# sDAN vs. Taranovsky’s ordinal notation

Here’re the comparisons between sDAN and Taranovsky’s ordinal notation.

Fundamental sequences: Let L(α) be the amount of C’s in standard representation of α, then ${\alpha[n]=\max\{\beta|\beta<\alpha\land L(\beta)\le L(\alpha)+n\}}$.

# Up to {1,,1,,2,,}

In last post, I showed that C(Ω22,0) works as the 1-separator in C(Ω2+____,β), but it’s C(Ω2+C(Ω2,C(Ω22,0)),0) that works as the 1-separator in C(Ω2+C(C(Ω22+____,0),C(Ω22,0)),β). And, C(Ω2,C(Ω22,0)) works as the 2-separator in C(Ω2+____,β).

## Up to a C(Ω2,C(Ω2,C(Ω22,0)))

Now let ordinal σ = C(Ω22,0). And α+ is shorthand for C(Ω2,α).

• {1{1,,2,,}2} has recursion level C(C(Ω22+C(Ω2+C(σ+++),0),0),0) (,, is the 1-separator on the double comma)
• {1{1{1,,2,,}2}2{1,,2,,}2} has recursion level C(C(Ω22+C(Ω2+C(σ+++),0)2,0),0)
• {1,,2{1,,2,,}2} has recursion level C(C(Ω22+C(C(Ω2+C(σ+++),0)+,C(Ω2+C(σ+++),0)),0),0)
• {1,,1,,2{1,,2,,}2} has recursion level C(C(Ω22+C(σ,C(Ω2+C(σ+++),0)),0),0)
• {1,,1,,1,,2{1,,2,,}2} has recursion level C(C(Ω22+C(Ω2++σ,C(Ω2+C(σ+++),0)),0),0)
• {1{1,,2,,}2{1,,2,,}2} has recursion level C(C(Ω22+C(Ω2+ω^(σ++σ),C(Ω2+C(σ+++),0)),0),0)
• {1{1,,1,,2,,}2{1,,2,,}2} has recursion level C(C(Ω22+C(Ω2+ω^ω^(σ++σ),C(Ω2+C(σ+++),0)),0),0)
• {1{1{1,,2,,}2,,}2{1,,2,,}2} has recursion level C(C(Ω22+C(Ω2+ω^ω^ω^(σ++σ),C(Ω2+C(σ+++),0)),0),0)
• {1{1,,2,,}3} has recursion level C(C(Ω22+C(Ω2+C(σ+++),C(Ω2+C(σ+++),0)),0),0)
• {1{1,,2,,}1,2} has recursion level C(C(Ω22+C(Ω2+C(σ+++)+1,0),0),0)
• {1{1,,2,,}1,,2} has recursion level C(C(Ω22+C(Ω2+C(σ+++)+σ,0),0),0)
• {1{1,,2,,}1,,1,,2} has recursion level C(C(Ω22+C(Ω2+C(σ+++)+σ++σ,0),0),0)
• {1{1,,2,,}1{1,,2,,}2} has recursion level C(C(Ω22+C(Ω2+C(σ+++)2,0),0),0)
• {1{2,,2,,}2} has recursion level C(C(Ω22+C(Ω2+ω^(C(σ+++)+1),0),0),0)
• {1,,2,,2,,} has recursion level C(C(Ω22+C(Ω2+ω^(C(σ+++)+σ),0),0),0)
• {1,,1,,2,,2,,} has recursion level C(C(Ω22+C(Ω2+ω^(C(σ+++)+σ++σ),0),0),0)
• {1{1,,2,,}2,,2,,} has recursion level C(C(Ω22+C(Ω2+ω^(C(σ+++)2),0),0),0)
• {1{1,,2,,}1{1,,2,,}2,,2,,} has recursion level C(C(Ω22+C(Ω2+ω^ω^(C(σ+++)2),0),0),0)
• {1{1{1,,2,,}2,,2,,}2,,2,,} has recursion level C(C(Ω22+C(Ω2+ω^ω^ω^(C(σ+++)2),0),0),0)
• {1,,3,,} has recursion level C(C(Ω22+C(Ω2+C(σ++,C(σ+++)),0),0),0)
• {1,,1,2,,} has recursion level C(C(Ω22+C(Ω2+C(σ+++1,σ+),0),0),0)
• {1,,1{1,,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(σ+++C(σ+++),σ+),0),0),0)
• {1,,1,,2,,} has recursion level C(C(Ω22+C(Ω2+C(σ++2,σ+),0),0),0)
• {1{2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(ω^(σ+++1),σ+),0),0),0)
• {1{1,,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(ω^(σ++2),σ+),0),0),0)
• {1{1{1,,2,,}2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(ω^ω^ω^(σ++2),σ+),0),0),0)
• {1{1 ,,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(σ++++),0),0),0) ( ,, is the 1-separator on the ,,)
• {1,,1,2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2+1,σ),σ+),0),0),0) (,, is the 2-separator on the double comma)
• {1,,1,,2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2+σ,σ),σ+),0),0),0)
• {1{1,,1{1{1,,1,,2,,}2}2,,}1,2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2+σ,σ),σ+)+1,0),0),0)
• {1{1,,1{1{1,,1,,2,,}2}2,,}1{1,,1,2,,}2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2+σ,σ),σ+)+C(C(Ω2+1,σ),σ+),0),0),0)
• {1{1,,1{1{1,,1,,2,,}2}2,,}1{1,,1{1{1,,1,,2,,}2}2,,}2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2+σ,σ),σ+)2,0),0),0)
• {1{2{1,,1{1{1,,1,,2,,}2}2,,}2,,}2} has recursion level C(C(Ω22+C(Ω2+ω^(C(C(Ω2+σ,σ),σ+)+1),0),0),0)
• {1,,2{1,,1{1{1,,1,,2,,}2}2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(σ++,C(C(Ω2+σ,σ),σ+)),0),0),0)
• {1{1,,1{1{1,,1,,2,,}2}2,,}1,2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2+σ,σ)+1,σ+),0),0),0)
• {1{1,,1,2,,}2,,1{1{1,,1,,2,,}2}2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2+σ,σ)+C(Ω2+1,σ),σ+),0),0),0)
• {1{1,,1{1{1,,1,,2,,}2}2,,}2,,1{1{1,,1,,2,,}2}2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2+σ,σ)2,σ+),0),0),0)
• {1,,2{1{1,,1,,2,,}2}2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2+σ,σ)++),0),0),0)
• {1,,1{1{1,,1,,2,,}2}1,2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2+σ+1,σ),σ+),0),0),0)
• {1,,1{1{1,,1,,2,,}2}1{1{1,,1,,2,,}2}2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2+σ2,σ),σ+),0),0),0)
• {1,,2{1,,1,,2,,}2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2+C(σ+,σ),σ),σ+),0),0),0)
• {1,,1,2{1,,1,,2,,}2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2+C(C(Ω2+1,σ),σ),σ),σ+),0),0),0)
• {1,,1{1{1,,1,,2,,}2}2{1,,1,,2,,}2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2+C(C(Ω2+σ,σ),σ),σ),σ+),0),0),0)
• {1,,1{1,,2{1,,1,,2,,}2}2{1,,1,,2,,}2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2+C(C(Ω2+,σ),σ),σ),σ+),0),0),0)
• {1,,1{1{1,,1,2{1,,1,,2,,}2}2,,2{1,,1,,2,,}2}2{1,,1,,2,,}2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2+C(C(Ω2+C(C(Ω2+1,σ),σ+),σ),σ),σ),σ+),0),0),0)
• {1,,1{1{1,,1{1{1,,1,,2,,}2}2{1,,1,,2,,}2}2,,2{1,,1,,2,,}2}2{1,,1,,2,,}2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2+C(C(Ω2+C(C(Ω2+σ,σ),σ+),σ),σ),σ),σ+),0),0),0)
• {1,,1{1{1,,1{1,,2{1,,1,,2,,}2}2{1,,1,,2,,}2}2,,2{1,,1,,2,,}2}2{1,,1,,2,,}2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2+C(C(Ω22,σ),σ),σ),σ+),0),0),0)
• {1,,2{1{1,,1{1,,2{1,,1,,2,,}2}2{1,,1,,2,,}2}2,,2{1,,1,,2,,}2}2{1,,1,,2,,}2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2+C(C(Ω22,σ)+,σ),σ),σ+),0),0),0)
• {1,,1{1{1,,1{1,,2{1,,1,,2,,}2}2{1,,1,,2,,}2}2,,2{1,,1,,2,,}2}1,2{1,,1,,2,,}2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2+C(C(Ω22+1,0),σ),σ),σ+),0),0),0)
• {1,,1{1{1,,1{1,,2{1,,1,,2,,}2}2{1,,1,,2,,}2}2,,2{1,,1,,2,,}2}1{1{1,,1{1{1,,1,,2,,}2}2,,}2}2{1,,1,,2,,}2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2+C(C(Ω22+C(Ω2+C(C(Ω2+σ,σ),σ+),0),0),σ),σ),σ+),0),0),0)
• {1,,1{1{1,,1{1,,2{1,,1,,2,,}2}2{1,,1,,2,,}2}2,,2{1,,1,,2,,}2}1{1{1,,1,,2,,}2}2{1,,1,,2,,}2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2+,σ),σ+),0),0),0)
• {1{1,,1,,2,,}1,2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2+,σ),σ+)+1,0),0),0)
• {1,,2{1,,1,,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(σ++,C(C(Ω2+,σ),σ+)),0),0),0)
• {1{1,,1,,2,,}1,2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2+,σ)+1,σ+),0),0),0)
• {1{1,,1,2,,}2,,1,,2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2+,σ)+C(Ω2+1,σ),σ+),0),0),0)
• {1,,2,,2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2+,σ)++),0),0),0)
• {1,,1,,1,2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2++1,σ),σ+),0),0),0)
• {1,,1{1,,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2+C(σ+++),σ),σ+),0),0),0)
• {1,,1{1{1,,1,2,,}2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2+C(C(Ω2+1,σ),σ+),σ),σ+),0),0),0)
• {1,,1,,2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22,σ),σ+),0),0),0)
• {1{1,,1,,2,,}1,2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22,σ),σ+)+1,0),0),0)
• {1,,2{1,,1,,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(σ++,C(C(Ω22,σ),σ+)),0),0),0)
• {1{1,,1,,2,,}3,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22,σ),C(C(Ω22,σ),σ+)),0),0),0)
• {1{1,,1,,2,,}1,2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22,σ)+1,σ+),0),0),0)
• {1{1,,1,,2,,}2,,1,,2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22,σ)2,σ+),0),0),0)
• {1,,2,,2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22,σ)++),0),0),0)
• {1,,1{1,,2,,2,,}1,,2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2+C(C(Ω22,σ)++),C(Ω22,σ)),σ+),0),0),0)
• {1,,1,,3,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22,C(Ω22,σ)),σ+),0),0),0)
• {1,,1,,1,2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22+1,0),σ+),0),0),0)
• {1,,1,,1{1{2,,}2}2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22+C(Ω2+ω^(σ++1),0),0),σ+),0),0),0)
• {1,,1,,1{1{1,,1,,1,2,,}2}2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22+C(Ω2+C(C(Ω22+1,0),σ+),0),0),σ+),0),0),0)
• {1,,1,,1,,2,,} has recursion level C(C(Ω22+C(Ω2++,0),0),0)

So here comes a further guess: does C(Ω2+C(Ω2,C(Ω2,C(Ω22,0))),0) work as the 2-separator in C(Ω2+C(C(Ω22+____,0),C(Ω22,0)),β)?

## Up to a C(Ω2,C(Ω2,C(Ω2,C(Ω22,0))))

Now let separator ♦ = {1{1,,1,,1,,2,,}2} and let ordinal σ = C(Ω22,0). And α+ is shorthand for C(Ω2,α).

• {1,,1{1{1{1,,1,,1♦2,,}2}2,,1{1,,1,,2}2}2{1,,1,,2}2} has recursion level C(C(Ω22+C(Ω2++,0),0),0)
• {1,,1{1{1{1,,1,,1♦2,,}2}2,,1{1,,1,,2}2}1{1{1{1,,1,,1♦2,,}2}2,,1{1,,1,,2}2}2{1,,1,,2}2} has recursion level C(C(Ω22+C(Ω2++,0)+C(C(Ω22+C(Ω2++,0),0),C(Ω2+σ,0)),0),0)
• {1,,1{1{1{1,,1,,1♦2,,}2}1,2,,1{1,,1,,2}2}2{1,,1,,2}2} has recursion level C(C(Ω22+C(Ω2++,0)+C(C(Ω22+C(Ω2++,0),0)+1,C(Ω2+σ,0)),0),0)
• {1{1,,1{1{1,,1,,1♦2,,}2}2}2,,1{1{1,,1,,1♦2,,}2}2} has recursion level C(C(Ω22+C(Ω2++,0)+C(C(Ω22+C(Ω2++,0),0)2,C(Ω2+σ,0)),0),0)
• {1,,2{1{1,,1,,1♦2,,}2}2} has recursion level C(C(Ω22+C(Ω2++,0)+C(C(Ω22+C(Ω2++,0),0)+,C(Ω2+σ,0)),0),0)
• {1,,1{1,,1{1{1,,1,,1♦2,,}2}2}1,2{1{1,,1,,1♦2,,}2}2} has recursion level C(C(Ω22+C(Ω2++,0)+C(C(Ω22+C(Ω2++,0)+1,0),C(Ω2+σ,0)),0),0)
• {1,,1{1,,1{1{1,,1,,1♦2,,}2}2}1{1,,2{1,,1,,2}2}2{1{1,,1,,1♦2,,}2}2} has recursion level C(C(Ω22+C(Ω2++,0)+C(Ω2+σ,0)+,0),0)
• {1,,1{1,,1{1{1,,1,,1♦2,,}2}2}1{1,,1{1{1,,1,,1♦2,,}2}2}2{1{1,,1,,1♦2,,}2}2} has recursion level C(C(Ω22+C(Ω2++,0)+C(Ω2+C(C(Ω22+C(Ω2++,0),0),σ),0),0),0)
• {1,,1{2{1,,1,,1♦2,,}2}2} has recursion level C(C(Ω22+C(Ω2++,0)+C(Ω2+ω^(C(C(Ω22+C(Ω2++,0),0),σ)+1),0),0),0)
• {1{1,,1,,2}2{1,,1,,1♦2,,}2} has recursion level C(C(Ω22+C(Ω2++,0)+C(Ω2+ω^(C(C(Ω22+C(Ω2++,0),0),σ)+σ),0),0),0)
• {1{1,,1,,1,2}2{1,,1,,1♦2,,}2} has recursion level C(C(Ω22+C(Ω2++,0)+C(Ω2++1,0),0),0)
• {1{1,,1,,1,,1,2}2{1,,1,,1♦2,,}2} has recursion level C(C(Ω22+C(Ω2++,0)+C(Ω2+2+1,0),0),0)
• {1{1{1,,2,,}2}2{1,,1,,1♦2,,}2} has recursion level C(C(Ω22+C(Ω2++,0)+C(Ω2+C(σ+++),0),0),0)
• {1{1{1,,1,2,,}2}2{1,,1,,1♦2,,}2} has recursion level C(C(Ω22+C(Ω2++,0)+C(Ω2+C(C(Ω2+1,σ),σ+),0),0),0)
• {1{1{1,,1,,2,,}2}2{1,,1,,1♦2,,}2} has recursion level C(C(Ω22+C(Ω2++,0)+C(Ω2+C(C(Ω22,σ),σ+),0),0),0)
• {1{1{1,,1,,1,2,,}2}2{1,,1,,1♦2,,}2} has recursion level C(C(Ω22+C(Ω2++,0)+C(Ω2+C(C(Ω22+1,0),σ+),0),0),0)
• {1{1{1,,1,,1♦2,,}2}2{1,,1,,1♦2,,}2} has recursion level C(C(Ω22+C(Ω2++,0)+C(Ω2+C(C(Ω22+C(Ω2++,0),0),σ+),0),0),0)
• {1{1{1,,1,,1♦2,,}2}1,2{1,,1,,1♦2,,}2} has recursion level C(C(Ω22+C(Ω2++,0)+ω^(C(Ω2+C(C(Ω22+C(Ω2++,0),0),σ+),0)+1),0),0)
• {1,,2{1,,1,,1♦2,,}2} has recursion level C(C(Ω22+C(Ω2++,0)+C(C(Ω2+C(C(Ω22+C(Ω2++,0),0),σ+),0)+,C(Ω2+C(C(Ω22+C(Ω2++,0),0),σ+),0)),0),0)
• {1,,1,,2{1,,1,,1♦2,,}2} has recursion level C(C(Ω22+C(Ω2++,0)+C(σ,C(Ω2+C(C(Ω22+C(Ω2++,0),0),σ+),0)),0),0)
• {1,,1,,1,,2{1,,1,,1♦2,,}2} has recursion level C(C(Ω22+C(Ω2++,0)+C(Ω2++σ,C(Ω2+C(C(Ω22+C(Ω2++,0),0),σ+),0)),0),0)
• {1{1,,1,,1,2,,}2{1,,1,,1♦2,,}2} has recursion level C(C(Ω22+C(Ω2++,0)+C(Ω2+C(C(Ω22+1,0),σ+),C(Ω2+C(C(Ω22+C(Ω2++,0),0),σ+),0)),0),0)
• {1{1,,1,,1{1{1,,1,,1♦2,,}2}2,,}2{1,,1,,1♦2,,}2} has recursion level C(C(Ω22+C(Ω2++,0)+C(Ω2+C(C(Ω22+C(Ω2+C(C(Ω22+C(Ω2++,0),0),σ+),0),0),σ+),C(Ω2+C(C(Ω22+C(Ω2++,0),0),σ+),0)),0),0)
• {1{1,,1,,1{1{1,,1,,1,2,,}2{1,,1,,1♦2,,}2}2,,}2{1,,1,,1♦2,,}2} has recursion level C(C(Ω22+C(Ω2++,0)+C(Ω2+C(C(Ω22+C(Ω2+C(C(Ω22+1,0),σ+),C(Ω2+C(C(Ω22+C(Ω2++,0),0),σ+),0)),0),σ+),C(Ω2+C(C(Ω22+C(Ω2++,0),0),σ+),0)),0),0)
• {1{1,,1,,1♦2,,}3} has recursion level C(C(Ω22+C(Ω2++,0)+C(Ω2+C(C(Ω22+C(Ω2++,0),0),σ+),C(Ω2+C(C(Ω22+C(Ω2++,0),0),σ+),0)),0),0)
• {1{1,,1,,1♦2,,}1,2} has recursion level C(C(Ω22+C(Ω2++,0)+C(Ω2+C(C(Ω22+C(Ω2++,0),0),σ+)+1,0),0),0)
• {1{1,,1,,1♦2,,}1{1,,1,,1,2,,}2} has recursion level C(C(Ω22+C(Ω2++,0)+C(Ω2+C(C(Ω22+C(Ω2++,0),0),σ+)+C(C(Ω22+1,0),σ+),0),0),0)
• {1{1,,1,,1♦2,,}1{1,,1,,1{1{1,,1,,1♦2,,}2}2,,}2} has recursion level C(C(Ω22+C(Ω2++,0)+C(Ω2+C(C(Ω22+C(Ω2++,0),0),σ+)+C(C(Ω22+C(Ω2+C(C(Ω22+C(Ω2++,0),0),σ+),0),0),σ+),0),0),0)
• {1{1,,1,,1♦2,,}1{1,,1,,1♦2,,}2} has recursion level C(C(Ω22+C(Ω2++,0)+C(Ω2+C(C(Ω22+C(Ω2++,0),0),σ+)2,0),0),0)
• {1{2{1,,1,,1♦2,,}2,,}2} has recursion level C(C(Ω22+C(Ω2++,0)+C(Ω2+ω^(C(C(Ω22+C(Ω2++,0),0),σ+)+1),0),0),0)
• {1,,2{1,,1,,1♦2,,}2,,} has recursion level C(C(Ω22+C(Ω2++,0)+C(Ω2+C(σ++,C(C(Ω22+C(Ω2++,0),0),σ+)),0),0),0)
• {1{1,,1,,1,2,,}2{1,,1,,1♦2,,}2,,} has recursion level C(C(Ω22+C(Ω2++,0)+C(Ω2+C(C(Ω2++1,σ),C(C(Ω22+C(Ω2++,0),0),σ+)),0),0),0)
• {1{1,,1,,2,,}2{1,,1,,1♦2,,}2,,} has recursion level C(C(Ω22+C(Ω2++,0)+C(Ω2+C(C(Ω22,σ),C(C(Ω22+C(Ω2++,0),0),σ+)),0),0),0)
• {1{1,,1,,1♦2,,}3,,} has recursion level C(C(Ω22+C(Ω2++,0)+C(Ω2+C(C(Ω22+C(Ω2++,0),0),C(C(Ω22+C(Ω2++,0),0),σ+)),0),0),0)
• {1{1,,1,,1♦2,,}1,2,,} has recursion level C(C(Ω22+C(Ω2++,0)+C(Ω2+C(C(Ω22+C(Ω2++,0),0)+1,σ+),0),0),0)
• {1{1,,1,2,,}2,,1,,1♦2,,} has recursion level C(C(Ω22+C(Ω2++,0)+C(Ω2+C(C(Ω22+C(Ω2++,0),0)+C(Ω2+1,σ),σ+),0),0),0)
• {1{1,,1,,1♦2,,}2,,1,,1♦2,,} has recursion level C(C(Ω22+C(Ω2++,0)+C(Ω2+C(C(Ω22+C(Ω2++,0),0)2,σ+),0),0),0)
• {1,,2,,1♦2,,} has recursion level C(C(Ω22+C(Ω2++,0)+C(Ω2+C(C(Ω22+C(Ω2++,0),0)++),0),0),0)
• {1,,1,,2♦2,,} has recursion level C(C(Ω22+C(Ω2++,0)+C(Ω2+C(C(Ω22,C(Ω22+C(Ω2++,0),0)),σ+),0),0),0)
• {1,,1,,1♦1,2,,} has recursion level C(C(Ω22+C(Ω2++,0)+C(Ω2+C(C(Ω22+C(Ω2++,0)+1,0),σ+),0),0),0)
• {1,,1,,1♦1{1{1,,1,,1♦2,,}2}2,,} has recursion level C(C(Ω22+C(Ω2++,0)+C(Ω2+C(C(Ω22+C(Ω2++,0)+C(Ω2+C(C(Ω22+C(Ω2++,0),0),σ+),0),0),σ+),0),0),0)
• {1,,1,,1♦1♦2,,} has recursion level C(C(Ω22+C(Ω2++,0)2,0),0)
• {1,,2{1,,1,,1,,2,,}2} has recursion level C(C(Ω22+C(C(Ω2++,0)+,C(Ω2++,0)),0),0)
• {1,,1,,2{1,,1,,1,,2,,}2} has recursion level C(C(Ω22+C(σ,C(Ω2++,0)),0),0)
• {1,,1,,1,,2{1,,1,,1,,2,,}2} has recursion level C(C(Ω22+C(Ω2++σ,C(Ω2++,0)),0),0)
• {1{1,,2,,}2{1,,1,,1,,2,,}2} has recursion level C(C(Ω22+C(Ω2+C(σ+++),C(Ω2++,0)),0),0)
• {1{1,,1,2,,}2{1,,1,,1,,2,,}2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2+1,σ),σ+),C(Ω2++,0)),0),0)
• {1{1,,1,,1,2,,}2{1,,1,,1,,2,,}2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2++1,σ),σ+),C(Ω2++,0)),0),0)
• {1{1,,1{1{1,,1,2,,}2,,}2,,}2{1,,1,,1,,2,,}2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2+C(C(Ω2+1,σ),σ+),σ),σ+),C(Ω2++,0)),0),0)
• {1{1,,1,,2,,}2{1,,1,,1,,2,,}2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22,σ),σ+),C(Ω2++,0)),0),0)
• {1{1,,1,,1,2,,}2{1,,1,,1,,2,,}2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22+1,0),σ+),C(Ω2++,0)),0),0)
• {1{1,,1,,1{1{1,,1,,1♦2,,}2}2,,}2{1,,1,,1,,2,,}2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22+C(Ω2+C(C(Ω22+C(Ω2++,0),0),σ+),0),0),σ+),C(Ω2++,0)),0),0)
• {1{1,,1,,1♦2,,}2{1,,1,,1,,2,,}2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22+C(Ω2++,0),0),σ+),C(Ω2++,0)),0),0)
• {1{1,,1,,1{1,,2{1,,1,,1,,2,,}2}2,,}2{1,,1,,1,,2,,}2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22+C(Ω2++,0)+,0),σ+),C(Ω2++,0)),0),0)
• {1{1,,1,,1,,2,,}3} has recursion level C(C(Ω22+C(Ω2++,C(Ω2++,0)),0),0)
• {1{1,,1,,1,,2,,}1,2} has recursion level C(C(Ω22+C(Ω2+++1,0),0),0)

As shown above, C(Ω2+C(Ω2,C(Ω2,C(Ω22,0))),0) is not so strong as I originally conjectured – it just works as the 1-separator in C(Ω2+C(C(Ω22+____,0),C(Ω2,C(Ω22,0))),β), and corresponds to {1{1,,1,,1,,2,,}2}.

