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,,2{1,,`1`,,1♦2,,}2}1{1,,1,,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(Ω22+C(Ω2+,C(Ω2+C(C(Ω22+C(Ω2++,0),0),σ+),0)),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+C(Ω22+C(Ω2++1,C(Ω2+C(C(Ω22+C(Ω2++,0),0),σ+),0)),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+C(Ω22+C(Ω2++σ,C(Ω2+C(C(Ω22+C(Ω2++,0),0),σ+),0)),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(Ω22+C(Ω2+C(C(Ω22+1,0),σ+),C(Ω2+C(C(Ω22+C(Ω2++,0),0),σ+),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`,,1♦2,,}3}2{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,,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♦2,,}1,,1,2} has recursion level C(C(Ω22+C(Ω2++,0)+C(Ω2+C(C(Ω22+C(Ω2++,0),0),σ+)+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,,2{1{1,,`1`,,1♦2,,}2■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`,,1♦2,,}2■2,,}2} has recursion level C(C(Ω22+C(Ω2+++,0)+C(Ω2+C(C(Ω22+C(Ω2++1,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,,}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++,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♦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`,,1♦2,,}2■2,,}3}2{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(C(Ω22+C(Ω2+C(Ω2,σ)+1,C(Ω2+C(C(Ω22+C(Ω2+C(Ω22),0),0),σ2),0)),0),σ),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,,`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),0)+C(Ω2+C(C(Ω22+C(Ω2+C(Ω22),0)+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),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)))

Beyond C(Ω22+C(Ω22,0),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 ♥ = {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♥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♥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♥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♥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♥2}2}2{1♥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♥2}2}1,2{1♥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♥2}2}1{1,,2{1,,1,,2}2}2{1♥2}2} has recursion level C(C(Ω22+σ+C(Ω2,C(Ω2+σ,0)),0),0)
  • {1,,1{1,,1{1♥2}2}1{1,,1{1,,1,,2}1,2}2{1♥2}2} has recursion level C(C(Ω22+σ+C(Ω2+σ+1,0),0),0)
  • {1,,1{1,,1{1♥2}2}1{1,,1{1,,1{1,,2,,2}1,2,,2}2}2{1♥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♥2}2}1{1,,1{1♥2}2}2{1♥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♥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♥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♥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♥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{1,,1,,2}2,,2}2,,1{1,,2,,2}1{1{1♥2}2,,1,,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,,3}2,,2}2,,1,,2}2,,2}2,,1{1,,2,,2}1{1{1♥2}2,,1,,2}2,,2} has recursion level C(C(Ω22+σ+C(Ω2+C(C(Ω22+σ,0)+C(Ω22+C(σ,C(Ω2+C(Ω2,σ),0)),0),σ),0),0),0)
