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)

pDAN vs. Taranovsky’s ordinal notation

Here’re the comparisons between pDAN and Taranovsky’s ordinal notation. Due to its high complexity, a separate page is needed. The fundamental sequences of Taranovsky’s notation can be easily defined. 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\}}.

Just warm up

  • {1,,1{1,,1,,2}2} has recursion level C(C(Ω22,0),0)
  • {1,,2{1,,1,,2}2} has recursion level C(C(Ω2,C(Ω22,0)),0)
  • {1,,3{1,,1,,2}2} has recursion level C(C(Ω2,C(Ω2,C(Ω22,0))),0)
  • {1,,1,2{1,,1,,2}2} has recursion level C(C(Ω2+1,C(Ω22,0)),0)
  • {1,,1{1,,2}2{1,,1,,2}2} has recursion level C(C(Ω21,C(Ω22,0)),0)
  • {1,,1{1,,3}2{1,,1,,2}2} has recursion level C(C(Ω2+C(Ω21),C(Ω22,0)),0)
  • {1,,1{1,,4}2{1,,1,,2}2} has recursion level C(C(Ω2+C(Ω2,C(Ω21)),C(Ω22,0)),0)
  • {1,,1{1,,1,2}2{1,,1,,2}2} has recursion level C(C(Ω2+C(Ω2+1,0),C(Ω22,0)),0)
  • {1,,1{1,,1,3}2{1,,1,,2}2} has recursion level C(C(Ω2+C(Ω2+2,0),C(Ω22,0)),0)
  • {1,,1{1,,1{1,,2}2}2{1,,1,,2}2} has recursion level C(C(Ω2+C(Ω21,0),C(Ω22,0)),0)
  • {1,,1{1,,1{1,,1,2}2}2{1,,1,,2}2} has recursion level C(C(Ω2+C(Ω2+C(Ω2+1,0),0),C(Ω22,0)),0)
  • {1,,1{1,,1{1,,1{1,,1,2}2}2}2{1,,1,,2}2} has recursion level C(C(Ω2+C(Ω2+C(Ω2+C(Ω2+1,0),0),0),C(Ω22,0)),0)
  • {1,,1{1,,1{1,,1,,2}2}2{1,,1,,2}2} has recursion level C(C(Ω22,C(Ω22,0)),0) (note that C(C(Ω2+C(Ω22,0),C(Ω22,0)),0) is not a valid term)
  • {1,,2{1,,1{1,,1,,2}2}2{1,,1,,2}2} has recursion level C(C(Ω2,C(Ω22,C(Ω22,0))),0)
  • {1,,1{1,,1{1,,1,,2}2}3{1,,1,,2}2} has recursion level C(C(Ω22,C(Ω22,C(Ω22,0))),0)
  • {1,,1{1,,1{1,,1,,2}2}4{1,,1,,2}2} has recursion level C(C(Ω22,C(Ω22,C(Ω22,C(Ω22,0)))),0)
  • {1,,1{1,,1{1,,1,,2}2}1,2{1,,1,,2}2} has recursion level C(C(Ω22+1,0),0) (note that if α is standard, and it doesn’t have subterms over Ω22, then C(C(Ω22+α,0),0) is valid)
  • {1,,1{1,,1{1,,1,,2}2}1{1,,2}2{1,,1,,2}2} has recursion level C(C(Ω22+Ω1,0),0)
  • {1,,1{1,,1{1,,1,,2}2}1{1,,1,2}2{1,,1,,2}2} has recursion level C(C(Ω22+C(Ω2+1,0),0),0)
  • {1,,1{1,,1{1,,1,,2}2}1{1,,1{1,,1,2}2}2{1,,1,,2}2} has recursion level C(C(Ω22+C(Ω2+C(Ω2+1,0),0),0),0)
  • {1,,1{1,,1{1,,1,,2}2}1{1,,1{1,,1,,2}2}2{1,,1,,2}2} has recursion level C(C(Ω22+C(Ω2+C(Ω22,0),0),0),0)

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

Now let separator ■ = {1,,1,,2} and let ordinal a = C(Ω2+C(Ω22,0),0).

  • {1,,1{1,,1■2}2{1,,1■2}2■2} has recursion level C(C(Ω22,C(Ω22+a,0)),0)
  • {1,,1{1,,1■2}1{1,,1■2}3■2} has recursion level C(C(Ω22+a,C(Ω22+a,0)),0)
  • {1,,1{1,,1■2}1{1,,1■2}1,2■2} has recursion level C(C(Ω22+a+1,0),0)
  • {1,,1{1,,1■2}1{1,,1■2}1{1,,1■2}2■2} has recursion level C(C(Ω22+a2,0),0)
  • {1,,1{2,,1■2}2■2} has recursion level C(C(Ω22+ω^(a+1),0),0)
  • {1,,1{1{1,,1■2}2,,1■2}2■2} has recursion level C(C(Ω22+ω^(a2),0),0)
  • {1,,1{1{1,,1■2}1{1,,1■2}2,,1■2}2■2} has recursion level C(C(Ω22+ω^ω^(a2),0),0)
  • {1,,1{1{1{1,,1■2}2,,1■2}2,,1■2}2■2} has recursion level C(C(Ω22+ω^ω^ω^(a2),0),0)
  • {1,,1{1{1,,2■2}2,,1■2}2■2} has recursion level C(C(Ω22+εa+1,0),0) = C(C(Ω22+C(C(Ω2,a),a),0),0)
  • {1,,1{1{1,,2■2}3,,1■2}2■2} has recursion level C(C(Ω22+εa+2,0),0) = C(C(Ω22+C(C(Ω2,a),C(C(Ω2,a),a)),0),0)
  • {1,,1{1{1,,2■2}1,2,,1■2}2■2} has recursion level C(C(Ω22+C(C(Ω2,a)+1,a),0),0)
  • {1,,1{1{1,,2■2}1{1,,2■2}2,,1■2}2■2} has recursion level C(C(Ω22+C(C(Ω2,a)2,a),0),0)
  • {1,,1{1{2,,2■2}2,,1■2}2■2} has recursion level C(C(Ω22+C(ω^(C(Ω2,a)+1),a),0),0)
  • {1,,1{1{1{1,,2■2}2,,2■2}2,,1■2}2■2} has recursion level C(C(Ω22+C(ω^(C(Ω2,a)2),a),0),0)
  • {1,,1{1{1,,3■2}2,,1■2}2■2} has recursion level C(C(Ω22+C(C(Ω2,C(Ω2,a)),a),0),0)
  • {1,,1{1{1,,1,2■2}2,,1■2}2■2} has recursion level C(C(Ω22+C(C(Ω2+1,a),a),0),0)
  • {1,,1{1{1,,1{1,,1■2}2■2}2,,1■2}2■2} has recursion level C(C(Ω22+C(C(Ω2+a,a),a),0),0)
  • {1,,1{1{1,,1{1{1,,1{1,,1■2}2■2}2,,1■2}2■2}2,,1■2}2■2} has recursion level C(C(Ω22+C(C(Ω2+C(C(Ω2+a,a),a),a),a),0),0)
  • {1,,1{1{1,,1{1,,2■2}2■2}2,,1■2}2■2} has recursion level C(C(Ω22+C(C(Ω2+C(Ω2,a),a),a),0),0)

So {1,,1{1,,2■2}2■2} has recursion level C(C(Ω22+C(C(Ω2+C(Ω2,a),a),a),0),0). (The recursion levels of {1,,1{1,,2■2}2■2}, {1{1,,1{1,,2■2}2■2}2,,1■2}, {1,,1{1{1,,1{1,,2■2}2■2}2,,1■2}2■2}, {1{1,,1{1{1,,1{1,,2■2}2■2}2,,1■2}2■2}2,,1■2}, etc. are equal. Separator {1,,1 A 2■2} (where A is any separator) reduces to an expression containing {1,,2■2}, then the {1,,2■2} acquires a layer of { ____ ,,1■2}.)

