As we introduced the First-order Array Notation and the Extended First-order Array Notation, we can apply the similar rules to the Second-order Array Notation right now.
Reuse the previous rules.
The <> separator indicates the first-order separator, and the <<>> separator indicates the second-order separator. The unicode form is «» (guillemet).
% indicates the latter of the array inside the separator.
The double slash (//) is the shorthand for the <<1>> separator, and the ~ (tilde) symbol is the alternative to //.
If there is a // before you, then:
For each 1 < t <= n where n is the layer of the // separator you are looking at, take A(t) to be the separator at that layer that contains that //.
Find the maximum t such that the A(t) is not the hyperseparators (e.g. /, <2>, <1 / 2>, //, etc.), and call it a.
Take the similar form of the A => S(n), then make the fundamental sequences of the hyperseparators that should play the role similarly to the FoAN and the XFoAN.
What this does is ensures that we use all normal separators first, then all separators with the singular <> separator, and only then we get to use the double «» separator. The limit of {1 <1 <...1 <1 <1 / 2> 2> 2...> 2> 2 // 2} is not {1 // 3}, but {1 <1 // 2> 2 // 2}.
As we'll seen below, the double «» hyperseparators have the same relation to Ω_2's that the single <> hyperseparators have to the lone Ω's.
1 // 2 has level ψ(ε(Ω+1)) = ψ(Ω_2) (Bachmann-Howard ordinal)
1 / 2 // 2 has level ψ(Ω_2+Ω) = ε(ψ(ε(Ω+1))+1)
1 / 1 / 2 // 2 has level ψ(Ω_2+Ω^2) = ζ(ψ(ε(Ω+1))+1)
1 <2> 2 // 2 has level ψ(Ω_2+Ω^ω) = φ(ω,ψ(ε(Ω+1))+1)
1 <1{1 // 2}2> 2 // 2 has level ψ(Ω_2+Ω^ψ(Ω_2)) = φ(ψ(ε(Ω+1)),1)
1 <1 / 2> 2 // 2 has level ψ(Ω_2+Ω^Ω) = Γ(ψ(ε(Ω+1))+1)
1 <1 / 3> 2 // 2 has level ψ(Ω_2+Ω^Ω^2) = φ(1,0,0,ψ(ε(Ω+1))+1)
1 <1 / 1,2> 2 // 2 has level ψ(Ω_2+Ω^Ω^ω) = ψ(ε(Ω+1)+Ω^Ω^ω)
1 <1 / 1 / 2> 2 // 2 has level ψ(Ω_2+Ω^Ω^Ω) = ψ(ε(Ω+1)+Ω^Ω^Ω)
1 <1 <1 / 2> 2> 2 // 2 has level ψ(Ω_2+Ω^Ω^Ω^Ω) = ψ(ε(Ω+1)+Ω^Ω^Ω^Ω)