  • {1{1,,1{1,,2,,2}1{1{1,,1,,1,2}2,,1,,2}2,,2}2,,1{1,,2,,2}1{1{1♥2}2,,1,,2}2,,2} has recursion level C(C(Ω22+σ+C(Ω2+C(C(Ω22+σ,0)+C(Ω22+C(Ω2+C(Ω2,σ)+1,0),0),σ),0),0),0)
  • {1{1,,1{1,,2,,2}1{1{1{1,,`1{1,,`1,,`2,,}2,,}2}2,,1,,2}2,,2}2,,1{1,,2,,2}1{1{1♥2}2,,1,,2}2,,2} has recursion level C(C(Ω22+σ+C(Ω2+C(C(Ω22+σ,0)+C(Ω22+C(Ω2+C(Ω22,σ),0),0),σ),0),0),0)
  • {1{1,,1{1,,2,,2}1{1{1♥2}2,,1,,2}2,,2}2,,1{1,,2,,2}1{1{1♥2}2,,1,,2}2,,2} has recursion level C(C(Ω22+σ+C(Ω2+C(C(Ω22+σ,0)2,σ),0),0),0)
  • {1,,2{1,,2,,2}1{1{1♥2}2,,1,,2}2,,2} has recursion level C(C(Ω22+σ+C(Ω2+C(C(Ω2,C(Ω22+σ,0)),σ),0),0),0)
  • {1,,1{1,,2,,2}1{1{1♥2}2,,1,,2}1,2,,2} has recursion level C(C(Ω22+σ+C(Ω2+C(C(Ω22+σ+1,0),σ),0),0),0)
  • {1,,1{1,,2,,2}1{1{1♥2}2,,1,,2}1{1,,1,,2}2,,2} has recursion level C(C(Ω22+σ+C(Ω2+C(Ω2,σ),0),0),0)
  • {1{1,,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♥2} has recursion level C(C(Ω22+σ+C(Ω2+C(C(Ω22,σ),C(Ω2,σ)),0),0),0)
  • {1{1{1,,`1`,,1{1♥2}2,,}2}2♥2} has recursion level C(C(Ω22+σ+C(Ω2+C(C(Ω22+σ,0),C(Ω2,σ)),0),0),0)
  • {1{1{1,,`1`,,1,,2,,}2}2♥2} has recursion level C(C(Ω22+σ+C(Ω2+C(Ω2,C(Ω2,σ)),0),0),0)
  • {1{1{1,,`1` `,,2,,}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♥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♥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♥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♥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♥2} has recursion level C(C(Ω22+σ+C(Ω2+C(Ω22+1,0),0),0),0)
  • {1{1♥2}2♥2} has recursion level C(C(Ω22+σ+C(Ω2+C(Ω22+σ,0),0),0),0)
  • {1,,1,,2♥2} has recursion level C(C(Ω22+σ+C(σ,C(Ω2+C(Ω22+σ,0),0)),0),0)
  • {1,,1{1,,2,,2♥2}1,2,,2♥2} has recursion level C(C(Ω22+σ+C(Ω2+C(C(Ω22+1,0),σ),C(Ω2+C(Ω22+σ,0),0)),0),0)
  • {1,,1{1,,2,,2♥2}1{1,,1,,2♥2}2,,2♥2} has recursion level C(C(Ω22+σ+C(Ω2+C(C(Ω22+C(Ω2+C(Ω2,σ),C(Ω2+C(Ω22+σ,0),0)),0),σ),C(Ω2+C(Ω22+σ,0),0)),0),0)
  • {1,,1,,1,2♥2} has recursion level C(C(Ω22+σ+C(Ω2+C(C(Ω22+C(Ω2+C(Ω2,σ)+1,C(Ω2+C(Ω22+σ,0),0)),0),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,,}2♥2} has recursion level C(C(Ω22+σ+C(Ω2+C(C(Ω22+C(Ω2+C(Ω22+1,0),C(Ω2+C(Ω22+σ,0),0)),0),σ),C(Ω2+C(Ω22+σ,0),0)),0),0)
  • {1♥3} has recursion level C(C(Ω22+σ+C(Ω2+C(C(Ω22+σ,0),σ),C(Ω2+C(Ω22+σ,0),0)),0),0)
  • {1{1♥3}2♥3} has recursion level C(C(Ω22+σ+C(Ω2+C(Ω22+σ,0),C(Ω2+C(Ω22+σ,0),0)),0),0)
  • {1♥1,2} has recursion level C(C(Ω22+σ+C(Ω2+C(Ω22+σ,0)+1,0),0),0)
  • {1♥1,,1,2} has recursion level C(C(Ω22+σ+C(Ω2+C(Ω22+σ,0)+C(Ω2,σ)+1,0),0),0)
  • {1♥1{1,,`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(C(Ω22+1,0),C(Ω2,σ)),0),0),0)
  • {1♥1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1{1♥2}2{1,,`1,,`2,,}2,,}2} has recursion level C(C(Ω22+σ+C(Ω2+C(Ω22+σ,0)+C(C(Ω22+C(Ω2+C(Ω22+σ,0),0),0),C(Ω2,σ)),0),0),0)
  • {1♥1{1,,`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(C(Ω22+σ,0),C(Ω2,σ)),0),0),0) (this time, the σ in C(Ω2+C(Ω22+σ,0)+C(C(Ω22+σ,0),C(Ω2,σ)),0) works as a 1-separator)