  • {1,,1{1,,1■2}2{1{1,,1{1,,2■2}2■2}2,,1■2}2■2} has recursion level C(C(Ω22,C(Ω22+C(C(Ω2+C(Ω2,a),a),a),0)),0)
  • {1,,1{1,,1■2}1{1,,1■2}2{1{1,,1{1,,2■2}2■2}2,,1■2}2■2} has recursion level C(C(Ω22+a,C(Ω22+C(C(Ω2+C(Ω2,a),a),a),0)),0)
  • {1,,1{1{1,,2■2}2,,1■2}2{1{1,,1{1,,2■2}2■2}2,,1■2}2■2} has recursion level C(C(Ω22+C(C(Ω2,a),a),C(Ω22+C(C(Ω2+C(Ω2,a),a),a),0)),0)
  • {1,,1{1{1,,1{1,,1■2}2■2}2,,1■2}2{1{1,,1{1,,2■2}2■2}2,,1■2}2■2} has recursion level C(C(Ω22+C(C(Ω2+a,a),a),C(Ω22+C(C(Ω2+C(Ω2,a),a),a),0)),0)
  • {1,,1{1{1,,1{1,,2■2}2■2}2,,1■2}3■2} has recursion level C(C(Ω22+C(C(Ω2+C(Ω2,a),a),a),C(Ω22+C(C(Ω2+C(Ω2,a),a),a),0)),0)
  • {1,,1{1{1,,1{1,,2■2}2■2}2,,1■2}1,2■2} has recursion level C(C(Ω22+C(C(Ω2+C(Ω2,a),a),a)+1,0),0)
  • {1,,1{2{1,,1{1,,2■2}2■2}2,,1■2}2■2} has recursion level C(C(Ω22+ω^(C(C(Ω2+C(Ω2,a),a),a)+1),0),0)
  • {1,,1{1{1,,2■2}2{1,,1{1,,2■2}2■2}2,,1■2}2■2} has recursion level C(C(Ω22+C(C(Ω2,a),C(C(Ω2+C(Ω2,a),a),a)),0),0)
  • {1,,1{1{1,,1,2■2}2{1,,1{1,,2■2}2■2}2,,1■2}2■2} has recursion level C(C(Ω22+C(C(Ω2+1,a),C(C(Ω2+C(Ω2,a),a),a)),0),0)
  • {1,,1{1{1,,1{1{1,,1{1,,2■2}2■2}2,,1■2}2■2}2{1,,1{1,,2■2}2■2}2,,1■2}2■2} has recursion level C(C(Ω22+C(C(Ω2+C(C(Ω2+C(Ω2,a),a),a),a),C(C(Ω2+C(Ω2,a),a),a)),0),0)
  • {1,,1{1{1,,1{1,,2■2}2■2}3,,1■2}2■2} has recursion level C(C(Ω22+C(C(Ω2+C(Ω2,a),a),C(C(Ω2+C(Ω2,a),a),a)),0),0)
  • {1,,1{1{1,,1{1,,2■2}2■2}1,2,,1■2}2■2} has recursion level C(C(Ω22+C(C(Ω2+C(Ω2,a),a)+1,a),0),0)
  • {1,,1{1{1,,1{1,,2■2}2■2}1{1,,1■2}2,,1■2}2■2} has recursion level C(C(Ω22+C(C(Ω2+C(Ω2,a),a)+a,a),0),0)
  • {1,,1{1{1,,1{1,,2■2}2■2}1{1{1,,1{1,,2■2}2■2}2,,1■2}2,,1■2}2■2} has recursion level C(C(Ω22+C(C(Ω2+C(Ω2,a),a)+C(C(Ω2+C(Ω2,a),a),a),a),0),0)
  • {1,,1{1{1,,1{1,,2■2}2■2}1{1,,2■2}2,,1■2}2■2} has recursion level C(C(Ω22+C(C(Ω2+C(Ω2,a),a)+C(Ω2,a),a),0),0)
  • {1,,1{1{1,,1{1,,2■2}2■2}1{1{1,,3■2}2,,2■2}2,,1■2}2■2} has recursion level C(C(Ω22+C(C(Ω2+C(Ω2,a),a)+C(C(Ω2,C(Ω2,a)),C(Ω2,a)),a),0),0)
  • {1,,1{1{1,,1{1,,2■2}2■2}1{1,,3■2}2,,1■2}2■2} has recursion level C(C(Ω22+C(C(Ω2+C(Ω2,a),a)+C(C(Ω2,C(Ω2,a)),C(Ω2,a)),a),0),0)
  • {1,,1{1{1,,1{1,,2■2}2■2}1{1{1,,4■2}2,,3■2}2,,1■2}2■2} has recursion level C(C(Ω22+C(C(Ω2+C(Ω2,a),a)+C(C(Ω2,C(Ω2,C(Ω2,a))),C(Ω2,a)),a),0),0)
  • {1,,1{1{1,,1{1,,2■2}2■2}1{1,,4■2}2,,1■2}2■2} has recursion level C(C(Ω22+C(C(Ω2+C(Ω2,a),a)+C(C(Ω2,C(Ω2,C(Ω2,a))),C(Ω2,a)),a),0),0) (when solving this term, by the adding rule, the {1,,4■2} will reduce to {1{1,,4■2}2,,3■2}, then {1{1{1,,4■2}2,,3■2}2,,2■2})
  • {1,,1{1{1,,1{1,,2■2}2■2}1{1,,1,2■2}2,,1■2}2■2} has recursion level C(C(Ω22+C(C(Ω2+C(Ω2,a),a)+C(C(Ω2+1,a),C(Ω2,a)),a),0),0)
  • {1,,1{1{1,,1{1,,2■2}2■2}1{1,,1{1,,1■2}2■2}2,,1■2}2■2} has recursion level C(C(Ω22+C(C(Ω2+C(Ω2,a),a)+C(C(Ω2+a,a),C(Ω2,a)),a),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■2}2■2}2,,1■2}2■2} has recursion level C(C(Ω22+C(C(Ω2+C(Ω2,a),a)+C(C(Ω2+C(C(Ω2+a,a),a),a),C(Ω2,a)),a),0),0)
  • {1,,1{1{1,,1{1,,2■2}2■2}1{1,,1{1{1,,1{1,,2■2}2■2}2,,1■2}2■2}2,,1■2}2■2} has recursion level C(C(Ω22+C(C(Ω2+C(Ω2,a),a)+C(C(Ω2+C(C(Ω2+C(Ω2,a),a),a),a),C(Ω2,a)),a),0),0)
  • {1,,1{1{1,,1{1,,2■2}2■2}1{1,,1{1,,2■2}2■2}2,,1■2}2■2} has recursion level C(C(Ω22+C(C(Ω2+C(Ω2,a),a)+C(C(Ω2+C(Ω2,a),a),C(Ω2,a)),a),0),0)
  • {1,,1{1{1,,1{1,,2■2}2■2}1{1,,1{1,,2■2}2■2}1,2,,1■2}2■2} has recursion level C(C(Ω22+C(C(Ω2+C(Ω2,a),a)+C(C(Ω2+C(Ω2,a),a),C(Ω2,a))+1,a),0),0)
  • {1,,1{1{1,,1{1,,2■2}2■2}1{1,,1{1,,2■2}2■2}1{1,,1{1,,2■2}2■2}2,,1■2}2■2} has recursion level C(C(Ω22+C(C(Ω2+C(Ω2,a),a)+C(C(Ω2+C(Ω2,a),a),C(Ω2,a))2,a),0),0)
  • {1,,1{1{2,,1{1,,2■2}2■2}2,,1■2}2■2} has recursion level C(C(Ω22+C(C(Ω2+C(Ω2,a),a)+ω^(C(C(Ω2+C(Ω2,a),a),C(Ω2,a))+1),a),0),0)
  • {1,,1{1{1{1,,1■2}2,,1{1,,2■2}2■2}2,,1■2}2■2} has recursion level C(C(Ω22+C(C(Ω2+C(Ω2,a),a)+ω^(C(C(Ω2+C(Ω2,a),a),C(Ω2,a))+a),a),0),0)
  • {1,,1{1{1{1,,2■2}2,,1{1,,2■2}2■2}2,,1■2}2■2} has recursion level C(C(Ω22+C(C(Ω2+C(Ω2,a),a)+ω^(C(C(Ω2+C(Ω2,a),a),C(Ω2,a))+C(Ω2,a)),a),0),0)

So {1{1,,2■2}2,,1{1,,2■2}2■2} has recursion level C(C(Ω22+C(C(Ω2+C(Ω2,a),a)+ω^(C(C(Ω2+C(Ω2,a),a),C(Ω2,a))+C(Ω2,a)),a),0),0). (the recursion levels of {1{1{1,,2■2}2,,1{1,,2■2}2■2}2,,1■2}, {1,,1{1{1{1,,2■2}2,,1{1,,2■2}2■2}2,,1■2}2■2} and {1{1,,1{1{1{1,,2■2}2,,1{1,,2■2}2■2}2,,1■2}2■2}2,,1■2} are equal)

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

So {1{1,,3■2}2,,1{1,,2■2}2■2} has recursion level C(C(Ω22+C(C(Ω2+C(Ω2,a),a)+ω^(C(C(Ω2+C(Ω2,a),a),C(Ω2,C(Ω2,a)))+C(Ω2,C(Ω2,a))),a),0),0). And further, the recursion level of {1{1,,k■2}2,,1{1,,2■2}2■2} is between C(C(Ω22+C(C(Ω2+C(Ω2,a),a)+C(Ω2,C(Ω2,…C(Ω2,a))),a),0),0) (with k-1 Ω2′s in the blue part) and C(C(Ω22+C(C(Ω2+C(Ω2,a),a)+C(Ω2,C(Ω2,…C(Ω2,a))),a),0),0) (with k Ω2′s in the blue part), so {1{1,,1,2■2}2,,1{1,,2■2}2■2} has recursion level C(C(Ω22+C(C(Ω2+C(Ω2,a),a)+C(Ω2+1,a),a),0),0).

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

So {1,,1{1,,1,,2}3} has recursion level C(C(Ω22+C(C(Ω22,0),C(Ω2+C(Ω22,0),0)),0),0). Here’re some approximations, to make the comparisons above more simple:

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

Up to {1,,2,,2}

Still, let separator ■ = {1,,1,,2} and let ordinal a = C(Ω2+C(Ω22,0),0).

  • {1,,1{1{1,,1■3}2,,1■2}2■2} has recursion level C(C(Ω22+C(C(Ω22,0),a),0),0) (the recursion levels of {1,,1■3}, {1,,1{1,,1■3}2■2}, {1{1,,1■3}2,,1■2}, {1{1,,1{1,,1■3}2■2}2,,1■2}, {1,,1{1{1,,1{1,,1■3}2■2}2,,1■2}2■2}, etc. are equal)
  • {1,,1{1{1,,1■3}2,,1■2}1,2■2} has recursion level C(C(Ω22+C(C(Ω22,0),a)+1,0),0)
  • {1,,1{1{1,,1■3}2,,1■2}1{1,,1■2}2■2} has recursion level C(C(Ω22+C(C(Ω22,0),a)+a,0),0)
  • {1,,1{1{1,,1■3}2,,1■2}1{1{1,,1,2■2}2,,1■2}2■2} has recursion level C(C(Ω22+C(C(Ω22,0),a)+C(C(Ω2+1,a),a),0),0)
  • {1,,1{1{1,,1■3}2,,1■2}1{1{1,,1{1,,1,2■2}2■2}2,,1■2}2■2} has recursion level C(C(Ω22+C(C(Ω22,0),a)+C(C(Ω2+C(Ω2+1,a),a),a),0),0)
  • {1,,1{1{1,,1■3}2,,1■2}1{1{1,,1■3}2,,1■2}2■2} has recursion level C(C(Ω22+C(C(Ω22,0),a)2,0),0)
  • {1,,1{2{1,,1■3}2,,1■2}2■2} has recursion level C(C(Ω22+ω^(C(C(Ω22,0),a)+1),0),0)
  • {1,,1{1{1,,2■2}2{1,,1■3}2,,1■2}2■2} has recursion level C(C(Ω22+C(C(Ω2,a),C(C(Ω22,0),a)),0),0)
  • {1,,1{1{1,,3■2}2{1,,1■3}2,,1■2}2■2} has recursion level C(C(Ω22+C(C(Ω2,C(Ω2,a)),C(C(Ω22,0),a)),0),0)
  • {1,,1{1{1,,1{1,,1■2}2■2}2{1,,1■3}2,,1■2}2■2} has recursion level C(C(Ω22+C(C(Ω2+a,a),C(C(Ω22,0),a)),0),0)
  • {1,,1{1{1,,1{1,,2■2}2■2}2{1,,1■3}2,,1■2}2■2} has recursion level C(C(Ω22+C(C(Ω2+C(Ω2,a),a),C(C(Ω22,0),a)),0),0)
  • {1,,1{1{1,,1{1,,1{1,,1■2}2■2}2■2}2{1,,1■3}2,,1■2}2■2} has recursion level C(C(Ω22+C(C(Ω2+C(Ω2+a,a),a),C(C(Ω22,0),a)),0),0)
  • {1,,1{1{1,,1■3}3,,1■2}2■2} has recursion level C(C(Ω22+C(C(Ω22,0),C(C(Ω22,0),a)),0),0)
  • {1,,1{1{1,,1■3}1,2,,1■2}2■2} has recursion level C(C(Ω22+C(C(Ω22,0)+1,a),0),0)
  • {1,,1{1{1,,1■3}1{1,,2■2}2,,1■2}2■2} has recursion level C(C(Ω22+C(C(Ω22,0)+C(Ω2,a),a),0),0)
  • {1,,1{1{1,,1■3}1{1,,1{1,,1■2}2■2}2,,1■2}2■2} has recursion level C(C(Ω22+C(C(Ω22,0)+C(C(Ω2+a,a),C(Ω2,a)),a),0),0)
  • {1,,1{1{1,,1■3}1{1,,1{1,,1{1,,1■2}2■2}2■2}2,,1■2}2■2} has recursion level C(C(Ω22+C(C(Ω22,0)+C(C(Ω2+C(Ω2+a,a),a),C(Ω2,a)),a),0),0)
  • {1,,1{1{1,,1■3}1{1,,1■3}2,,1■2}2■2} has recursion level C(C(Ω22+C(C(Ω22,0)+C(C(Ω22,0),C(Ω2,a)),a),0),0)
  • {1,,1{1{2,,1■3}2,,1■2}2■2} has recursion level C(C(Ω22+C(C(Ω22,0)+ω^(C(C(Ω22,0),C(Ω2,a))+1),a),0),0)
  • {1,,1{1{1{1,,2■2}2,,1■3}2,,1■2}2■2} has recursion level C(C(Ω22+C(C(Ω22,0)+ω^(C(C(Ω22,0),C(Ω2,a))+C(Ω2,a)),a),0),0)
  • {1,,1{1{1{1,,1■3}2,,2■2}2,,1■2}2■2} has recursion level C(C(Ω22+C(C(Ω22,0),a),0),0)
  • {1,,1{1{1{1,,1■3}2,,2■2}1,2,,1■2}2■2} has recursion level C(C(Ω22+C(C(Ω22,0)+1,a),0),0)
  • {1,,1{1{1{1,,1■3}2,,2■2}1{1{1,,1■3}2,,2■2}2,,1■2}2■2} has recursion level C(C(Ω22+C(C(Ω22,0)+C(C(Ω22,0),C(Ω2,a)),a),0),0)
  • {1,,1{1{2{1,,1■3}2,,2■2}2,,1■2}2■2} has recursion level C(C(Ω22+C(C(Ω22,0)+ω^(C(C(Ω22,0),C(Ω2,a))+1),a),0),0)
  • {1,,1{1{1{1,,3■2}2{1,,1■3}2,,2■2}2,,1■2}2■2} has recursion level C(C(Ω22+C(C(Ω22,0)+C(C(Ω2,C(Ω2,a)),C(C(Ω22,0),C(Ω2,a))),a),0),0)
  • {1,,1{1{1{1,,1{1,,1■2}2■2}2{1,,1■3}2,,2■2}2,,1■2}2■2} has recursion level C(C(Ω22+C(C(Ω22,0)+C(C(Ω2+a,a),C(C(Ω22,0),C(Ω2,a))),a),0),0)
  • {1,,1{1{1{1,,1■3}3,,2■2}2,,1■2}2■2} has recursion level C(C(Ω22+C(C(Ω22,0)+C(C(Ω22,0),C(C(Ω22,0),C(Ω2,a))),a),0),0)
  • {1,,1{1{1{1,,1■3}1,2,,2■2}2,,1■2}2■2} has recursion level C(C(Ω22+C(C(Ω22,0)+C(C(Ω22,0)+1,C(Ω2,a)),a),0),0)
  • {1,,1{1{1{1,,1■3}1{1,,3■2}2,,2■2}2,,1■2}2■2} has recursion level C(C(Ω22+C(C(Ω22,0)+C(Ω2,C(Ω2,a)),a),0),0)
  • {1,,1{1{1{1,,1■3}1{1,,3■2}1,2,,2■2}2,,1■2}2■2} has recursion level C(C(Ω22+C(C(Ω22,0)+C(Ω2,C(Ω2,a))+1,a),0),0)
  • {1,,1{1{1{1,,1■3}1{2,,3■2}2,,2■2}2,,1■2}2■2} has recursion level C(C(Ω22+C(C(Ω22,0)+ω^(C(Ω2,C(Ω2,a))+1),a),0),0)
  • {1,,1{1{1{1,,1■3}1{1,,1■3}2,,2■2}2,,1■2}2■2} has recursion level C(C(Ω22+C(C(Ω22,0)+C(C(Ω22,0),C(Ω2,C(Ω2,a))),a),0),0)
  • {1,,1{1{1{2,,1■3}2,,2■2}2,,1■2}2■2} has recursion level C(C(Ω22+C(C(Ω22,0)+ω^(C(C(Ω22,0),C(Ω2,C(Ω2,a)))+1),a),0),0)
  • {1,,1{1{1{1{1,,3■2}2,,1■3}2,,2■2}2,,1■2}2■2} has recursion level C(C(Ω22+C(C(Ω22,0)+ω^(C(C(Ω22,0),C(Ω2,C(Ω2,a)))+C(Ω2,C(Ω2,a))),a),0),0)

