^(I2)) = ψ(Φ(2,0,0))

• 1{1 {2 {1 \ 1 \ 2} 2 \ 1 \ 2} 2 \ 1 \ 2}2 has level ψ(I^(Iω)) = ψ(Φ(ω,0,0))

• 1{1 {1 {1 \ 1 \ 2} 3 \ 1 \ 2} 2 \ 1 \ 2}2 has level ψ(I^I^2) = ψ(Φ(1,0,0,0))

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

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

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

Moving on...

• 1{1 \ 2 \ 2}2 has level ψ(Ω(I+1)) = ψ(ε(I+1)) = ψ(M+ψ_M(M+1))

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

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

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

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

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

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

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

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

• 1{1 {1 {1 \ 2 \ 2} 2 \ 1 \ 2} 3 \ 2 \ 2}2 has level ψ(Ω(I+1)+ψ_(I+1)(Ω(I+1))2)

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

• 1{1 {2 {1 \ 2 \ 2} 2 \ 1 \ 2} 2 \ 2 \ 2}2 has level ψ(Ω(I+1)+ψ_(I+1)(Ω(I+1))^ω)

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

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

• 1{1 {1 \ 2 \ 2} 3 \ 2 \ 2}2 has level ψ(Ω(I+1)3)

• 1{1 {1 \ 2 \ 2} 1,2 \ 2 \ 2}2 has level ψ(Ω(I+1)ω)

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

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

• 1{1 {1 \ 2 \ 2} 1 {1 \ 1 \ 2} 3 \ 2 \ 2}2 has level ψ(Ω(I+1)I2)

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

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

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

• 1{1 {1 \ 2 \ 2} 1 {1 \ 2 \ 2} 1 {1 \ 2 \ 2} 2 \ 2 \ 2}2 has level ψ(Ω(I+1)^3)

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

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

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

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

• 1{1 \ 3 \ 2}2 has level ψ(Ω(I+2)) = ψ(ε(Ω(I+1)+1)) = ψ(M+ψ_M(M+2))

• 1{1 {1 \ 3 \ 2} 2 \ 3 \ 2}2 has level ψ(Ω(I+2)2)

• 1{1 \ 4 \ 2}2 has level ψ(Ω(I+3)) = ψ(ε(Ω(I+2)+1)) = ψ(M+ψ_M(M+3))

• 1{1 \ 1,2 \ 2}2 has level ψ(Ω(I+ω))

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

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

• 1{1 \ 1 {1 \ 1 \ 2} 2 \ 2}2 has level ψ(Ω(I2)) = ψ(M+ψ_M(M+ψ_M(M)))

• 1{1 \ 2 {1 \ 1 \ 2} 2 \ 2}2 has level ψ(Ω(I2+1))

• 1{1 \ 1 {1 \ 1 \ 2} 3 \ 2}2 has level ψ(Ω(I3))

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

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

• 1{1 \ 1 {1 \ 2 \ 2} 2 \ 2}2 has level ψ(Ω(Ω(I+1))) = ψ(M+ψ_M(M+ψ_M(M+1)))

• 1{1 \ 1 {1 \ 1 {1 \ 1 \ 2} 2 \ 2} 2 \ 2}2 has level ψ(Ω(Ω(I2))) = ψ(M+ψ_M(M+ψ_M(M+ψ_M(M))))

Then...