Now let ordinal σ = C(Ω22,0). And α+ is shorthand for C(Ω2,α).

• {1{1,,1,,1,,2,,}1,,2} has recursion level C(C(Ω22+C(Ω2+++σ,0),0),0)
• {1{1,,1,,1,,2,,}1{1{1,,1,,1,,2,,}1,,2}1{1{1,,1,,1,,2,,}1,,2}2} has recursion level C(C(Ω22+C(Ω2+++σ2,0),0),0)
• {1,,2{1,,1,,1,,2,,}1,,2} has recursion level C(C(Ω22+C(Ω2+++C(σ+,σ),0),0),0)
• {1,,1{1{1,,1,,1,,2,,}1,,2}2{1,,1,,1,,2,,}1,,2} has recursion level C(C(Ω22+C(Ω2+++C(C(Ω2+σ,σ),σ),0),0),0)
• {1,,1{1,,2{1,,1,,1,,2,,}1,,2}2{1,,1,,1,,2,,}1,,2} has recursion level C(C(Ω22+C(Ω2+++C(C(Ω2+,σ),σ),0),0),0)
• {1,,1{1,,3{1,,1,,1,,2,,}1,,2}2{1,,1,,1,,2,,}1,,2} has recursion level C(C(Ω22+C(Ω2+++C(C(Ω2++,σ),σ),0),0),0)
• {1,,1{1,,3{1,,1,,1,,2,,}1,,2}1{1,,2{1,,1,,1,,2,,}1,,2}2{1,,1,,1,,2,,}1,,2} has recursion level C(C(Ω22+C(Ω2+++C(C(Ω22,σ),σ),0),0),0)
• {1,,1{1,,3{1,,1,,1,,2,,}1,,2}1{1,,2{1,,1,,1,,2,,}1,,2}1,2{1,,1,,1,,2,,}1,,2} has recursion level C(C(Ω22+C(Ω2+++C(C(Ω22+1,0),σ),0),0),0)
• {1,,1{1,,3{1,,1,,1,,2,,}1,,2}1{1,,2{1,,1,,1,,2,,}1,,2}1{1{1,,1,,1,,2,,}1,,2}2{1,,1,,1,,2,,}1,,2} has recursion level C(C(Ω22+C(Ω2+++,0),0),0)
• {1{1,,1,,1,,2,,}1,,1,2} has recursion level C(C(Ω22+C(Ω2++++1,0),0),0)
• {1{1,,1,,1,,2,,}1{1,,1,,2,,}2} has recursion level C(C(Ω22+C(Ω2+++C(C(Ω2+σ,σ),σ+),0),0),0)
• {1{1,,1,,1,,2,,}1{1,,1,,1,2,,}2} has recursion level C(C(Ω22+C(Ω2+++C(C(Ω2++1,σ),σ+),0),0),0)
• {1{1,,1,,1,,2,,}1{1,,1,,2,,}2} has recursion level C(C(Ω22+C(Ω2+++C(C(Ω2++,σ),σ+),0),0),0)
• {1{1,,1,,1,,2,,}1{1,,1,,1,,2,,}2} has recursion level C(C(Ω22+C(Ω2+++C(C(Ω2+++σ,σ),σ+),0),0),0) (here Taranovsky’s ordinal notation has an unexpected growth boost because C(Ω2+++C(C(Ω2++,σ),σ+),0) is valid)
• {1,,1,,1,,1,2,,} has recursion level C(C(Ω22+C(Ω2+++C(C(Ω2++++1,σ),σ+),0),0),0)
• {1,,1,,1,,2,,} has recursion level C(C(Ω22+C(Ω2+++C(C(Ω22,σ),σ+),0),0),0)
• {1,,1,,1,,1,2,,} has recursion level C(C(Ω22+C(Ω2+++C(C(Ω22+1,0),σ+),0),0),0)
• {1,,1,,1,,1,,2,,} has recursion level C(C(Ω22+C(Ω2++2,0),0),0)
• {1{1,,1,,1,,1,,2,,}1,2} has recursion level C(C(Ω22+C(Ω2++2+1,0),0),0)
• {1{1,,1,,1,,1,,2,,}1{1,,1,,1,2,,}2} has recursion level C(C(Ω22+C(Ω2++2+C(C(Ω2++1,σ),σ+),0),0),0)
• {1{1,,1,,1,,1,,2,,}1{1,,1,,2,,}2} has recursion level C(C(Ω22+C(Ω2++2+C(C(Ω2++,σ),σ+),0),0),0)
• {1{1,,1,,1,,1,,2,,}1{1,,1,,1,,2,,}2} has recursion level C(C(Ω22+C(Ω2++2+C(C(Ω2++2,σ),σ+),0),0),0)
• {1{1,,1,,1,,1,,2,,}1{1,,1,,1,,1,,2,,}2} has recursion level C(C(Ω22+C(Ω2++2+C(C(Ω2++2+σ,σ),σ+),0),0),0)
• {1,,1,,1,,1,,2,,} has recursion level C(C(Ω22+C(Ω2++2+C(C(Ω22,σ),σ+),0),0),0)
• {1,,1,,1,,1,,1,,2,,} has recursion level C(C(Ω22+C(Ω2++3,0),0),0)
• {1,,1,,1,,1,,1,,1,,2,,} has recursion level C(C(Ω22+C(Ω2++4,0),0),0)
• {1,,1{2,,}2,,} has recursion level C(C(Ω22+C(Ω2+ω^(σ+++1),0),0),0)
• {1,,1{1,,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+ω^(σ+++σ),0),0),0)
• {1{1,,1{1{1{1,,1{1,,2,,}2,,}2}2,,}2,,}1,2} has recursion level C(C(Ω22+C(Ω2+ω^(σ+++σ)+1,0),0),0)
• {1{1,,1{1,,2,,}2,,}1,2} has recursion level C(C(Ω22+C(Ω2+ω^(σ+++)+1,0),0),0)
• {1{1,,1{1,,2,,}2,,}1,,1,2} has recursion level C(C(Ω22+C(Ω2+ω^(σ+++)+σ++1,0),0),0)
• {1{1,,1{1,,2,,}2,,}1{1,,1,,2,,}2} has recursion level C(C(Ω22+C(Ω2+ω^(σ+++)+C(C(Ω2++,σ),σ+),0),0),0)
• {1{1,,1{1,,2,,}2,,}1{1,,1{1,,2,,}2,,}2} has recursion level C(C(Ω22+C(Ω2+ω^(σ+++)+C(C(Ω2+ω^(σ+++σ),σ),σ+),0),0),0)
• {1,,1{1,,2,,}1,,2,,} has recursion level C(C(Ω22+C(Ω2+ω^(σ+++)+C(C(Ω22,σ),σ+),0),0),0)
• {1,,1{1,,2,,}1,,1,,2,,} has recursion level C(C(Ω22+C(Ω2+ω^(σ+++)+σ++,0),0),0)
• {1,,1{2,,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+ω^(σ++++1),0),0),0)
• {1,,1{1,,3,,}2,,} has recursion level C(C(Ω22+C(Ω2+ω^(σ++++σ),0),0),0)
• {1,,1{1{1{1,,1,,2,,}2,,}2,,}2,,} has recursion level C(C(Ω22+C(Ω2+ω^(σ+++C(C(Ω2++,σ),σ+)),0),0),0)
• {1,,1{1{1{1,,1{2,,2,,}2,,}2,,}2,,}2,,} has recursion level C(C(Ω22+C(Ω2+ω^(σ+++C(C(Ω2+ω^(σ++++1),σ),σ+)),0),0),0)
• {1,,1{1,,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+ω^(σ+++C(C(Ω22,σ),σ+)),0),0),0)
• {1{1,,1{1,,2,,}2,,}1,2} has recursion level C(C(Ω22+C(Ω2+ω^(σ+++C(C(Ω22,σ),σ+))+1,0),0),0)
• {1{1,,1{1,,2,,}2,,}1{1,,1{2,,}2,,}2} has recursion level C(C(Ω22+C(Ω2+ω^(σ+++C(C(Ω22,σ),σ+))+C(C(Ω2+ω^(σ+++1),σ),σ+),0),0),0)
• {1{1,,1{1,,2,,}2,,}1{1,,1{1{1{1,,1{2,,}2,,}2,,}2,,}2,,}2} has recursion level C(C(Ω22+C(Ω2+ω^(σ+++C(C(Ω22,σ),σ+))+C(C(Ω2+ω^(σ+++C(C(Ω2+ω^(σ+++1),σ),σ+)),σ),σ+),0),0),0)
• {1{1,,1{1,,2,,}2,,}1{1,,1{1,,2,,}2,,}2} has recursion level C(C(Ω22+C(Ω2+ω^(σ+++C(C(Ω22,σ),σ+))+C(C(Ω22,σ),σ+),0),0),0) (here an erratic thing comes because C(C(Ω2+ω^(σ+++C(C(Ω22,σ),σ+)),σ),σ+) is not valid)
• {1{2{1,,1{1,,2,,}2,,}2,,}2} has recursion level C(C(Ω22+C(Ω2+ω^(σ+++C(C(Ω22,σ),σ+))+ω^(C(C(Ω22,σ),σ+)+1),0),0),0)
• {1,,2{1,,1{1,,2,,}2,,}2,,} has recursion level C(C(Ω22+C(Ω2+ω^(σ+++C(C(Ω22,σ),σ+))+C(σ++,C(C(Ω22,σ),σ+)),0),0),0)
• {1{1,,1{2,,}2,,}2{1,,1{1,,2,,}2,,}2,,} has recursion level C(C(Ω22+C(Ω2+ω^(σ+++C(C(Ω22,σ),σ+))+C(C(Ω2+ω^(σ+++1),σ),C(C(Ω22,σ),σ+)),0),0),0)
• {1{1,,1{1{1{1,,1{2,,}2,,}2,,}2,,}2,,}2{1,,1{1,,2,,}2,,}2,,} has recursion level C(C(Ω22+C(Ω2+ω^(σ+++C(C(Ω22,σ),σ+))+C(C(Ω2+ω^(σ+++C(C(Ω2+ω^(σ+++1),σ),σ+)),σ),C(C(Ω22,σ),σ+)),0),0),0)
• {1{1,,1{1{1{1,,1{1,,2,,}2,,}2,,}2,,}2,,}2{1,,1{1,,2,,}2,,}2,,} has recursion level C(C(Ω22+C(Ω2+ω^(σ+++C(C(Ω22,σ),σ+))+C(C(Ω2+ω^(σ+++C(C(Ω22,σ),σ+)),σ),C(C(Ω22,σ),σ+)),0),0),0)
• {1{1,,1{1{1{1,,1{1,,2,,}2,,}2,,}2,,}1,2,,}2{1,,1{1,,2,,}2,,}2,,} has recursion level C(C(Ω22+C(Ω2+ω^(σ+++C(C(Ω22,σ),σ+))+C(C(Ω2+ω^(σ+++C(C(Ω22,σ),σ+))+1,σ),C(C(Ω22,σ),σ+)),0),0),0)
• {1{1,,1{1{1{1,,1{1,,2,,}2,,}2,,}2,,}1{1{1,,1{2,,}2,,}2,,}2,,}2{1,,1{1,,2,,}2,,}2,,} has recursion level C(C(Ω22+C(Ω2+ω^(σ+++C(C(Ω22,σ),σ+))+C(C(Ω2+ω^(σ+++C(C(Ω22,σ),σ+))+C(C(Ω2+ω^(σ+++1),σ),σ+),σ),C(C(Ω22,σ),σ+)),0),0),0)
• {1{1,,1{1{1{1,,1{1,,2,,}2,,}2,,}2,,}1{1{1,,1{1,,2,,}2,,}2,,}2,,}2{1,,1{1,,2,,}2,,}2,,} has recursion level C(C(Ω22+C(Ω2+ω^(σ+++C(C(Ω22,σ),σ+))+C(C(Ω2+ω^(σ+++C(C(Ω22,σ),σ+))+C(C(Ω22,σ),σ+),σ),C(C(Ω22,σ),σ+)),0),0),0)
• {1{1,,1{1{1{1,,1{1,,2,,}2,,}2,,}2,,}1{2{1,,1{1,,2,,}2,,}2,,}2,,}2{1,,1{1,,2,,}2,,}2,,} has recursion level C(C(Ω22+C(Ω2+ω^(σ+++C(C(Ω22,σ),σ+))+C(C(Ω2+ω^(σ+++C(C(Ω22,σ),σ+))+ω^(C(C(Ω22,σ),σ+)+1),σ),C(C(Ω22,σ),σ+)),0),0),0)
• {1{1,,1{1{1{1,,1{1,,2,,}2,,}2,,}2,,}1,,2,,}2{1,,1{1,,2,,}2,,}2,,} has recursion level C(C(Ω22+C(Ω2+ω^(σ+++C(C(Ω22,σ),σ+))+C(C(Ω22,σ),C(C(Ω22,σ),σ+)),0),0),0)
• {1{1,,1{1{1{1,,1{1,,2,,}2,,}2,,}2,,}1,,2,,}1,2{1,,1{1,,2,,}2,,}2,,} has recursion level C(C(Ω22+C(Ω2+ω^(σ+++C(C(Ω22,σ),σ+))+C(C(Ω22,σ)+1,σ+),0),0),0)
• {1{1{1,,1,2,,}2,,1{1{1{1,,1{1,,2,,}2,,}2,,}2,,}1,,2,,}2{1,,1{1,,2,,}2,,}2,,} has recursion level C(C(Ω22+C(Ω2+ω^(σ+++C(C(Ω22,σ),σ+))+C(C(Ω22,σ)+C(Ω2+1,σ),σ+),0),0),0)
• {1{1{1,,1{1{1{1,,1{1,,2,,}2,,}2,,}2,,}2,,}2,,1{1{1{1,,1{1,,2,,}2,,}2,,}2,,}1,,2,,} 2{1,,1{1,,2,,}2,,}2,,} has recursion level C(C(Ω22+C(Ω2+ω^(σ+++C(C(Ω22,σ),σ+))+C(C(Ω22,σ)+C(Ω2+ω^(σ+++C(C(Ω22,σ),σ+)),σ),σ+),0),0),0)
• {1{1{1,,1{1{1{1,,1{1,,2,,}2,,}2,,}2,,}1,,2,,}2,,1{1{1{1,,1{1,,2,,}2,,}2,,}2,,}1,,2,,} 2{1,,1{1,,2,,}2,,}2,,} has recursion level C(C(Ω22+C(Ω2+ω^(σ+++C(C(Ω22,σ),σ+))+C(C(Ω22,σ)2,σ+),0),0),0)
• {1{1,,2{1{1{1,,1{1,,2,,}2,,}2,,}2,,}1,,2,,}2{1,,1{1,,2,,}2,,}2,,} has recursion level C(C(Ω22+C(Ω2+ω^(σ+++C(C(Ω22,σ),σ+))+C(C(Ω22,σ)++),0),0),0)
• {1{1,,1{1{1{1,,1{1,,2,,}2,,}2,,}2,,}2,,2,,}2{1,,1{1,,2,,}2,,}2,,} has recursion level C(C(Ω22+C(Ω2+ω^(σ+++C(C(Ω22,σ),σ+))+C(C(Ω2+ω^(σ+++C(C(Ω22,σ),σ+)),C(Ω22,σ)),σ+),0),0),0)
• {1{1,,1{1{1{1,,1{1,,2,,}2,,}2,,}2,,}1,,3,,}2{1,,1{1,,2,,}2,,}2,,} has recursion level C(C(Ω22+C(Ω2+ω^(σ+++C(C(Ω22,σ),σ+))+C(C(Ω22,C(Ω22,σ)),σ+),0),0),0)
• {1{1,,1{1{1{1,,1{1,,2,,}2,,}2,,}2,,}1,,1,2,,}2{1,,1{1,,2,,}2,,}2,,} has recursion level C(C(Ω22+C(Ω2+ω^(σ+++C(C(Ω22,σ),σ+))+C(C(Ω22+1,0),σ+),0),0),0)
• {1{1,,1{1{1{1,,1{1,,2,,}2,,}2,,}2,,}1,,1,,2,,}2{1,,1{1,,2,,}2,,}2,,} has recursion level C(C(Ω22+C(Ω2+ω^(σ+++C(C(Ω22,σ),σ+))+σ++,0),0),0)
• {1{1,,1{1{1{1,,1{1,,2,,}2,,}2,,}2,,}1{2,,}2,,}2{1,,1{1,,2,,}2,,}2,,} has recursion level C(C(Ω22+C(Ω2+ω^(σ+++C(C(Ω22,σ),σ+))+ω^(σ+++1),0),0),0)
• {1{1,,1{1{1{1,,1{1,,2,,}2,,}2,,}2,,}1{1{1{1,,1{1,,2,,}2,,}2,,}2,,}2,,}2{1,,1{1,,2,,}2,,}2,,} has recursion level C(C(Ω22+C(Ω2+ω^(σ+++C(C(Ω22,σ),σ+))2,0),0),0)
• {1{1,,1{2{1{1,,1{1,,2,,}2,,}2,,}2,,}2,,}2{1,,1{1,,2,,}2,,}2,,} has recursion level C(C(Ω22+C(Ω2+ω^(σ+++C(C(Ω22,σ),σ+)+1),0),0),0)
• {1{1,,1{1{1{1,,1{1,,2,,}2,,}2,,}3,,}2,,}2{1,,1{1,,2,,}2,,}2,,} has recursion level C(C(Ω22+C(Ω2+ω^(σ+++C(C(Ω22,σ),σ+)2),0),0),0)
• {1{1,,1{1{1{1,,1{2,,}2,,}2{1,,1{1,,2,,}2,,}2,,}2,,}2,,}2{1,,1{1,,2,,}2,,}2,,} has recursion level C(C(Ω22+C(Ω2+ω^(σ+++C(C(Ω2+ω^(σ+++1),σ),C(C(Ω22,σ),σ+))),0),0),0)
• {1{1,,1{1,,2,,}2,,}3,,} has recursion level C(C(Ω22+C(Ω2+ω^(σ+++C(C(Ω22,σ),C(C(Ω22,σ),σ+))),0),0),0) (now it returns to the normal)
• {1{1,,1{1,,2,,}2,,}1,2,,} has recursion level C(C(Ω22+C(Ω2+ω^(σ+++C(C(Ω22,σ)+1,σ+)),0),0),0)
• {1,,2{1,,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+ω^(σ+++C(C(Ω22,σ)++)),0),0),0)
• {1,,1{1,,2,,}1,2,,} has recursion level C(C(Ω22+C(Ω2+ω^(σ+++C(C(Ω22+1,0),σ+)),0),0),0)
• {1,,1{1,,2,,}1,,2,,} has recursion level C(C(Ω22+C(Ω2+ω^(σ++2),0),0),0)
• {1,,1{1,,2,,}1{2,,}2,,} has recursion level C(C(Ω22+C(Ω2+ω^(σ++2)+ω^(σ+++1),0),0),0)
• {1,,1{1,,2,,}1{1,,2,,}1,,2,,} has recursion level C(C(Ω22+C(Ω2+ω^(σ++2)2,0),0),0)
• {1,,1{2,,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+ω^(σ++2+1),0),0),0)
• {1,,1{1,,1,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+ω^ω^(σ+++1),0),0),0)
• {1,,1{1 ,,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(σ+++++),0),0),0)
• {1,,1{1{1,,1,2,,}2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2+1,σ),σ++),0),0),0)
• {1,,1{1{1,,1,,1,2,,}2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2++1,σ),σ++),0),0),0)
• {1,,1{1{1,,1{1{1,,1,2,,}2,,}2,,}2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2+C(C(Ω2+1,σ),σ+),σ),σ++),0),0),0)
• {1,,1{1{1,,1{1{1,,1,,2,,}2,,}2,,}2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2+C(C(Ω2++,σ),σ+),σ),σ++),0),0),0)
• {1,,1{1{1,,1{1{1,,1{1{1,,1,2,,}2,,}2,,}2,,}2,,}2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2+C(C(Ω2+C(C(Ω2+1,σ),σ++),σ),σ+),σ),σ++),0),0),0)
• {1,,1{1{1,,1,,2,,}2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2+C(C(Ω22,σ),σ+),σ),σ++),0),0),0)
• {1,,2{1{1,,1,,2,,}2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2+C(C(Ω22,σ)++),σ),σ++),0),0),0)
• {1,,1{1{1,,1,,2,,}2,,}1,2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2+C(C(Ω22+1,0),σ+),σ),σ++),0),0),0)
• {1,,1{1{1,,1,,2,,}2,,}1,,2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2++,σ),σ++),0),0),0)
• {1,,1{2{1,,1,,2,,}2,,}2,,} has recursion level C(C(Ω22+C(Ω2+ω^(C(C(Ω2++,σ),σ++)+1),0),0),0)
• {1,,1{1{1,,1,,2,,}1,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2++,σ)+1,σ++),0),0),0)
• {1,,1{1{1,,2,,2,,}2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2++,σ)+++),0),0),0)
• {1,,1{1{1,,1,,1,2,,}2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2+++1,σ),σ++),0),0),0)
• {1,,1{1{1,,1{1{1,,1,2,,}2,,}2,,}2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2+C(C(Ω2+1,σ),σ++),σ),σ++),0),0),0)
• {1,,1{1{1,,1{1{1,,1,,2,,}2,,}2,,}2,,}1,,2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2+C(C(Ω2++,σ),σ++),σ),σ++),0),0),0)
• {1,,1 ,,2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22,σ),σ++),0),0),0)
• {1{1,,1 ,,2,,}1,2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22,σ),σ++)+1,0),0),0)
• {1{1,,1 ,,2,,}1{1,,1,,2,,}2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22,σ),σ++)+C(C(Ω2++,σ),σ+),0),0),0)
• {1{1,,1 ,,2,,}1{1,,1{1{1,,1,,2,,}2,,}2,,}2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22,σ),σ++)+C(C(Ω2+C(C(Ω2++,σ),σ++),σ),σ+),0),0),0)
• {1{1,,1 ,,2,,}1{1,,1 ,,2,,}2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22,σ),σ++)+C(C(Ω22,σ),σ+),0),0),0) (here’s another similar yet erratic thing)
• {1{1,,1,,2,,}2{1,,1 ,,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22,σ),σ++)+C(C(Ω2++,σ),C(C(Ω22,σ),σ+)),0),0),0)
• {1{1,,1 ,,2,,}3,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22,σ),σ++)+C(C(Ω22,σ),C(C(Ω22,σ),σ+)),0),0),0)
• {1{1,,1 ,,2,,}1,2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22,σ),σ++)+C(C(Ω22,σ)+1,σ+),0),0),0)
• {1,,2 ,,2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22,σ),σ++)+C(C(Ω22,σ)++),0),0),0)
• {1,,1,,2 ,,2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22,σ),σ++)+C(C(Ω2++,C(Ω22,σ)),σ+),0),0),0)
• {1,,1{1{1,,1 ,,2,,}2,,}2 ,,2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22,σ),σ++)+C(C(Ω22,C(Ω22,σ)),σ+),0),0),0)
• {1,,1{1{1,,1 ,,2,,}2,,}1,2 ,,2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22,σ),σ++)+C(C(Ω22+1,0),σ+),0),0),0)
• {1,,1{1{1,,1 ,,2,,}2,,}1,,2 ,,2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22,σ),σ++)+σ++,0),0),0)
• {1{1,,1{1{1,,1 ,,2,,}2,,}1,,2 ,,2,,}1,2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22,σ),σ++)+σ+++1,0),0),0)
• {1{1,,1{1{1,,1 ,,2,,}2,,}1,,2 ,,2,,}1{1,,1,,2,,}2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22,σ),σ++)+σ+++C(C(Ω2++,σ),σ+),0),0),0)
• {1{1,,1{1{1,,1 ,,2,,}2,,}1,,2 ,,2,,}1{1,,1 ,,2,,}2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22,σ),σ++)+σ+++C(C(Ω22,σ),σ+),0),0),0)
• {1{1,,1{1{1,,1 ,,2,,}2,,}1,,2 ,,2,,}1{1,,1{1{1,,1 ,,2,,}2,,}1,2 ,,2,,}2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22,σ),σ++)+σ+++C(C(Ω22+1,0),σ+),0),0),0)
• {1{1,,1{1{1,,1 ,,2,,}2,,}1,,2 ,,2,,}1{1,,1{1{1,,1 ,,2,,}2,,}1,,2 ,,2,,}2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22,σ),σ++)+σ++2,0),0),0)
• {1{2{1,,1{1{1,,1 ,,2,,}2,,}1,,2 ,,2,,}2,,}2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22,σ),σ++)+ω^(σ+++1),0),0),0)
• {1{1,,1,2,,}2{1,,1{1{1,,1 ,,2,,}2,,}1,,2 ,,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22,σ),σ++)+C(C(Ω2+1,σ),σ++),0),0),0)
• {1{1,,1{1{1,,1 ,,2,,}2,,}2,,}2{1,,1{1{1,,1 ,,2,,}2,,}1,,2 ,,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22,σ),σ++)+C(C(Ω2+C(C(Ω22,σ),σ+),σ),σ++),0),0),0)
• {1{1,,1{1{1,,1{1{1,,1 ,,2,,}2,,}1,2 ,,2,,}2,,}2,,}2{1,,1{1{1,,1 ,,2,,}2,,}1,,2 ,,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22,σ),σ++)+C(C(Ω2+C(C(Ω22+1,0),σ+),σ),σ++),0),0),0)
• {1{1,,1{1{1,,1{1{1,,1 ,,2,,}2,,}1,,2 ,,2,,}2,,}2,,}2{1,,1{1{1,,1 ,,2,,}2,,}1,,2 ,,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22,σ),σ++)+C(C(Ω2++,σ),σ++),0),0),0)
• {1{1,,2{1{1,,1{1{1,,1 ,,2,,}2,,}1,,2 ,,2,,}2,,}2,,}2{1,,1{1{1,,1 ,,2,,}2,,}1,,2 ,,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22,σ),σ++)+C(C(Ω2++,σ)+++),0),0),0)
• {1{1,,1{1{1,,1{1{1,,1 ,,2,,}2,,}1,,2 ,,2,,}2,,}1,2,,}2{1,,1{1{1,,1 ,,2,,}2,,}1,,2 ,,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22,σ),σ++)+C(C(Ω2+++1,σ),σ++),0),0),0)
• {1{1,,1{1{1,,1,2,,}2{1,,1{1{1,,1 ,,2,,}2,,}1,,2 ,,2,,}2,,}2,,}2{1,,1{1{1,,1 ,,2,,}2,,}1,,2 ,,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22,σ),σ++)+C(C(Ω2+C(C(Ω2+1,σ),σ++),σ),σ++),0),0),0)
• {1{1,,1,,2,,}2{1,,1{1{1,,1 ,,2,,}2,,}1,,2 ,,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22,σ),σ++)2,0),0),0)
• {1{1{1,,1,,2,,}2{1,,1{1{1,,1 ,,2,,}2,,}1,,2 ,,2,,}2,,}1,2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22,σ),σ++)2+1,0),0),0)
• {1{1{1,,1,,2,,}2{1,,1{1{1,,1 ,,2,,}2,,}1,,2 ,,2,,}2,,}1{1{1,,1 ,,2,,}2,,}2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22,σ),σ++)2+C(C(Ω22,σ),σ+),0),0),0)
• {1{1{1,,1,,2,,}2{1,,1{1{1,,1 ,,2,,}2,,}1,,2 ,,2,,}2,,}1{1{1,,1{1{1,,1 ,,2,,}2,,}1,,2 ,,2,,}2,,}2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22,σ),σ++)2+σ++,0),0),0)
• {1{1{1,,1,,2,,}2{1,,1{1{1,,1 ,,2,,}2,,}1,,2 ,,2,,}2,,}1{1{1,,1,,2,,}2{1,,1{1{1,,1 ,,2,,}2,,}1,,2 ,,2,,}2,,}2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22,σ),σ++)3,0),0),0)
• {1{2{1,,1,,2,,}2{1,,1{1{1,,1 ,,2,,}2,,}1,,2 ,,2,,}2,,}2} has recursion level C(C(Ω22+C(Ω2+ω^(C(C(Ω22,σ),σ++)+1),0),0),0)
• {1{1,,1,2,,}2{1,,1,,2,,}2{1,,1{1{1,,1 ,,2,,}2,,}1,,2 ,,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2+1,σ),C(C(Ω22,σ),σ++)),0),0),0)
• {1{1,,1{1{1,,1 ,,2,,}2,,}2,,}2{1,,1,,2,,}2{1,,1{1{1,,1 ,,2,,}2,,}1,,2 ,,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2+C(C(Ω22,σ),σ+),σ),C(C(Ω22,σ),σ++)),0),0),0)
• {1{1,,1{1{1,,1{1{1,,1 ,,2,,}2,,}1,,2 ,,2,,}2,,}2,,}2{1,,1,,2,,}2{1,,1{1{1,,1 ,,2,,}2,,}1,,2 ,,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2++,σ),C(C(Ω22,σ),σ++)),0),0),0)
• {1{1,,1{1{1,,1,2,,}2{1,,1{1{1,,1 ,,2,,}2,,}1,,2 ,,2,,}2,,}2,,}2{1,,1,,2,,}2{1,,1{1{1,,1 ,,2,,}2,,}1,,2 ,,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2+C(C(Ω2+1,σ),σ++),σ),C(C(Ω22,σ),σ++)),0),0),0)
• {1{1,,1{1{1,,1{1{1,,1{1{1,,1 ,,2,,}2,,}1,,2 ,,2,,}2,,}2,,}2{1,,1{1{1,,1 ,,2,,}2,,}1,,2 ,,2,,}2,,}2,,}2{1,,1,,2,,}2{1,,1{1{1,,1 ,,2,,}2,,}1,,2 ,,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2+C(C(Ω2++,σ),σ++),σ),C(C(Ω22,σ),σ++)),0),0),0)
• {1{1,,1{1{1,,1,,2,,}2{1,,1{1{1,,1 ,,2,,}2,,}1,,2 ,,2,,}2,,}2,,}2{1,,1,,2,,}2{1,,1{1{1,,1 ,,2,,}2,,}1,,2 ,,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2+C(C(Ω22,σ),σ++),σ),C(C(Ω22,σ),σ++)),0),0),0)
• {1{1,,1{1{1,,1,2,,}2{1,,1,,2,,}2{1,,1{1{1,,1 ,,2,,}2,,}1,,2 ,,2,,}2,,}2,,}2{1,,1,,2,,}2{1,,1{1{1,,1 ,,2,,}2,,}1,,2 ,,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2+C(C(Ω2+1,σ),C(C(Ω22,σ),σ++)),σ),C(C(Ω22,σ),σ++)),0),0),0)
• {1{1,,1,,2,,}3{1,,1{1{1,,1 ,,2,,}2,,}1,,2 ,,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22,σ),C(C(Ω22,σ),σ++)),0),0),0)
• {1{1,,1,,2,,}1,2{1,,1{1{1,,1 ,,2,,}2,,}1,,2 ,,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22,σ)+1,σ++),0),0),0)
• {1{1{1,,1,2,,}2,,1,,2,,}2{1,,1{1{1,,1 ,,2,,}2,,}1,,2 ,,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22,σ)+C(Ω2+1,σ),σ++),0),0),0)
• {1{1{1,,1{1{1,,1{1{1,,1 ,,2,,}2,,}1,,2 ,,2,,}2,,}2,,}2,,1,,2,,}2{1,,1{1{1,,1 ,,2,,}2,,}1,,2 ,,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22,σ)+C(Ω2++,σ),σ++),0),0),0)
• {1{1{1,,1,,2,,}2,,1,,2,,}2{1,,1{1{1,,1 ,,2,,}2,,}1,,2 ,,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22,σ)2,σ++),0),0),0)
• {1{1,,2,,2,,}2{1,,1{1{1,,1 ,,2,,}2,,}1,,2 ,,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22,σ)+++),0),0),0)
• {1{1,,1{1{1,,1{1{1,,1 ,,2,,}2,,}1,,2 ,,2,,}2,,}2,,2,,}2{1,,1{1{1,,1 ,,2,,}2,,}1,,2 ,,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2++,C(Ω22,σ)),σ++),0),0),0)
• {1{1,,1,,3,,}2{1,,1{1{1,,1 ,,2,,}2,,}1,,2 ,,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22,C(Ω22,σ)),σ++),0),0),0)
• {1{1,,1,,1,2,,}2{1,,1{1{1,,1 ,,2,,}2,,}1,,2 ,,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22+1,0),σ++),0),0),0)
• {1{1,,1,,1{1{1{1,,1,,1,2,,}2{1,,1{1{1,,1 ,,2,,}2,,}1,,2 ,,2,,}2,,}2}2,,}2{1,,1{1{1,,1 ,,2,,}2,,}1,,2 ,,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22+C(Ω2+C(C(Ω22+1,0),σ++),0),0),σ++),0),0),0)
• {1{1,,1,,1,,2,,}2{1,,1{1{1,,1 ,,2,,}2,,}1,,2 ,,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+++,0),0),0)