So {1,,1{1{1{1,,3■2}2,,1■3}2,,1■2}2■2} has recursion level C(C(Ω22+C(C(Ω22,0)+ω^(C(C(Ω22,0),C(Ω2,C(Ω2,a)))+C(Ω2,C(Ω2,a))),a),0),0). And further, the recursion level of {1,,1{1{1{1,,k■2}2,,1■3}2,,1■2}2■2} is between C(C(Ω22+C(C(Ω22,0)+C(Ω2,C(Ω2,…C(Ω2,a))),a),0),0) (with k-1 Ω2′s in the blue part) and C(C(Ω22+C(C(Ω22,0)+C(Ω2,C(Ω2,…C(Ω2,a))),a),0),0) (with k Ω2′s in the blue part), so {1,,1{1{1{1,,1,2■2}2,,1■3}2,,1■2}2■2} has recursion level C(C(Ω22+C(C(Ω22,0)+C(Ω2+1,a),a),0),0).

  • {1,,1{1{1{1,,1{1,,1■2}2■2}2,,1■3}2,,1■2}2■2} has recursion level C(C(Ω22+C(C(Ω22,0)+C(Ω2+a,a),a),0),0)
  • {1{1,,1{1,,2■2}2■2}2,,1■3} has recursion level C(C(Ω22+C(C(Ω22,0)+C(Ω2+C(Ω2,a),a),a),0),0)
  • {1{1,,1{1,,1{1,,1■2}2■2}2■2}2,,1■3} has recursion level C(C(Ω22+C(C(Ω22,0)+C(Ω2+C(Ω2+a,a),a),a),0),0)
  • {1{1,,1■3}2,,1■3} has recursion level C(C(Ω22+C(C(Ω22,0)2,a),0),0)
  • {1,,1{1{1{1,,1,2■2}2{1,,1■3}2,,1■3}2,,1■2}2■2} has recursion level C(C(Ω22+C(C(Ω22,0)2+C(Ω2+1,a),a),0),0)
  • {1{1,,1■3}3,,1■3} has recursion level C(C(Ω22+C(C(Ω22,0)3,a),0),0)
  • {1{1,,1■3}1,2,,1■3} has recursion level C(C(Ω22+C(ω^(C(Ω22,0)+1),a),0),0)
  • {1{1,,1■3}1{1,,1■3}2,,1■3} has recursion level C(C(Ω22+C(ω^(C(Ω22,0)2),a),0),0)
  • {1{2,,1■3}2,,1■3} has recursion level C(C(Ω22+C(ω^ω^(C(Ω22,0)+1),a),0),0)
  • {1{1,,2■3}2,,1■3} has recursion level C(C(Ω22+C(C(Ω2,C(Ω22,0)),a),0),0)
  • also, {1,,2■3} has recursion level C(C(Ω22+C(C(Ω2,C(Ω22,0)),a),0),0)
  • {1,,1{1{1{1,,1,2■2}2{1,,2■3}2,,1■3}2,,1■2}2■2} has recursion level C(C(Ω22+C(C(Ω2,C(Ω22,0))+C(Ω2+1,a),a),0),0)
  • {1{1,,1■3}2{1,,2■3}2,,1■3} has recursion level C(C(Ω22+C(C(Ω2,C(Ω22,0))+C(Ω22,0),a),0),0)
  • {1{1{1,,2■3}2,,1■3}2{1,,2■3}2,,1■3} has recursion level C(C(Ω22+C(C(Ω2,C(Ω22,0))+C(C(Ω2,C(Ω22,0)),C(Ω22,0)),a),0),0)
  • {1{1,,2■3}3,,1■3} has recursion level C(C(Ω22+C(C(Ω2,C(Ω22,0))2,a),0),0)
  • {1{1,,2■3}1,2,,1■3} has recursion level C(C(Ω22+C(ω^(C(Ω2,C(Ω22,0))+1),a),0),0)
  • {1{2,,2■3}2,,1■3} has recursion level C(C(Ω22+C(ω^ω^(C(Ω2,C(Ω22,0))+1),a),0),0)
  • {1,,3■3} has recursion level C(C(Ω22+C(C(Ω2,C(Ω2,C(Ω22,0))),a),0),0)
  • {1,,1,2■3} has recursion level C(C(Ω22+C(C(Ω2+1,C(Ω22,0)),a),0),0)
  • {1,,1{1,,1■2}2■3} has recursion level C(C(Ω22+C(C(Ω2+a,C(Ω22,0)),a),0),0)
  • {1,,1{1,,2■2}2■3} has recursion level C(C(Ω22+C(C(Ω2+C(Ω2,a),C(Ω22,0)),a),0),0)
  • {1,,1{1,,1{1,,1■2}2■2}2■3} has recursion level C(C(Ω22+C(C(Ω2+C(Ω2+a,a),C(Ω22,0)),a),0),0)
  • {1,,1{1,,1■3}2■3} has recursion level C(C(Ω22+C(C(Ω22,C(Ω22,0)),a),0),0) (here again, C(C(Ω2+C(Ω22,0),C(Ω22,0)),a) is not a valid term)
  • {1,,2{1,,1■3}2■3} has recursion level C(C(Ω22+C(C(Ω2,C(Ω22,C(Ω22,0))),a),0),0)
  • {1,,1{1,,1■3}3■3} has recursion level C(C(Ω22+C(C(Ω22,C(Ω22,C(Ω22,0))),a),0),0)
  • {1,,1{1,,1■3}1,2■3} has recursion level C(C(Ω22+C(C(Ω22+1,0),a),0),0)
  • {1,,1{1,,1■3}1{1,,1■2}2■3} has recursion level C(C(Ω22+C(C(Ω22+a,0),a),0),0)
  • {1,,1{1,,1■3}1{1{1,,1{1,,1■2}2■2}2,,1■2}2■3} has recursion level C(C(Ω22+C(C(Ω22+C(C(Ω2+a,a),a),0),a),0),0)
  • {1,,1{1,,1■3}1{1{1,,1■3}2,,1■2}2■3} has recursion level C(C(Ω22+C(C(Ω22+C(C(Ω22,0),a),0),a),0),0)
  • {1,,1{1,,1■3}1{1{1,,1{1,,1■3}1,2■3}2,,1■2}2■3} has recursion level C(C(Ω22+C(C(Ω22+C(C(Ω22+1,0),a),0),a),0),0)
  • {1,,1{1,,1■3}1{1{1,,1{1,,1■3}1{1,,1■2}2■3}2,,1■2}2■3} has recursion level C(C(Ω22+C(C(Ω22+C(C(Ω22+a,0),a),0),a),0),0)
  • {1,,1{1,,1■3}1{1,,2■2}2■3} has recursion level C(C(Ω22+C(Ω2,a),0),0) (here, it hits the line of “C(C(Ω22+α,0),0) is valid”)
  • also, {1,,1{1{1,,1{1,,1■3}1{1,,2■2}2■3}2,,1■2}2■2} has recursion level C(C(Ω22+C(Ω2,a),0),0)
  • {1,,1{1{1,,1{1,,1■3}1{1,,2■2}2■3}2,,1■2}1,2■2} has recursion level C(C(Ω22+C(Ω2,a)+1,0),0)
  • {1,,1{1{1,,1{1,,1■3}1{1,,2■2}2■3}2,,1■2}1{1,,1■2}2■2} has recursion level C(C(Ω22+C(Ω2,a)+a,0),0)
  • {1,,1{1{1,,1{1,,1■3}1{1,,2■2}2■3}2,,1■2}1{1{1,,1■3}2,,1■2}2■2} has recursion level C(C(Ω22+C(Ω2,a)+C(C(Ω22,0),a),0),0)
  • {1,,1{1{1,,1{1,,1■3}1{1,,2■2}2■3}2,,1■2}1{1{1,,1{1,,1■3}1{1,,1■2}2■3}2,,1■2}2■2} has recursion level C(C(Ω22+C(Ω2,a)+C(C(Ω22+a,0),a),0),0)
  • {1,,1{1{1,,1{1,,1■3}1{1,,2■2}2■3}2,,1■2}1{1{1,,1{1,,1■3}1{1,,2■2}2■3}2,,1■2}2■2} has recursion level C(C(Ω22+C(Ω2,a)+C(C(Ω22+C(Ω2,a),0),a),0),0)
  • {1,,1{2{1,,1{1,,1■3}1{1,,2■2}2■3}2,,1■2}2■2} has recursion level C(C(Ω22+C(Ω2,a)+ω^(C(C(Ω22+C(Ω2,a),0),a)+1),0),0)
  • {1,,1{1{1,,1■3}2{1,,1{1,,1■3}1{1,,2■2}2■3}2,,1■2}2■2} has recursion level C(C(Ω22+C(Ω2,a)+C(C(Ω22,0),C(C(Ω22+C(Ω2,a),0),a)),0),0)
  • {1,,1{1{1,,1{1,,1■3}1{1,,1■2}2■3}2{1,,1{1,,1■3}1{1,,2■2}2■3}2,,1■2}2■2} has recursion level C(C(Ω22+C(Ω2,a)+C(C(Ω22+a,0),C(C(Ω22+C(Ω2,a),0),a)),0),0)
  • {1,,1{1{1,,1{1,,1■3}1{1,,2■2}2■3}3,,1■2}2■2} has recursion level C(C(Ω22+C(Ω2,a)+C(C(Ω22+C(Ω2,a),0),C(C(Ω22+C(Ω2,a),0),a)),0),0)
  • {1,,1{1{1,,1{1,,1■3}1{1,,2■2}2■3}1,2,,1■2}2■2} has recursion level C(C(Ω22+C(Ω2,a)+C(C(Ω22+C(Ω2,a),0)+1,a),0),0)
  • {1,,1{1{1,,1{1,,1■3}1{1,,2■2}2■3}1{1,,2■2}2,,1■2}2■2} has recursion level C(C(Ω22+C(Ω2,a)+C(C(Ω22+C(Ω2,a),0)+C(Ω2,a),a),0),0)
  • {1,,1{1{1{1,,1,2■2}2,,1{1,,1■3}1{1,,2■2}2■3}2,,1■2}2■2} has recursion level C(C(Ω22+C(Ω2,a)+C(C(Ω22+C(Ω2,a),0)+C(Ω2+1,a),a),0),0)
  • {1{1,,1■3}2,,1{1,,1■3}1{1,,2■2}2■3} has recursion level C(C(Ω22+C(Ω2,a)+C(C(Ω22+C(Ω2,a),0)+C(Ω22,0),a),0),0)
  • {1{1,,1{1,,1■3}1{1,,1■2}2■3}2,,1{1,,1■3}1{1,,2■2}2■3} has recursion level C(C(Ω22+C(Ω2,a)+C(C(Ω22+C(Ω2,a),0)+C(Ω22+a,0),a),0),0)
  • {1{1,,1{1,,1■3}1{1,,2■2}2■3}2,,1{1,,1■3}1{1,,2■2}2■3} has recursion level C(C(Ω22+C(Ω2,a)+C(C(Ω22+C(Ω2,a),0)2,a),0),0)
  • {1{1,,1{1,,1■3}1{1,,2■2}2■3}1,2,,1{1,,1■3}1{1,,2■2}2■3} has recursion level C(C(Ω22+C(Ω2,a)+C(ω^(C(Ω22+C(Ω2,a),0)+1),a),0),0)
  • {1{2,,1{1,,1■3}1{1,,2■2}2■3}2,,1{1,,1■3}1{1,,2■2}2■3} has recursion level C(C(Ω22+C(Ω2,a)+C(ω^ω^(C(Ω22+C(Ω2,a),0)+1),a),0),0)
  • {1,,2{1,,1■3}1{1,,2■2}2■3} has recursion level C(C(Ω22+C(Ω2,a)+C(C(Ω2,C(Ω22+C(Ω2,a),0)),a),0),0)
  • {1,,1{1,,1■2}2{1,,1■3}1{1,,2■2}2■3} has recursion level C(C(Ω22+C(Ω2,a)+C(C(Ω2+a,C(Ω22+C(Ω2,a),0)),a),0),0)
  • {1,,1{1,,1{1,,1■2}2■2}2{1,,1■3}1{1,,2■2}2■3} has recursion level C(C(Ω22+C(Ω2,a)+C(C(Ω2+C(Ω2+a,a),C(Ω22+C(Ω2,a),0)),a),0),0)
  • {1,,1{1,,1■3}2{1,,2■2}2■3} has recursion level C(C(Ω22+C(Ω2,a)+C(C(Ω22,C(Ω22+C(Ω2,a),0)),a),0),0)
  • {1,,1{1,,1■3}1{1,,1■2}2{1,,2■2}2■3} has recursion level C(C(Ω22+C(Ω2,a)+C(C(Ω22+a,C(Ω22+C(Ω2,a),0)),a),0),0)
  • {1,,1{1,,1■3}1{1,,2■2}3■3} has recursion level C(C(Ω22+C(Ω2,a)+C(C(Ω22+C(Ω2,a),C(Ω22+C(Ω2,a),0)),a),0),0)
  • {1,,1{1,,1■3}1{1,,2■2}1,2■3} has recursion level C(C(Ω22+C(Ω2,a)+C(C(Ω22+C(Ω2,a)+1,0),a),0),0)
  • {1,,1{1,,1■3}1{1,,2■2}1{1,,1■2}2■3} has recursion level C(C(Ω22+C(Ω2,a)+C(C(Ω22+C(Ω2,a)+a,0),a),0),0)
  • {1,,1{1,,1■3}1{1,,2■2}1{1{1,,1{1,,1■3}1{1,,2■2}1{1,,1■2}2■3}2,,1■2}2■3} has recursion level C(C(Ω22+C(Ω2,a)+C(C(Ω22+C(Ω2,a)+C(C(Ω22+C(Ω2,a)+a,0),a),0),a),0),0)
  • {1,,1{1,,1■3}1{1,,2■2}1{1,,2■2}2■3} has recursion level C(C(Ω22+C(Ω2,a)2,0),0)
  • {1,,1{1,,1■3}1{2,,2■2}2■3} has recursion level C(C(Ω22+ω^(C(Ω2,a)+1),0),0)
  • {1,,1{1,,1■3}1{1{1,,3■2}2,,2■2}2■3} has recursion level C(C(Ω22+C(C(Ω2,C(Ω2,a)),C(Ω2,a)),0),0)
  • {1,,1{1,,1■3}1{1{1,,1,2■2}2,,2■2}2■3} has recursion level C(C(Ω22+C(C(Ω2+1,a),C(Ω2,a)),0),0)
  • {1,,1{1,,1■3}1{1{1,,1{1,,1,2■2}2■2}2,,2■2}2■3} has recursion level C(C(Ω22+C(C(Ω2+C(Ω2+1,a),a),C(Ω2,a)),0),0)
  • {1,,1{1,,1■3}1{1{1,,1■3}2,,2■2}2■3} has recursion level C(C(Ω22+C(C(Ω22,0),C(Ω2,a)),0),0)
  • {1,,1{1,,1■3}1{1{1,,1{1,,1■3}1{1,,1■2}2■3}2,,2■2}2■3} has recursion level C(C(Ω22+C(C(Ω22+a,0),C(Ω2,a)),0),0)
  • {1,,1{1,,1■3}1{1{1,,1{1,,1■3}1{1,,2■2}2■3}2,,2■2}2■3} has recursion level C(C(Ω22+C(C(Ω22+C(Ω2,a),0),C(Ω2,a)),0),0)
  • {1,,1{1,,1■3}1{1{1,,1{1,,1■3}1{1{1,,1{1,,1■3}1{1,,2■2}2■3}2,,2■2}2■3}2,,2■2}2■3} has recursion level C(C(Ω22+C(C(Ω22+C(C(Ω22+C(Ω2,a),0),C(Ω2,a)),0),C(Ω2,a)),0),0)
  • {1,,1{1,,1■3}1{1,,3■2}2■3} has recursion level C(C(Ω22+C(Ω2,C(Ω2,a)),0),0)
  • {1,,1{1,,1■3}1{1,,4■2}2■3} has recursion level C(C(Ω22+C(Ω2,C(Ω2,C(Ω2,a))),0),0)
  • {1,,1{1,,1■3}1{1,,1,2■2}2■3} has recursion level C(C(Ω22+C(Ω2+1,a),0),0)
  • {1,,1{1,,1■3}1{1,,1{1,,1■2}2■2}2■3} has recursion level C(C(Ω22+C(Ω2+a,a),0),0)
  • {1,,1{1,,1■3}1{1,,1{1,,1{1,,1■2}2■2}2■2}2■3} has recursion level C(C(Ω22+C(Ω2+C(Ω2+a,a),a),0),0)
  • {1,,1{1,,1■3}1{1,,1■3}2■3} has recursion level C(C(Ω22+C(Ω2+C(Ω22,0),a),0),0)