  • {1♥1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1 {1♥1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}2} 2{1,,`1,,`2,,}2,,}1,2} has recursion level C(C(Ω22+σ+C(Ω2+C(Ω22+σ,0)+C(C(Ω22+σ,0),C(Ω2,σ))+1,0),0),0)
  • {1♥1{2{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1 {1♥1{1,,`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+σ+C(Ω2+C(Ω22+σ,0)+ω^(C(C(Ω22+σ,0),C(Ω2,σ))+1),0),0),0)
  • {1♥1{1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1 {1♥1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}2} 2{1,,`1,,`2,,}2,,}1,2,,}2} has recursion level C(C(Ω22+σ+C(Ω2+C(Ω22+σ,0)+C(C(Ω22+σ,0)+1,C(Ω2,σ)),0),0),0)
  • {1♥1{1,,`2{1,,`1{1,,`1,,`2,,}2,,}1 {1♥1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}2} 2{1,,`1,,`2,,}2,,}2} has recursion level C(C(Ω22+σ+C(Ω2+C(Ω22+σ,0)+C(C(Ω2,C(Ω22+σ,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,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}2} 2{1,,`1,,`2,,}2,,}2} has recursion level C(C(Ω22+σ+C(Ω2+C(Ω22+σ,0)+C(C(Ω22,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{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}2} 1,2{1,,`1,,`2,,}2,,}2} has recursion level C(C(Ω22+σ+C(Ω2+C(Ω22+σ,0)+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{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}2}1 {1♥1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}2} 2{1,,`1,,`2,,}2,,}2} has recursion level C(C(Ω22+σ+C(Ω2+C(Ω22+σ,0)+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} has recursion level C(C(Ω22+σ+C(Ω2+C(Ω22+σ,0)+C(Ω2,C(Ω2,σ))+1,0),0),0) (however, the extra 1-separator doesn’t affect the strength much)
  • {1♥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+1,0),σ),0),0),0)
  • {1♥1♥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{2♥2,,}2} has recursion level C(C(Ω22+σ+C(Ω2+C(Ω22+σ,0)+ω^(C(Ω2+C(Ω22+σ,0),σ)+1),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+σ+C(Ω2+C(Ω22+σ,0)+C(C(Ω22+1,0),C(Ω2+C(Ω22+σ,0),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+σ+C(Ω2+C(Ω22+σ,0)+C(C(Ω22+σ,0),C(Ω2+C(Ω22+σ,0),σ)),0),0),0)
  • {1{1{1,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}2♥2,,}1,2} has recursion level C(C(Ω22+σ+C(Ω2+C(Ω22+σ,0)+C(Ω2,C(Ω2+C(Ω22+σ,0),σ))+1,0),0),0)
  • {1♥1,2,,} has recursion level C(C(Ω22+σ+C(Ω2+C(Ω22+σ,0)+C(Ω2+C(Ω22+σ,0)+1,σ),0),0),0)
  • {1♥1`,,2,,} has recursion level C(C(Ω22+σ+C(Ω2+C(Ω22+σ,0)+C(Ω22,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)

Now let separator ◊ = {1{1,,`1,,2,,}2,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,} and ♦ = {1,,1{1,,1{1,,1,,2◊2}2◊2}1{1,,2◊2}2{1,,1,,2◊2}2◊2} and ♥ = {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♥2}2,,1{1,,1,,2}2}1,2{1,,1,,2}2} has recursion level C(C(Ω22+C(Ω2,σ)+1,0),0)
  • {1,,1{1{1♥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(Ω22+C(Ω2,σ)+C(C(Ω22+C(Ω2,σ),0),C(Ω2+σ,0)),0),0)
  • {1,,1{2{1,,1{1♥2}2}2,,1{1,,1,,2}2}2{1,,1,,2}2} has recursion level C(C(Ω22+C(Ω2,σ)+ω^(C(C(Ω22+C(Ω2,σ),0),C(Ω2+σ,0))+1),0),0)