Up to here, it seems that C(Ω2,C(Ω22,0)) corresponds to the double comma, and C(Ω2,C(Ω2,C(Ω22,0))) corresponds to {1{1,,1{1{1,,1 ,,2,,}2,,}1,,2 ,,2,,}2,,}.

## Up to a C(Ω2+C(Ω22,0),C(Ω22,0))

Now let separator ■ = {1,,1{1{1,,1 ,,2,,}2,,}1,,2 ,,2,,} and separator ♦ = {1{1{1,,1,,1,,2,,}2■2,,}2} = {1{1{1,,1,,1,,2,,}2{1,,1{1{1,,1 ,,2,,}2,,}1,,2 ,,2,,}2,,}2} and let ordinal σ = C(Ω22,0). And α+ is shorthand for C(Ω2,α).

• {1{1{1,,1,,1♦2,,}2■2,,}2} has recursion level C(C(Ω22+C(Ω2+++,0),0),0)
• {1{1,,1,,1,2}2{1{1,,1,,1♦2,,}2■2,,}2} has recursion level C(C(Ω22+C(Ω2+++,0)+C(Ω2++1,0),0),0)
• {1{1{1,,1,,1{1{1,,1,,1,,2,,}2}2,,}2}2{1{1,,1,,1♦2,,}2■2,,}2} has recursion level C(C(Ω22+C(Ω2+++,0)+C(Ω2+C(C(Ω22+C(Ω2++,0),0),σ+),0),0),0)
• {1{1{1,,1,,1{1{1{1,,1,,1♦2,,}2■2,,}2}2,,}2}2{1{1,,1,,1♦2,,}2■2,,}2} has recursion level C(C(Ω22+C(Ω2+++,0)+C(Ω2+C(C(Ω22+C(Ω2+++,0),0),σ+),0),0),0)
• {1{1{1,,1,,1,,2,,}2}2{1{1,,1,,1♦2,,}2■2,,}2} has recursion level C(C(Ω22+C(Ω2+++,0)+C(Ω2++,0),0),0)
• {1{1{1,,1 ,,2,,}2}2{1{1,,1,,1♦2,,}2■2,,}2} has recursion level C(C(Ω22+C(Ω2+++,0)+C(Ω2+C(C(Ω22,σ),σ++),0),0),0)
• {1{1{1{1,,1,,1♦2,,}2■2,,}2}2{1{1,,1,,1♦2,,}2■2,,}2} has recursion level C(C(Ω22+C(Ω2+++,0)+C(Ω2+C(C(Ω22+C(Ω2+++,0),0),σ++),0),0),0)
• {1,,1,,1,2{1{1,,1,,1♦2,,}2■2,,}2} has recursion level C(C(Ω22+C(Ω2+++,0)+C(Ω2++1,C(Ω2+C(C(Ω22+C(Ω2+++,0),0),σ++),0)),0),0)
• {1{1,,1,,1,,2,,}2{1{1,,1,,1♦2,,}2■2,,}2} has recursion level C(C(Ω22+C(Ω2+++,0)+C(Ω2++,C(Ω2+C(C(Ω22+C(Ω2+++,0),0),σ++),0)),0),0)
• {1{1{1,,1,,1♦2,,}2■2,,}3} has recursion level C(C(Ω22+C(Ω2+++,0)+C(Ω2+C(C(Ω22+C(Ω2+++,0),0),σ++),C(Ω2+C(C(Ω22+C(Ω2+++,0),0),σ++),0)),0),0)
• {1{1{1,,1,,1♦2,,}2■2,,}1,2} has recursion level C(C(Ω22+C(Ω2+++,0)+C(Ω2+C(C(Ω22+C(Ω2+++,0),0),σ++)+1,0),0),0)
• {1{1{1,,1,,1♦2,,}2■2,,}1{1,,1 ,,2,,}2} has recursion level C(C(Ω22+C(Ω2+++,0)+C(Ω2+C(C(Ω22+C(Ω2+++,0),0),σ++)+C(C(Ω22,σ),σ+),0),0),0)
• {1{1{1,,1,,1♦2,,}2■2,,}1{1,,1{1{1,,1 ,,2,,}2,,}1,2 ,,2,,}2} has recursion level C(C(Ω22+C(Ω2+++,0)+C(Ω2+C(C(Ω22+C(Ω2+++,0),0),σ++)+C(C(Ω22+1,0),σ+),0),0),0)
• {1{1{1,,1,,1♦2,,}2■2,,}1{1,,1{1{1,,1 ,,2,,}2,,}1{1{1{1,,1,,1♦2,,}2■2,,}2}2 ,,2,,}2} has recursion level C(C(Ω22+C(Ω2+++,0)+C(Ω2+C(C(Ω22+C(Ω2+++,0),0),σ++)+C(C(Ω22+C(Ω2+++,0),0),σ+),0),0),0)
• {1{1{1,,1,,1♦2,,}2■2,,}1■2} has recursion level C(C(Ω22+C(Ω2+++,0)+C(Ω2+C(C(Ω22+C(Ω2+++,0),0),σ++)+σ++,0),0),0)
• {1{1{1,,1,,1♦2,,}2■2,,}1{1{1,,1,,2,,}2■2,,}2} has recursion level C(C(Ω22+C(Ω2+++,0)+C(Ω2+C(C(Ω22+C(Ω2+++,0),0),σ++)+C(C(Ω22,σ),σ++),0),0),0)
• {1{1{1,,1,,1♦2,,}2■2,,}1{1{1,,1,,2,,}1,2■2,,}2} has recursion level C(C(Ω22+C(Ω2+++,0)+C(Ω2+C(C(Ω22+C(Ω2+++,0),0),σ++)+C(C(Ω22,σ)+1,σ++),0),0),0)
• {1{1{1,,1,,1♦2,,}2■2,,}1{1{1,,1,,3,,}2■2,,}2} has recursion level C(C(Ω22+C(Ω2+++,0)+C(Ω2+C(C(Ω22+C(Ω2+++,0),0),σ++)+C(C(Ω22,C(Ω22,σ)),σ++),0),0),0)
• {1{1{1,,1,,1♦2,,}2■2,,}1{1{1,,1,,1,2,,}2■2,,}2} has recursion level C(C(Ω22+C(Ω2+++,0)+C(Ω2+C(C(Ω22+C(Ω2+++,0),0),σ++)+C(C(Ω22+1,0),σ++),0),0),0)
• {1{1{1,,1,,1♦2,,}2■2,,}1{1{1,,1,,1♦2,,}2■2,,}2} has recursion level C(C(Ω22+C(Ω2+++,0)+C(Ω2+C(C(Ω22+C(Ω2+++,0),0),σ++)2,0),0),0)
• {1{2{1,,1,,1♦2,,}2■2,,}2} has recursion level C(C(Ω22+C(Ω2+++,0)+C(Ω2+ω^(C(C(Ω22+C(Ω2+++,0),0),σ++)+1),0),0),0)
• {1{1,,1,2,,}2{1,,1,,1♦2,,}2■2,,} has recursion level C(C(Ω22+C(Ω2+++,0)+C(Ω2+C(C(Ω2+1,σ),C(C(Ω22+C(Ω2+++,0),0),σ++)),0),0),0)
• {1{1,,1{1■2,,}2,,}2{1,,1,,1♦2,,}2■2,,} has recursion level C(C(Ω22+C(Ω2+++,0)+C(Ω2+C(C(Ω2++,σ),C(C(Ω22+C(Ω2+++,0),0),σ++)),0),0),0)
• {1{1,,1{1{1,,1,,1♦2,,}2■2,,}2,,}2{1,,1,,1♦2,,}2■2,,} has recursion level C(C(Ω22+C(Ω2+++,0)+C(Ω2+C(C(Ω2+C(C(Ω22+C(Ω2+++,0),0),σ++),σ),C(C(Ω22+C(Ω2+++,0),0),σ++)),0),0),0)
• {1{1,,1,,2,,}2{1,,1,,1♦2,,}2■2,,} has recursion level C(C(Ω22+C(Ω2+++,0)+C(Ω2+C(C(Ω22,σ),C(C(Ω22+C(Ω2+++,0),0),σ++)),0),0),0)
• {1{1,,1,,2,,}1,2{1,,1,,1♦2,,}2■2,,} has recursion level C(C(Ω22+C(Ω2+++,0)+C(Ω2+C(C(Ω22,σ)+1,C(C(Ω22+C(Ω2+++,0),0),σ++)),0),0),0)
• {1{1,,2,,2,,}2{1,,1,,1♦2,,}2■2,,} has recursion level C(C(Ω22+C(Ω2+++,0)+C(Ω2+C(C(Ω22,σ)+,C(C(Ω22+C(Ω2+++,0),0),σ++)),0),0),0)
• {1{1,,1,,3,,}2{1,,1,,1♦2,,}2■2,,} has recursion level C(C(Ω22+C(Ω2+++,0)+C(Ω2+C(C(Ω22,C(Ω22,σ)),C(C(Ω22+C(Ω2+++,0),0),σ++)),0),0),0)
• {1{1,,1,,1,2,,}2{1,,1,,1♦2,,}2■2,,} has recursion level C(C(Ω22+C(Ω2+++,0)+C(Ω2+C(C(Ω22+1,0),C(C(Ω22+C(Ω2+++,0),0),σ++)),0),0),0)
• {1{1,,1,,1♦2,,}3■2,,} has recursion level C(C(Ω22+C(Ω2+++,0)+C(Ω2+C(C(Ω22+C(Ω2+++,0),0),C(C(Ω22+C(Ω2+++,0),0),σ++)),0),0),0)
• {1{1,,1,,1♦2,,}1,2■2,,} has recursion level C(C(Ω22+C(Ω2+++,0)+C(Ω2+C(C(Ω22+C(Ω2+++,0),0)+1,σ++),0),0),0)
• {1{1,,2,,1♦2,,}2■2,,} has recursion level C(C(Ω22+C(Ω2+++,0)+C(Ω2+C(C(Ω22+C(Ω2+++,0),0)+++),0),0),0)
• {1{1,,1,,1♦1,2,,}2■2,,} has recursion level C(C(Ω22+C(Ω2+++,0)+C(Ω2+C(C(Ω22+C(Ω2+++,0)+1,0),σ++),0),0),0)
• {1{1,,1,,1♦1♦2,,}2■2,,} has recursion level C(C(Ω22+C(Ω2+++,0)2,0),0)
• {1{1,,1,,1,,2,,}2{1{1,,1,,1,,2,,}2■2,,}2} has recursion level C(C(Ω22+C(Ω2++,C(Ω2+++,0)),0),0)
• {1{1{1,,1,,1{1{1{1,,1,,1♦2,,}2■2,,}2}2,,}2■2,,}2{1{1,,1,,1,,2,,}2■2,,}2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22+C(Ω2+C(C(Ω22+C(Ω2+++,0),0),σ++),0),0),σ++),C(Ω2+++,0)),0),0)
• {1{1{1,,1,,1♦2,,}2■2,,}2{1{1,,1,,1,,2,,}2■2,,}2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22+C(Ω2+++,0),0),σ++),C(Ω2+++,0)),0),0)
• {1{1{1,,1,,1,,2,,}2■2,,}3} has recursion level C(C(Ω22+C(Ω2+++,C(Ω2+++,0)),0),0)
• {1{1{1,,1,,1,,2,,}2■2,,}1,2} has recursion level C(C(Ω22+C(Ω2++++1,0),0),0)
• {1{1{1,,1,,1,,2,,}2■2,,}1{1,,1,2,,}2} has recursion level C(C(Ω22+C(Ω2++++C(C(Ω2+1,σ),σ+),0),0),0)
• {1{1{1,,1,,1,,2,,}2■2,,}1{1,,1,,2,,}2} has recursion level C(C(Ω22+C(Ω2++++C(C(Ω2++,σ),σ+),0),0),0)
• {1{1{1,,1,,1,,2,,}2■2,,}1{1,,1 ,,2,,}2} has recursion level C(C(Ω22+C(Ω2++++C(C(Ω2+++,σ),σ+),0),0),0)
• {1{1{1,,1,,1,,2,,}2■2,,}1{1,,2 ,,2,,}2} has recursion level C(C(Ω22+C(Ω2++++C(C(Ω2+++,σ)++),0),0),0)
• {1,,1 ,,3,,} has recursion level C(C(Ω22+C(Ω2++++C(C(Ω2+++,C(Ω2+++,σ)),σ+),0),0),0) (the growth boost appears again)
• {1,,1 ,,1,2,,} has recursion level C(C(Ω22+C(Ω2++++C(C(Ω2++++1,σ),σ+),0),0),0)
• {1,,1 ,,1{1{1,,1 ,,2,,}2,,}2,,} has recursion level C(C(Ω22+C(Ω2++++C(C(Ω2++++C(C(Ω2+++,σ),σ+),σ),σ+),0),0),0)
• {1,,1 ,,1,,2,,} has recursion level C(C(Ω22+C(Ω2++++C(C(Ω22,σ),σ+),0),0),0)
• {1{1,,1 ,,2,,}2{1,,1 ,,1,,2,,}2,,} has recursion level C(C(Ω22+C(Ω2++++C(C(Ω2+++,σ),C(C(Ω22,σ),σ+)),0),0),0)
• {1{1,,1 ,,1{1{1,,1 ,,1,,2,,}2,,}2,,}2{1,,1 ,,1,,2,,}2,,} has recursion level C(C(Ω22+C(Ω2++++C(C(Ω2++++C(C(Ω22,σ),σ+),σ),C(C(Ω22,σ),σ+)),0),0),0)
• {1{1,,1 ,,1,,2,,}3,,} has recursion level C(C(Ω22+C(Ω2++++C(C(Ω22,σ),C(C(Ω22,σ),σ+)),0),0),0)
• {1{1,,1 ,,1,,2,,}1,2,,} has recursion level C(C(Ω22+C(Ω2++++C(C(Ω22,σ)+1,σ+),0),0),0)
• {1{1,,1 ,,1,,2,,}2,,1 ,,1,,2,,} has recursion level C(C(Ω22+C(Ω2++++C(C(Ω22,σ)2,σ+),0),0),0)
• {1,,2 ,,1,,2,,} has recursion level C(C(Ω22+C(Ω2++++C(C(Ω22,σ)++),0),0),0)
• {1,,1 ,,1,,3,,} has recursion level C(C(Ω22+C(Ω2++++C(C(Ω22,C(Ω22,σ)),σ+),0),0),0)
• {1,,1 ,,1,,1,2,,} has recursion level C(C(Ω22+C(Ω2++++C(C(Ω22+1,0),σ+),0),0),0)
• {1,,1 ,,1,,1,,2,,} has recursion level C(C(Ω22+C(Ω2+++++,0),0),0)
• {1,,1 ,,1,,1,,1,,2,,} has recursion level C(C(Ω22+C(Ω2+++++2,0),0),0)
• {1,,1 ,,1{2,,}2,,} has recursion level C(C(Ω22+C(Ω2++++ω^(σ+++1),0),0),0)
• {1,,1 ,,1{1{1,,1,2,,}2,,}2,,} has recursion level C(C(Ω22+C(Ω2++++C(C(Ω2+1,σ),σ++),0),0),0)
• {1,,1 ,,1{1{1,,1{1{1,,1 ,,2,,}2,,}2,,}2,,}2,,} has recursion level C(C(Ω22+C(Ω2++++C(C(Ω2+C(C(Ω2+++,σ),σ+),σ),σ++),0),0),0)
• {1,,1 ,,1{1{1,,1{1{1,,1 ,,1,,1,,2,,}2,,}2,,}2,,}2,,} has recursion level C(C(Ω22+C(Ω2++++C(C(Ω2+C(C(Ω2+++++,σ),σ+),σ),σ++),0),0),0)
• {1,,1 ,,1{1{1,,1,,2,,}2,,}2,,} has recursion level C(C(Ω22+C(Ω2++++C(C(Ω2+C(C(Ω22,σ),σ+),σ),σ++),0),0),0)
• {1{1,,1 ,,1{1{1,,1,,2,,}2,,}2,,}1,2,,} has recursion level C(C(Ω22+C(Ω2++++C(C(Ω2+C(C(Ω22,σ)+1,σ+),σ),σ++),0),0),0)
• {1,,2 ,,1{1{1,,1,,2,,}2,,}2,,} has recursion level C(C(Ω22+C(Ω2++++C(C(Ω2+C(C(Ω22,σ)++),σ),σ++),0),0),0)
• {1,,1 ,,1{1{1,,1,,2,,}2,,}1,2,,} has recursion level C(C(Ω22+C(Ω2++++C(C(Ω2+C(C(Ω22+1,0),σ+),σ),σ++),0),0),0)
• {1,,1 ,,1{1{1,,1,,2,,}2,,}1,,2,,} has recursion level C(C(Ω22+C(Ω2++++C(C(Ω2++,σ),σ++),0),0),0)
• {1{1,,1 ,,1{1{1,,1,,2,,}2,,}1,,2,,}1{1,,1 ,,1{1{1,,1,,2,,}2,,}2,,}2} has recursion level C(C(Ω22+C(Ω2++++C(C(Ω2++,σ),σ++)+C(C(Ω2++++C(C(Ω2++,σ),σ++),σ),σ+),0),0),0)
• {1,,1 ,,1{1{1,,1,,2,,}2,,}1,,2,,} has recursion level C(C(Ω22+C(Ω2++++C(C(Ω2++,σ),σ++)+C(C(Ω22,σ),σ+),0),0),0)
• {1,,1 ,,1{1{1,,1,,2,,}2,,}1,,1,,2,,} has recursion level C(C(Ω22+C(Ω2++++C(C(Ω2++,σ),σ++)+σ++,0),0),0)
• {1,,1 ,,1{2{1,,1,,2,,}2,,}2,,} has recursion level C(C(Ω22+C(Ω2++++ω^(C(C(Ω2++,σ),σ++)+1),0),0),0)
• {1,,1 ,,1{1{1,,1,,2,,}1,2,,}2,,} has recursion level C(C(Ω22+C(Ω2++++C(C(Ω2++,σ)+1,σ++),0),0),0)
• {1,,1 ,,1{1{1,,2,,2,,}2,,}2,,} has recursion level C(C(Ω22+C(Ω2++++C(C(Ω2++,σ)+++),0),0),0)
• {1,,1 ,,1{1{1,,1,,1,2,,}2,,}2,,} has recursion level C(C(Ω22+C(Ω2++++C(C(Ω2+++1,σ),σ++),0),0),0)
• {1,,1 ,,1{1{1,,1{1{1,,1,,1,2,,}2,,}2,,}2,,}2,,} has recursion level C(C(Ω22+C(Ω2++++C(C(Ω2+C(C(Ω2+++1,σ),σ++),σ),σ++),0),0),0)
• {1,,1 ,,1{1{1,,1 ,,2,,}2,,}2,,} has recursion level C(C(Ω22+C(Ω2++++C(C(Ω2+++,σ),σ++),0),0),0)
• {1,,1 ,,1{1{1,,1 ,,2,,}2,,}1,,1,,2,,} has recursion level C(C(Ω22+C(Ω2++++C(C(Ω2+++,σ),σ++)+σ++,0),0),0)
• {1,,1 ,,1{1{1,,1 ,,2,,}1,2,,}2,,} has recursion level C(C(Ω22+C(Ω2++++C(C(Ω2+++,σ)+1,σ++),0),0),0)
• {1,,1 ,,1{1{1,,2 ,,2,,}2,,}2,,} has recursion level C(C(Ω22+C(Ω2++++C(C(Ω2+++,σ)+++),0),0),0)
• {1,,1 ,,1{1{1,,1 ,,1,2,,}2,,}2,,} has recursion level C(C(Ω22+C(Ω2++++C(C(Ω2++++1,σ),σ++),0),0),0)
• {1,,1 ,,1{1{1,,1 ,,1{1{1,,1 ,,1,2,,}2,,}2,,}2,,}2,,} has recursion level C(C(Ω22+C(Ω2++++C(C(Ω2++++C(C(Ω2++++1,σ),σ++),σ),σ++),0),0),0)
• {1,,1 ,,1 ,,2,,} has recursion level C(C(Ω22+C(Ω2++++C(C(Ω22,σ),σ++),0),0),0)
• {1{1,,1 ,,1 ,,2,,}1{1,,1 ,,1 ,,2,,}2} has recursion level C(C(Ω22+C(Ω2++++C(C(Ω22,σ),σ++)+C(C(Ω22,σ),σ+),0),0),0) (erratic thing again)