It seems that {1,,1■3} approximately corresponds to C(Ω22,0), but it’s not so in following comparisons. Now let ordinal a = C(Ω2+C(Ω22,0),0) and b = C(Ω2+C(Ω22,0),a) = C(Ω2+C(Ω22,0),C(Ω2+C(Ω22,0),0)).

  • {1,,1{1,,1■3}1{1,,1■3}2■3} has recursion level C(C(Ω22+b,0),0)
  • {1,,1{1{1,,1{1,,1■3}1{1,,1■3}2■3}2,,1■2}1,2■2} has recursion level C(C(Ω22+b+1,0),0)
  • {1,,1{1{1,,1{1,,1■3}1{1,,1■3}2■3}2,,1■2}1{1,,1■2}2■2} has recursion level C(C(Ω22+b+a,0),0)
  • {1,,1{1{1,,1{1,,1■3}1{1,,1■3}2■3}2,,1■2}1{1{1,,1{1,,1■3}1{1,,1■3}2■3}2,,1■2}2■2} has recursion level C(C(Ω22+b+C(C(Ω22+b,0),a),0),0)
  • {1,,1{2{1,,1{1,,1■3}1{1,,1■3}2■3}2,,1■2}2■2} has recursion level C(C(Ω22+b+ω^(C(C(Ω22+b,0),a)+1),0),0)
  • {1,,1{1{1,,1{1,,1■3}1{1,,1■3}2■3}1,2,,1■2}2■2} has recursion level C(C(Ω22+b+C(C(Ω22+b,0)+1,a),0),0)
  • {1{1,,1■3}2,,1{1,,1■3}1{1,,1■3}2■3} has recursion level C(C(Ω22+b+C(C(Ω22+b,0)+C(Ω22,0),a),0),0)
  • {1{1,,1{1,,1■3}1{1,,1■3}2■3}2,,1{1,,1■3}1{1,,1■3}2■3} has recursion level C(C(Ω22+b+C(C(Ω22+b,0)2,a),0),0)
  • {1,,2{1,,1■3}1{1,,1■3}2■3} has recursion level C(C(Ω22+b+C(C(Ω2,C(Ω22+b,0)),a),0),0)
  • {1,,1{1,,1■3}2{1,,1■3}2■3} has recursion level C(C(Ω22+b+C(C(Ω22,C(Ω22+b,0)),a),0),0)
  • {1,,1{1,,1■3}1{1,,1■3}1,2■3} has recursion level C(C(Ω22+b+C(C(Ω22+b+1,0),a),0),0)
  • {1,,1{1,,1■3}1{1,,1■3}1{1,,1■2}2■3} has recursion level C(C(Ω22+b+C(C(Ω22+b+a,0),a),0),0)
  • {1,,1{1,,1■3}1{1,,1■3}1{1,,2■2}2■3} has recursion level C(C(Ω22+b+C(Ω2,a),0),0)
  • {1,,1{1,,1■3}1{1,,1■3}1{1,,1{1,,2■2}2■2}2■3} has recursion level C(C(Ω22+b+C(Ω2+C(Ω2,a),a),0),0)
  • {1,,1{1,,1■3}1{1,,1■3}1{1,,1■3}2■3} has recursion level C(C(Ω22+b2,0),0)
  • {1,,1{2,,1■3}2■3} has recursion level C(C(Ω22+ω^(b+1),0),0)
  • {1,,1{1{1,,1■2}2,,1■3}2■3} has recursion level C(C(Ω22+ω^(b+a),0),0)
  • {1,,1{1{1,,2■2}2,,1■3}2■3} has recursion level C(C(Ω22+ω^(b+C(Ω2,a)),0),0)
  • {1,,1{1{1,,1{1,,2■2}2■2}2,,1■3}2■3} has recursion level C(C(Ω22+ω^(b+C(Ω2+C(Ω2,a),a)),0),0)
  • {1,,1{1{1,,1■3}2,,1■3}2■3} has recursion level C(C(Ω22+ω^(b2),0),0)
  • {1,,1{1{1,,2■3}2,,1■3}2■3} has recursion level C(C(Ω22+C(C(Ω2,b),b),0),0)
  • {1,,1{1{1,,1,2■3}2,,1■3}2■3} has recursion level C(C(Ω22+C(C(Ω2+1,b),b),0),0)
  • {1,,1{1{1,,1{1,,1■3}2■3}2,,1■3}2■3} has recursion level C(C(Ω22+C(C(Ω2+b,b),b),0),0)
  • {1,,1{1{1,,1{2,,1■3}2■3}2,,1■3}2■3} has recursion level C(C(Ω22+C(C(Ω2+ω^(b+1),b),b),0),0)
  • {1,,1{1{1,,1{1{1,,1{1,,1■3}2■3}2,,1■3}2■3}2,,1■3}2■3} has recursion level C(C(Ω22+C(C(Ω2+C(C(Ω2+b,b),b),b),b),0),0)
  • {1,,1{1,,2■3}2■3} has recursion level C(C(Ω22+C(C(Ω2+C(Ω2,b),b),b),0),0)
  • {1,,1{1,,3■3}2■3} has recursion level C(C(Ω22+C(C(Ω2+C(Ω2,C(Ω2,b)),b),b),0),0)
  • {1,,1{1,,1,2■3}2■3} has recursion level C(C(Ω22+C(C(Ω2+C(Ω2+1,b),b),b),0),0)
  • {1,,1{1,,1{1,,1■3}2■3}2■3} has recursion level C(C(Ω22+C(C(Ω2+C(Ω2+b,b),b),b),0),0)
  • {1,,1■4} has recursion level C(C(Ω22+C(C(Ω22,0),b),0),0)
  • {1,,1{1{1,,1■4}2,,1■3}1{1,,1■3}2■3} has recursion level C(C(Ω22+C(C(Ω22,0),b)+b,0),0)
  • {1,,1{1{1,,1■4}2,,1■3}1{1{1,,2■3}2,,1■3}2■3} has recursion level C(C(Ω22+C(C(Ω22,0),b)+C(C(Ω2,b),b),0),0)
  • {1,,1{1{1,,1■4}2,,1■3}1{1{1,,1{1,,1■3}2■3}2,,1■3}2■3} has recursion level C(C(Ω22+C(C(Ω22,0),b)+C(C(Ω2+b,b),b),0),0)
  • {1,,1{1{1,,1■4}2,,1■3}1{1{1,,1■4}2,,1■3}2■3} has recursion level C(C(Ω22+C(C(Ω22,0),b)2,0),0)
  • {1,,1{2{1,,1■4}2,,1■3}2■3} has recursion level C(C(Ω22+ω^(C(C(Ω22,0),b)+1),0),0)
  • {1,,1{1{1,,2■3}2{1,,1■4}2,,1■3}2■3} has recursion level C(C(Ω22+C(C(Ω2,b),C(C(Ω22,0),b)),0),0)
  • {1,,1{1{1,,1{1,,1■3}2■3}2{1,,1■4}2,,1■3}2■3} has recursion level C(C(Ω22+C(C(Ω2+b,b),C(C(Ω22,0),b)),0),0)
  • {1,,1{1{1,,1■4}3,,1■3}2■3} has recursion level C(C(Ω22+C(C(Ω22,0),C(C(Ω22,0),b)),0),0)
  • {1,,1{1{1,,1■4}1,2,,1■3}2■3} has recursion level C(C(Ω22+C(C(Ω22,0)+1,b),0),0)
  • {1,,1{1{1,,1■4}1{1,,2■3}2,,1■3}2■3} has recursion level C(C(Ω22+C(C(Ω22,0)+C(Ω2,b),b),0),0)
  • {1,,1{1{1,,1■4}1{1,,1{1,,1■3}2■3}2,,1■3}2■3} has recursion level C(C(Ω22+C(C(Ω22,0)+C(C(Ω2+b,b),C(Ω2,b)),b),0),0)
  • {1,,1{1{1,,1■4}1{1,,1■4}2,,1■3}2■3} has recursion level C(C(Ω22+C(C(Ω22,0)+C(C(Ω22,0),C(Ω2,b)),b),0),0)
  • {1,,1{1{2,,1■4}2,,1■3}2■3} has recursion level C(C(Ω22+C(C(Ω22,0)+ω^(C(C(Ω22,0),C(Ω2,b))+1),b),0),0)
  • {1,,1{1{1{1,,2■3}2,,1■4}2,,1■3}2■3} has recursion level C(C(Ω22+C(C(Ω22,0)+ω^(C(C(Ω22,0),C(Ω2,b))+C(Ω2,b)),b),0),0)
  • {1,,1{1{1{1,,3■3}2,,1■4}2,,1■3}2■3} has recursion level C(C(Ω22+C(C(Ω22,0)+ω^(C(C(Ω22,0),C(Ω2,C(Ω2,b)))+C(Ω2,C(Ω2,b))),b),0),0)
  • {1,,1{1{1{1,,1,2■3}2,,1■4}2,,1■3}2■3} has recursion level C(C(Ω22+C(C(Ω22,0)+C(Ω2+1,b),b),0),0)
  • {1{1,,1■4}2,,1■4} has recursion level C(C(Ω22+C(C(Ω22,0)2,b),0),0)
  • {1{1,,1■4}1,2,,1■4} has recursion level C(C(Ω22+C(ω^(C(Ω22,0)+1),b),0),0)
  • {1{1,,1■4}1{1,,1■4}2,,1■4} has recursion level C(C(Ω22+C(ω^(C(Ω22,0)2),b),0),0)
  • {1{2,,1■4}2,,1■4} has recursion level C(C(Ω22+C(ω^ω^(C(Ω22,0)+1),b),0),0)
  • {1,,2■4} has recursion level C(C(Ω22+C(C(Ω2,C(Ω22,0)),b),0),0)
  • {1,,1,2■4} has recursion level C(C(Ω22+C(C(Ω2+1,C(Ω22,0)),b),0),0)
  • {1,,1{1,,1■2}2■4} has recursion level C(C(Ω22+C(C(Ω2+a,C(Ω22,0)),b),0),0)
  • {1,,1{1,,2■2}2■4} has recursion level C(C(Ω22+C(C(Ω2+C(Ω2,a),C(Ω22,0)),b),0),0)
  • {1,,1{1,,1■3}2■4} has recursion level C(C(Ω22+C(C(Ω2+b,C(Ω22,0)),b),0),0)
  • {1,,1{1,,2■3}2■4} has recursion level C(C(Ω22+C(C(Ω2+C(Ω2,b),C(Ω22,0)),b),0),0)
  • {1,,1{1,,1■4}2■4} has recursion level C(C(Ω22+C(C(Ω22,C(Ω22,0)),b),0),0)
  • {1,,1{1,,1■4}1,2■4} has recursion level C(C(Ω22+C(C(Ω22+1,0),b),0),0)
  • {1,,1{1,,1■4}1{1,,1■3}2■4} has recursion level C(C(Ω22+C(C(Ω22+b,0),b),0),0)
  • {1,,1{1,,1■4}1{1,,2■3}2■4} has recursion level C(C(Ω22+C(Ω2,b),0),0)
  • {1,,1{1,,1■4}1{1,,3■3}2■4} has recursion level C(C(Ω22+C(Ω2,C(Ω2,b)),0),0)
  • {1,,1{1,,1■4}1{1,,1,2■3}2■4} has recursion level C(C(Ω22+C(Ω2+1,b),0),0)
  • {1,,1{1,,1■4}1{1,,1{1,,1■3}2■3}2■4} has recursion level C(C(Ω22+C(Ω2+b,b),0),0)
  • {1,,1{1,,1■4}1{1,,1{1,,1{1,,1■3}2■3}2■3}2■4} has recursion level C(C(Ω22+C(Ω2+C(Ω2+b,b),b),0),0)
  • {1,,1{1,,1■4}1{1,,1■4}2■4} has recursion level C(C(Ω22+C(Ω2+C(Ω22,0),b),0),0)
  • {1,,1{1{1,,2■4}2,,1■4}2■4} has recursion level C(C(Ω22+C(C(Ω2,C(Ω2+C(Ω22,0),b)),C(Ω2+C(Ω22,0),b)),0),0)
  • {1,,1■5} has recursion level C(C(Ω22+C(C(Ω22,0),C(Ω2+C(Ω22,0),b)),0),0)
  • {1,,1{1,,1■5}1{1,,2■4}2■5} has recursion level C(C(Ω22+C(Ω2,C(Ω2+C(Ω22,0),b)),0),0)
  • {1,,1{1,,1■5}1{1,,1■5}2■5} has recursion level C(C(Ω22+C(Ω2+C(Ω22,0),C(Ω2+C(Ω22,0),b)),0),0)
  • {1,,1■6} has recursion level C(C(Ω22+C(C(Ω22,0),C(Ω2+C(Ω22,0),C(Ω2+C(Ω22,0),b))),0),0)

Generally, the recursion level of {1,,1■k} is between C(C(Ω22+C(Ω2+C(Ω22,0),C(Ω2+C(Ω22,0),…C(Ω2+C(Ω22,0),0))),0),0) (with k-2 Ω2+C(Ω22,0)′s in the blue part) and C(C(Ω22+C(Ω2+C(Ω22,0),C(Ω2+C(Ω22,0),…C(Ω2+C(Ω22,0),0))),0),0) (with k-1 Ω2+C(Ω22,0)′s in the blue part), so {1,,1■1,2} has recursion level C(C(Ω22+C(Ω2+C(Ω22,0)+1,0),0),0). Now let ordinal a = C(Ω2+C(Ω22,0),0) and b = C(Ω2+C(Ω22,0)+1,0).