• 1{1 \ 1 \ 3}2 has level ψ(I_2) = ψ(M2)

• 1{1 {1 \ 1 \ 2} 2 \ 1 \ 3}2 has level ψ(I_2+I)

• 1{1 {1 \ 2 \ 2} 2 \ 1 \ 3}2 has level ψ(I_2+Ω(I+1))

• 1{1 {1 \ 1 {1 \ 1 \ 2} 2 \ 2} 2 \ 1 \ 3}2 has level ψ(I_2+Ω(I2))

• 1{1 {1 \ 1 {1 \ 1 \ 3} 2 \ 2} 2 \ 1 \ 3}2 has level ψ(I_2+ψ_{I_2}(I2))

• 1{1 {1 \ 1 \ 3} 2 \ 1 \ 3}2 has level ψ(I_2*2)

• 1{1 {1 \ 1 \ 3} 1,2 \ 1 \ 3}2 has level ψ(I_2*ω)

• 1{1 {1 \ 1 \ 3} 1 {1 \ 2} 2 \ 1 \ 3}2 has level ψ(I_2*Ω)

• 1{1 {1 \ 1 \ 3} 1 {1 \ 1 \ 2} 2 \ 1 \ 3}2 has level ψ(I_2*I)

• 1{1 {1 \ 1 \ 3} 1 {1 \ 1 {1 \ 1 \ 3} 2 \ 2} 2} 2 \ 1 \ 3}2 has level ψ(I_2*ψ_{I_2}(I2))

• 1{1 {1 \ 1 \ 3} 1 {1 \ 1 \ 3} 2 \ 1 \ 3}2 has level ψ(I_2^2)

• 1{1 {1 \ 1 \ 3} 1 {1 \ 1 \ 3} 1 {1 \ 1 \ 3} 2 \ 1 \ 3}2 has level ψ(I_2^3)

• 1{1 {2 \ 1 \ 3} 2 \ 1 \ 3}2 has level ψ(I_2^ω)

• 1{1 {1 {1 \ 1 \ 3} 2 \ 1 \ 3} 2 \ 1 \ 3}2 has level ψ(I_2^I_2)

• 1{1 \ 2 \ 3}2 has level ψ(Ω(I_2+1)) = ψ(M2+ψ_M(M2+1))

• 1{1 \ 1,2 \ 3}2 has level ψ(Ω(I_2+ω))

• 1{1 \ 1 {1 \ 1 \ 2} 2 \ 3}2 has level ψ(Ω(I_2+I)) = ψ(M2+ψ_M(M2+ψ_M(M)))

• 1{1 \ 1 {1 \ 1 \ 3} 2 \ 3}2 has level ψ(Ω(I_2*2)) = ψ(M2+ψ_M(M2+ψ_M(M2)))

• 1{1 \ 1 {1 \ 2 \ 3} 2 \ 3}2 has level ψ(Ω(Ω(I_2+1))) = ψ(M2+ψ_M(M2+ψ_M(M2+1)))

• 1{1 \ 1 \ 4}2 has level ψ(I_3) = ψ(M3)

• 1{1 \ 2 \ 4}2 has level ψ(Ω(I_3+1)) = ψ(M3+ψ_M(M3+1))

• 1{1 \ 1 \ 5}2 has level ψ(I_4) = ψ(M4)

• 1{1 \ 1 \ 6}2 has level ψ(I_5) = ψ(M5)

• 1{1 \ 1 \ 1,2}2 has level ψ(I_ω) = ψ(Mω)

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

• 1{1 {1 \ 1 \ 2} 2 \ 1 \ 1,2}2 has level ψ(I_ω+I)

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

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

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

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

• 1{1 \ 1 \ 2,2}2 has level ψ(I_(ω+1)) = ψ(Mω+M)

• 1{1 \ 1 \ 3,2}2 has level ψ(I_(ω+2)) = ψ(Mω+M2)

• 1{1 \ 1 \ 1,3}2 has level ψ(I_(ω2)) = ψ(Mω2)

• 1{1 \ 1 \ 1,1,2}2 has level ψ(I_(ω^2)) = ψ(Mω^2)

• 1{1 \ 1 \ 1{2}2}2 has level ψ(I_(ω^ω)) = ψ(Mω^ω)

• 1{1 \ 1 \ 1 {1 \ 2} 2}2 has level ψ(I_(Ω)) = ψ(MΩ)

• 1{1 \ 1 \ 1 {1 \ 3} 2}2 has level ψ(I_(Ω_2)) = ψ(MΩ_2)