1 <1 <1 / 1 / 2> 2> 2 // 2 has level ψ(Ω_2+Ω^Ω^Ω^Ω^Ω) = ψ(ε(Ω+1)+Ω^Ω^Ω^Ω^Ω)
1 <1 // 2> 2 // 2 has level ψ(Ω_2+ψ_1(Ω_2)) = ψ(ε(Ω+1)*2)
1 <1 // 2> 3 // 2 has level ψ(Ω_2+ψ_1(Ω_2)2) = ψ(ε(Ω+1)*3)
1 <1 // 2> 1,2 // 2 has level ψ(Ω_2+ψ_1(Ω_2+1)) = ψ(ε(Ω+1)*ω)
1 <1 // 2> 1 / 2 // 2 has level ψ(Ω_2+ψ_1(Ω_2+Ω)) = ψ(ε(Ω+1)*Ω)
1 <1 // 2> 1 <1 / 2> 2 // 2 has level ψ(Ω_2+ψ_1(Ω_2+Ω^2)) = ψ(ε(Ω+1)*Ω^Ω)
1 <1 // 2> 1 <1 // 2> 2 // 2 has level ψ(Ω_2+ψ_1(Ω_2+ψ_1(Ω_2))) = ψ(ε(Ω+1)^2)
1 <2 // 2> 2 // 2 has level ψ(Ω_2+ψ_1(Ω_2+ψ_1(Ω_2+1))) = ψ(ε(Ω+1)^ω)
1 <1 / 2 // 2> 2 // 2 has level ψ(Ω_2+ψ_1(Ω_2+ψ_1(Ω_2+Ω))) = ψ(ε(Ω+1)^Ω)
1 <1 <1 // 2> 2 // 2> 2 // 2 has level ψ(Ω_2+ψ_1(Ω_2+ψ_1(Ω_2+ψ_1(Ω_2)))) = ψ(ε(Ω+1)^ε(Ω+1))
1 <1 <1 // 2> 1 <1 // 2> 2 // 2> 2 // 2 has level ψ(Ω_2+ψ_1(Ω_2+ψ_1(Ω_2+ψ_1(Ω_2+ψ_1(Ω_2))))) = ψ(ε(Ω+1)^ε(Ω+1)^ε(Ω+1))
1 // 3 has level ψ(Ω_2*2) = ψ(ε(Ω+2))
1 // 4 has level ψ(Ω_2*3) = ψ(ε(Ω+3))
1 // 1,2 has level ψ(Ω_2*ω) = ψ(ε(Ω+ω))
1 // 1{1 // 2}2 has level ψ(Ω_2*ψ(Ω_2)) = ψ(ε(Ω+ψ(ε(Ω+1))))
1 // 1 / 2 has level ψ(Ω_2*Ω) = ψ(ε(Ω2))
1 // 2 / 2 has level ψ(Ω_2*Ω+Ω_2) = ψ(ε(Ω2+1))
1 // 1 / 3 has level ψ(Ω_2*Ω2) = ψ(ε(Ω3))
1 // 1 / 1,2 has level ψ(Ω_2*Ωω) = ψ(ε(Ωω))
1 // 1 / 1 / 2 has level ψ(Ω_2*Ω^2) = ψ(ε(Ω^2))
1 // 1 <2> 2 has level ψ(Ω_2*Ω^ω) = ψ(ε(Ω^ω))
1 // 1 <1 / 2> 2 has level ψ(Ω_2*Ω^Ω) = ψ(ε(Ω^Ω))
1 // 1 <1 // 2> 2 has level ψ(Ω_2*ψ_1(Ω_2)) = ψ(ε(ε(Ω+1)))
1 // 1 // 2 has level ψ(Ω_2^2) = ψ(ζ(Ω+1))
1 <1 // 1 // 2> 2 // 1 // 2 has level ψ(Ω_2^2+ψ_1(Ω_2^2)) = ψ(ζ(Ω+1)*2)
1 // 2 // 2 has level ψ(Ω_2^2+Ω_2) = ψ(ε(ζ(Ω+1)+1))
1 // 1 / 2 // 2 has level ψ(Ω_2^2+Ω_2*Ω) = ψ(ε(ζ(Ω+1)+Ω))
1 // 1 <1 // 1 // 2> 2 // 2 has level ψ(Ω_2^2+Ω_2*ψ_1(Ω_2^2)) = ψ(ε(ζ(Ω+1)*2))
1 // 1 // 3 has level ψ(Ω_2^2*2) = ψ(ζ(Ω+2))
1 // 1 // 1,2 has level ψ(Ω_2^2*ω) = ψ(ζ(Ω+ω))
1 // 1 // 1 / 2 has level ψ(Ω_2^2*Ω) = ψ(ζ(Ω2))
1 // 1 // 1 // 2 has level ψ(Ω_2^3) = ψ(η(Ω+1))
1 // 1 // 1 // 1 // 2 has level ψ(Ω_2^4) = ψ(φ(4,Ω+1))
1 «2» 2 has level ψ(Ω_2^ω) = ψ(φ(ω,Ω+1))