  • {1,,1{1{1,,1■1,2}2,,1■2}1{1,,1■2}2■2} has recursion level C(C(Ω22+b+a,0),0)
  • {1,,1{1{1,,1■1,2}2,,1■2}1{1{1,,1■3}2,,1■2}2■2} has recursion level C(C(Ω22+b+C(C(Ω22,0),a),0),0)
  • {1,,1{1{1,,1■1,2}2,,1■2}1{1{1,,1■1,2}2,,1■2}2■2} has recursion level C(C(Ω22+b+C(C(Ω22+b,0),a),0),0)
  • {1,,1{1{1,,1■1,2}1,2,,1■2}2■2} has recursion level C(C(Ω22+b+C(C(Ω22+b,0)+1,a),0),0)
  • {1{1,,1■1,2}2,,1■1,2} has recursion level C(C(Ω22+b+C(C(Ω22+b,0)2,a),0),0)
  • {1{1{1,,1■1,2}2,,1■1,2}2,,1■1,2} has recursion level C(C(Ω22+b+C(ω^ω^(C(Ω22+b,0)2),a),0),0)
  • {1,,2■1,2} has recursion level C(C(Ω22+b+C(C(Ω2,C(Ω22+b,0)),a),0),0)
  • {1,,1{1,,1■2}2■1,2} has recursion level C(C(Ω22+b+C(C(Ω2+a,C(Ω22+b,0)),a),0),0)
  • {1,,1{1,,1■3}2■1,2} has recursion level C(C(Ω22+b+C(C(Ω22,C(Ω22+b,0)),a),0),0)
  • {1,,1{1,,1■1,2}2■1,2} has recursion level C(C(Ω22+b+C(C(Ω22+b,C(Ω22+b,0)),a),0),0)
  • {1,,1{1,,1■1,2}1,2■1,2} has recursion level C(C(Ω22+b+C(C(Ω22+b+1,0),a),0),0)
  • {1,,1{1,,1■1,2}1{1,,1■2}2■1,2} has recursion level C(C(Ω22+b+C(C(Ω22+b+a,0),a),0),0)
  • {1,,1{1,,1■1,2}1{1,,2■2}2■1,2} has recursion level C(C(Ω22+b+C(Ω2,a),0),0)
  • {1,,1{1,,1■1,2}1{1,,1■3}2■1,2} has recursion level C(C(Ω22+b+C(Ω2+(Ω22,0),a),0),0)
  • {1,,1{1,,1■1,2}1{1,,1■1,2}2■1,2} has recursion level C(C(Ω22+b2,0),0)
  • {1,,1{1{1,,2■1,2}2,,1■1,2}2■1,2} has recursion level C(C(Ω22+C(C(Ω2,b),b),0),0)
  • {1,,1{1{1,,1{1,,1■1,2}2■1,2}2,,1■1,2}2■1,2} has recursion level C(C(Ω22+C(C(Ω2+b,b),b),0),0)
  • {1,,1{1,,2■1,2}2■1,2} has recursion level C(C(Ω22+C(C(Ω2+C(Ω2,b),b),b),0),0)
  • {1,,1{1,,1{1,,2■1,2}2■1,2}2■1,2} has recursion level C(C(Ω22+C(C(Ω2+C(Ω2+C(Ω2,b),b),b),b),0),0)
  • {1,,1■2,2} has recursion level C(C(Ω22+C(C(Ω22,0),b),0),0)
  • {1{1,,1■2,2}2,,1■2,2} has recursion level C(C(Ω22+C(C(Ω22,0)2,b),0),0)
  • {1,,2■2,2} has recursion level C(C(Ω22+C(C(Ω2,C(Ω22,0)),b),0),0)
  • {1,,1{1,,1■2,2}2■2,2} has recursion level C(C(Ω22+C(C(Ω22,C(Ω22,0)),b),0),0)
  • {1,,1{1,,1■2,2}1{1,,1■1,2}2■2,2} has recursion level C(C(Ω22+C(C(Ω22+b,0),b),0),0)
  • {1,,1{1,,1■2,2}1{1,,2■1,2}2■2,2} has recursion level C(C(Ω22+C(Ω2,b),0),0)
  • {1,,1■3,2} has recursion level C(C(Ω22+C(C(Ω22,0),C(Ω2+C(Ω22,0),b)),0),0)
  • {1,,1■1,3} has recursion level C(C(Ω22+C(Ω2+C(Ω22,0)+1,b),0),0)
  • {1,,1■1,1,2} has recursion level C(C(Ω22+C(Ω2+C(Ω22,0)+2,0),0),0)
  • {1,,1■1{1,,1■2}2} has recursion level C(C(Ω22+C(Ω2+C(Ω22,0)+a,0),0),0)
  • {1,,1■1{1,,1■1,2}2} has recursion level C(C(Ω22+C(Ω2+C(Ω22,0)+b,0),0),0)
  • {1,,1■1■2} has recursion level C(C(Ω22+C(Ω2+C(Ω22,0)2,0),0),0)
  • {1,,1■1,2■2} has recursion level C(C(Ω22+C(Ω2+C(Ω22,0)+1,C(Ω2+C(Ω22,0)2,0)),0),0)
  • {1,,1■1{1,,1■1■2}2■2} has recursion level C(C(Ω22+C(Ω2+C(Ω22,0)+C(Ω2+C(Ω22,0)2,0),C(Ω2+C(Ω22,0)2,0)),0),0)
  • {1,,1■1■3} has recursion level C(C(Ω22+C(Ω2+C(Ω22,0)2,C(Ω2+C(Ω22,0)2,0)),0),0)
  • {1,,1■1■1,2} has recursion level C(C(Ω22+C(Ω2+C(Ω22,0)2+1,0),0),0)
  • {1,,1■1■1■2} has recursion level C(C(Ω22+C(Ω2+C(Ω22,0)3,0),0),0)
  • {1,,1{2,,1,,2}2} has recursion level C(C(Ω22+C(Ω2+ω^(C(Ω22,0)+1),0),0),0)
  • {1,,1{1{1,,1{2,,1,,2}2}2,,1,,2}2} has recursion level C(C(Ω22+C(Ω2+ω^(C(Ω22,0)+C(Ω2+ω^(C(Ω22,0)+1),0)),0),0),0)
  • {1,,1{1■2,,1,,2}2} has recursion level C(C(Ω22+C(Ω2+ω^(C(Ω22,0)2),0),0),0)
  • {1,,1{1■1,2,,1,,2}2} has recursion level C(C(Ω22+C(Ω2+ω^ω^(C(Ω22,0)+1),0),0),0)
  • {1,,1{1{2,,1,,2}2,,1,,2}2} has recursion level C(C(Ω22+C(Ω2+ω^ω^ω^(C(Ω22,0)+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,C(Ω22,0)),C(Ω22,0)),0),0),0)

So {1,,2,,2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2,C(Ω22,0)),C(Ω22,0)),0),0),0). Here’re some approximations, to make the comparisons above more simple:

  • {1,,1■2} (where separator ■ = {1,,1,,2}) approximately corresponds to C(Ω2+C(Ω22,0),0)
  • {1,,2■2} approximately corresponds to C(Ω2,C(Ω2+C(Ω22,0),0))
  • {1,,1■3} approximately corresponds to C(Ω2+C(Ω22,0),C(Ω2+C(Ω22,0),0))
  • {1,,2■3} approximately corresponds to C(Ω2,C(Ω2+C(Ω22,0),C(Ω2+C(Ω22,0),0)))
  • {1,,1■4} approximately corresponds to C(Ω2+C(Ω22,0),C(Ω2+C(Ω22,0),C(Ω2+C(Ω22,0),0)))
  • {1,,1■1,2} approximately corresponds to C(Ω2+C(Ω22,0)+1,0)
  • {1,,1■2,2} approximately corresponds to C(Ω2+C(Ω22,0),C(Ω2+C(Ω22,0)+1,0))
  • {1,,1■1,1,2} approximately corresponds to C(Ω2+C(Ω22,0)+2,0)
  • {1,,1■1■2} approximately corresponds to C(Ω2+C(Ω22,0)2,0)
  • {1,,1{2,,1,,2}2} approximately corresponds to C(Ω2+ω^(C(Ω22,0)+1),0)
  • {1,,1{1■2,,1,,2}2} approximately corresponds to C(Ω2+ω^(C(Ω22,0)2),0)
  • {1,,1{1■1■2,,1,,2}2} approximately corresponds to C(Ω2+ω^ω^(C(Ω22,0)2),0)
  • {1,,1{1{1■2,,1,,2}2,,1,,2}2} approximately corresponds to C(Ω2+ω^ω^ω^(C(Ω22,0)2),0)
  • {1,,1{1,,2,,2}2} approximately corresponds to C(Ω2+C(C(Ω2,C(Ω22,0)),C(Ω22,0)),0)

However, the ordinal C(Ω22,0) in Taranovsky’s ordinal notation behaves in an irregular way.

Up to {1,,1,,3}

  • {1,,1{1,,1,,2}1,2{1{1,,2,,2}2,,1,,2}2} has recursion level C(C(Ω22+C(Ω2+C(Ω22,0)+1,C(Ω2+C(C(Ω2,C(Ω22,0)),C(Ω22,0)),0)),0),0)
  • {1,,1{1,,1,,2}1{1,,1{1{1,,2,,2}2,,1,,2}2}2{1{1,,2,,2}2,,1,,2}2} has recursion level C(C(Ω22+C(Ω2+C(Ω22,0)+C(Ω2+C(C(Ω2,C(Ω22,0)),C(Ω22,0)),0),C(Ω2+C(C(Ω2,C(Ω22,0)),C(Ω22,0)),0)),0),0)
  • {1,,1{1,,1,,2}1{1,,1,,2}2{1{1,,2,,2}2,,1,,2}2} has recursion level C(C(Ω22+C(Ω2+C(Ω22,0)2,C(Ω2+C(C(Ω2,C(Ω22,0)),C(Ω22,0)),0)),0),0)
  • {1,,1{1{1,,1,,2}2,,1,,2}2{1{1,,2,,2}2,,1,,2}2} has recursion level C(C(Ω22+C(Ω2+ω^(C(Ω22,0)2),C(Ω2+C(C(Ω2,C(Ω22,0)),C(Ω22,0)),0)),0),0)