• 1{1 \ 1 \ 1 {1 \ 1,2} 2}2 has level ψ(I_(Ω_ω)) = ψ(MΩ_ω)

• 1{1 \ 1 \ 1 {1 \ 1 \ 2} 2}2 has level ψ(I_I) = ψ(Mψ_M(M))

• 1{1 \ 1 \ 1 {1 \ 1 \ 2} 2}2 has level ψ(I_Ω(I+1)) = ψ(Mψ_M(M+1))

• 1{1 \ 1 \ 1 {1 \ 1 \ 3} 2}2 has level ψ(I_(I_2)) = ψ(Mψ_M(M2))

• 1{1 \ 1 \ 1 {1 \ 1 \ 1,2} 2}2 has level ψ(I_(I_ω)) = ψ(Mψ_M(Mω))

• 1{1 \ 1 \ 1 {1 \ 1 \ 1 {1 \ 1 \ 2} 2} 2}2 has level ψ(I_(I_I)) = ψ(Mψ_M(Mψ_M(M)))

Then...

• 1{1 \ 1 \ 1 \ 2}2 has level ψ(I(2,0)) = ψ(M^2).

• 1{1 {1 \ 2} 2 \ 1 \ 1 \ 2}2 has level ψ(I(2,0)+Ω)

• 1{1 {1 \ 1 \ 2} 2 \ 1 \ 1 \ 2}2 has level ψ(I(2,0)+I)

• 1{1 {1 \ 1 \ 3} 2 \ 1 \ 1 \ 2}2 has level ψ(I(2,0)+I_2)

• 1{1 {1 \ 1 \ 1 {1 \ 1 \ 1 \ 2} 2} 2 \ 1 \ 1 \ 2}2 has level ψ(I(2,0)+ψ_{I(2,0)}(I(2,0)))

• 1{1 {1 \ 1 \ 1 \ 2} 2 \ 1 \ 1 \ 2}2 has level ψ(I(2,0)*2)

• 1{1 {1 \ 1 \ 1 \ 2} 1,2 \ 1 \ 1 \ 2}2 has level ψ(I(2,0)*ω)

• 1{1 {1 \ 1 \ 1 \ 2} 1 {1 \ 1 \ 1 \ 2} 2 \ 1 \ 1 \ 2}2 has level ψ(I(2,0)^2)

• 1{1 {2 \ 1 \ 1 \ 2} 2 \ 1 \ 1 \ 2}2 has level ψ(I(2,0)^ω)

• 1{1 {1 {1 \ 1 \ 1 \ 2} 2 \ 1 \ 1 \ 2} 2 \ 1 \ 1 \ 2}2 has level ψ(I(2,0)^I(2,0))

• 1{1 \ 2 \ 1 \ 2}2 has level ψ(Ω(I(2,0)+1)) = ψ(M^2+ψ_M(M^2+1))

• 1{1 \ 1,2 \ 1 \ 2}2 has level ψ(Ω(I(2,0)+ω)) = ψ(M^2+ψ_M(M^2+ω))

• 1{1 \ 1 {1 \ 1 \ 1 \ 2} 2 \ 1 \ 2}2 has level ψ(Ω(I(2,0)*2)) = ψ(M^2+ψ_M(M^2+ψ_M(M^2)))

• 1{1 \ 1 \ 2 \ 2}2 has level ψ(I(1,I(2,0)+1)) = ψ(M^2+M)

• 1{1 \ 2 \ 2 \ 2}2 has level ψ(Ω(I(1,I(2,0)+1)+1)) = ψ(M^2+M+ψ_M(M^2+M+1))

• 1{1 \ 1 \ 3 \ 2}2 has level ψ(I(1,I(2,0)+2)) = ψ(M^2+M2)

• 1{1 \ 1 \ 1,2 \ 2}2 has level ψ(I(1,I(2,0)+ω)) = ψ(M^2+Mω)