1 «2» 3 has level ψ(Ω_2^ω*2) = ψ(φ(ω,Ω+2))
1 «2» 1 / 2 has level ψ(Ω_2^ω*Ω) = ψ(φ(ω,Ω2))
1 «2» 1 // 2 has level ψ(Ω_2^(ω+1)) = ψ(φ(ω+1,Ω+1))
1 «2» 1 «2» 2 has level ψ(Ω_2^(ω2)) = ψ(φ(ω2,Ω+1))
1 «3» 2 has level ψ(Ω_2^(ω^2)) = ψ(φ(ω^2,Ω+1))
1 «1,2» 2 has level ψ(Ω_2^(ω^ω)) = ψ(φ(ω^ω,Ω+1))
1 «1{1 // 2}2» 2 has level ψ(Ω_2^ψ(Ω_2)) = ψ(φ(ψ(ε(Ω+1)),Ω+1))
1 «1 / 2» 2 has level ψ(Ω_2^Ω) = ψ(φ(Ω,1))
1 «2 / 2» 2 has level ψ(Ω_2^(Ω+1)) = ψ(φ(Ω+1,0))
1 «1 / 3» 2 has level ψ(Ω_2^(Ω2)) = ψ(φ(Ω2,0))
1 «1 / 1 / 2» 2 has level ψ(Ω_2^(Ω^2)) = ψ(φ(Ω^2,0))
1 «1 <2> 2» 2 has level ψ(Ω_2^(Ω^ω)) = ψ(φ(Ω^ω,0))
1 «1 <1 / 2> 2» 2 has level ψ(Ω_2^(Ω^Ω)) = ψ(φ(Ω^Ω,0))
1 «1 <1 // 2> 2» 2 has level ψ(Ω_2^ψ_1(Ω_2)) = ψ(φ(ε(Ω+1),0))
1 «1 // 2» 2 has level ψ(Ω_2^Ω_2) = ψ(Γ(Ω+1))
1 «1 // 2» 3 has level ψ(Ω_2^Ω_2*2) = ψ(Γ(Ω+2))
1 «1 // 2» 1 / 2 has level ψ(Ω_2^Ω_2*Ω) = ψ(Γ(Ω2))
1 «1 // 2» 1 // 2 has level ψ(Ω_2^(Ω_2+1)) = ψ(φ(1,1,Ω+1))
1 «1 // 2» 1 «2» 2 has level ψ(Ω_2^(Ω_2+ω)) = ψ(φ(1,ω,Ω+1))
1 «1 // 2» 1 «1 / 2» 2 has level ψ(Ω_2^(Ω_2+Ω)) = ψ(φ(1,Ω,1))
1 «1 // 2» 1 «1 // 2» 2 has level ψ(Ω_2^(Ω_2*2)) = ψ(φ(2,0,Ω+1))
1 «2 // 2» 2 has level ψ(Ω_2^(Ω_2*ω)) = ψ(φ(ω,0,Ω+1))
1 «1 / 2 // 2» 2 has level ψ(Ω_2^(Ω_2*Ω)) = ψ(φ(Ω,0,1))
1 «1 // 3» 2 has level ψ(Ω_2^Ω_2^2) = ψ(φ(1,0,0,Ω+1))
1 «1 // 4» 2 has level ψ(Ω_2^Ω_2^3) = ψ(φ(1,0,0,0,Ω+1))
1 «1 // 1,2» 2 has level ψ(Ω_2^Ω_2^ω)
1 «1 // 1 / 2» 2 has level ψ(Ω_2^Ω_2^Ω)
1 «1 // 1 // 2» 2 has level ψ(Ω_2^Ω_2^Ω_2)
1 «1 // 2 // 2» 2 has level ψ(Ω_2^Ω_2^(Ω_2+1))
1 «1 // 1 // 3» 2 has level ψ(Ω_2^Ω_2^(Ω_2*2))
1 «1 // 1 // 1 // 2» 2 has level ψ(Ω_2^Ω_2^Ω_2^2)
1 «1 «2» 2» 2 has level ψ(Ω_2^Ω_2^Ω_2^ω)
1 «1 «1 // 2» 2» 2 has level ψ(Ω_2^Ω_2^Ω_2^Ω_2)
1 «1 «1 // 1 // 2» 2» 2 has level ψ(Ω_2^Ω_2^Ω_2^Ω_2^Ω_2)
1 «1 «1 «1 // 2» 2» 2» 2 has level ψ(Ω_2^Ω_2^Ω_2^Ω_2^Ω_2^Ω_2)
Similarly, each layer of separators adds two Ω_2's into the ordinal power of the second uncountable cardinal, and the limit of the level is ψ(ε(Ω_2+1)) = ψ(Ω_3).