  • {1,,1{1{1,,2,,2}2,,1,,2}3} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2,C(Ω22,0)),C(Ω22,0)),C(Ω2+C(C(Ω2,C(Ω22,0)),C(Ω22,0)),0)),0),0)
  • {1,,1{1{1,,2,,2}2,,1,,2}1,2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2,C(Ω22,0)),C(Ω22,0))+1,0),0),0)
  • {1,,1{1{1,,2,,2}2,,1,,2}1{1,,1,,2}2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2,C(Ω22,0)),C(Ω22,0))+C(Ω22,0),0),0),0)
  • {1,,1{1{1,,2,,2}2,,1,,2}1{1{1,,2,,2}2,,1,,2}2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2,C(Ω22,0)),C(Ω22,0))2,0),0),0)
  • {1{1{1,,2,,2}2,,1,,2}2{1,,2,,2}2,,1,,2} has recursion level C(C(Ω22+C(Ω2+ω^(C(C(Ω2,C(Ω22,0)),C(Ω22,0))2),0),0),0)
  • {1{1,,2,,2}3,,1,,2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2,C(Ω22,0)),C(C(Ω2,C(Ω22,0)),C(Ω22,0))),0),0),0)
  • {1{1,,2,,2}1,2,,1,,2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2,C(Ω22,0))+1,C(Ω22,0)),0),0),0)
  • {1{1,,2,,2}1{1,,2,,2}2,,1,,2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2,C(Ω22,0))2,C(Ω22,0)),0),0),0)
  • {1,,3,,2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2,C(Ω2,C(Ω22,0))),C(Ω22,0)),0),0),0)
  • {1,,1,2,,2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2+1,C(Ω22,0)),C(Ω22,0)),0),0),0)
  • {1,,1{1,,1{1,,1,,2}1,2}2,,2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2+C(Ω2+C(Ω22,0)+1,0),C(Ω22,0)),C(Ω22,0)),0),0),0)
  • {1,,1{1,,1,,2}2,,2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2+C(Ω22,0),C(Ω22,0)),C(Ω22,0)),0),0),0)
  • {1,,1{1{1,,2,,2}2,,1,,2}2,,2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2+C(C(Ω2,C(Ω22,0)),C(Ω22,0)),C(Ω22,0)),C(Ω22,0)),0),0),0)
  • {1,,1{1{1,,1{1,,1,,2}2,,2}2,,1,,2}2,,2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2+C(C(Ω2+C(Ω22,0),C(Ω22,0)),C(Ω22,0)),C(Ω22,0)),C(Ω22,0)),0),0),0)
  • {1,,1{1,,2,,2}2,,2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22,C(Ω22,0)),C(Ω22,0)),0),0),0) (It’s a bit erratic here. Things like C(Ω2+C(C(Ω2+C(Ω2,C(Ω22,0)),C(Ω22,0)),C(Ω22,0)),0) are not valid)
  • {1,,1{1,,1,,2}1,2{1{1,,1{1,,2,,2}2,,2}2,,1,,2}2} has recursion level C(C(Ω22+C(Ω2+C(Ω22,0)+1,C(Ω2+C(C(Ω22,C(Ω22,0)),C(Ω22,0)),0)),0),0)
  • {1,,1{1{1,,2,,2}2,,1,,2}2{1{1,,1{1,,2,,2}2,,2}2,,1,,2}2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2,C(Ω22,0)),C(Ω22,0)),C(Ω2+C(C(Ω22,C(Ω22,0)),C(Ω22,0)),0)),0),0)
  • {1,,1{1{1,,1{1,,1,,2}2,,2}2,,1,,2}2{1{1,,1{1,,2,,2}2,,2}2,,1,,2}2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2+C(Ω22,0),C(Ω22,0)),C(Ω22,0)),C(Ω2+C(C(Ω22,C(Ω22,0)),C(Ω22,0)),0)),0),0)
  • {1,,1{1{1,,1{1,,2,,2}2,,2}2,,1,,2}3} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22,C(Ω22,0)),C(Ω22,0)),C(Ω2+C(C(Ω22,C(Ω22,0)),C(Ω22,0)),0)),0),0)
  • {1,,1{1{1,,1{1,,2,,2}2,,2}2,,1,,2}1,2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22,C(Ω22,0)),C(Ω22,0))+1,0),0),0)
  • {1,,1{1{1,,1{1,,2,,2}2,,2}2,,1,,2}1{1{1,,1{1,,2,,2}2,,2}2,,1,,2}2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22,C(Ω22,0)),C(Ω22,0))2,0),0),0)
  • {1{1,,2,,2}2{1,,1{1,,2,,2}2,,2}2,,1,,2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2,C(Ω22,0)),C(C(Ω22,C(Ω22,0)),C(Ω22,0))),0),0),0)
  • {1{1,,1{1,,1,,2}2,,2}2{1,,1{1,,2,,2}2,,2}2,,1,,2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2+C(Ω22,0),C(Ω22,0)),C(C(Ω22,C(Ω22,0)),C(Ω22,0))),0),0),0)
  • {1{1,,1{1{1,,1{1,,1,,2}2,,2}2,,1,,2}2,,2}2{1,,1{1,,2,,2}2,,2}2,,1,,2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2+C(C(Ω2+C(Ω22,0),C(Ω22,0)),C(Ω22,0)),C(Ω22,0)),C(C(Ω22,C(Ω22,0)),C(Ω22,0))),0),0),0)
  • {1{1,,1{1{1,,1{1,,2,,2}2,,2}2,,1,,2}2,,2}2{1,,1{1,,2,,2}2,,2}2,,1,,2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2+C(C(Ω22,C(Ω22,0)),C(Ω22,0)),C(Ω22,0)),C(C(Ω22,C(Ω22,0)),C(Ω22,0))),0),0),0)
  • {1{1,,1{1{1,,1{1,,1,,2}2,,2}2{1,,1{1,,2,,2}2,,2}2,,1,,2}2,,2}2{1,,1{1,,2,,2}2,,2}2,,1,,2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2+C(C(Ω2+C(Ω22,0),C(Ω22,0)),C(C(Ω22,C(Ω22,0)),C(Ω22,0))),C(Ω22,0)),C(C(Ω22,C(Ω22,0)),C(Ω22,0))),0),0),0)
  • {1{1,,1{1,,2,,2}2,,2}3,,1,,2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22,C(Ω22,0)),C(C(Ω22,C(Ω22,0)),C(Ω22,0))),0),0),0)
  • {1{1,,1{1,,2,,2}2,,2}1,2,,1,,2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22,C(Ω22,0))+1,C(Ω22,0)),0),0),0)
  • {1{1,,1{1,,2,,2}2,,2}1{1,,1,,2}2,,1,,2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22,C(Ω22,0))+C(Ω22,0),C(Ω22,0)),0),0),0)
  • {1{1,,1{1,,2,,2}2,,2}1{1,,2,,2}2,,1,,2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22,C(Ω22,0))+C(Ω2,C(Ω22,0)),C(Ω22,0)),0),0),0)
  • {1{1,,1{1,,2,,2}2,,2}1{1,,3,,2}2,,1,,2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22,C(Ω22,0))+C(C(Ω2,C(Ω2,C(Ω22,0))),C(Ω2,C(Ω22,0))),C(Ω22,0)),0),0),0)
  • {1{1,,1{1,,2,,2}2,,2}1{1,,1{1,,1,,2}2,,2}2,,1,,2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22,C(Ω22,0))+C(C(Ω2+C(Ω22,0),C(Ω22,0)),C(Ω2,C(Ω22,0))),C(Ω22,0)),0),0),0)
  • {1{1,,1{1,,2,,2}2,,2}1{1,,1{1,,2,,2}2,,2}1,2,,1,,2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22,C(Ω22,0))+C(C(Ω22,C(Ω22,0)),C(Ω2,C(Ω22,0)))+1,C(Ω22,0)),0),0),0)
  • {1{2,,1{1,,2,,2}2,,2}2,,1,,2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22,C(Ω22,0))+ω^(C(C(Ω22,C(Ω22,0)),C(Ω2,C(Ω22,0)))+1),C(Ω22,0)),0),0),0)
  • {1{1{1,,2,,2}2,,1{1,,2,,2}2,,2}2,,1,,2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22,C(Ω22,0))+ω^(C(C(Ω22,C(Ω22,0)),C(Ω2,C(Ω22,0)))+C(Ω2,C(Ω22,0))),C(Ω22,0)),0),0),0)