• 1{1 \ 1 \ 1 {1 \ 1 \ 1 \ 2} 2 \ 2}2 has level ψ(I(1,I(2,0)*2)) = ψ(M^2+Mψ(M^2))

• 1{1 \ 1 \ 1 \ 3}2 has level ψ(I(2,1)) = ψ(M^2*2)

• 1{1 \ 1 \ 1 \ 4}2 has level ψ(I(2,2)) = ψ(M^2*3)

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

• 1{1 \ 1 \ 1 \ 1 {1 \ 2} 2}2 has level ψ(I(2,Ω)) = ψ(M^2*Ω)

• 1{1 \ 1 \ 1 \ 1 {1 \ 1 \ 2} 2}2 has level ψ(I(2,I(1,0))) = ψ(M^2*ψ_M(M))

• 1{1 \ 1 \ 1 \ 1 {1 \ 1 \ 1 \ 2} 2}2 has level ψ(I(2,I(2,0))) = ψ(M^2*ψ_M(M^2))

• 1{1 \ 1 \ 1 \ 1 \ 2}2 has level ψ(I(3,0)) = ψ(M^3)

• 1{1 \ 1 \ 1 \ 1 \ 3}2 has level ψ(I(3,1)) = ψ(M^3*2)

• 1{1 \ 1 \ 1 \ 1 \ 1 \ 2}2 has level ψ(I(4,0)) = ψ(M^4)

• 1{1 \ 1 \ 1 \ 1 \ 1 \ 1 \ 2}2 has level ψ(I(5,0)) = ψ(M^5)

We can have dimensional \ arrays like (for example, nesting the previous level of \ separators, we can even have something like {1 \ 2}\, {2 \ 2}\, {1 \ 3}\, {1 \ 1 \ 2}\, {1 {2}\ 2}\, and so on):

• 1{1 {2}\ 2}2 has level ψ(I(ω,0)) = ψ(M^ω)

• 1{1 \ 2 {2}\ 2}2 has level ψ(Ω(I(ω,0)+1)) = ψ(M^ω+ψ_M(M^ω+1))

• 1{1 \ 1 \ 2 {2}\ 2}2 has level ψ(I(1,I(ω,0)+1)) = ψ(M^ω+M)

• 1{1 {2}\ 3}2 has level ψ(I(ω,1)) = ψ(M^ω*2)

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

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

• 1{1 {2}\ 1 \ 2}2 has level ψ(I(ω+1,0)) = ψ(M^(ω+1))

• 1{1 {2}\ 1 \ 1 \ 2}2 has level ψ(I(ω+2,0)) = ψ(M^(ω+2))

• 1{1 {2}\ 1 {2}\ 2}2 has level ψ(I(ω2,0)) = ψ(M^(ω2))

• 1{1 {3}\ 2}2 has level ψ(I(ω^2,0)) = ψ(M^(ω^2))

• 1{1 {4}\ 2}2 has level ψ(I(ω^3,0)) = ψ(M^(ω^3))

• 1{1 {1,2}\ 2}2 has level ψ(I(ω^ω,0)) = ψ(M^(ω^ω))

• 1{1 {1,1,2}\ 2}2 has level ψ(I(ω^ω^2,0)) = ψ(M^(ω^ω^2))

• 1{1 {1{2}2}\ 2}2 has level ψ(I(ω^ω^ω,0)) = ψ(M^(ω^ω^ω))

• 1{1 {1 {1 {1 \ 2} 2} 2}\ 2}2 has level ψ(I(ε0,0)) = ψ(M^(ψ(Ω)))

• 1{1 {1 {1 \ 2} 2}\ 2}2 has level ψ(I(Ω,0)) = ψ(M^Ω)

• 1{1 {1 {1 \ 3} 2}\ 2}2 has level ψ(I(Ω_2,0)) = ψ(M^(Ω_2))