Despite of the erratic thing, which contributes little to the strength of TON (vs. sDAN), the C(Ω2,C(Ω2,C(Ω2,C(Ω22,0)))) approximately corresponds to {1,,3,,} (or the  ,,), then approximately

• {1,,1{2 ,,}2,,} has recursion level C(C(Ω22+C(Ω2+ω^(σ++++1),0),0),0)
• {1,,1{1 ,,2 ,,}2,,} has recursion level C(C(Ω22+C(Ω2+ω^(σ++++C(C(Ω22,σ),σ++)),0),0),0)
• {1,,1{1 ,,1,2 ,,}2,,} has recursion level C(C(Ω22+C(Ω2+ω^ω^(σ++++1),0),0),0)
• {1,,1{1{2 ,,}2 ,,}2,,} has recursion level C(C(Ω22+C(Ω2+ω^ω^ω^(σ++++1),0),0),0)
• {1,,1{1  ,,2 ,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(σ+++++++),0),0),0)
• {1,,1{1{1,,1,2,,}2 ,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2+1,σ),σ+++),0),0),0)
• {1,,1{1{1,,1,,2,,}2 ,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2+σ,σ),σ+++),0),0),0)
• {1,,1{1{1,,1{2,,}2,,}2 ,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2+ω^(σ++1),σ),σ+++),0),0),0)
• {1,,1{1{1,,1{2,,}2,,}2 ,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2+ω^(σ+++1),σ),σ+++),0),0),0)
• {1,,1{1{1,,1{2 ,,}2,,}2 ,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2+ω^(σ++++1),σ),σ+++),0),0),0)
• {1,,1  ,,2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22,σ),σ+++),0),0),0)

As shown above, the recursion level of {1,,1,,2,,} is C(C(Ω22+C(Ω2+C(C(Ω22,C(Ω22,0)),C(Ω2,C(Ω22,0))),0),0),0), the recursion level of {1,,1 ,,2,,} is C(C(Ω22+C(Ω2+C(C(Ω22,C(Ω22,0)),C(Ω2,C(Ω2,C(Ω22,0)))),0),0),0), the recursion level of {1,,1  ,,2,,} is C(C(Ω22+C(Ω2+C(C(Ω22,C(Ω22,0)),C(Ω2,C(Ω2,C(Ω2,C(Ω22,0))))),0),0),0), and so on. Generally, the recursion level of {1,,1{1,,k,,}2,,} (k > 1) is between C(C(Ω22+C(Ω2+C(Ω2,C(Ω2,…C(Ω2,C(Ω22,0))…)),0),0),0) (with k-1 Ω2′s in the blue part) and C(C(Ω22+C(Ω2+C(Ω2,C(Ω2,…C(Ω2,C(Ω22,0))…)),0),0),0) (with k Ω2′s in the blue part), so {1,,1{1,,1,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω2+1,C(Ω22,0)),0),0),0).

Now let ordinal σ = C(Ω22,0). And α+ is shorthand for C(Ω2,α).

• {1{1,,1{1,,1,2,,}2,,}2} has recursion level C(C(Ω22+C(Ω2+C(Ω2+1,σ),0),0),0)
• {1{1,,1,,1,2}2{1,,1{1,,1,2,,}2,,}2} has recursion level C(C(Ω22+C(Ω2+C(Ω2+1,σ),0)+C(Ω2++1,0),0),0)
• {1{1{1,,1,,2,,}2}2{1,,1{1,,1,2,,}2,,}2} has recursion level C(C(Ω22+C(Ω2+C(Ω2+1,σ),0)+C(Ω2+C(C(Ω22,σ)+),0),0),0)
• {1{1{1,,1 ,,2,,}2}2{1,,1{1,,1,2,,}2,,}2} has recursion level C(C(Ω22+C(Ω2+C(Ω2+1,σ),0)+C(Ω2+C(C(Ω22,σ)++),0),0),0)
• {1{1{1,,1{1,,1,2,,}2,,}2}2{1,,1{1,,1,2,,}2,,}2} has recursion level C(C(Ω22+C(Ω2+C(Ω2+1,σ),0)2,0),0)
• {1,,1,,1,2{1,,1{1,,1,2,,}2,,}2} has recursion level C(C(Ω22+C(Ω2++1,C(Ω2+C(Ω2+1,σ),0)),0),0)
• {1{1,,1,,2,,}2{1,,1{1,,1,2,,}2,,}2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22,σ)+),C(Ω2+C(Ω2+1,σ),0)),0),0)
• {1{1,,1 ,,2,,}2{1,,1{1,,1,2,,}2,,}2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22,σ)++),C(Ω2+C(Ω2+1,σ),0)),0),0)
• {1{1,,1{1,,1,2,,}2,,}3} has recursion level C(C(Ω22+C(Ω2+C(Ω2+1,σ),C(Ω2+C(Ω2+1,σ),0)),0),0)
• {1{1,,1{1,,1,2,,}2,,}1,2} has recursion level C(C(Ω22+C(Ω2+C(Ω2+1,σ)+1,0),0),0)
• {1{1,,1{1,,1,2,,}2,,}1{1,,1,2,,}2} has recursion level C(C(Ω22+C(Ω2+C(Ω2+1,σ)+C(C(Ω2+1,σ),σ+),0),0),0)
• {1{1,,1{1,,1,2,,}2,,}1{1,,1,,2,,}2} has recursion level C(C(Ω22+C(Ω2+C(Ω2+1,σ)+C(C(Ω2++,σ),σ+),0),0),0)
• {1{1,,1{1,,1,2,,}2,,}1{1,,1 ,,2,,}2} has recursion level C(C(Ω22+C(Ω2+C(Ω2+1,σ)+C(C(Ω2+++,σ),σ+),0),0),0)
• {1{1,,1{1,,1,2,,}2,,}1{1,,1{1,,1,2,,}2,,}2} has recursion level C(C(Ω22+C(Ω2+C(Ω2+1,σ)+C(C(Ω2+C(Ω2+1,σ),σ),σ+),0),0),0)
• {1{1,,1,2,,}1{1,,1{1,,1,2,,}2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω2+1,σ)+C(C(Ω2+1,σ),C(C(Ω2+C(Ω2+1,σ),σ),σ+)),0),0),0)
• {1{1,,1{1,,1,2,,}2,,}3,,} has recursion level C(C(Ω22+C(Ω2+C(Ω2+1,σ)+C(C(Ω2+C(Ω2+1,σ),σ),C(C(Ω2+C(Ω2+1,σ),σ),σ+)),0),0),0)
• {1{1,,1{1,,1,2,,}2,,}1,2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω2+1,σ)+C(C(Ω2+C(Ω2+1,σ),σ)+1,σ+),0),0),0)
• {1,,2{1,,1,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω2+1,σ)+C(C(Ω2+C(Ω2+1,σ),σ)++),0),0),0)
• {1,,1{1,,1,2,,}3,,} has recursion level C(C(Ω22+C(Ω2+C(Ω2+1,σ)+C(C(Ω2+C(Ω2+1,σ),C(Ω2+C(Ω2+1,σ),σ)),σ+),0),0),0)
• {1,,1{1,,1,2,,}1,2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω2+1,σ)+C(C(Ω2+C(Ω2+1,σ)+1,σ),σ+),0),0),0)
• {1,,1{1,,1,2,,}1,,2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω2+1,σ)+C(C(Ω22,σ),σ+),0),0),0)
• {1,,2{1,,1,2,,}1,,2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω2+1,σ)+C(C(Ω22,σ)++),0),0),0)
• {1,,1{1,,1,2,,}1,,1,2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω2+1,σ)+C(C(Ω22+1,0),σ+),0),0),0)
• {1,,1{1,,1,2,,}1,,1,,2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω2+1,σ)+σ++,0),0),0)
• {1,,1{1,,1,2,,}1{2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω2+1,σ)+ω^(σ+++1),0),0),0)
• {1,,1{1,,1,2,,}1{1{1,,1,2,,}2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω2+1,σ)+C(C(Ω2+1,σ),σ++),0),0),0)
• {1,,1{1,,1,2,,}1{1{1,,1{1,,1,2,,}2,,}2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω2+1,σ)+C(C(Ω2+C(Ω2+1,σ),σ),σ++),0),0),0)
• {1,,1{1,,1,2,,}1 ,,2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω2+1,σ)+C(C(Ω22,σ),σ++),0),0),0)
• {1,,1{1,,1,2,,}1{2 ,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω2+1,σ)+σ+++,0),0),0)
• {1,,1{1,,1,2,,}1  ,,2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω2+1,σ)+C(C(Ω22,σ),σ+++),0),0),0)
• {1,,1{1,,1,2,,}1{1,,1,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω2+1,σ)2,0),0),0)
• {1,,1{1{1,,2,2,,}2,,1,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2+1,σ)+,C(Ω2+1,σ)),0),0),0)
• {1,,1{1{1,,1{1,,1,2,,}2,,}2,,1,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2+C(Ω2+1,σ),σ),C(Ω2+1,σ)),0),0),0)
• {1,,1{1{1,,1{1{1,,1{1,,1,2,,}2,,}2,,1,2,,}2,,}2,,1,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2+C(C(Ω2+C(Ω2+1,σ),σ),C(Ω2+1,σ)),σ),C(Ω2+1,σ)),0),0),0)
• {1,,1{1,,2,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22,σ),C(Ω2+1,σ)),0),0),0)
• {1,,1{1,,3,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22,σ),C(Ω2+1,σ)+),0),0),0)
• {1,,1{1,,1,3,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω2+1,C(Ω2+1,σ)),0),0),0)
• {1,,1{1,,1,1,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω2+2,σ),0),0),0)
• {1,,1{1,,1{1{1,,1{1,,1,2,,}2,,}2}2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω2+C(Ω2+C(Ω2+1,σ),0),σ),0),0),0)
• {1,,1{1,,1,,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω2+σ,σ),0),0),0)

Due to the “growth boosts”, {1,,k,,} approximately corresponds to C(Ω2,C(Ω2,…C(Ω2,C(Ω22,0))…)) (with k Ω2′s in the blue part), and {1,,1,2,,} approximately corresponds to C(Ω2+1,C(Ω22,0)).

## Beyond C(Ω2+C(Ω22,0),C(Ω22,0))

Now let separator ♦ = {1{1,,1{1,,1,,2,,}2,,}2} and let ordinal σ = C(Ω22,0). And α+ is shorthand for C(Ω2,α).

• {1{1,,1{1,,1♦2,,}2,,}2} has recursion level C(C(Ω22+C(Ω2+C(Ω2+σ,σ),0),0),0)
• {1{1,,1,,1,2}2{1,,1{1,,1♦2,,}2,,}2} has recursion level C(C(Ω22+C(Ω2+C(Ω2+σ,σ),0)+C(Ω2++1,0),0),0)
• {1{1{1,,1,,2,,}2}2{1,,1{1,,1♦2,,}2,,}2} has recursion level C(C(Ω22+C(Ω2+C(Ω2+σ,σ),0)+C(Ω2+C(C(Ω22,σ),σ+),0),0),0)
• {1{1{1,,1{1,,1,2,,}2,,}2}2{1,,1{1,,1♦2,,}2,,}2} has recursion level C(C(Ω22+C(Ω2+C(Ω2+σ,σ),0)+C(Ω2+C(Ω2+1,σ),0),0),0)
• {1{1{1,,1{1,,1♦2,,}2,,}2}2{1,,1{1,,1♦2,,}2,,}2} has recursion level C(C(Ω22+C(Ω2+C(Ω2+σ,σ),0)2,0),0)
• {1,,1,,1,2{1,,1{1,,1♦2,,}2,,}2} has recursion level C(C(Ω22+C(Ω2++1,C(Ω2+C(Ω2+σ,σ),0)),0),0)
• {1{1,,1{1,,1,2,,}2,,}2{1,,1{1,,1♦2,,}2,,}2} has recursion level C(C(Ω22+C(Ω2+C(Ω2+1,σ),C(Ω2+C(Ω2+σ,σ),0)),0),0)
• {1{1,,1{1,,1{1{1,,1{1,,1♦2,,}2,,}2}2,,}2,,}2{1,,1{1,,1♦2,,}2,,}2} has recursion level C(C(Ω22+C(Ω2+C(Ω2+C(Ω2+C(Ω2+σ,σ),0),σ),C(Ω2+C(Ω2+σ,σ),0)),0),0)
• {1{1,,1{1,,1♦2,,}2,,}3} has recursion level C(C(Ω22+C(Ω2+C(Ω2+σ,σ),C(Ω2+C(Ω2+σ,σ),0)),0),0)
• {1{1,,1{1,,1♦2,,}2,,}1,2} has recursion level C(C(Ω22+C(Ω2+C(Ω2+σ,σ)+1,0),0),0)
• {1{1,,1{1,,1♦2,,}2,,}1{1,,1{1,,1,2,,}2,,}2} has recursion level C(C(Ω22+C(Ω2+C(Ω2+σ,σ)+C(C(Ω2+C(Ω2+1,σ),σ),σ+),0),0),0)
• {1{1,,1{1,,1♦2,,}2,,}1{1,,1{1,,1♦2,,}2,,}2} has recursion level C(C(Ω22+C(Ω2+C(Ω2+σ,σ)+C(C(Ω2+C(Ω2+σ,σ),σ),σ+),0),0),0)
• {1{1,,1{1,,1♦2,,}2,,}1,2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω2+σ,σ)+C(C(Ω2+C(Ω2+σ,σ),σ)+1,σ+),0),0),0)
• {1,,2{1,,1♦2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω2+σ,σ)+C(C(Ω2+C(Ω2+σ,σ),σ)++),0),0),0)
• {1,,1{1,,1♦2,,}1,2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω2+σ,σ)+C(C(Ω2+C(Ω2+σ,σ)+1,σ),σ+),0),0),0)
• {1,,1{1,,1♦2,,}1,,2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω2+σ,σ)+C(C(Ω22,σ),σ+),0),0),0)
• {1,,1{1,,1♦2,,}1,,1,,2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω2+σ,σ)+σ++,0),0),0)
• {1,,1{1,,1♦2,,}1 ,,2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω2+σ,σ)+C(C(Ω22,σ),σ++),0),0),0)
• {1,,1{1,,1♦2,,}1{1,,1,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω2+σ,σ)+C(Ω2+1,σ),0),0),0)
• {1,,1{1,,1♦2,,}1{1,,1♦2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω2+σ,σ)2,0),0),0)
• {1,,1{1{1,,1{1,,1,2,,}2,,}2,,1♦2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2+C(Ω2+1,σ),σ),C(Ω2+σ,σ)),0),0),0)
• {1,,1{1{1,,1{1,,1♦2,,}2,,}2,,1♦2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2+C(Ω2+σ,σ),σ),C(Ω2+σ,σ)),0),0),0)
• {1,,1{1,,2♦2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22,σ),C(Ω2+σ,σ)),0),0),0)
• {1,,1{1,,1,2♦2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω2+1,C(Ω2+σ,σ)),0),0),0)
• {1,,1{1,,1♦3,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω2+σ,C(Ω2+σ,σ)),0),0),0)
• {1,,1{1,,1♦1,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω2+σ+1,σ),0),0),0)
• {1,,1{1,,1♦1♦2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω2+σ2,σ),0),0),0)
• {1,,1{1,,1{2{1,,1{1,,1,,2,,}2,,}2}2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω2+ω^(σ+1),σ),0),0),0)
• {1,,2{1,,1{1,,1,,2,,}2,,}2} has recursion level C(C(Ω22+C(Ω2+C(Ω2+C(σ+,σ),σ),0),0),0)
• {1,,1,2{1,,1{1,,1,,2,,}2,,}2} has recursion level C(C(Ω22+C(Ω2+C(Ω2+C(C(Ω2+1,σ),σ),σ),0),0),0)
• {1,,1{1,,1,2{1,,1{1,,1,,2,,}2,,}2}2{1,,1{1,,1,,2,,}2,,}2} has recursion level C(C(Ω22+C(Ω2+C(Ω2+C(C(Ω2+C(Ω2+1,σ),σ),σ),σ),0),0),0)
• {1,,1{1,,1♦2{1,,1{1,,1,,2,,}2,,}2}2{1,,1{1,,1,,2,,}2,,}2} has recursion level C(C(Ω22+C(Ω2+C(Ω2+C(C(Ω2+C(Ω2+σ,σ),σ),σ),σ),0),0),0)
• {1,,1{1,,1{1,,2{1,,1{1,,1,,2,,}2,,}2}2{1,,1{1,,1,,2,,}2,,}2}2{1,,1{1,,1,,2,,}2,,}2} has recursion level C(C(Ω22+C(Ω2+C(Ω2+C(C(Ω22,σ),σ),σ),0),0),0)
• {1,,2{1,,1{1,,2{1,,1{1,,1,,2,,}2,,}2}2{1,,1{1,,1,,2,,}2,,}2}2{1,,1{1,,1,,2,,}2,,}2} has recursion level C(C(Ω22+C(Ω2+C(Ω2+C(C(Ω22,σ)+,σ),σ),0),0),0)
• {1,,1{1,,1{1,,2{1,,1{1,,1,,2,,}2,,}2}2{1,,1{1,,1,,2,,}2,,}2}1,2{1,,1{1,,1,,2,,}2,,}2} has recursion level C(C(Ω22+C(Ω2+C(Ω2+C(C(Ω22+1,0),σ),σ),0),0),0)
• {1,,1{1,,1{1,,2{1,,1{1,,1,,2,,}2,,}2}2{1,,1{1,,1,,2,,}2,,}2}1♦2{1,,1{1,,1,,2,,}2,,}2} has recursion level C(C(Ω22+C(Ω2+C(Ω2+,σ),0),0),0)
• {1,,1,,1,2{1,,1{1,,1,,2,,}2,,}2} has recursion level C(C(Ω22+C(Ω2++1,C(Ω2+C(Ω2+,σ),0)),0),0)
• {1{1,,1{1,,1,,2,,}2,,}1,2} has recursion level C(C(Ω22+C(Ω2+C(Ω2+,σ)+1,0),0),0)
• {1{1,,1{1,,1,,2,,}2,,}1{1,,1,2,,}2} has recursion level C(C(Ω22+C(Ω2+C(Ω2+,σ)+C(C(Ω2+1,σ),σ+),0),0),0)
• {1{1,,1{1,,1,,2,,}2,,}1{1,,1{1,,1,,2,,}2,,}2} has recursion level C(C(Ω22+C(Ω2+C(Ω2+,σ)+C(C(Ω2+C(Ω2+σ,σ),σ),σ+),0),0),0)
• {1{1,,1{1,,1,,2,,}2,,}1{1,,1{1,,1,,2,,}2,,}1,2} has recursion level C(C(Ω22+C(Ω2+C(Ω2+,σ)+C(C(Ω2+C(Ω2+,σ),σ),σ+)+1,0),0),0)
• {1{1,,1{1,,1,,2,,}2,,}1,2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω2+,σ)+C(C(Ω2+C(Ω2+,σ),σ)+1,σ+),0),0),0)
• {1,,2{1,,1,,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω2+,σ)+C(C(Ω2+C(Ω2+,σ),σ)++),0),0),0)
• {1,,1{1,,1,,2,,}1,2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω2+,σ)+C(C(Ω2+C(Ω2+,σ)+1,σ),σ+),0),0),0)
• {1,,1{1,,1,,2,,}1,,2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω2+,σ)+C(C(Ω22,σ),σ+),0),0),0)
• {1,,1{1,,1,,2,,}1,,1,,2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω2+,σ)+σ++,0),0),0)
• {1,,1{1,,1,,2,,}1{1,,1,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω2+,σ)+C(Ω2+1,σ),0),0),0)
• {1,,1{1,,1,,2,,}1{1,,1,,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω2+,σ)+C(Ω2+σ,σ),0),0),0)
• {1{1,,1{1,,1,,2,,}1{1,,1,,2,,}2,,}1,2} has recursion level C(C(Ω22+C(Ω2+C(Ω2+,σ)2+1,0),0),0)
• {1,,1{2,,1,,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+ω^(C(Ω2+,σ)+1),0),0),0)
• {1,,1{1{1,,2,,2,,}2,,1,,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2+,σ)+,C(Ω2+,σ)),0),0),0)
• {1,,1{1{1,,1{1,,1,,2,,}2,,}2,,1,,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2+C(Ω2+σ,σ),σ),C(Ω2+,σ)),0),0),0)
• {1,,1{1,,2,,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22,σ),C(Ω2+,σ)),0),0),0)
• {1,,1{1,,1,2,,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω2+1,C(Ω2+,σ)),0),0),0)
• {1,,1{1,,1,,3,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω2+σ,C(Ω2+,σ)),0),0),0)
• {1,,1{1,,1,,1,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω2++1,σ),0),0),0)
• {1,,1{1,,1{2,,}2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω2+ω^(σ++1),σ),0),0),0)
• {1,,1{1,,1{1{1,,1{1,,1,,2,,}2,,}2,,}2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω2+C(C(Ω2+C(Ω2+σ,σ),σ),σ+),σ),0),0),0)
• {1,,1{1,,1{1{1,,1{1,,1{2,,}2,,}2,,}2,,}2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω2+C(C(Ω2+C(Ω2+ω^(σ++1),σ),σ),σ+),σ),0),0),0)
• {1,,1{1,,1,,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω2+C(C(Ω22,σ),σ+),σ),0),0),0)
• {1,,1{1,,1 ,,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω2+C(C(Ω22,σ),σ++),σ),0),0),0)
• {1,,1{1,,1{1,,1,2,,}2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω2+C(Ω2+1,σ),σ),0),0),0)
• {1,,1{1,,1{1,,1{1,,1,2,,}2,,}2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω2+C(Ω2+C(Ω2+1,σ),σ),σ),0),0),0)
• {1,,1{1,,1,,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω22,σ),0),0),0)

So {1,,1,,2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω22,C(Ω22,0)),0),0),0).