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

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

So {1{1,,3,,2}2,,1{1,,2,,2}2,,2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22,C(Ω22,0))+ω^(C(C(Ω22,C(Ω22,0)),C(Ω2,C(Ω2,C(Ω22,0))))+C(Ω2,C(Ω2,C(Ω22,0)))),C(Ω22,0)),0),0),0). And further, the recursion level of {1{1,,k,,2}2,,1{1,,2,,2}2,,2} is between C(C(Ω22+C(Ω2+C(C(Ω22,C(Ω22,0))+C(Ω2,C(Ω2,…C(Ω2,C(Ω22,0)))),C(Ω22,0)),0),0),0) (with k-1 Ω2′s in the blue part) and C(C(Ω22+C(Ω2+C(C(Ω22,C(Ω22,0))+C(Ω2,C(Ω2,…C(Ω2,C(Ω22,0)))),C(Ω22,0)),0),0),0) (with k Ω2′s in the blue part), so {1{1,,1,2,,2}2,,1{1,,2,,2}2,,2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22,C(Ω22,0))+C(Ω2+1,C(Ω22,0)),C(Ω22,0)),0),0),0).

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

Now let separator ◊ = {1,,1{1,,2,,2}1{1,,1,,2}2,,2} and let ordinal a = C(Ω2+C(Ω22,0),0), and b = C(Ω2+C(Ω2,C(Ω22,0)),0).

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

Now let separator ◊ = {1,,1{1,,2,,2}1{1,,1,,2}2,,2} and let ordinal b = C(Ω2+C(Ω2,C(Ω22,0)),0), and a = C(Ω2+C(C(Ω22+b,0),C(Ω22,0)),0) = C(Ω2+C(C(Ω22+C(Ω2+C(Ω2,C(Ω22,0)),0),0),C(Ω22,0)),0).