  • {1,,2{1♥2}2} has recursion level C(C(Ω22+C(Ω2,σ)+C(C(Ω2,C(Ω22+C(Ω2,σ),0)),C(Ω2+σ,0)),0),0)
  • {1,,1{1,,1{1♥2}2}1,2{1♥2}2} has recursion level C(C(Ω22+C(Ω2,σ)+C(C(Ω22+C(Ω2,σ)+1,0),C(Ω2+σ,0)),0),0)
  • {1,,1{1,,1{1♥2}2}1{1,,2{1,,1,,2}2}2{1♥2}2} has recursion level C(C(Ω22+C(Ω2,σ)+C(Ω2,C(Ω2+σ,0)),0),0)
  • {1,,1{1,,1{1♥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{1♥2}2} has recursion level C(C(Ω22+C(Ω2,σ)+C(Ω2+C(C(Ω22+σ,0),σ),0),0),0)
  • {1,,1{1,,1{1♥2}2}1{1,,1{1♥2}2}2{1♥2}2} has recursion level C(C(Ω22+C(Ω2,σ)+C(Ω2+C(C(Ω22+C(Ω2,σ),0),σ),0),0),0)
  • {1,,1{1♥2}1,2} has recursion level C(C(Ω22+C(Ω2,σ)+C(Ω2+C(C(Ω22+C(Ω2,σ),0),σ)+1,0),0),0)
  • {1{1,,1,,2}2♥2} has recursion level C(C(Ω22+C(Ω2,σ)+C(Ω2+ω^(C(C(Ω22+C(Ω2,σ),0),σ)+1),0),0),0)
  • {1{1,,1,,1,2}2♥2} has recursion level C(C(Ω22+C(Ω2,σ)+C(Ω2+C(Ω2,σ)+1,0),0),0)
  • {1{1{1,,`1`,,1{1♥2}2,,}2}2♥2} has recursion level C(C(Ω22+C(Ω2,σ)+C(Ω2+C(C(Ω22+C(Ω2,σ),0),C(Ω2,σ)),0),0),0)
  • {1{1{1,,`1`,,1,,2,,}2}2♥2} has recursion level C(C(Ω22+C(Ω2,σ)+C(Ω2+C(Ω2,C(Ω2,σ)),0),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♥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◊2}2,,}2,,`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,σ)+C(Ω2+C(Ω22+σ,0)+1,0),0),0)
  • {1 {1{1{1,,`1{1◊2}2,,}2,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1 {1,,`1{1,,`1{1,,`1,,`2,,}2,,}1 {1{1{1,,`1{1◊2}2,,}2,,`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} 2{1,,`1,,`2,,}2,,}2} 2♥2} has recursion level C(C(Ω22+C(Ω2,σ)+C(Ω2+C(Ω22+σ,0)+C(C(Ω22+σ,0),C(Ω2,σ)),0),0),0)
  • {1 {1{1{1,,`1{1◊2}2,,}2,,`1{1,,`1{1,,`1,,`2,,}2,,}1,,2{1,,`1,,`2,,}2,,}1 {1,,`1{1,,`1{1,,`1,,`2,,}2,,}1 {1{1{1,,`1{1◊2}2,,}2,,`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} 1,2{1,,`1,,`2,,}2,,}2} 2♥2} has recursion level C(C(Ω22+C(Ω2,σ)+C(Ω2+C(Ω22+σ,0)+C(C(Ω22+σ+1,0),C(Ω2,σ)),0),0),0)
  • {1 {1{1{1,,`1{1◊2}2,,}2,,`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,,`1{1◊2}2,,}2,,`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} 2{1,,`1,,`2,,}2,,}2} 2♥2} has recursion level C(C(Ω22+C(Ω2,σ)+C(Ω2+C(Ω22+σ,0)+C(C(Ω22+ω^(σ+1),0),C(Ω2,σ)),0),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)
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13 thoughts on “sDAN vs. Taranovsky’s ordinal notation

      • Alemagno12 says:

        According to my approximation method, C(Ω_2*2+C(Ω_2,C(Ω_2*2,0)),0)C(Ω_2+C(Ω_2,C(Ω_2*2,0)),C(Ω_2*2,0)) approximately corresponds to {1{1,,`1,,2^,,}2,,`1{1,,`1{1,,`1,,`2^,,}2^,,}1,,2{1,,`1,,`2^,,}2^,,}

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  1. 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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  2. 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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  3. 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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