As we introduced the Second-order Array Notation, we can apply the similar rules to the Higher-order Array Notation right now.
Reuse the previous rules (including similar rules).
The <> separator indicates the first-order separator, and the <<>> separator indicates the second-order separator. The unicode form is «» (guillemet). Each slashes (/), or pairs of angle brackets (<>), indicate the n-th order separator.
% indicates the latter of the array inside the separator.
The slashes ///.../// (with n slashes) is the shorthand for the <<<...<<<1>>>...>>> separator (with n pairs of angle brackets).
We can continue further with further copies of hyperseparators. The general rule is the same as that in the last section, but with each "2" replaced by "n", and each "1" replaced by "n-1".
As we'll seen below, the multiple hyperseparators (in fact, with n pairs of angle brackets) have the same relation to Ω_n's.
1 /// 2 has level ψ(Ω_3) = ψ(ε(Ω_2+1))
1 / 2 /// 2 has level ψ(Ω_3+Ω) = ε(ψ(ε(Ω_2+1))+1)
1 <1 /// 2> 2 /// 2 has level ψ(Ω_3+ψ_1(Ω_3)) = ψ(ε(Ω_2+1)+ψ_1(ε(Ω_2+1)))
1 // 2 /// 2 has level ψ(Ω_3+Ω_2) = ψ(ε(Ω_2+1)+Ω_2)
1 <<1 /// 2>> 2 /// 2 has level ψ(Ω_3+ψ_2(Ω_3)) = ψ(ε(Ω_2+1)*2)
1 /// 3 has level ψ(Ω_3*2) = ψ(ε(Ω_2+2))
1 /// 1{1 /// 2}2 has level ψ(Ω_3*ψ(Ω_3)) = ψ(ε(Ω_2+ψ(ε(Ω_2+1))))
1 /// 1 / 2 has level ψ(Ω_3*Ω) = ψ(ε(Ω_2+Ω))
1 /// 1 // 2 has level ψ(Ω_3*Ω_2) = ψ(ε(Ω_2*2))
1 /// 1 <<1 /// 2>> 2 has level ψ(Ω_3*ψ_2(Ω_3)) = ψ(ε(ε(Ω_2+1)))
1 /// 1 /// 2 has level ψ(Ω_3^2) = ψ(ζ(Ω_2+1))
1 /// 1 /// 1 /// 2 has level ψ(Ω_3^3) = ψ(η(Ω_2+1))
1 <<<2>>> 2 has level ψ(Ω_3^ω) = ψ(φ(ω,Ω_2+1))
1 <<<1 / 2>>> 2 has level ψ(Ω_3^Ω) = ψ(φ(Ω,Ω_2+1))
1 <<<1 // 2>>> 2 has level ψ(Ω_3^Ω_2) = ψ(φ(Ω_2,1))
1 <<<1 /// 2>>> 2 has level ψ(Ω_3^Ω_3) = ψ(Γ(Ω_2+1))
1 <<<2 /// 2>>> 2 has level ψ(Ω_3^(Ω_3*ω)) = ψ(φ(ω,0,Ω_2+1))
1 <<<1 /// 3>>> 2 has level ψ(Ω_3^Ω_3^2) = ψ(φ(1,0,0,Ω_2+1))
1 <<<1 /// 1,2>>> 2 has level ψ(Ω_3^Ω_3^ω)
1 <<<1 /// 1 /// 2>>> 2 has level ψ(Ω_3^Ω_3^Ω_3)
1 <<<1 <<<1 /// 2>>> 2>>> 2 has level ψ(Ω_3^Ω_3^Ω_3^Ω_3)
1 //// 2 has level ψ(Ω_4) = ψ(ε(Ω_3+1))
1 //// 3 has level ψ(Ω_4*2) = ψ(ε(Ω_3+2))
1 //// 1 //// 2 has level ψ(Ω_4^2) = ψ(ζ(Ω_3+2))
1 <<<<2>>>> 2 has level ψ(Ω_4^ω) = ψ(φ(ω,Ω_3+1))
1 <<<<1 //// 2>>>> 2 has level ψ(Ω_4^Ω_4) = ψ(Γ(Ω_3+1))
1 ///// 2 has level ψ(Ω_5) = ψ(ε(Ω_4+1))
1 ////// 2 has level ψ(Ω_6) = ψ(ε(Ω_5+1))
1 /////// 2 has level ψ(Ω_7) = ψ(ε(Ω_6+1))
The limit of this level is ψ_0(Ω_ω), known as Buchholz's ordinal. It is a large countable ordinal that is the proof theoretic ordinal of Π1^1-CA0 subsystem. It is believed to be the first ordinal α which g_α(n) in the slow-growing hierarchy "catches up" with f_α(n) in the fast-growing hierarchy.