# Up to {1,,1{1,,1,,2,,}3,,}

## Up to a C(Ω2,C(Ω22,C(Ω22,0)))

Now let ordinal σ = σ1 = C(Ω22,0), and σn+1 = C(Ω22,σn).

• {1{1,,1{1,,1,,2,,}2,,}2} has recursion level C(C(Ω22+C(Ω22,0),0),0)
• {1{1,,1,,1,2}2{1,,1{1,,1,,2,,}2,,}2} has recursion level C(C(Ω22+C(Ω22,0)+C(Ω2+C(Ω2,σ)+1,0),0),0)
• {1{1{1,,1,,2,,}2}2{1,,1{1,,1,,2,,}2,,}2} has recursion level C(C(Ω22+C(Ω22,0)+C(Ω2+C(σ2,C(Ω2,σ)),0),0),0)
• {1{1{1,,1{1,,1,2,,}2,,}2}2{1,,1{1,,1,,2,,}2,,}2} has recursion level C(C(Ω22+C(Ω22,0)+C(Ω2+C(Ω2+1,σ),0),0),0)
• {1{1{1,,1{1,,1{1,,1,2,,}2,,}2,,}2}2{1,,1{1,,1,,2,,}2,,}2} has recursion level C(C(Ω22+C(Ω22,0)+C(Ω2+C(Ω2+C(Ω2+1,σ),σ),0),0),0)
• {1{1{1,,1{1,,1,,2,,}2,,}2}2{1,,1{1,,1,,2,,}2,,}2} has recursion level C(C(Ω22+C(Ω22,0)2,0),0)
• {1,,2{1,,1{1,,1,,2,,}2,,}2} has recursion level C(C(Ω22+C(C(Ω2,C(Ω22,0)),C(Ω22,0)),0),0)
• {1,,1,,1,2{1,,1{1,,1,,2,,}2,,}2} has recursion level C(C(Ω22+C(Ω2+C(Ω2,σ)+1,C(Ω22,0)),0),0)
• {1{1,,1{1,,1,2,,}2,,}2{1,,1{1,,1,,2,,}2,,}2} has recursion level C(C(Ω22+C(Ω2+C(Ω2+1,σ),C(Ω22,0)),0),0)
• {1{1,,1{1,,1,,2,,}2,,}3} has recursion level C(C(Ω22+C(Ω22,C(Ω22,0)),0),0)
• {1{1,,1{1,,1,,2,,}2,,}1,2} has recursion level C(C(Ω22+C(Ω22+1,0),0),0)
• {1{1,,1{1,,1,,2,,}2,,}1,,2} has recursion level C(C(Ω22+C(Ω22+σ,0),0),0)
• {1{1,,1{1,,1,,2,,}2,,}1,,1,2} has recursion level C(C(Ω22+C(Ω22+C(Ω2,σ)+1,0),0),0)
• {1{1,,1{1,,1,,2,,}2,,}1{1,,2,,}2} has recursion level C(C(Ω22+C(Ω22+C(C(Ω2,C(Ω2,σ)),C(Ω2,σ)),0),0),0)
• {1{1,,1{1,,1,,2,,}2,,}1{1,,1,2,,}2} has recursion level C(C(Ω22+C(Ω22+C(C(Ω2+1,σ),C(Ω2,σ)),0),0),0)
• {1{1,,1{1,,1,,2,,}2,,}1{1,,1,,2,,}2} has recursion level C(C(Ω22+C(Ω22+C(C(Ω2+C(Ω2,C(Ω2,σ)),σ),C(Ω2,σ)),0),0),0)
• {1{1,,1{1,,1,,2,,}2,,}1{1,,1{1,,1,2,,}2,,}2} has recursion level C(C(Ω22+C(Ω22+C(C(Ω2+C(Ω2+1,σ),σ),C(Ω2,σ)),0),0),0)
• {1{1,,1{1,,1,,2,,}2,,}1{1,,1{1,,1,,2,,}2,,}2} has recursion level C(C(Ω22+C(Ω22+C(σ2,C(Ω2,σ)),0),0),0) (from here on, the growth slows down. C(C(Ω22,σ),C(Ω2,σ)) is not valid)
• {1{1,,1,2,,}2{1,,1{1,,1,,2,,}2,,}2,,} has recursion level C(C(Ω22+C(Ω22+C(C(Ω2+1,σ),C(σ2,C(Ω2,σ))),0),0),0)
• {1{1,,1{1,,1,,2,,}2,,}3,,} has recursion level C(C(Ω22+C(Ω22+C(σ2,C(σ2,C(Ω2,σ))),0),0),0)
• {1{1,,1{1,,1,,2,,}2,,}1,2,,} has recursion level C(C(Ω22+C(Ω22+C(σ2+1,C(Ω2,σ)),0),0),0)
• {1{1,,1{1,,1,,2,,}2,,}2,,1{1,,1,,2,,}2,,} has recursion level C(C(Ω22+C(Ω22+C(σ22,C(Ω2,σ)),0),0),0)
• {1,,2{1,,1,,2,,}2,,} has recursion level C(C(Ω22+C(Ω22+C(C(Ω22),C(Ω2,σ)),0),0),0)
• {1,,1,2{1,,1,,2,,}2,,} has recursion level C(C(Ω22+C(Ω22+C(C(Ω2+1,σ2),C(Ω2,σ)),0),0),0)
• {1,,1{1,,1,2,,}2{1,,1,,2,,}2,,} has recursion level C(C(Ω22+C(Ω22+C(C(Ω2+C(Ω2+1,σ),σ2),C(Ω2,σ)),0),0),0)
• {1,,1{1,,1{1,,1,,2,,}2,,}2{1,,1,,2,,}2,,} has recursion level C(C(Ω22+C(Ω22+C(σ3,C(Ω2,σ)),0),0),0)
• {1,,1{1,,1{1,,1,,2,,}2,,}3{1,,1,,2,,}2,,} has recursion level C(C(Ω22+C(Ω22+C(σ4,C(Ω2,σ)),0),0),0)
• {1,,1{1,,1{1,,1,,2,,}2,,}1,2{1,,1,,2,,}2,,} has recursion level C(C(Ω22+C(Ω22+C(C(Ω22+1,0),C(Ω2,σ)),0),0),0)
• {1,,1{1,,1{1,,1,,2,,}2,,}1{1{1,,1{1,,1,,2,,}2,,}1,2}2{1,,1,,2,,}2,,} has recursion level C(C(Ω22+C(Ω22+C(C(Ω22+C(Ω22+1,0),0),C(Ω2,σ)),0),0),0)
• {1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,} has recursion level C(C(Ω22+C(Ω22+C(Ω2,C(Ω2,σ)),0),0),0)
• {1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1,2} has recursion level C(C(Ω22+C(Ω22+C(Ω2,C(Ω2,σ))+1,0),0),0)
• {1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1{1,,1,2,,}2} has recursion level C(C(Ω22+C(Ω22+C(Ω2,C(Ω2,σ))+C(C(Ω2+1,σ),C(Ω2,σ)),0),0),0)
• {1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1{1,,1{1,,1,,2,,}2,,}2} has recursion level C(C(Ω22+C(Ω22+C(Ω2,C(Ω2,σ))+C(σ2,C(Ω2,σ)),0),0),0)
• {1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}2} has recursion level C(C(Ω22+C(Ω22+C(Ω2,C(Ω2,σ))2,0),0),0)
• {1{1,,1,2,,}2{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}2,,} has recursion level C(C(Ω22+C(Ω22+C(C(Ω2+1,σ),C(Ω2,C(Ω2,σ))),0),0),0)
• {1{1,,1{1,,1,,2,,}2,,}2{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}2,,} has recursion level C(C(Ω22+C(Ω22+C(σ2,C(Ω2,C(Ω2,σ))),0),0),0)
• {1{1,,1{1,,1{1,,1,,2,,}2,,}1,2{1,,1,,2,,}2,,}2{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}2,,} has recursion level C(C(Ω22+C(Ω22+C(C(Ω22+1,0),C(Ω2,C(Ω2,σ))),0),0),0)
• {1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}3,,} has recursion level C(C(Ω22+C(Ω22+C(Ω2,C(Ω2,C(Ω2,σ))),0),0),0)
• {1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}4,,} has recursion level C(C(Ω22+C(Ω22+C(Ω2,C(Ω2,C(Ω2,C(Ω2,σ)))),0),0),0)
• {1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1,2,,} has recursion level C(C(Ω22+C(Ω22+C(Ω2+1,σ),0),0),0)
• {1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1,,2,,} has recursion level C(C(Ω22+C(Ω22+C(Ω2+σ,σ),0),0),0)
• {1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1,,1,2,,} has recursion level C(C(Ω22+C(Ω22+C(Ω2+C(Ω2,σ)+1,σ),0),0),0)
• {1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1{1{1,,1,2,,}2,,}2,,} has recursion level C(C(Ω22+C(Ω22+C(Ω2+C(C(Ω2+1,σ),C(Ω2,σ)),σ),0),0),0)
• {1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1{1{1,,1{1,,1,,2,,}2,,}2,,}2,,} has recursion level C(C(Ω22+C(Ω22+C(Ω2+C(σ2,C(Ω2,σ)),σ),0),0),0)
• {1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1 {1{1,,1{1,,1{1,,1,,2,,}2,,}1,2{1,,1,,2,,}2,,}2,,}2,,} has recursion level C(C(Ω22+C(Ω22+C(Ω2+C(C(Ω22+1,0),C(Ω2,σ)),σ),0),0),0)
• {1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1 {1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}2,,}2,,} has recursion level C(C(Ω22+C(Ω22+C(Ω2+C(Ω2,C(Ω2,σ)),σ),0),0),0)
• {1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1 {1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1,2,,}2,,} has recursion level C(C(Ω22+C(Ω22+C(Ω2+C(Ω2+1,σ),σ),0),0),0)
• {1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1{1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1 {1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1,2,,}2,,}2,,} has recursion level C(C(Ω22+C(Ω22+C(Ω2+C(Ω2+C(Ω2+1,σ),σ),σ),0),0),0)
• {1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1,,2,,} has recursion level C(C(Ω22+C(Ω22+C(Ω22,σ),0),0),0)
• {1{1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1,,2,,}1,2} has recursion level C(C(Ω22+C(Ω22+C(Ω22,σ)+1,0),0),0)
• {1{1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1,,2,,}1 {1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}2} has recursion level C(C(Ω22+C(Ω22+C(Ω22,σ)+C(Ω2,C(Ω2,σ)),0),0),0)
• {1{1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1,,2,,}1 {1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1,,2,,}2} has recursion level C(C(Ω22+C(Ω22+C(Ω22,σ)2,0),0),0)
• {1{2{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1,,2,,}2} has recursion level C(C(Ω22+C(Ω22+ω^(C(Ω22,σ)+1),0),0),0)
• {1{1,,1,2,,}2{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1,,2,,} has recursion level C(C(Ω22+C(Ω22+C(C(Ω2+1,C(Ω22,σ)),C(Ω22,σ)),0),0),0)
• {1{1,,1{1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1,,2,,}2,,}2 {1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1,,2,,} has recursion level C(C(Ω22+C(Ω22+C(C(Ω2+C(Ω22,σ),C(Ω22,σ)),C(Ω22,σ)),0),0),0)
• {1{1,,1,,2,,}2{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1,,2,,} has recursion level C(C(Ω22+C(Ω22+C(C(Ω2+C(Ω2,C(Ω22,σ)),C(Ω22,σ)),C(Ω22,σ)),0),0),0)
• {1{1,,1{1,,1,2,,}2,,}2{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1,,2,,} has recursion level C(C(Ω22+C(Ω22+C(C(Ω2+C(Ω2+1,C(Ω22,σ)),C(Ω22,σ)),C(Ω22,σ)),0),0),0)
• {1{1,,1{1,,1,,2,,}2,,}2{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1,,2,,} has recursion level C(C(Ω22+C(Ω22+C(σ2,C(Ω22,σ)),0),0),0)
• {1{1,,1{1,,1{1,,1,,2,,}2,,}1,2{1,,1,,2,,}2,,}2{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1,,2,,} has recursion level C(C(Ω22+C(Ω22+C(C(Ω22+1,0),C(Ω22,σ)),0),0),0)
• {1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}2,,2,,} has recursion level C(C(Ω22+C(Ω22+C(Ω2,C(Ω22,σ)),0),0),0)
• {1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1 {1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1,,2,,}2,,2,,} has recursion level C(C(Ω22+C(Ω22+C(Ω2+C(Ω22,σ),C(Ω22,σ)),0),0),0)
• {1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1,,3,,} has recursion level C(C(Ω22+C(Ω22+C(Ω22,C(Ω22,σ)),0),0),0)
• {1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1,,1,2,,} has recursion level C(C(Ω22+C(Ω22+C(Ω22+1,σ),0),0),0)
• {1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1,,1 {1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1,,2,,}2,,} has recursion level C(C(Ω22+C(Ω22+C(Ω22+C(Ω22,σ),σ),0),0),0)
• {1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1,,1,,2,,} has recursion level C(C(Ω22+C(Ω222,0),0),0)
• {1{1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1,,1,,2,,}1 {1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}2} has recursion level C(C(Ω22+C(Ω222+C(Ω2,C(Ω2,σ)),0),0),0)
• {1{1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1,,1,,2,,}1 {1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1,,2,,}2} has recursion level C(C(Ω22+C(Ω222+C(Ω22,σ),0),0),0)
• {1{1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1,,1,,2,,}1 {1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1,,1,,2,,}2} has recursion level C(C(Ω22+C(Ω222+C(Ω222,σ),0),0),0)
• {1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}2,,1,,2,,} has recursion level C(C(Ω22+C(Ω222+C(Ω2,C(Ω222,σ)),0),0),0)
• {1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1,,2,,2,,} has recursion level C(C(Ω22+C(Ω222+C(Ω22,C(Ω222,σ)),0),0),0)
• {1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1,,1,,3,,} has recursion level C(C(Ω22+C(Ω222+C(Ω222,C(Ω222,σ)),0),0),0)
• {1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1,,1,,1,2,,} has recursion level C(C(Ω22+C(Ω222+C(Ω222+1,σ),0),0),0)
• {1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1,,1,,1,,2,,} has recursion level C(C(Ω22+C(Ω223,0),0),0)
• {1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1{2,,}2,,} has recursion level C(C(Ω22+C(Ω2+ω^(σ2+1),0),0),0)
• {1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1{1,,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+ω^(σ22),0),0),0)
• {1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1 ,,2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22),σ2),0),0),0)
• {1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1 ,,1 ,,2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22),σ2)2,0),0),0)
• {1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1{2 ,,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+ω^(C(C(Ω22),σ2)+1),0),0),0)
• {1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1{1 ,,3,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22),C(C(Ω22),σ2)),0),0),0)
• {1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1{1 ,,1 ,,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22)2,σ2),0),0),0)
• {1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1  ,,2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2,C(Ω22)),σ2),0),0),0)
• {1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1{1,,1,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2+1,σ2),σ2),0),0),0)
• {1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1{1,,1,,1,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2+C(Ω2,σ)+1,σ2),σ2),0),0),0)
• {1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1 {1,,1{1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}2,,}2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2+C(Ω2,C(Ω2,σ)),σ2),σ2),0),0),0)
• {1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1 {1,,1{1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1,,2,,}2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2+C(Ω22,σ),σ2),σ2),0),0),0)
• {1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1 {1,,1{1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1 ,,2,,}2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2+C(Ω2+C(C(Ω22),σ2),σ),σ2),σ2),0),0),0)
• {1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1{1,,1,,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω222),σ2),0),0),0)
• {1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1{1,,1{1 ,,2,,}2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2+C(C(Ω22),σ2),σ2),σ2),0),0),0)
• {1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1{1,,1{1{1,,1,,2,,}2,,}2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2+C(C(Ω222),σ2),σ2),σ2),0),0),0)
• {1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1{1,,1 ,,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(σ32),0),0),0)
• {1{1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1{1,,1 ,,2,,}2,,}1 {1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1{1,,1 ,,2,,}2,,}2} has recursion level C(C(Ω22+C(Ω2+C(σ32)+C(Ω2+C(σ32),σ),0),0),0)
• {1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1{1,,1 ,,2,,}1,2,,} has recursion level C(C(Ω22+C(Ω2+C(σ32)+C(Ω2+C(σ32)+1,σ),0),0),0)
• {1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1{1,,1 ,,2,,}1,,2,,} has recursion level C(C(Ω22+C(Ω2+C(σ32)+σ2,0),0),0)
• {1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1{1,,1 ,,2,,}1{1,,1 ,,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(σ32)2,0),0),0)
• {1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1{1 ,,2{1,,1 ,,2,,}2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22),C(σ32)),0),0),0)
• {1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1{1{1,,1,,2,,}2{1,,1 ,,2,,}2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω222),C(σ32)),0),0),0)
• {1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1{1{1,,1 ,,2,,}3,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(σ3,C(σ32)),0),0),0)
• {1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1{1{1,,1 ,,2,,}1,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(σ3+1,σ2),0),0),0)
• {1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1{1,,2 ,,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω23),σ2),0),0),0)
• {1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1{1,,1 ,,3,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(σ42),0),0),0)
• {1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1{1,,1 ,,1,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22+1,0),σ2),0),0),0)
• {1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1{1,,1 ,,1,,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω22),0),0),0)

Here’re some approximations, to make the comparisons above more simple:

• The double comma approximately corresponds to C(Ω2,C(Ω22,0))
• {1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}2,,} approximately corresponds to C(Ω2,C(Ω2,C(Ω22,0)))
• {1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}3,,} approximately corresponds to C(Ω2,C(Ω2,C(Ω2,C(Ω22,0))))
• {1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1,2,,} approximately corresponds to C(Ω2+1,C(Ω22,0))
• {1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1,,2,,} approximately corresponds to C(Ω2+C(Ω2,C(Ω22,0)),C(Ω22,0))
• {1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1 {1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1,,2,,}2,,} approximately corresponds to C(Ω2+C(Ω2+C(Ω2,C(Ω22,0)),C(Ω22,0)),C(Ω22,0))
• {1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1,,2,,} approximately corresponds to C(Ω2+C(Ω22,C(Ω22,0)),C(Ω22,0))
• {1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1 ,,2,,} approximately corresponds to C(Ω2+C(C(Ω2,C(Ω22,C(Ω22,0))),C(Ω22,C(Ω22,0))),C(Ω22,0))
• {1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1{1,,1,,2,,}2,,} approximately corresponds to C(Ω2+C(C(Ω2+C(Ω22,C(Ω22,0)),C(Ω22,C(Ω22,0))),C(Ω22,C(Ω22,0))),C(Ω22,0))
• {1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1{1,,1 ,,2,,}2,,} approximately corresponds to C(Ω2+C(C(Ω22,C(Ω22,C(Ω22,0))),C(Ω22,C(Ω22,0))),C(Ω22,0))
• {1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1{1,,1 ,,3,,}2,,} approximately corresponds to C(Ω2+C(C(Ω22,C(Ω22,C(Ω22,C(Ω22,0)))),C(Ω22,C(Ω22,0))),C(Ω22,0))
• {1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1{1,,1 ,,1,2,,}2,,} approximately corresponds to C(Ω2+C(C(Ω22+1,0),C(Ω22,C(Ω22,0))),C(Ω22,0))

## Up to a C(Ω22,C(Ω22,C(Ω22,0)))

Now let separator ♦ = {1{1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1{1,,1 ,,1,,2,,}2,,}2} and let ordinal σ = σ1 = C(Ω22,0), and σn+1 = C(Ω22,σn).