• 1{1 {1 {1 \ 1 \ 2} 2}\ 2}2 has level ψ(I(I(1,0),0)) = ψ(M^(ψ_M(M)))

• 1{1 {1 {1 \ 1 \ 1 \ 2} 2}\ 2}2 has level ψ(I(I(2,0),0)) = ψ(M^(ψ_M(M^2)))

• 1{1 {1 {1 {2}\ 2} 2}\ 2}2 has level ψ(I(I(ω,0),0)) = ψ(M^(ψ_M(M^2)))

• 1{1 {1 {1 {1 {1 \ 2} 2}\ 2} 2}\ 2}2 has level ψ(I(I(Ω,0),0)) = ψ(M^(ψ_M(M^Ω)))

• 1{1 {1 {1 {1 {1 \ 1 \ 2} 2}\ 2} 2}\ 2}2 has level ψ(I(I(I(1,0),0),0)) = ψ(M^(ψ_M(M^(ψ_M(M)))))

Moving on...

• 1{1 {1 \ 2}\ 2}2 has level ψ(I(1,0,0)) = ψ(M^M)

• 1{1 \ 2 {1 \ 2}\ 2}2 has level ψ(Ω(I(1,0,0)+1)) = ψ(M^M+ψ_M(M^M+1))

• 1{1 \ 1 \ 2 {1 \ 2}\ 2}2 has level ψ(I(1,I(1,0,0)+1)) = ψ(M^M+M)

• 1{1 {2}\ 2 {1 \ 2}\ 2}2 has level ψ(I(ω,I(1,0,0)+1)) = ψ(M^M+M^ω)

• 1{1 {1 {1 \ 2} 2}\ 2 {1 \ 2}\ 2}2 has level ψ(I(Ω,I(1,0,0)+1)) = ψ(M^M+M^Ω)

• 1{1 {1 {1 {1 \ 2}\ 2} 2}\ 2 {1 \ 2}\ 2}2 has level ψ(I(I(1,0,0),1)) = ψ(M^M+M^(ψ_M(M^M)))

• 1{1 {1 \ 2}\ 3}2 has level ψ(I(1,0,1)) = ψ(M^M*2)

• 1{1 {1 \ 2}\ 1,2}2 has level ψ(I(1,0,ω)) = ψ(M^M*ω)

• 1{1 {1 \ 2}\ 1 {1 \ 2} 2}2 has level ψ(I(1,0,Ω)) = ψ(M^M*Ω)

• 1{1 {1 \ 2}\ 1 {1 \ 1 \ 2} 2}2 has level ψ(I(1,0,I)) = ψ(M^M*I)

• 1{1 {1 \ 2}\ 1 \ 2}2 has level ψ(I(1,1,0)) = ψ(M^(M+1))

• 1{1 {1 \ 2}\ 1 \ 1 \ 2}2 has level ψ(I(1,2,0)) = ψ(M^(M+2))

• 1{1 {1 \ 2}\ 1 {2}\ 2}2 has level ψ(I(1,ω,0)) = ψ(M^(M+ω))

• 1{1 {1 \ 2}\ 1 {1 {1 \ 1 \ 2} 2}\ 2}2 has level ψ(I(1,I,0)) = ψ(M^(M+I))

• 1{1 {1 \ 2}\ 1 {1 \ 2}\ 2}2 has level ψ(I(2,0,0)) = ψ(M^(M2))

• 1{1 {1 \ 2}\ 1 {1 \ 2}\ 1 {1 \ 2}\ 2}2 has level ψ(I(3,0,0)) = ψ(M^(M3))

• 1{1 {2 \ 2}\ 2}2 has level ψ(I(ω,0,0)) = ψ(M^(Mω))

• 1{1 {1 {1 \ 2} 2 \ 2}\ 2}2 has level ψ(I(Ω,0,0)) = ψ(M^(MΩ))

• 1{1 {1 {1 \ 1 \ 2} 2 \ 2}\ 2}2 has level ψ(I(I,0,0)) = ψ(M^(MI))