• {1{1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1{1,,1 ,,1♦2,,}2,,}2} has recursion level C(C(Ω22+C(Ω2+C(Ω22),0),0),0)
• {1{1,,1,,1,2}2{1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1{1,,1 ,,1♦2,,}2,,}2} has recursion level C(C(Ω22+C(Ω2+C(Ω22),0)+C(Ω2+C(Ω2,σ)+1,0),0),0)
• {1{1{1,,1{1,,1,,2,,}2,,}2}2{1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1{1,,1 ,,1♦2,,}2,,}2} has recursion level C(C(Ω22+C(Ω2+C(Ω22),0)+C(Ω22,0),0),0)
• {1{1{1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1{1,,1 ,,1,2,,}2,,}2}2 {1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1{1,,1 ,,1♦2,,}2,,}2} has recursion level C(C(Ω22+C(Ω2+C(Ω22),0)+C(Ω2+C(C(Ω22+1,0),σ2),0),0),0)
• {1{1{1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1{1,,1 ,,1♦2,,}2,,}2}2 {1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1{1,,1 ,,1♦2,,}2,,}2} has recursion level C(C(Ω22+C(Ω2+C(Ω22),0)+C(Ω2+C(C(Ω22+C(Ω2+C(Ω22),0),0),σ2),0),0),0)
• {1,,1,,1,2{1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1{1,,1 ,,1♦2,,}2,,}2} has recursion level C(C(Ω22+C(Ω2+C(Ω22),0)+C(Ω2+C(Ω2,σ)+1,C(Ω2+C(C(Ω22+C(Ω2+C(Ω22),0),0),σ2),0)),0),0)
• {1{1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1{1,,1 ,,1♦2,,}2,,}1,2} has recursion level C(C(Ω22+C(Ω2+C(Ω22),0)+C(Ω2+C(C(Ω22+C(Ω2+C(Ω22),0),0),σ2)+1,0),0),0)
• {1{1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1{1,,1 ,,1♦1,2,,}2,,}2} has recursion level C(C(Ω22+C(Ω2+C(Ω22),0)+C(Ω2+C(C(Ω22+C(Ω2+C(Ω22),0)+1,0),σ2),0),0),0)
• {1{1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1{1,,1 ,,1♦1♦2,,}2,,}2} has recursion level C(C(Ω22+C(Ω2+C(Ω22),0)2,0),0)
• {1,,1,,1,2{1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1{1,,1 ,,1,,2,,}2,,}2} has recursion level C(C(Ω22+C(Ω2+C(Ω2,σ)+1,C(Ω2+C(Ω22),0)),0),0)
• {1{1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1{1,,1 ,,1,2,,}2,,}2 {1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1{1,,1 ,,1,,2,,}2,,}2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22+1,0),σ2),C(Ω2+C(Ω22),0)),0),0)
• {1{1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1{1,,1 ,,1,,2,,}2,,}3} has recursion level C(C(Ω22+C(Ω2+C(Ω22),C(Ω2+C(Ω22),0)),0),0)
• {1{1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1{1,,1 ,,1,,2,,}2,,}1,2} has recursion level C(C(Ω22+C(Ω2+C(Ω22)+1,0),0),0)
• {1{1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1{1,,1 ,,1,,2,,}2,,}1{1,,1{1,,1,,2,,}2,,}2} has recursion level C(C(Ω22+C(Ω2+C(Ω22)+C(σ2,C(Ω2,σ)),0),0),0)
• {1{1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1{1,,1 ,,1,,2,,}2,,}1 {1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}2} has recursion level C(C(Ω22+C(Ω2+C(Ω22)+C(Ω2,C(Ω2,σ)),0),0),0)
• {1{1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1{1,,1 ,,1,,2,,}2,,}1 {1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1,2,,}2} has recursion level C(C(Ω22+C(Ω2+C(Ω22)+C(Ω2+1,σ),0),0),0)
• {1{1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1{1,,1 ,,1,,2,,}2,,}1 {1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1,,2,,}2} has recursion level C(C(Ω22+C(Ω2+C(Ω22)+C(Ω22,σ),0),0),0)
• {1{1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1{1,,1 ,,1,,2,,}2,,}1 {1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1{1,,1,2,,}2,,}2} has recursion level C(C(Ω22+C(Ω2+C(Ω22)+C(Ω2+C(C(Ω2+1,σ2),σ2),σ),0),0),0)
• {1{1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1{1,,1 ,,1,,2,,}2,,}1 {1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1{1,,1,,2,,}2,,}2} has recursion level C(C(Ω22+C(Ω2+C(Ω22)+C(Ω2+C(C(Ω222),σ2),σ),0),0),0)
• {1{1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1{1,,1 ,,1,,2,,}2,,}1 {1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1{1,,1 ,,2,,}2,,}2} has recursion level C(C(Ω22+C(Ω2+C(Ω22)+C(Ω2+C(σ32),σ),0),0),0)
• {1{1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1{1,,1 ,,1,,2,,}2,,}1 {1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1{1,,1 ,,1,2,,}2,,}2} has recursion level C(C(Ω22+C(Ω2+C(Ω22)+C(Ω2+C(C(Ω22+1,0),σ2),σ),0),0),0)
• {1{1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1{1,,1 ,,1,,2,,}2,,}1 {1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1{1,,1 ,,1,,2,,}2,,}2} has recursion level C(C(Ω22+C(Ω2+C(Ω22)+C(Ω2+C(Ω22),σ),0),0),0)
• {1{1,,1{1,,1,,2,,}2,,}2{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1{1,,1 ,,1,,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω22)+C(σ2,C(Ω2+C(Ω22),σ)),0),0),0)
• {1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}2{1,,1 ,,1,,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω22)+C(Ω2,C(Ω2+C(Ω22),σ)),0),0),0)
• {1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1,,2{1,,1 ,,1,,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω22)+C(Ω22,C(Ω2+C(Ω22),σ)),0),0),0)
• {1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1{1,,1 ,,2,,}2{1,,1 ,,1,,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω22)+C(Ω2+C(σ32),C(Ω2+C(Ω22),σ)),0),0),0)
• {1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1{1,,1 ,,1,,2,,}3,,} has recursion level C(C(Ω22+C(Ω2+C(Ω22)+C(Ω2+C(Ω22),C(Ω2+C(Ω22),σ)),0),0),0)
• {1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1{1,,1 ,,1,,2,,}1,2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω22)+C(Ω2+C(Ω22)+1,σ),0),0),0)
• {1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1{1,,1 ,,1,,2,,}1,,2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω22)+σ2,0),0),0)
• {1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1{1,,1 ,,1,,2,,}1{1,,1,,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω22)+C(C(Ω222),σ2),0),0),0)
• {1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1{1,,1 ,,1,,2,,}1{1,,1 ,,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω22)+C(C(Ω2+C(Ω22),σ2),σ2),0),0),0)
• {1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1{1,,1 ,,1,,2,,}1{1,,1 ,,1,,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω22)+C(C(Ω2+C(Ω22)+σ,σ2),σ2),0),0),0)
• {1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1{1,,1 ,,1,,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω22)+C(C(Ω2+C(Ω22)+σ22),σ2),0),0),0)
• {1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1{1,,1 ,,1{1{1,,1 ,,2,,}2,,}2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω22)+C(C(Ω2+C(Ω22)+C(C(Ω2+C(Ω22),σ2),σ2),σ2),σ2),0),0),0)
• {1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1{1,,1 ,,1 ,,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω22)+C(σ32),0),0),0)
• {1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1{1,,1 ,,1 ,,1,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω22)+C(C(Ω22+1,0),σ2),0),0),0)
• {1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1{1,,1 ,,1 ,,1,,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω22)2,0),0),0)
• {1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1{1,,1 ,,1 ,,1 ,,1,,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω22)3,0),0),0)
• {1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1{1,,1{2 ,,}2,,}2,,} has recursion level C(C(Ω22+C(Ω2+ω^(C(Ω22)+1),0),0),0)
• {1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1{1,,1{1{1,,1,2,,}2 ,,}2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2+1,σ2),C(Ω22)),0),0),0)
• {1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1{1,,1{1{1,,1,,2,,}2 ,,}2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω222),C(Ω22)),0),0),0)
• {1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1{1,,1{1{1,,1 ,,2,,}2 ,,}2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2+C(σ32),σ2),C(Ω22)),0),0),0)
• {1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1{1,,1{1{1,,1 ,,1,,2,,}2 ,,}2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2+C(Ω22),σ2),C(Ω22)),0),0),0)
• {1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1{1,,1{1{1,,1{1{1,,1 ,,1,,2,,}2 ,,}2,,}2 ,,}2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2+C(C(Ω2+C(Ω22),σ2),C(Ω22)),σ2),C(Ω22)),0),0),0)
• {1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1{1,,1  ,,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(σ3,C(Ω22)),0),0),0)
• {1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1{1,,1   ,,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(σ3,C(Ω2,C(Ω22))),0),0),0)
• {1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1{1,,1{1,,1,2,,}2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω2+1,σ2),0),0),0)
• {1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1{1,,1{1,,1{1{1,,1{1,,1,,2,,}2,,}2,,}2,,}2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω2+C(Ω22,σ),σ2),0),0),0)
• {1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1 {1,,1{1,,1{1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1{1,,1{1,,1,2,,}2,,}2,,}2,,}2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω2+C(Ω2+C(Ω2+1,σ2),σ),σ2),0),0),0)
• {1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1{1,,1{1,,1,,2,,}2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω222),0),0),0)
• {1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1{1,,1{1,,1{1{1,,1,2,,}2,,}2,,}2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω2+C(C(Ω2+1,σ2),σ2),σ2),0),0),0)
• {1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1{1,,1{1,,1{1{1,,1,,2,,}2,,}2,,}2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω2+C(C(Ω222),σ2),σ2),0),0),0)
• {1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1{1,,1{1,,1{1{1,,1{1,,1,2,,}2,,}2,,}2,,}2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω2+C(C(Ω2+C(Ω2+1,σ2),σ2),σ2),σ2),0),0),0)
• {1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1{1,,1{1,,1{1{1,,1{1,,1,,2,,}2,,}2,,}2,,}2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω2+C(C(Ω2+C(Ω222),σ2),σ2),σ2),0),0),0)
• {1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1{1,,1{1,,1 ,,2,,}2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω2+C(σ32),σ2),0),0),0)
• {1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1{1,,1{1,,1  ,,2,,}2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω2+C(σ3,C(Ω22)),σ2),0),0),0)
• {1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1{1,,1{1,,1{1,,1,2,,}2,,}2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω2+C(Ω2+1,σ2),σ2),0),0),0)
• {1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1{1,,1{1,,1{1,,1{1,,1,2,,}2,,}2,,}2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω2+C(Ω2+C(Ω2+1,σ2),σ2),σ2),0),0),0)
• {1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1{1,,1{1,,1,,2,,}2,,}2,,} has recursion level C(C(Ω22+C(Ω23,0),0),0)

Here’re some approximations, to make the comparisons above more simple:

• {1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1{1,,1 ,,1,,2,,}2,,} approximately corresponds to C(Ω2+C(Ω2,C(Ω22,C(Ω22,0))),C(Ω22,0))
• {1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1{1,,1 ,,1 ,,1,,2,,}2,,} approximately corresponds to C(Ω2+C(Ω2,C(Ω22,C(Ω22,0)))2,C(Ω22,0))
• {1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1{1,,1  ,,2,,}2,,} approximately corresponds to C(Ω2+C(C(Ω22,C(Ω22,C(Ω22,0))),C(Ω2,C(Ω22,C(Ω22,0)))),C(Ω22,0))
• {1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1{1,,1   ,,2,,}2,,} approximately corresponds to C(Ω2+C(C(Ω22,C(Ω22,C(Ω22,0))),C(Ω2,C(Ω2,C(Ω22,C(Ω22,0))))),C(Ω22,0))
• {1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1{1,,1{1,,1,2,,}2,,}2,,} approximately corresponds to C(Ω2+C(Ω2+1,C(Ω22,C(Ω22,0))),C(Ω22,0))
• {1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1{1,,1{1,,1,,2,,}2,,}2,,} approximately corresponds to C(Ω2+C(Ω2+C(Ω22,C(Ω22,0)),C(Ω22,C(Ω22,0))),C(Ω22,0))
• {1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1{1,,1{1,,1{1,,1,2,,}2,,}2,,}2,,} approximately corresponds to C(Ω2+C(Ω2+C(Ω2+1,C(Ω22,C(Ω22,0))),C(Ω22,C(Ω22,0))),C(Ω22,0))

The C(Ω22,C(Ω22,0)) seems to correspond to the ,, (i.e. {1,,2,,}) in {1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1{ ____ ,,}2,,}. And C(Ω2,C(Ω22,C(Ω22,0))) seems to correspond to {1,,3,,}, C(Ω2,C(Ω2,C(Ω22,C(Ω22,0)))) seems to correspond to {1,,4,,}, C(Ω2+1,C(Ω22,C(Ω22,0))) seems to correspond to {1,,1,2,,}, and C(Ω2+C(Ω22,C(Ω22,0)),C(Ω22,C(Ω22,0))) seems to correspond to {1,,1,,2,,}. But those will be weaken in next comparisons.

## Up to a C(Ω22+C(Ω22,0),0)

Now let ordinal σ = σ1 = C(Ω22,0), and σn+1 = C(Ω22,σn).

• {1{1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1{1,,1{1,,1,,2,,}2,,}2,,}1,2} has recursion level C(C(Ω22+C(Ω23+1,0),0),0)
• {1{1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1{1,,1{1,,1,,2,,}2,,}2,,}1 {1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1,,2,,}2} has recursion level C(C(Ω22+C(Ω23+C(Ω22,σ),0),0),0)
• {1{1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1{1,,1{1,,1,,2,,}2,,}2,,}1 {1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1{1,,1{1,,1,,2,,}2,,}2,,}2} has recursion level C(C(Ω22+C(Ω23+C(Ω23,σ),0),0),0)
• {1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}2{1,,1{1,,1,,2,,}2,,}2,,} has recursion level C(C(Ω22+C(Ω23+C(Ω2,C(Ω23,σ)),0),0),0)
• {1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1{1,,1{1,,1,,2,,}2,,}3,,} has recursion level C(C(Ω22+C(Ω23+C(Ω23,C(Ω23,σ)),0),0),0)
• {1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1{1,,1{1,,1,,2,,}2,,}1,2,,} has recursion level C(C(Ω22+C(Ω23+C(Ω23+1,σ),0),0),0)
• {1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1{1,,1{1,,1,,2,,}2,,}1,,2,,} has recursion level C(C(Ω22+C(Ω232,0),0),0)
• {1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1{1,,1{1,,1,,2,,}2,,}1{1,,1,2,,}2,,} has recursion level C(C(Ω22+C(Ω23+C(C(Ω2+1,σ2),σ2),0),0),0)
• {1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1{1,,1{1,,1,,2,,}2,,}1{1,,1{1,,1,2,,}2,,}2,,} has recursion level C(C(Ω22+C(Ω23+C(C(Ω2+C(Ω2+1,σ2),σ2),σ2),0),0),0)
• {1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1{1,,1{1,,1,,2,,}2,,}1{1,,1{1,,1,,2,,}2,,}2,,} has recursion level C(C(Ω22+C(Ω23+C(σ32),0),0),0)
• {1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1 {1,,1{1,,1,,2,,}2,,}1{1,,1{1,,1,,2,,}2,,}1{1,,1{1,,1,,2,,}2,,}2,,} has recursion level C(C(Ω22+C(Ω23+C(σ32)2,0),0),0)
• {1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1{2{1,,1{1,,1,,2,,}2,,}2,,}2,,} has recursion level C(C(Ω22+C(Ω23+ω^(C(σ32)+1),0),0),0)
• {1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1{1 ,,2{1,,1{1,,1,,2,,}2,,}2,,}2,,} has recursion level C(C(Ω22+C(Ω23+C(C(Ω22),C(σ32)),0),0),0)
• {1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1{1{1,,1{1,,1,,2,,}2,,}3,,}2,,} has recursion level C(C(Ω22+C(Ω23+C(σ3,C(σ32)),0),0),0)
• {1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1{1{1,,1{1,,1,,2,,}2,,}1,2,,}2,,} has recursion level C(C(Ω22+C(Ω23+C(σ3+1,σ2),0),0),0)
• {1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1{1{1,,1{1,,1,,2,,}2,,}2,,1{1,,1,,2,,}2,,}2,,} has recursion level C(C(Ω22+C(Ω23+C(σ32,σ2),0),0),0)
• {1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1{1,,2{1,,1,,2,,}2,,}2,,} has recursion level C(C(Ω22+C(Ω23+C(C(Ω23),σ2),0),0),0)
• {1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1{1,,1{1,,1,2,,}2{1,,1,,2,,}2,,}2,,} has recursion level C(C(Ω22+C(Ω23+C(C(Ω2+C(Ω2+1,σ2),σ3),σ2),0),0),0)
• {1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1{1,,1{1,,1{1,,1,,2,,}2,,}2{1,,1,,2,,}2,,}2,,} has recursion level C(C(Ω22+C(Ω23+C(σ42),0),0),0)
• {1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1{1,,1{1,,1{1,,1,,2,,}2,,}1,2{1,,1,,2,,}2,,}2,,} has recursion level C(C(Ω22+C(Ω23+C(C(Ω22+1,0),σ2),0),0),0)
• {1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}2,,} has recursion level C(C(Ω22+C(Ω23+C(Ω22),0),0),0)
• {1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1,,2,,} has recursion level C(C(Ω22+C(Ω23+C(Ω22)+σ2,0),0),0)
• {1{2{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}2,,}2,,} has recursion level C(C(Ω22+C(Ω23+ω^(C(Ω22)+1),0),0),0)
• {1,,2,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,} has recursion level C(C(Ω22+C(Ω23+ω^(C(Ω22)+σ2),0),0),0)
• {1{1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}2,,}2 {1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}2,,} has recursion level C(C(Ω22+C(Ω23+ω^(C(Ω22)2),0),0),0)
• {1 ,,2{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}2,,} has recursion level C(C(Ω22+C(Ω23+C(C(Ω2,C(Ω22)),C(Ω22)),0),0),0)
• {1{1,,1,,2,,}2{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}2,,} has recursion level C(C(Ω22+C(Ω23+C(C(Ω222),C(Ω22)),0),0),0)
• {1{1,,1{1,,1,,2,,}2,,}2{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}2,,} has recursion level C(C(Ω22+C(Ω23+C(σ3,C(Ω22)),0),0),0)
• {1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}3,,} has recursion level C(C(Ω22+C(Ω23+C(Ω2,C(Ω22)),0),0),0)
• {1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1,2,,} has recursion level C(C(Ω22+C(Ω23+C(Ω2+1,σ2),0),0),0)
• {1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1,,2,,} has recursion level C(C(Ω22+C(Ω23+C(Ω222),0),0),0)
• {1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1 {1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}2,,}2,,} has recursion level C(C(Ω22+C(Ω23+C(Ω2+C(Ω22),σ2),0),0),0)
• {1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1 {1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1,,2,,}2,,} has recursion level C(C(Ω22+C(Ω23+C(Ω2+C(Ω222),σ2),0),0),0)
• {1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1 ,,2,,} has recursion level C(C(Ω22+C(Ω23+C(Ω232),0),0),0)
• {1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1 ,,1 {1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1 ,,2,,}2,,} has recursion level C(C(Ω22+C(Ω23+C(Ω23+C(Ω232),σ2),0),0),0)
• {1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1 ,,1 ,,2,,} has recursion level C(C(Ω22+C(Ω232,0),0),0)
• {1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1{1,,1,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2+1,σ3),σ3),0),0),0)
• {1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1{1,,1,,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω223),σ3),0),0),0)
• {1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1{1,,1 ,,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω233),σ3),0),0),0)
• {1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1{1,,1{1{1,,1 ,,2,,}2 ,,}2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2+C(C(Ω233),σ3),σ3),σ3),0),0),0)
• {1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1{1,,1  ,,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(σ43),0),0),0)
• {1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1{1,,1  ,,1,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22+1,0),σ3),0),0),0)
• {1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1{1,,1  ,,1,,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω23),0),0),0)
• {1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1{1,,1{1,,1,2,,}2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω2+1,σ3),0),0),0)
• {1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1{1,,1{1,,1 ,,2,,}2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω233),0),0),0)
• {1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1{1,,1{1,,1{1,,1 ,,2,,}2,,}2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω2+C(Ω233),σ3),0),0),0)
• {1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1{1,,1{1,,1,,2,,}2,,}2,,} has recursion level C(C(Ω22+C(Ω24,0),0),0)
• {1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}2,,} has recursion level C(C(Ω22+C(Ω24+C(Ω23),0),0),0)
• {1{2,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}2,,} has recursion level C(C(Ω22+C(Ω24+ω^(C(Ω23)+1),0),0),0)
• {1 ,,2,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,} has recursion level C(C(Ω22+C(Ω24+ω^(C(Ω23)+σ3),0),0),0)

Generally, the recursion level of {1{1,,k,,}2,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,} is between C(C(Ω22+C(Ω2k+1,0),0),0) and C(C(Ω22+C(Ω2k+2,0),0),0) (where σ1 = C(Ω22,0), and σn+1 = C(Ω22,σn)), so {1{1,,1,2,,}2,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω22+1,0),0),0),0).

• {1{1{1,,1,2,,}2,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1,2} has recursion level C(C(Ω22+C(Ω2+C(Ω22+1,0)+1,0),0),0)
• {1{1{1,,1,2,,}2,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1,,1,2} has recursion level C(C(Ω22+C(Ω2+C(Ω22+1,0)+C(Ω2,C(Ω22,0))+1,0),0),0)
• {1{1{1,,1,2,,}2,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1 {1{1,,1,2,,}2,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}2} has recursion level C(C(Ω22+C(Ω2+C(Ω22+1,0)+C(Ω2+C(Ω22+1,0),C(Ω22,0)),0),0),0)
• {1,,2{1,,1,2,,}2,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω22+1,0)+ω^(C(Ω2+C(Ω22+1,0),C(Ω22,0))+C(Ω2,C(Ω22,0))),0),0),0)
• {1,,2{1,,1,2,,}2,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω22+1,0)+ω^(C(Ω2+C(Ω22+1,0),C(Ω22,C(Ω22,0)))+C(Ω22,C(Ω22,0))),0),0),0)
• {1 ,,2{1,,1,2,,}2,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω22+1,0)+ω^(C(Ω2+C(Ω22+1,0),C(Ω22,C(Ω22,C(Ω22,0))))+C(Ω22,C(Ω22,C(Ω22,0)))),0),0),0)
• {1{1,,1,2,,}3,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω22+1,0)2,0),0),0)
• {1{1{1,,2,2,,}2,,1,2,,}2,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2,C(Ω22+1,0)),C(Ω22+1,0)),0),0),0)
• {1{1{1,,1{1,,1,2,,}2,,}2,,1,2,,}2,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2+C(Ω22+1,0),C(Ω22+1,0)),C(Ω22+1,0)),0),0),0)
• {1{1{1,,1{1{1,,1{1,,1,2,,}2,,}2,,1,2,,}2,,}2,,1,2,,}2,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2+C(C(Ω2+C(Ω22+1,0),C(Ω22+1,0)),C(Ω22+1,0)),C(Ω22+1,0)),C(Ω22+1,0)),0),0),0)
• {1{1{1,,1{1,,2,2,,}2,,}2,,1,2,,}2,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22,C(Ω22+1,0)),C(Ω22+1,0)),0),0),0)
• {1{1{1,,1{1,,2,2,,}1,2,,}2,,1,2,,}2,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22+1,C(Ω22+1,0)),C(Ω22+1,0)),0),0),0)
• {1{1{1,,1{1,,2,2,,}1,1,2,,}2,,1,2,,}2,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22+2,0),C(Ω22+1,0)),0),0),0)
• {1{1{1,,1{1,,2,2,,}1,,2,,}2,,1,2,,}2,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω2,C(Ω22+1,0)),0),0),0)
• {1{1{1,,1{1,,3,2,,}2,,}2,,1,2,,}2,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22,C(Ω22+1,0)),C(Ω2,C(Ω22+1,0))),0),0),0)
• {1{1{1,,1{1,,1,3,,}2,,}2,,1,2,,}2,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω2+1,C(Ω22+1,0)),0),0),0)
• {1{1{1,,1{1,,1{1,,1,2,,}2,,}2,,}2,,1,2,,}2,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω2+C(Ω22+1,0),C(Ω22+1,0)),0),0),0)
• {1{1{1,,1{1,,1{1,,1{1,,1,2,,}2,,}2,,}2,,}2,,1,2,,}2,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω2+C(Ω2+C(Ω22+1,0),C(Ω22+1,0)),C(Ω22+1,0)),0),0),0)
• {1{1{1,,1{1,,1,,2,,}2,,}2,,1,2,,}2,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω22,C(Ω22+1,0)),0),0),0)
• {1{1,,1,2,,}2{1{1,,1{1,,1,,2,,}2,,}2,,1,2,,}2,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω22,C(Ω22+1,0))+C(Ω22+1,0),0),0),0)
• {1{1{1,,1{1,,1,,2,,}2,,}2,,1,2,,}3,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω22,C(Ω22+1,0))+C(C(Ω22,C(Ω22+1,0)),C(Ω22+1,0)),0),0),0)
• {1{1{1,,1{1,,1{1,,1,,2,,}2,,}2{1,,1,,2,,}2,,}2,,1,2,,} 2,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω22,C(Ω22+1,0))+C(C(Ω22,C(Ω22,C(Ω22+1,0))),C(Ω22+1,0)),0),0),0)
• {1{1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}2,,1,2,,} 2,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω22,C(Ω22+1,0))+C(Ω2,C(Ω22+1,0)),0),0),0)
• {1{1,,1,2,,}2{1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}2,,1,2,,} 2,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω22,C(Ω22+1,0))+C(Ω2,C(Ω22+1,0))+C(Ω22+1,0),0),0),0)
• {1{1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}2,,1,2,,} 3,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω22,C(Ω22+1,0))+C(Ω2,C(Ω22+1,0))2,0),0),0)
• {1{2{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}2,,1,2,,} 2,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω22,C(Ω22+1,0))+ω^ω^(C(Ω2,C(Ω22+1,0))+1),0),0),0)
• {1{1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}3,,1,2,,} 2,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω22,C(Ω22+1,0))+C(Ω2,C(Ω2,C(Ω22+1,0))),0),0),0)
• {1{1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1{1,,1,2,,}2,,1,2,,} 2,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω22,C(Ω22+1,0))+C(Ω2+C(Ω22+1,0),C(Ω22+1,0)),0),0),0)
• {1{1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1 {1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}2,,1,2,,}2,,1,2,,} 2,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω22,C(Ω22+1,0))+C(Ω2+C(Ω2,C(Ω22+1,0)),C(Ω22+1,0)),0),0),0)
• {1{1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1{1,,2,2,,}2,,1,2,,} 2,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω22,C(Ω22+1,0))2,0),0),0)
• {1{1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1{1,,1{1,,3,2,,}2,,}2,,1,2,,} 2,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22,C(Ω22,C(Ω22+1,0))),C(Ω22,C(Ω22+1,0))),0),0),0)
• {1{1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1{1,,1{1,,4,2,,}2,,}2,,1,2,,} 2,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22,C(Ω22,C(Ω22+1,0))),C(Ω2,C(Ω22,C(Ω22+1,0)))),0),0),0)
• {1{1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1{1,,1{1,,1,3,,}2,,}2,,1,2,,} 2,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω2+1,C(Ω22,C(Ω22+1,0))),0),0),0)
• {1{1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1{1,,1{1,,1,,2,,}2,,}2,,1,2,,} 2,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω22,C(Ω22,C(Ω22+1,0))),0),0),0)
• {1{1{2,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}2,,1,2,,} 2,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+ω^(C(Ω22,C(Ω22,C(Ω22+1,0)))+1),0),0),0)
• {1{1,,2,2,,}2,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+ω^(C(Ω22,C(Ω22,C(Ω22+1,0)))+C(Ω22,C(Ω22+1,0))),0),0),0)
• {1{1,,3,2,,}2,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+ω^(C(Ω22,C(Ω22,C(Ω22,C(Ω22+1,0))))+C(Ω22,C(Ω22,C(Ω22+1,0)))),0),0),0)
• {1{1,,1,3,,}2,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω22+1,C(Ω22+1,0)),0),0),0)
• {1{1,,1,1,2,,}2,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω22+2,0),0),0),0)
• {1{1,,1{1{1{1,,1,2,,}2,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}2}2,,}2,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,} has recursion level C(C(Ω22+C(Ω2+C(Ω22+C(Ω2+C(Ω22+1,0),0),0),0),0),0)
• {1{1,,1,,2,,}2,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,} has recursion level C(C(Ω22+C(Ω22,0),0),0)