• 1{1 {1 \ 3}\ 2}2 has level ψ(I(1,0,0,0)) = ψ(M^M^2)

• 1{1 {1 \ 3}\ 1 {1 \ 3}\ 2}2 has level ψ(I(2,0,0,0)) = ψ(M^(M^2*2))

• 1{1 {1 \ 4}\ 2}2 has level ψ(I(1,0,0,0,0)) = ψ(M^M^3)

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

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

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

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

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

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

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

So the limit of the nested \ arrays is approximately ψ(ε(M+1)) = ψ(Ω(M+1)).

At this point, we need to define a separator that ranks higher than all the {x}\ separators. To do this, just use (\n) to diagonalize over the previous {x}\ separators. Of course, after \{{1}} comes \{{2}}, \{{1,2}}, \{{1{1 \ 2}2}}, \{{1 \ 2}}, and so on.

• 1{1 \{{1}} 2}2 has level ψ(Ω(M+1))

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

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

• 1{1 \{{1}} 3}2 has level ψ(Ω(M+1)2)

• 1{1 \{{1}} 1,2}2 has level ψ(Ω(M+1)ω)

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

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

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

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

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

• 1{1 \{{2}} 2}2 has level ψ(Ω(M+2))

• 1{1 \{{3}} 2}2 has level ψ(Ω(M+3))

• 1{1 \{{4}} 2}2 has level ψ(Ω(M+4))

• 1{1 \{{1,2}} 2}2 has level ψ(Ω(M+ω))

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

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

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

• 1{1 \{{1 \ 2}} 2}2 has level ψ(Ω(M2))

• 1{1 \{{2 \ 2}} 2}2 has level ψ(Ω(M2+1))

• 1{1 \{{1 \ 3}} 2}2 has level ψ(Ω(M3))

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

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

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

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

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

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

• 1{1 \{{1 \{{1 \ 2}} 2}} 2}2 has level ψ(Ω(Ω(M2)))

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

The limit is ψ(I_(M+1)) = ψ(M_2)

And finally, moving on the multiple \ separators! Use the ^ symbol to reach even more. The array 1{1 \\ x}2 diagonalizes over the sequence 1{1 \{{x+1}} 2}2.

• 1{1 \\ 1 \\ 2}2 has level ψ(M_2) = ψ(I_(M+1))

• 1{1 \ 2 \\ 1 \\ 2}2 has level ψ(M_2+M) = ψ(I_(M+1)+M)

• 1{1 \{{1}} 2 \\ 1 \\ 2}2 = 1{1 {1 \\ 2} 2 \\ 1 \\ 2}2 has level ψ(M_2+Ω(M+1)) = ψ(I_(M+1)+Ω(M+1))

• 1{1 {1 \\ 1,2} 2 \\ 1 \\ 2}2 has level ψ(M_2+Ω(M+ω))

• 1{1 {1 \\ 1 {1 \\ 2} 2} 2 \\ 1 \\ 2}2 has level ψ(M_2+Ω(M2))

• 1{1 {1 \\ 1 {1 \\ 1 \\ 2} 2} 2 \\ 1 \\ 2}2 has level ψ(M_2+ψ_{I_(M+1)}(M_2))

• 1{1 {1 \\ 1 \\ 2} 2 \\ 1 \\ 2}2 has level ψ(M_2+ψ_{M_2}(M_2))

• 1{1 \\ 2 \\ 2}2 has level ψ(M_2+ψ_{M_2}(M_2+1))

• 1{1 \\ 1 \\ 3}2 has level ψ(M_2*2)

• 1{1 \\ 1 \\ 1,2}2 has level ψ(M_2*ω)

• 1{1 \\ 1 \\ 1 \ 2}2 has level ψ(M_2*M)

• 1{1 \\ 1 \\ 1 {1 \\ 2} 2}2 has level ψ(M_2*Ω(M+1))

• 1{1 \\ 1 \\ 1 \\ 2}2 has level ψ(M_2^2)

• 1{1 {2}\\ 2}2 has level ψ(M_2^ω)

• 1{1 {1 \ 2}\\ 2}2 has level ψ(M_2^M)

• 1{1 {1 \\ 2}\\ 2}2 has level ψ(M_2^M_2)

• 1{1 {1 \\ 1 \\ 2}\\ 2}2 has level ψ(M_2^M_2^M_2)

• 1{1 \\{{1}} 2}2 has level ψ(Ω(M_2+1))

• 1{1 \\{{1}} 3}2 has level ψ(Ω(M_2+1)*2)

• 1{1 \\{{1}} 1,2}2 has level ψ(Ω(M_2+1)*ω)

• 1{1 \\{{1}} 1 \\{{1}} 2}2 has level ψ(Ω(M_2+1)^2)

• 1{1 \\{{2}} 2}2 has level ψ(Ω(M_2+2))

• 1{1 \\{{1,2}} 2}2 has level ψ(Ω(M_2+ω))

• 1{1 \\{{1 \ 2}} 2}2 has level ψ(Ω(M_2+M))

• 1{1 \\{{1 \\ 2}} 2}2 has level ψ(Ω(M_2*2))

• 1{1 \\{{1 \\{{1}} 2}} 2}2 has level ψ(Ω(Ω(M_2+1)))

• 1{1 \\\ 1 \\\ 2}2 has level ψ(M_3) = ψ(I_(M_2+1))

• 1{1 \\\ 1 \\\ 3}2 has level ψ(M_3*2)

• 1{1 \\\ 1 \\\ 1 \\\ 2}2 has level ψ(M_3^2)

• 1{1 \\\{{1}} 2}2 has level ψ(Ω(M_3+1))

• 1{1 \\\\ 1 \\\\ 2}2 has level ψ(M_4)

• 1{1 \\\\\ 1 \\\\\ 2}2 has level ψ(M_5)

Now we need a way to generalize the last section. Here, I use a caret (^) symbol after the \ to denote the number of slashes. So \^1 = \, \^2 = \\, \^3 = \\\, and so on. Of course, you can even have \^(a){{b}} but these aren't necessary. To finish up, we can nest the slashes after the ^.

• 1{1 \^(1,2) 2}2 has level ψ(M_ω)

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

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

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

• 1{1 \^(1,3) 2}2 has level ψ(M_(ω2))

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

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

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

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

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

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

• 1{1 \^(1 \ 2) 2}2 has level ψ(M_M)

• 1{1 \^(1 \\ 2) 2}2 has level ψ(M_M_2)

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

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

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

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

• 1{1 \^(1 \^(1 \\ 2) 2) 2}2 has level ψ(M_M_M_2)

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

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

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

So the limit of the Extended Expanding Array Notation is ψ(M(2,0)), which is the Mahlo fixed point (ψ(M_M_M_...)). It is also equal to ψ(N) = ψ(M(2;0)).

---<| THE END |>---

Update 1st January 2023, following the introduction of New Year's Numbers 2023:

I define the special function that diagonalizes over the limit of the notation, called M function, to be:

M(a, 0) = a[1, 2]a = a[a]a

M(a, 1) = a[1 {1 \ 2} 2]a = a[1 {1 / 2} 2]a

M(a, 2) = a[1 {1 \^(1 \ 2) 2} 2]a

M(a, 3) = a[1 {1 \^(1 \^(1 \ 2) 2) 2} 2]a

M(a, 4) = a[1 {1 \^(1 \^(1 \^(1 \ 2) 2) 2) 2} 2]a

M(a, b) = a[1 {1 \^(1 \^(1 \^(... 1 \^(1 \^(1 \^(1 \ 2) 2) 2) 2 ...) 2) 2) 2} 2]a w/ b backslashes (\), for b >= 1