Up to here, some approximations make the comparisons above more simple:

• The double comma approximately corresponds to C(Ω2,C(Ω22,0))
• ,, approximately corresponds to C(Ω22,C(Ω22,0))
• {1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}2,,} approximately corresponds to C(Ω2,C(Ω22,C(Ω22,0)))
•  ,, approximately corresponds to C(Ω22,C(Ω22,C(Ω22,0)))
• {1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}2 ,,} approximately corresponds to C(Ω2,C(Ω22,C(Ω22,C(Ω22,0))))
• {1,,1,2,,} approximately corresponds to C(Ω22+1,0)
• {1,,2,2,,} approximately corresponds to C(Ω22,C(Ω22+1,0))
• {1,,3,2,,} approximately corresponds to C(Ω22,C(Ω22,C(Ω22+1,0)))

## Up to a C(Ω23,0)

Now let separator ♦ = {1{1{1,,1,,2,,}2,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}2} and let ordinal σ = C(Ω22,0).

• {1,,1{1{1,,1{1{1{1,,1♦2,,}2,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}2}2}2,,1{1,,1,,2}2}2 {1,,1,,2}2} has recursion level C(C(Ω22+σ,0),0)
• {1,,1{1{1,,1{1{1{1,,1♦2,,}2,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}2}2}2,,1{1,,1,,2}2}1,2 {1,,1,,2}2} has recursion level C(C(Ω22+σ+1,0),0)
• {1,,1{2{1,,1{1{1{1,,1♦2,,}2,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}2}2}2,,1{1,,1,,2}2}2 {1,,1,,2}2} has recursion level C(C(Ω22+σ+ω^(C(C(Ω22+σ,0),C(Ω2+σ,0))+1),0),0)
• {1,,2{1{1{1,,1♦2,,}2,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}2}2} has recursion level C(C(Ω22+σ+C(C(Ω2,C(Ω22+σ,0)),C(Ω2+σ,0)),0),0)
• {1,,1{1,,1{1{1{1,,1♦2,,}2,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}2}2}2 {1{1{1,,1♦2,,}2,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}2}2} has recursion level C(C(Ω22+σ+C(C(Ω22+σ,C(Ω22+σ,0)),C(Ω2+σ,0)),0),0)
• {1,,1{1,,1{1{1{1,,1♦2,,}2,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}2}2}1,2 {1{1{1,,1♦2,,}2,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}2}2} has recursion level C(C(Ω22+σ+C(C(Ω22+σ+1,0),C(Ω2+σ,0)),0),0)
• {1,,1{1,,1{1{1{1,,1♦2,,}2,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}2}2}1{1,,2{1,,1,,2}2}2 {1{1{1,,1♦2,,}2,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}2}2} has recursion level C(C(Ω22+σ+C(Ω2,C(Ω2+σ,0)),0),0)
• {1,,1{1,,1{1{1{1,,1♦2,,}2,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}2}2}1{1,,1{1,,1,,2}1,2}2 {1{1{1,,1♦2,,}2,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}2}2} has recursion level C(C(Ω22+σ+C(Ω2+σ+1,0),0),0)
• {1,,1{1,,1{1{1{1,,1♦2,,}2,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}2}2}1{1,,1{1,,1{1,,2,,2}1,2,,2}2} 2{1{1{1,,1♦2,,}2,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}2}2} has recursion level C(C(Ω22+σ+C(Ω2+C(C(Ω22+1,0),σ),0),0),0)
• {1,,1{1,,1{1{1{1,,1♦2,,}2,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}2}2}1 {1,,1{1{1{1,,1♦2,,}2,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}2}2}2 {1{1{1,,1♦2,,}2,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}2}2} has recursion level C(C(Ω22+σ+C(Ω2+C(C(Ω22+σ,0),σ),0),0),0)
• {1,,1{1,,1,,2}2{1{1{1,,1♦2,,}2,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}2}2} has recursion level C(C(Ω22+σ+C(σ,C(Ω2+C(C(Ω22+σ,0),σ),0)),0),0)
• {1,,1{1{1{1,,1♦2,,}2,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}2}3} has recursion level C(C(Ω22+σ+C(C(Ω22+σ,0),C(Ω2+C(C(Ω22+σ,0),σ),0)),0),0)
• {1,,1{1{1{1,,1♦2,,}2,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}2}1,2} has recursion level C(C(Ω22+σ+C(Ω2+C(C(Ω22+σ,0),σ)+1,0),0),0)
• {1{1,,1,,2}2{1{1,,1♦2,,}2,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}2} has recursion level C(C(Ω22+σ+C(Ω2+ω^(C(C(Ω22+σ,0),σ)+σ),0),0),0)
• {1{1,,1{1,,2,,2}1,2,,2}2,,1{1{1{1,,1♦2,,}2,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}2}2,,2} has recursion level C(C(Ω22+σ+C(Ω2+C(C(Ω22+σ,0)+C(Ω22+1,0),σ),0),0),0)
• {1{1,,1{1,,2,,2}1{1,,1,,2}2,,2}2,,1{1{1{1,,1♦2,,}2,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}2}2,,2} has recursion level C(C(Ω22+σ+C(Ω2+C(C(Ω22+σ,0)+C(Ω22+C(Ω2+C(Ω2,σ),0),0),σ),0),0),0)
• {1{1,,1{1,,2,,2}1{1{1,,1{1,,1{1,,1,,3}2,,2}1{1,,2,,2}2{1,,1,,3}2,,2}2,,1,,2}2,,2}2 ,,1{1{1{1,,1♦2,,}2,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}2}2,,2} has recursion level C(C(Ω22+σ+C(Ω2+C(C(Ω22+σ,0)+C(Ω22+C(Ω2,C(Ω2+C(Ω2,σ),0)),0),σ),0),0),0)
• {1{1,,1{1,,2,,2}1{1{1,,1{1{1{1,,1,2,,}2,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}2}2,,2}2,,1,,2}2,,2}2 ,,1{1{1{1,,1♦2,,}2,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}2}2,,2} has recursion level C(C(Ω22+σ+C(Ω2+C(C(Ω22+σ,0)+C(Ω22+C(Ω2+C(Ω22+1,0),0),0),σ),0),0),0)
• {1{1,,1{1,,2,,2}1{1{1,,1{1{1{1,,1♦2,,}2,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}2}2,,2}2,,1,,2}2,,2}2 ,,1{1{1{1,,1♦2,,}2,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}2}2,,2} has recursion level C(C(Ω22+σ+C(Ω2+C(C(Ω22+σ,0)2,σ),0),0),0)
• {1{1,,1,,1,2}2{1{1,,1♦2,,}2,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}2} has recursion level C(C(Ω22+σ+C(Ω2+C(Ω2,σ)+1,0),0),0)
• {1{1{1,,1,,2,,}2}2{1{1,,1♦2,,}2,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}2} has recursion level C(C(Ω22+σ+C(Ω2+C(C(Ω22,σ),C(Ω2,σ)),0),0),0)
• {1{1{1,,1 ,,2,,}2}2{1{1,,1♦2,,}2,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}2} has recursion level C(C(Ω22+σ+C(Ω2+C(C(Ω22,σ),C(Ω2,C(Ω2,σ))),0),0),0)
• {1{1{1,,1{1,,1,2,,}2,,}2}2{1{1,,1♦2,,}2,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}2} has recursion level C(C(Ω22+σ+C(Ω2+C(Ω2+1,σ),0),0),0)
• {1{1{1,,1{1,,1,,2,,}2,,}2}2{1{1,,1♦2,,}2,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}2} has recursion level C(C(Ω22+σ+C(Ω2+C(Ω22,σ),0),0),0)
• {1{1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}2}2 {1{1,,1♦2,,}2,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}2} has recursion level C(C(Ω22+σ+C(Ω2+C(Ω22,σ)+C(Ω2,C(Ω2,σ)),0),0),0)
• {1{1{1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1,,1,,2,,}2}2 {1{1,,1♦2,,}2,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}2} has recursion level C(C(Ω22+σ+C(Ω2+C(Ω22,σ)2,0),0),0)
• {1{1{1{1,,1,2,,}2,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}2}2 {1{1,,1♦2,,}2,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}2} has recursion level C(C(Ω22+σ+C(Ω2+C(Ω22+1,0),0),0),0)
• {1{1{1{1,,1♦2,,}2,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}2}2 {1{1,,1♦2,,}2,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}2} has recursion level C(C(Ω22+σ+C(Ω2+C(Ω22+σ,0),0),0),0)
• {1,,1,,2{1{1,,1♦2,,}2,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}2} has recursion level C(C(Ω22+σ+C(σ,C(Ω2+C(Ω22+σ,0),0)),0),0)
• {1{1{1,,1♦2,,}2,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1,2} has recursion level C(C(Ω22+σ+C(Ω2+C(Ω22+σ,0)+1,0),0),0)
• {1{1{1,,1♦2,,}2,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1 {1{1,,1♦2,,}2,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}2} has recursion level C(C(Ω22+σ+C(Ω2+C(Ω22+σ,0)+C(Ω2+C(Ω22+σ,0),σ),0),0),0)
• {1{1,,1,2,,}2{1,,1♦2,,}2,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,} has recursion level C(C(Ω22+σ+C(Ω2+C(Ω22+σ,0)+C(Ω22+1,0),0),0),0)
• {1{1,,1♦2,,}3,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,} has recursion level C(C(Ω22+σ+C(Ω2+C(Ω22+σ,0)2,0),0),0)
• {1{1,,1,2♦2,,}2,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,} has recursion level C(C(Ω22+σ+C(Ω2+C(Ω22+1,C(Ω22+σ,0)),0),0),0)
• {1{1,,1♦3,,}2,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,} has recursion level C(C(Ω22+σ+C(Ω2+C(Ω22+σ,C(Ω22+σ,0)),0),0),0)
• {1{1,,1♦1,2,,}2,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,} has recursion level C(C(Ω22+σ+C(Ω2+C(Ω22+σ+1,0),0),0),0)
• {1{1,,1♦1♦2,,}2,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,} has recursion level C(C(Ω22+σ2,0),0)
• {1,,2{1{1,,1,,2,,}2,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}2} has recursion level C(C(Ω22+C(C(Ω2,σ),σ),0),0)
• {1,,1,2{1{1,,1,,2,,}2,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}2} has recursion level C(C(Ω22+C(C(Ω2+1,σ),σ),0),0)
• {1,,1,,2{1{1,,1,,2,,}2,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}2} has recursion level C(C(Ω22+C(C(Ω22,σ),σ),0),0)

At this stage, C(Ω2+____,0) ordinals correspond to {1{1{1,,1 A 2,,}2,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}2} with lv(A) < lv(,,); and C(Ω22,0) is the supremum of them, so it corresponds to {1{1{1,,1,,2,,}2,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}2}. When it reduces, the double comma (red) is the 2-separator, {1{1{1,,1,,2,,}2,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}2} is where it drops down to (1-separator).

Now let separator ◊ = {1{1,,1,,2,,}2,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,} and let ordinal σ = C(Ω22,0).

• {1{1,,1{1,,1,,2◊2}2◊2}2◊2} has recursion level C(C(Ω22+C(C(Ω22,σ),σ),0),0)
• {1{1,,2◊2}2{1,,1{1,,1,,2◊2}2◊2}2◊2} has recursion level C(C(Ω22+C(C(Ω2,σ),C(C(Ω22,σ),σ)),0),0)
• {1{1,,1{1,,1,,2◊2}2◊2}3◊2} has recursion level C(C(Ω22+C(C(Ω22,σ),C(C(Ω22,σ),σ)),0),0)
• {1{1,,1{1,,1,,2◊2}2◊2}1,2◊2} has recursion level C(C(Ω22+C(C(Ω22,σ)+1,σ),0),0)
• {1{1,,1{1,,1,,2◊2}2◊2}2,,1{1,,1,,2◊2}2◊2} has recursion level C(C(Ω22+C(C(Ω22,σ)2,σ),0),0)
• {1,,2{1,,1,,2◊2}2◊2} has recursion level C(C(Ω22+C(C(Ω2,C(Ω22,σ)),σ),0),0)
• {1,,1{1,,1,2◊2}2{1,,1,,2◊2}2◊2} has recursion level C(C(Ω22+C(C(Ω2+C(Ω2+1,σ),C(Ω22,σ)),σ),0),0)
• {1,,1{1,,1{1,,1,,2◊2}2◊2}2{1,,1,,2◊2}2◊2} has recursion level C(C(Ω22+C(C(Ω22,C(Ω22,σ)),σ),0),0)
• {1,,1{1,,1{1,,1,,2◊2}2◊2}1,2{1,,1,,2◊2}2◊2} has recursion level C(C(Ω22+C(C(Ω22+1,0),σ),0),0)
• {1,,1{1,,1{1,,1,,2◊2}2◊2}1{1{1{1,,1{1◊2}2,,}2,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}2}2 {1,,1,,2◊2}2◊2} has recursion level C(C(Ω22+C(C(Ω22+C(Ω2+C(Ω22+σ,0),0),0),σ),0),0)
• {1,,1{1,,1{1,,1,,2◊2}2◊2}1{1◊2}2{1,,1,,2◊2}2◊2} has recursion level C(C(Ω22+C(C(Ω22+σ,0),σ),0),0)
• {1,,1{1,,1{1,,1,,2◊2}2◊2}1{1{1,,1{1,,1{1,,1,,2◊2}2◊2}1{1◊2}2{1,,1,,2◊2}2◊2}2◊2}2{1,,1,,2◊2}2◊2} has recursion level C(C(Ω22+C(C(Ω22+C(C(Ω22+σ,0),σ),0),σ),0),0)
• {1,,1{1,,1{1,,1,,2◊2}2◊2}1{1,,2◊2}2{1,,1,,2◊2}2◊2} has recursion level C(C(Ω22+C(Ω2,σ),0),0)
• {1{1{1{1,,1{1◊2}2,,}2,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}2}2 {1{1,,1{1,,1{1,,1{1,,1,,2◊2}2◊2}1{1,,2◊2}2{1,,1,,2◊2}2◊2}2,,} 2,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}2} has recursion level C(C(Ω22+C(Ω2,σ)+C(Ω2+C(Ω22+σ,0),0),0),0)
• {1{1{1{1,,1{1,,1{1,,1{1,,1,,2◊2}2◊2}1{1,,2◊2}2{1,,1,,2◊2}2◊2} 2,,}2,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}2}2 {1{1,,1{1,,1{1,,1{1,,1,,2◊2}2◊2}1{1,,2◊2}2{1,,1,,2◊2}2◊2}2,,} 2,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}2} has recursion level C(C(Ω22+C(Ω2,σ)+C(Ω2+C(Ω22+C(Ω2,σ),0),0),0),0)
• {1,,1,,2{1{1,,1{1,,1{1,,1{1,,1,,2◊2}2◊2}1{1,,2◊2}2{1,,1,,2◊2}2◊2}2,,} 2,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}2} has recursion level C(C(Ω22+C(Ω2,σ)+C(σ,C(Ω2+C(Ω22+C(Ω2,σ),0),0)),0),0)
• {1{1{1,,1{1,,1{1,,1{1,,1,,2◊2}2◊2}1{1,,2◊2}2{1,,1,,2◊2}2◊2}2,,} 2,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1,2} has recursion level C(C(Ω22+C(Ω2,σ)+C(Ω2+C(Ω22+C(Ω2,σ),0)+1,0),0),0)
• {1{1,,1{1,,1{1,,1{1,,1,,2◊2}2◊2}1{1,,2◊2}2{1,,1,,2◊2}2◊2}1,2,,} 2,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,} has recursion level C(C(Ω22+C(Ω2,σ)+C(Ω2+C(Ω22+C(Ω2,σ)+1,0),0),0),0)
• {1{1,,1{1,,1{1,,1{1,,1,,2◊2}2◊2}1{1,,2◊2}2{1,,1,,2◊2}2◊2}1{1◊2}2,,} 2,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,} has recursion level C(C(Ω22+C(Ω2,σ)+σ,0),0)
• {1{1,,1{1,,1{1,,1{1,,1,,2◊2}2◊2}1{1,,2◊2}2{1,,1,,2◊2}2◊2}1 {1,,1{1,,1{1,,1,,2◊2}2◊2}1,2{1,,1,,2◊2}2◊2} 2,,}2,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,} has recursion level C(C(Ω22+C(Ω2,σ)+C(C(Ω22+1,0),σ),0),0)
• {1{1,,1{1,,1{1,,1{1,,1,,2◊2}2◊2}1{1,,2◊2}2{1,,1,,2◊2}2◊2}1 {1,,1{1,,1{1,,1,,2◊2}2◊2}1{1◊2}2{1,,1,,2◊2}2◊2} 2,,}2,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,} has recursion level C(C(Ω22+C(Ω2,σ)+C(C(Ω22+σ,0),σ),0),0)
• {1{1,,1{1,,1{1,,1{1,,1,,2◊2}2◊2}1{1,,2◊2}2{1,,1,,2◊2}2◊2}1 {1,,1{1,,1{1,,1,,2◊2}2◊2}1{1,,2◊2}2{1,,1,,2◊2}2◊2} 2,,}2,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,} has recursion level C(C(Ω22+C(Ω2,σ)+C(C(Ω22+C(Ω2,σ),0),σ),0),0)
• {1{1,,2◊2}2{1,,1{1,,1{1,,1,,2◊2}2◊2}1{1,,2◊2}2{1,,1,,2◊2}2◊2}2◊2} has recursion level C(C(Ω22+C(Ω2,σ)+C(C(Ω2,σ),C(C(Ω22+C(Ω2,σ),0),σ)),0),0)
• {1{1,,1{1,,1{1,,1,,2◊2}2◊2}1,2{1,,1,,2◊2}2◊2}2 {1,,1{1,,1{1,,1,,2◊2}2◊2}1{1,,2◊2}2{1,,1,,2◊2}2◊2}2◊2} has recursion level C(C(Ω22+C(Ω2,σ)+C(C(Ω22+1,0),C(C(Ω22+C(Ω2,σ),0),σ)),0),0)
• {1{1,,1{1,,1{1,,1,,2◊2}2◊2}1{1◊2}2{1,,1,,2◊2}2◊2}2 {1,,1{1,,1{1,,1,,2◊2}2◊2}1{1,,2◊2}2{1,,1,,2◊2}2◊2}2◊2} has recursion level C(C(Ω22+C(Ω2,σ)+C(C(Ω22+σ,0),C(C(Ω22+C(Ω2,σ),0),σ)),0),0)
• {1{1,,1{1,,1{1,,1,,2◊2}2◊2}1{1,,2◊2}2{1,,1,,2◊2}2◊2}3◊2} has recursion level C(C(Ω22+C(Ω2,σ)+C(C(Ω22+C(Ω2,σ),0),C(C(Ω22+C(Ω2,σ),0),σ)),0),0)
• {1{1,,1{1,,1{1,,1,,2◊2}2◊2}1{1,,2◊2}2{1,,1,,2◊2}2◊2}1,2◊2} has recursion level C(C(Ω22+C(Ω2,σ)+C(C(Ω22+C(Ω2,σ),0)+1,σ),0),0)
• {1,,2{1,,1{1,,1,,2◊2}2◊2}1{1,,2◊2}2{1,,1,,2◊2}2◊2} has recursion level C(C(Ω22+C(Ω2,σ)+C(C(Ω2,C(Ω22+C(Ω2,σ),0)),σ),0),0)
• {1,,1{1,,1{1,,1,,2◊2}2◊2}2{1,,2◊2}2{1,,1,,2◊2}2◊2} has recursion level C(C(Ω22+C(Ω2,σ)+C(C(Ω22,C(Ω22+C(Ω2,σ),0)),σ),0),0)
• {1,,1{1,,1{1,,1,,2◊2}2◊2}1{1,,2◊2}1,2{1,,1,,2◊2}2◊2} has recursion level C(C(Ω22+C(Ω2,σ)+C(C(Ω22+C(Ω2,σ)+1,0),σ),0),0)
• {1,,1{1,,1{1,,1,,2◊2}2◊2}1{1,,2◊2}1{1,,2◊2}2{1,,1,,2◊2}2◊2} has recursion level C(C(Ω22+C(Ω2,σ)2,0),0)
• {1,,1{1,,1{1,,1,,2◊2}2◊2}1{1{1,,3◊2}2,,2◊2}2{1,,1,,2◊2}2◊2} has recursion level C(C(Ω22+C(C(Ω2,C(Ω2,σ)),C(Ω2,σ)),0),0)
• {1,,1{1,,1{1,,1,,2◊2}2◊2}1{1,,3◊2}2{1,,1,,2◊2}2◊2} has recursion level C(C(Ω22+C(Ω2,C(Ω2,σ)),0),0)
• {1,,1{1,,1{1,,1,,2◊2}2◊2}1{1,,1,2◊2}2{1,,1,,2◊2}2◊2} has recursion level C(C(Ω22+C(Ω2+1,σ),0),0)
• {1,,1{1,,1{1,,1,,2◊2}2◊2}1{1,,1{1{1{1,,1{1◊2}2,,}2,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}2} 2◊2}2{1,,1,,2◊2}2◊2} has recursion level C(C(Ω22+C(Ω2+C(Ω2+C(Ω22+σ,0),0),σ),0),0)
• {1,,1{1,,1{1,,1,,2◊2}2◊2}1{1,,1{1{1{1,,1{1,,1{1,,1{1,,1,,2◊2}2◊2}1{1,,2◊2}2{1,,1,,2◊2}2◊2} 2,,}2,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}2}2◊2}2{1,,1,,2◊2}2◊2} has recursion level C(C(Ω22+C(Ω2+C(Ω2+C(Ω22+C(Ω2,σ),0),0),σ),0),0)
• {1,,1{1,,1{1,,1,,2◊2}2◊2}1{1,,1{1◊2}2◊2}2{1,,1,,2◊2}2◊2} has recursion level C(C(Ω22+C(Ω2+σ,σ),0),0)
• {1,,1{1,,1{1,,1,,2◊2}2◊2}1{1,,1{1{1,,1{1,,1{1,,1,,2◊2}2◊2}1{1◊2}2{1,,1,,2◊2}2◊2}2◊2}2◊2} 2{1,,1,,2◊2}2◊2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22+σ,0),σ),σ),0),0)
• {1,,1{1,,1{1,,1,,2◊2}2◊2}1{1,,1{1{1,,1{1,,1{1,,1,,2◊2}2◊2}1{1,,2◊2}2{1,,1,,2◊2}2◊2}2◊2}2◊2} 2{1,,1,,2◊2}2◊2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22+C(Ω2,σ),0),σ),σ),0),0)
• {1,,1{1,,1{1,,1,,2◊2}2◊2}1{1,,1{1,,2◊2}2◊2}2{1,,1,,2◊2}2◊2} has recursion level C(C(Ω22+C(Ω2+C(Ω2,σ),σ),0),0)
• {1,,1{1,,1{1,,1,,2◊2}2◊2}1{1,,1{1,,1,2◊2}2◊2}2{1,,1,,2◊2}2◊2} has recursion level C(C(Ω22+C(Ω2+C(Ω2+1,σ),σ),0),0)
• {1,,1{1,,1{1,,1,,2◊2}2◊2}1{1,,1{1,,1{1,,1,2◊2}2◊2}2◊2}2{1,,1,,2◊2}2◊2} has recursion level C(C(Ω22+C(Ω2+C(Ω2+C(Ω2+1,σ),σ),σ),0),0)
• {1,,1{1,,1{1,,1,,2◊2}2◊2}1{1,,1{1,,1,,2◊2}2◊2}2{1,,1,,2◊2}2◊2} has recursion level C(C(Ω22+C(Ω2+C(Ω22,σ),σ),0),0)
• {1,,1{2,,1{1,,1,,2◊2}2◊2}2{1,,1,,2◊2}2◊2} has recursion level C(C(Ω22+ω^(C(Ω2+C(Ω22,σ),σ)+1),0),0)
• {1,,1{1{1,,2{1,,1,,2◊2}2◊2}2,,1{1,,1,,2◊2}2◊2}2{1,,1,,2◊2}2◊2} has recursion level C(C(Ω22+C(C(Ω2,C(Ω2+C(Ω22,σ),σ)),C(Ω2+C(Ω22,σ),σ)),0),0)
• {1,,1{1{1,,1{1,,1{1,,1,,2◊2}2◊2}2{1,,1,,2◊2}2◊2}2,,1{1,,1,,2◊2}2◊2}2{1,,1,,2◊2}2◊2} has recursion level C(C(Ω22+C(C(Ω2+C(Ω2+C(Ω22,σ),σ),C(Ω2+C(Ω22,σ),σ)),C(Ω2+C(Ω22,σ),σ)),0),0)
• {1,,1{1,,2{1,,1,,2◊2}2◊2}2{1,,1,,2◊2}2◊2} has recursion level C(C(Ω22+C(C(Ω2+C(Ω2,C(Ω2+C(Ω22,σ),σ)),C(Ω2+C(Ω22,σ),σ)),C(Ω2+C(Ω22,σ),σ)),0),0)
• {1,,1{1,,1,,2◊2}3◊2} has recursion level C(C(Ω22+C(C(Ω22,σ),C(Ω2+C(Ω22,σ),σ)),0),0)
• {1,,1{1,,1{1,,1,,2◊2}3◊2}1{1,,1{1,,1,,2◊2}3◊2}2{1,,1,,2◊2}3◊2} has recursion level C(C(Ω22+C(Ω2+C(Ω22,σ),C(Ω2+C(Ω22,σ),σ)),0),0)
• {1,,1{1,,1,,2◊2}4◊2} has recursion level C(C(Ω22+C(C(Ω22,σ),C(Ω2+C(Ω22,σ),C(Ω2+C(Ω22,σ),σ))),0),0)
• {1,,1{1,,1,,2◊2}1,2◊2} has recursion level C(C(Ω22+C(Ω2+C(Ω22,σ)+1,σ),0),0)
• {1,,1{1,,1,,2◊2}1{1◊2}2◊2} has recursion level C(C(Ω22+C(Ω2+C(Ω22,σ)+σ,σ),0),0)
• {1,,1{1,,1,,2◊2}1{1,,2◊2}2◊2} has recursion level C(C(Ω22+C(Ω2+C(Ω22,σ)+C(Ω2,σ),σ),0),0)
• {1,,1{1,,1,,2◊2}1{1,,1{1,,1,,2◊2}2◊2}2◊2} has recursion level C(C(Ω22+C(Ω2+C(Ω22,σ)+C(Ω2+C(Ω22,σ),σ),σ),0),0)
• {1,,1{1,,1,,2◊2}1{1,,1,,2◊2}2◊2} has recursion level C(C(Ω22+C(Ω2+C(Ω22,σ)2,σ),0),0)
• {1,,2,,2◊2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2,C(Ω22,σ)),C(Ω22,σ)),σ),0),0)
• {1,,1{1,,1,,2◊2}2,,2◊2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2+C(Ω22,σ),C(Ω22,σ)),C(Ω22,σ)),σ),0),0)
• {1,,1{1{1,,2,,2◊2}2,,1,,2◊2}2,,2◊2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2+C(C(Ω2,C(Ω22,σ)),C(Ω22,σ)),C(Ω22,σ)),C(Ω22,σ)),σ),0),0)
• {1,,1{1,,2,,2◊2}2,,2◊2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22,C(Ω22,σ)),C(Ω22,σ)),σ),0),0)
• {1,,1{1,,2,,2◊2}1,2,,2◊2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22+1,0),C(Ω22,σ)),σ),0),0)
• {1,,1{1,,2,,2◊2}1{1◊2}2,,2◊2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22+σ,0),C(Ω22,σ)),σ),0),0)
• {1,,1{1,,2,,2◊2}1{1,,1,,2◊2}2,,2◊2} has recursion level C(C(Ω22+C(Ω2+C(Ω2,C(Ω22,σ)),σ),0),0)
• {1,,1,,1,2◊2} has recursion level C(C(Ω22+C(Ω2+C(Ω2,C(Ω22,σ))+1,σ),0),0)
• {1,,1,,1,,2◊2} has recursion level C(C(Ω22+C(Ω2+C(Ω2,C(Ω22,σ))+C(Ω22,σ),σ),0),0)
• {1,,1,,1,,1,,2◊2} has recursion level C(C(Ω22+C(Ω2+C(Ω2,C(Ω22,σ))2+C(Ω22,σ),σ),0),0)
• {1{2,,}2◊2} has recursion level C(C(Ω22+C(Ω2+ω^(C(Ω2,C(Ω22,σ))+1),σ),0),0)
• {1{1,,2,,}2◊2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2,C(Ω2,C(Ω22,σ))),C(Ω2,C(Ω22,σ))),σ),0),0)
• {1{1,,1,,2,,}2◊2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2+C(Ω22,σ),C(Ω22,σ)),C(Ω2,C(Ω22,σ))),σ),0),0)
• {1{1,,1,,2,,}2◊2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22,C(Ω22,σ)),C(Ω2,C(Ω22,σ))),σ),0),0)
• {1{1,,1 ,,2,,}2◊2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22,C(Ω22,σ)),C(Ω2,C(Ω2,C(Ω22,σ)))),σ),0),0)
• {1{1,,1{1,,1,2,,}2,,}2◊2} has recursion level C(C(Ω22+C(Ω2+C(Ω2+1,C(Ω22,σ)),σ),0),0)
• {1{1,,1{1,,1,,2,,}2,,}2◊2} has recursion level C(C(Ω22+C(Ω2+C(Ω2+C(Ω22,σ),C(Ω22,σ)),σ),0),0)
• {1{1,,1{1,,1,,2,,}2,,}2◊2} has recursion level C(C(Ω22+C(Ω2+C(Ω2+C(C(Ω22,C(Ω22,σ)),C(Ω2,C(Ω22,σ))),C(Ω22,σ)),σ),0),0)
• {1{1,,1{1,,1{1,,1,2,,}2,,}2,,}2◊2} has recursion level C(C(Ω22+C(Ω2+C(Ω2+C(Ω2+1,C(Ω22,σ)),C(Ω22,σ)),σ),0),0)
• {1{1,,1{1,,1,,2,,}2,,}2◊2} has recursion level C(C(Ω22+C(Ω2+C(Ω22,C(Ω22,σ)),σ),0),0)
• {1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}2◊2} has recursion level C(C(Ω22+C(Ω2+C(Ω22,C(Ω22,σ))+C(Ω2,C(Ω2,C(Ω22,σ))),σ),0),0)
• {1{1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1,2,,}2◊2} has recursion level C(C(Ω22+C(Ω2+C(Ω22,C(Ω22,σ))+C(Ω2+1,C(Ω22,σ)),σ),0),0)
• {1{1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1,,2,,}2◊2} has recursion level C(C(Ω22+C(Ω2+C(Ω22,C(Ω22,σ))+C(Ω2+C(Ω22,C(Ω22,σ)),C(Ω22,σ)),σ),0),0)
• {1{1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1,,1,,2,,}2◊2} has recursion level C(C(Ω22+C(Ω2+C(Ω22,C(Ω22,σ))2,σ),0),0)
• {1{1{1,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1 ,,1 ,,2,,}2◊2} has recursion level C(C(Ω22+C(Ω2+C(Ω22,C(Ω22,C(Ω22,σ)))2,σ),0),0)
• {1{1{1,,1,2,,}2,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}2◊2} has recursion level C(C(Ω22+C(Ω2+C(Ω22+1,0),σ),0),0)
• {1{1{1,,1{1◊2}2,,}2,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}2◊2} has recursion level C(C(Ω22+C(Ω2+C(Ω22+σ,0),σ),0),0)
• {1{1{1,,1{1,,2◊2}2,,}2,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}2◊2} has recursion level C(C(Ω22+C(Ω2+C(Ω22+C(Ω2,σ),0),σ),0),0)
• {1{1{1,,1{1{1{1,,1,2,,}2,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}2◊2} 2,,}2,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}2◊2} has recursion level C(C(Ω22+C(Ω2+C(Ω22+C(Ω2+C(Ω22+1,0),σ),0),σ),0),0)
• {1◊3} has recursion level C(C(Ω22+C(Ω22,σ),0),0)
• {1◊4} has recursion level C(C(Ω22+C(Ω22,C(Ω22,σ)),0),0)
• {1◊1,2} has recursion level C(C(Ω22+C(Ω22+1,0),0),0)
• {1◊1{1◊2}2} has recursion level C(C(Ω22+C(Ω22+σ,0),0),0)
• {1◊1{1◊1,2}2} has recursion level C(C(Ω22+C(Ω22+C(Ω22+1,0),0),0),0)
• {1◊1,,2} has recursion level C(C(Ω23,0),0)

So {1{1{1,,1,,2,,}2,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}1,,2} has recursion level C(C(Ω23,0),0).

Here’re some approximations, to make the comparisons above more simple:

• {1◊2} (where ◊ = {1{1,,1,,2,,}2,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}) approximately corresponds to C(Ω22,0)
• {1{1{1,,1{1◊2}2,,}2,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}2} approximately corresponds to C(Ω2+C(Ω22+C(Ω22,0),0),0)
• {1,,2◊2} approximately corresponds to C(Ω2,C(Ω22,0))
• {1{1{1,,1{1,,2◊2}2,,}2,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}2} approximately corresponds to C(Ω2+C(Ω22+C(Ω2,C(Ω22,0)),0),0)
• {1,,1{1◊2}2◊2} approximately corresponds to C(Ω2+C(Ω22,0),C(Ω22,0))
• {1,,1{1,,2◊2}2◊2} approximately corresponds to C(Ω2+C(Ω2,C(Ω22,0)),C(Ω22,0))
• {1,,1{1,,1,,2◊2}2◊2} approximately corresponds to C(Ω2+C(Ω22,C(Ω22,0)),C(Ω22,0))
• {1,,1{1,,1,,2◊2}1,2◊2} approximately corresponds to C(Ω2+C(Ω22,C(Ω22,0))+1,C(Ω22,0))
• {1,,1,,2◊2} approximately corresponds to C(Ω2+C(Ω2,C(Ω22,C(Ω22,0))),C(Ω22,0))
• {1{1,,1{1,,1,2,,}2,,}2◊2} approximately corresponds to C(Ω2+C(Ω2+1,C(Ω22,C(Ω22,0))),C(Ω22,0))
• {1{1,,1{1,,1,,2,,}2,,}2◊2} approximately corresponds to C(Ω2+C(Ω22,C(Ω22,C(Ω22,0))),C(Ω22,0))
• {1◊3} approximately corresponds to C(Ω22,C(Ω22,0))
• {1◊1,2} approximately corresponds to C(Ω22+1,0)
• {1{1{1,,1{1◊1,2}2,,}2,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}2} approximately corresponds to C(Ω2+C(Ω22+C(Ω22+1,0),0),0)
• {1{1{1,,1{1◊1{1◊2}2}2,,}2,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,}2} approximately corresponds to C(Ω2+C(Ω22+C(Ω22+C(Ω22,0),0),0),0)

## Beyond C(Ω23,0)

Now let separator ◊ = {1{1,,1,,2,,}2,,1{1,,1{1,,1,,2,,}2,,}1,,2{1,,1,,2,,}2,,} and let ordinal σ = C(Ω22,0).

• {1,,1{1{1◊1{1◊1,,2}2}2,,1{1,,1,,2}2}2{1,,1,,2}2} has recursion level C(C(Ω23,0),0)
• {1{1,,1{1{1◊1{1◊1,,2}2}2,,1{1,,1,,2}2}2{1,,1,,2}2}3} has recursion level C(C(Ω23,0),C(C(Ω23,0),0))
• {1{1,,1{1{1◊1{1◊1,,2}2}2,,1{1,,1,,2}2}2{1,,1,,2}2}1,2} has recursion level C(C(Ω23,0)+1,0)
• {1{1,,1{1{1◊1{1◊1,,2}2}2,,1{1,,1,,2}2}2{1,,1,,2}2}2,,1{1{1◊1{1◊1,,2}2}2,,1{1,,1,,2}2}2{1,,1,,2}2} has recursion level C(C(Ω23,0)2,0)
• {1,,2{1{1◊1{1◊1,,2}2}2,,1{1,,1,,2}2}2{1,,1,,2}2} has recursion level C(C(Ω2,C(Ω23,0)),0)
• {1,,1{1,,1{1,,1,,2}2}2{1{1◊1{1◊1,,2}2}2,,1{1,,1,,2}2}2{1,,1,,2}2} has recursion level C(C(Ω22,C(Ω23,0)),0)
• {1,,1{1{1◊1{1◊1,,2}2}2,,1{1,,1,,2}2}3{1,,1,,2}2} has recursion level C(C(Ω23,C(Ω23,0)),0)
• {1,,1{1{1◊1{1◊1,,2}2}2,,1{1,,1,,2}2}1,2{1,,1,,2}2} has recursion level C(C(Ω23+1,0),0)
• {1,,1{1{1◊1{1◊1,,2}2}2,,1{1,,1,,2}2}1{1,,1{1,,1,,2}2}2{1,,1,,2}2} has recursion level C(C(Ω23+C(Ω2+σ,0),0),0)
• {1,,1{1{1◊1{1◊1,,2}2}2,,1{1,,1,,2}2}1{1{1,,1{1,,1,,2}3}2,,1{1,,1,,2}2}2{1,,1,,2}2} has recursion level C(C(Ω23+C(σ,C(Ω2+σ,0)),0),0)
• {1,,1{1{1◊1{1◊1,,2}2}2,,1{1,,1,,2}2}1{1{1◊2}2,,1{1,,1,,2}2}2{1,,1,,2}2} has recursion level C(C(Ω23+C(C(Ω22+σ,0),C(Ω2+σ,0)),0),0)
• {1,,1{1{1◊1{1◊1,,2}2}2,,1{1,,1,,2}2}1{1{1◊1{1◊1,,2}2}2,,1{1,,1,,2}2}2{1,,1,,2}2} has recursion level C(C(Ω23+C(C(Ω23,0),C(Ω2+σ,0)),0),0)
• {1,,1{2{1◊1{1◊1,,2}2}2,,1{1,,1,,2}2}2{1,,1,,2}2} has recursion level C(C(Ω23+ω^(C(C(Ω23,0),C(Ω2+σ,0))+1),0),0)
• {1,,1{1{1◊1{1◊1,,2}2}1,2,,1{1,,1,,2}2}2{1,,1,,2}2} has recursion level C(C(Ω23+C(C(Ω23,0)+1,C(Ω2+σ,0)),0),0)
• {1{1,,1{1,,1,,2}3}2,,1{1◊1{1◊1,,2}2}2} has recursion level C(C(Ω23+C(C(Ω23,0)+σ,C(Ω2+σ,0)),0),0)
• {1{1,,1{1◊2}2}2,,1{1◊1{1◊1,,2}2}2} has recursion level C(C(Ω23+C(C(Ω23,0)+C(Ω22+σ,0),C(Ω2+σ,0)),0),0)
• {1{1,,1{1◊1{1◊1,,2}2}2}2,,1{1◊1{1◊1,,2}2}2} has recursion level C(C(Ω23+C(C(Ω23,0)2,C(Ω2+σ,0)),0),0)
• {1,,2{1◊1{1◊1,,2}2}2} has recursion level C(C(Ω23+C(C(Ω2,C(Ω23,0)),C(Ω2+σ,0)),0),0)
• {1,,1{1,,1{1,,1,,2}3}2{1◊1{1◊1,,2}2}2} has recursion level C(C(Ω23+C(C(Ω22,C(Ω23,0)),C(Ω2+σ,0)),0),0)
• {1,,1{1,,1{1◊2}2}2{1◊1{1◊1,,2}2}2} has recursion level C(C(Ω23+C(C(Ω22+σ,C(Ω23,0)),C(Ω2+σ,0)),0),0)
• {1,,1{1,,1{1◊1{1◊1,,2}2}2}2{1◊1{1◊1,,2}2}2} has recursion level C(C(Ω23+C(C(Ω23,C(Ω23,0)),C(Ω2+σ,0)),0),0)
• {1,,1{1,,1{1◊1{1◊1,,2}2}2}1,2{1◊1{1◊1,,2}2}2} has recursion level C(C(Ω23+C(C(Ω23+1,0),C(Ω2+σ,0)),0),0)
• {1,,1{1,,1{1◊1{1◊1,,2}2}2}1{1,,2{1,,1,,2}2}2{1◊1{1◊1,,2}2}2} has recursion level C(C(Ω23+C(Ω2,C(Ω2+σ,0)),0),0)
• {1,,1{1,,1{1◊1{1◊1,,2}2}2}1{1,,1{1,,1,,2}1,2}2{1◊1{1◊1,,2}2}2} has recursion level C(C(Ω23+C(Ω2+σ+1,0),0),0)
• {1,,1{1,,1{1◊1{1◊1,,2}2}2}1{1,,1{1,,1,,2}1{1,,1,,2}2}2{1◊1{1◊1,,2}2}2} has recursion level C(C(Ω23+C(Ω2+σ2,0),0),0)
• {1,,1{1,,1{1◊1{1◊1,,2}2}2}1{1,,1{1,,1{1,,2,,2}1,2,,2}2}2{1◊1{1◊1,,2}2}2} has recursion level C(C(Ω23+C(Ω2+C(C(Ω22+1,0),σ),0),0),0)
• {1,,1{1,,1{1◊1{1◊1,,2}2}2}1{1,,1{1◊2}2}2{1◊1{1◊1,,2}2}2} has recursion level C(C(Ω23+C(Ω2+C(C(Ω22+σ,0),σ),0),0),0)
• {1,,1{1,,1{1◊1{1◊1,,2}2}2}1{1,,1{1◊1{1◊1,,2}2}2}2{1◊1{1◊1,,2}2}2} has recursion level C(C(Ω23+C(Ω2+C(C(Ω23,0),σ),0),0),0)
• {1,,1{1,,1,,2}2{1◊1{1◊1,,2}2}2} has recursion level C(C(Ω23+C(σ,C(Ω2+C(C(Ω23,0),σ),0)),0),0)
• {1,,1{1◊1{1◊1,,2}2}1,2} has recursion level C(C(Ω23+C(Ω2+C(C(Ω23,0),σ)+1,0),0),0)
• {1,,1{2◊1{1◊1,,2}2}2} has recursion level C(C(Ω23+C(Ω2+ω^(C(C(Ω23,0),σ)+1),0),0),0)
• {1{1,,1,,2}2◊1{1◊1,,2}2} has recursion level C(C(Ω23+C(Ω2+ω^(C(C(Ω23,0),σ)+σ),0),0),0)
• {1,,2{1◊1{1◊1,,2}2}2,,2} has recursion level C(C(Ω23+C(Ω2+C(C(Ω2,C(Ω23,0)),σ),0),0),0)
• {1,,1{1,,2,,2}2{1◊1{1◊1,,2}2}2,,2} has recursion level C(C(Ω23+C(Ω2+C(C(Ω22,C(Ω23,0)),σ),0),0),0)
• {1,,1{1,,2,,2}1{1,,1,,2}2{1◊1{1◊1,,2}2}2,,2} has recursion level C(C(Ω23+C(Ω2+C(C(Ω22+C(Ω2+C(Ω2,σ),0),C(Ω23,0)),σ),0),0),0)

## 9 thoughts on “sDAN vs. Taranovsky’s ordinal notation”

1. Can you tell me how much is C(C(W_2*2+C(W_2*2,0),0),0) roughly in SAN?
I don’t find it.

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• Aarex Tiaokhiao says:

I guess he didn’t reach yet. He is at C(C(C(Ω_2,Ω_2)+C(Ω_2+C(Ω_2,C(C(Ω_2,Ω_2),C(C(Ω_2,Ω_2)))),0)+C(Ω_2+C(C(Ω_2,Ω_2),C(C(Ω_2,Ω_2))),0),0),0) for now.

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• Aarex Tiaokhiao says:

I don’t know.

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2. What is the fixed point of α in C(C(Ω_2*2,α),0)?
Isn’t C(C(Ω_2+1,0),0) equal to the limit of C(C(Ω_2,C(Ω_2,C(Ω_2,C(Ω_2,……)))),0)?

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• And doesn’t the same apply to C(C(Ω_2*2+1,0),0)

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• Aarex Tiaokhiao says:

It is C(C(Ω_2*2+1,0),0) = C(C(C(0,C(Ω_2,Ω_2)),0),0).

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3. Aarex Tiaokhiao says:

And the limit of C(C(Ω_2*2+C(Ω_2+C(C(Ω_2*2+C(Ω_2+C(C(…),C(Ω_2,C(Ω_2,C(Ω_2*2)))))),C(Ω_2,C(Ω_2,C(Ω_2*2))))))) is approx {1{1{1,,,3}1{1{1{1,,,3}1{1{1,,,3}3}2,,,2}2{1,,,3}2}1{1,,,2}2{1{1,,,3}3}2,,,2}2,,,2}.

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4. Aarex Tiaokhiao says:

So the limit of “…,, is C(C(Ω_2*2+C(Ω_2+C(C(Ω_2+1,C(Ω_2*2,0)),C(Ω_2,C(Ω_2*2,0))),0),0),0)

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