  • {1,,1{1{1,,1◊2}2,,1◊2}2◊2} has recursion level C(C(Ω22+b+ω^(a2),0),0)
  • {1,,1{1{1,,2◊2}2,,1◊2}2◊2} has recursion level C(C(Ω22+b+C(C(Ω2,a),a),0),0)
  • {1,,1{1{1,,1,2◊2}2,,1◊2}2◊2} has recursion level C(C(Ω22+b+C(C(Ω2+1,a),a),0),0)
  • {1,,1{1{1,,1{1,,1{1,,1{1,,2,,2}1,2,,2}2}2◊2}2,,1◊2}2◊2} has recursion level C(C(Ω22+b+C(C(Ω2+C(Ω2+C(C(Ω22+1,0),C(Ω22,0)),0),a),a),0),0)
  • {1,,1{1{1,,1{1,,1◊2}2◊2}2,,1◊2}2◊2} has recursion level C(C(Ω22+b+C(C(Ω2+a,a),a),0),0)
  • {1,,1{1,,2◊2}2◊2} has recursion level C(C(Ω22+b+C(C(Ω2+C(Ω2,a),a),a),0),0)
  • {1,,1{1,,1{1,,2◊2}2◊2}2◊2} has recursion level C(C(Ω22+b+C(C(Ω2+C(Ω2+C(Ω2,a),a),a),a),0),0)
  • {1,,1{1,,1,,2}2◊2} has recursion level C(C(Ω22+b+C(C(Ω22,0),a),0),0)
  • {1,,1{1{1,,1{1,,1,,2}2◊2}2,,1{1,,1,,2}2}1,2{1,,1,,2}2} has recursion level C(C(Ω22+b+C(C(Ω22,0),a)+1,0),0)
  • {1,,1{1{1,,1{1,,1,,2}2◊2}2,,1{1,,1,,2}2}1{1,,1{1,,1,,2}2}2{1,,1,,2}2} has recursion level C(C(Ω22+b+C(C(Ω22,0),a)+C(Ω2+C(Ω22,0),0),0),0)
  • {1,,1{1{1,,1{1,,1,,2}2◊2}2,,1{1,,1,,2}2}1{1{1,,1{1,,1,,2}2◊2}2,,1{1,,1,,2}2}2{1,,1,,2}2} has recursion level C(C(Ω22+b+C(C(Ω22,0),a)+C(C(Ω22+b+C(C(Ω22,0),a),0),C(Ω2+C(Ω22,0),0)),0),0)
  • {1,,1{2{1,,1{1,,1,,2}2◊2}2,,1{1,,1,,2}2}2{1,,1,,2}2} has recursion level C(C(Ω22+b+C(C(Ω22,0),a)+ω^(C(C(Ω22+b+C(C(Ω22,0),a),0),C(Ω2+C(Ω22,0),0))+1),0),0)
  • {1,,1{1{1,,1{1,,1,,2}2◊2}1,2,,1{1,,1,,2}2}2{1,,1,,2}2} has recursion level C(C(Ω22+b+C(C(Ω22,0),a)+C(C(Ω22+b+C(C(Ω22,0),a),0)+1,C(Ω2+C(Ω22,0),0)),0),0)
  • {1{1,,1{1,,1,,2}3}2,,1{1,,1,,2}2◊2} has recursion level C(C(Ω22+b+C(C(Ω22,0),a)+C(C(Ω22+b+C(C(Ω22,0),a),0)+C(Ω22,0),C(Ω2+C(Ω22,0),0)),0),0)
  • {1{1,,1◊2}2,,1{1,,1,,2}2◊2} has recursion level C(C(Ω22+b+C(C(Ω22,0),a)+C(C(Ω22+b+C(C(Ω22,0),a),0)+C(Ω22+b,0),C(Ω2+C(Ω22,0),0)),0),0)
  • {1{1{1,,2◊2}2,,1◊2}2,,1{1,,1,,2}2◊2} has recursion level C(C(Ω22+b+C(C(Ω22,0),a)+C(C(Ω22+b+C(C(Ω22,0),a),0)+C(C(Ω2,C(Ω22+b,0)),C(Ω22+b,0)),C(Ω2+C(Ω22,0),0)),0),0)
  • {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+b+C(C(Ω22,0),a)+C(C(Ω22+b+C(C(Ω22,0),a),0)+C(C(Ω22+b+C(C(Ω22,0),a),0),C(Ω22+b,0)),C(Ω2+C(Ω22,0),0)),0),0)
  • {1{1,,2◊2}2,,1{1,,1,,2}2◊2} has recursion level C(C(Ω22+b+C(C(Ω22,0),a)+C(C(Ω22+b+C(C(Ω22,0),a),0)+C(Ω2,C(Ω22+b,0)),C(Ω2+C(Ω22,0),0)),0),0)
  • {1{1,,1{1,,1◊2}1,2◊2}2,,1{1,,1,,2}2◊2} has recursion level C(C(Ω22+b+C(C(Ω22,0),a)+C(C(Ω22+b+C(C(Ω22,0),a),0)+C(Ω22+b+1,0),C(Ω2+C(Ω22,0),0)),0),0)
  • {1{1,,1{1,,1,,2}2◊2}2,,1{1,,1,,2}2◊2} has recursion level C(C(Ω22+b+C(C(Ω22,0),a)+C(C(Ω22+b+C(C(Ω22,0),a),0)2,C(Ω2+C(Ω22,0),0)),0),0)
  • {1,,2{1,,1,,2}2◊2} has recursion level C(C(Ω22+b+C(C(Ω22,0),a)+C(C(Ω2,C(Ω22+b+C(C(Ω22,0),a),0)),C(Ω2+C(Ω22,0),0)),0),0)
  • {1,,1{1,,1{1,,1,,2}2◊2}1,2{1,,1,,2}2◊2} has recursion level C(C(Ω22+b+C(C(Ω22,0),a)+C(C(Ω22+b+C(C(Ω22,0),a)+1,0),C(Ω2+C(Ω22,0),0)),0),0)
  • {1,,1{1,,1{1,,1,,2}2◊2}1{1,,1{1,,1,,2}2}2{1,,1,,2}2◊2} has recursion level C(C(Ω22+b+C(C(Ω22,0),a)+C(C(Ω22+b+C(C(Ω22,0),a)+C(Ω2+C(Ω22,0),0),0),C(Ω2+C(Ω22,0),0)),0),0)
  • {1,,1{1,,1{1,,1,,2}2◊2}1{1,,2{1,,1,,2}2}2{1,,1,,2}2◊2} has recursion level C(C(Ω22+b+C(C(Ω22,0),a)+C(Ω2,C(Ω2+C(Ω22,0),0)),0),0)
  • {1,,1{1,,1{1,,1,,2}2◊2}1{1,,1{1,,1,,2}1,2}2{1,,1,,2}2◊2} has recursion level C(C(Ω22+b+C(C(Ω22,0),a)+C(Ω2+C(Ω22,0)+1,0),0),0)
  • {1,,1{1,,1{1,,1,,2}2◊2}1{1,,1◊2}2{1,,1,,2}2◊2} has recursion level C(C(Ω22+b+C(C(Ω22,0),a)+a,0),0)
  • {1,,1{1,,1{1,,1,,2}2◊2}1{1{1,,2◊2}2,,1◊2}2{1,,1,,2}2◊2} has recursion level C(C(Ω22+b+C(C(Ω22,0),a)+C(C(Ω2,a),a),0),0)
  • {1,,1{1,,1{1,,1,,2}2◊2}1{1{1,,1{1,,1◊2}2◊2}2,,1◊2}2{1,,1,,2}2◊2} has recursion level C(C(Ω22+b+C(C(Ω22,0),a)+C(C(Ω2+a,a),a),0),0)
  • {1,,1{1,,1{1,,1,,2}2◊2}1{1{1,,1{1,,1,,2}2◊2}2,,1◊2}2{1,,1,,2}2◊2} has recursion level C(C(Ω22+b+C(C(Ω22,0),a)2,0),0)
  • {1,,1{1,,1{1,,1,,2}2◊2}1{2{1,,1{1,,1,,2}2◊2}2,,1◊2}2{1,,1,,2}2◊2} has recursion level C(C(Ω22+b+ω^(C(C(Ω22,0),a)+1),0),0)
  • {1,,1{1,,1{1,,1,,2}2◊2}1{1{1,,1{1,,1,,2}2◊2}1,2,,1◊2}2{1,,1,,2}2◊2} has recursion level C(C(Ω22+b+C(C(Ω22,0)+1,a),0),0)
  • {1,,1{1,,1{1,,1,,2}2◊2}1{1{1{1,,1{1,,1,,2}2◊2}2,,1{1,,1,,2}2◊2}2,,1◊2}2{1,,1,,2}2◊2} has recursion level C(C(Ω22+b+C(C(Ω22,0)2,a),0),0)
  • {1,,1{1,,1{1,,1,,2}2◊2}1{1{1,,2{1,,1,,2}2◊2}2,,1◊2}2{1,,1,,2}2◊2} has recursion level C(C(Ω22+b+C(C(Ω2,C(Ω22,0)),a),0),0)
  • {1,,1{1,,1{1,,1,,2}2◊2}1{1{1,,1{1,,1{1,,1,,2}2◊2}2{1,,1,,2}2◊2}2,,1◊2}2{1,,1,,2}2◊2} has recursion level C(C(Ω22+b+C(C(Ω22,C(Ω22,0)),a),0),0)
  • {1,,1{1,,1{1,,1,,2}2◊2}1{1{1,,1{1,,1{1,,1,,2}2◊2}1,2{1,,1,,2}2◊2}2,,1◊2}2{1,,1,,2}2◊2} has recursion level C(C(Ω22+b+C(C(Ω22+1,0),a),0),0)
  • {1,,1{1,,1{1,,1,,2}2◊2}1{1{1,,1{1,,1{1,,1,,2}2◊2}1{1,,1◊2}2{1,,1,,2}2◊2}2,,1◊2}2{1,,1,,2}2◊2} has recursion level C(C(Ω22+b+C(C(Ω22+a,0),a),0),0)
  • {1,,1{1,,1{1,,1,,2}2◊2}1{1,,2◊2}2{1,,1,,2}2◊2} has recursion level C(C(Ω22+b+C(C(Ω22+C(Ω2,a),0),a),0),0) (it’s different to previous comparisons on {1,,1{1,,1{1,,1,,2}k+1}1{1,,2{1,,1,,2}k}2{1,,1,,2}k+1}′s)
  • {1,,1{1,,1{1,,1,,2}2◊2}1{1,,1{1,,2◊2}2◊2}2{1,,1,,2}2◊2} has recursion level C(C(Ω22+b+C(C(Ω22+C(Ω2+C(Ω2,a),a),0),a),0),0)
  • {1,,1{1,,1{1,,1,,2}2◊2}1{1,,1{1,,1,,2}2◊2}2{1,,1,,2}2◊2} has recursion level C(C(Ω22+b+C(C(Ω22+C(Ω2+C(Ω22,0),a),0),a),0),0)
  • {1,,1{1{1,,1{1,,1{1,,1,,2}2◊2}1{1,,1{1,,1,,2}2◊2}2{1,,1,,2}2◊2}2,,1{1,,1,,2}2}1,2{1,,1,,2}2} has recursion level C(C(Ω22+b+C(C(Ω22+C(Ω2+C(Ω22,0),a),0),a)+1,0),0)
  • {1,,1{1{1,,1{1,,1{1,,1,,2}2◊2}1{1,,1{1,,1,,2}2◊2}2{1,,1,,2}2◊2}2,,1{1,,1,,2}2} 1{1,,1{1,,1,,2}2}2{1,,1,,2}2} has recursion level C(C(Ω22+b+C(C(Ω22+C(Ω2+C(Ω22,0),a),0),a)+C(Ω2+C(Ω22,0),0),0),0)
  • {1,,1{1{1,,1{1,,1{1,,1,,2}2◊2}1{1,,1{1,,1,,2}2◊2}2{1,,1,,2}2◊2}2,,1{1,,1,,2}2} 1{1{1,,1{1,,1{1,,1,,2}2◊2}1{1,,1{1,,1,,2}2◊2}2{1,,1,,2}2◊2}2,,1{1,,1,,2}2}2{1,,1,,2}2} has recursion level C(C(Ω22+b+C(C(Ω22+C(Ω2+C(Ω22,0),a),0),a)+C(C(Ω22+b+C(C(Ω22+C(Ω2+C(Ω22,0),a),0),a),0),C(Ω2+C(Ω22,0),0)),0),0)
  • {1,,1{1{1,,1{1,,1{1,,1,,2}2◊2}1{1,,1{1,,1,,2}2◊2}2{1,,1,,2}2◊2}1,2,,1{1,,1,,2}2}2{1,,1,,2}2} has recursion level C(C(Ω22+b+C(C(Ω22+C(Ω2+C(Ω22,0),a),0),a)+C(C(Ω22+b+C(C(Ω22+C(Ω2+C(Ω22,0),a),0),a),0)+1,C(Ω2+C(Ω22,0),0)),0),0)
  • {1{1,,1{1,,1,,2}3}2,,1{1,,1{1,,1,,2}2◊2}1{1,,1{1,,1,,2}2◊2}2{1,,1,,2}2◊2} has recursion level C(C(Ω22+b+C(C(Ω22+C(Ω2+C(Ω22,0),a),0),a)+C(C(Ω22+b+C(C(Ω22+C(Ω2+C(Ω22,0),a),0),a),0)+C(Ω22,0),C(Ω2+C(Ω22,0),0)),0),0)
  • {1{1,,1◊2}2,,1{1,,1{1,,1,,2}2◊2}1{1,,1{1,,1,,2}2◊2}2{1,,1,,2}2◊2} has recursion level C(C(Ω22+b+C(C(Ω22+C(Ω2+C(Ω22,0),a),0),a)+C(C(Ω22+b+C(C(Ω22+C(Ω2+C(Ω22,0),a),0),a),0)+C(Ω22+b,0),C(Ω2+C(Ω22,0),0)),0),0)
  • {1{1,,1{1,,1{1,,1,,2}2◊2}1{1,,1{1,,1,,2}2◊2}2{1,,1,,2}2◊2} 2,,1{1,,1{1,,1,,2}2◊2}1{1,,1{1,,1,,2}2◊2}2{1,,1,,2}2◊2} has recursion level C(C(Ω22+b+C(C(Ω22+C(Ω2+C(Ω22,0),a),0),a)+C(C(Ω22+b+C(C(Ω22+C(Ω2+C(Ω22,0),a),0),a),0)2,C(Ω2+C(Ω22,0),0)),0),0)
  • {1,,2{1,,1{1,,1,,2}2◊2}1{1,,1{1,,1,,2}2◊2}2{1,,1,,2}2◊2} has recursion level C(C(Ω22+b+C(C(Ω22+C(Ω2+C(Ω22,0),a),0),a)+C(C(Ω2,C(Ω22+b+C(C(Ω22+C(Ω2+C(Ω22,0),a),0),a),0)),C(Ω2+C(Ω22,0),0)),0),0)
  • {1,,1{1,,1{1,,1,,2}2◊2}1{1,,1{1,,1,,2}2◊2}3{1,,1,,2}2◊2} has recursion level C(C(Ω22+b+C(C(Ω22+C(Ω2+C(Ω22,0),a),0),a)+C(C(Ω22+b+C(C(Ω22+C(Ω2+C(Ω22,0),a),0),a),C(Ω22+b+C(C(Ω22+C(Ω2+C(Ω22,0),a),0),a),0)),C(Ω2+C(Ω22,0),0)),0),0)
  • {1,,1{1,,1{1,,1,,2}2◊2}1{1,,1{1,,1,,2}2◊2}1,2{1,,1,,2}2◊2} has recursion level C(C(Ω22+b+C(C(Ω22+C(Ω2+C(Ω22,0),a),0),a)+C(C(Ω22+b+C(C(Ω22+C(Ω2+C(Ω22,0),a),0),a)+1,0),C(Ω2+C(Ω22,0),0)),0),0)
  • {1,,1{1,,1{1,,1,,2}2◊2}1{1,,1{1,,1,,2}2◊2}1{1,,1{1,,1,,2}2}2{1,,1,,2}2◊2} has recursion level C(C(Ω22+b+C(C(Ω22+C(Ω2+C(Ω22,0),a),0),a)+C(C(Ω22+b+C(C(Ω22+C(Ω2+C(Ω22,0),a),0),a)+C(Ω2+C(Ω22,0),0),0),C(Ω2+C(Ω22,0),0)),0),0)
  • {1,,1{1,,1{1,,1,,2}2◊2}1{1,,1{1,,1,,2}2◊2}1{1,,2{1,,1,,2}2}2{1,,1,,2}2◊2} has recursion level C(C(Ω22+b+C(C(Ω22+C(Ω2+C(Ω22,0),a),0),a)+C(Ω2,C(Ω2+C(Ω22,0),0)),0),0)
  • {1,,1{1,,1{1,,1,,2}2◊2}1{1,,1{1,,1,,2}2◊2}1{1,,1{1,,1,,2}1,2}2{1,,1,,2}2◊2} has recursion level C(C(Ω22+b+C(C(Ω22+C(Ω2+C(Ω22,0),a),0),a)+C(Ω2+C(Ω22,0)+1,0),0),0)
  • {1,,1{1,,1{1,,1,,2}2◊2}1{1,,1{1,,1,,2}2◊2}1{1,,1◊2}2{1,,1,,2}2◊2} has recursion level C(C(Ω22+b+C(C(Ω22+C(Ω2+C(Ω22,0),a),0),a)+a,0),0)
  • {1,,1{1,,1{1,,1,,2}2◊2}1{1,,1{1,,1,,2}2◊2}1 {1{1,,1{1,,1{1,,1,,2}2◊2}1{1,,1{1,,1,,2}2◊2}2{1,,1,,2}2◊2}2,,1◊2}2{1,,1,,2}2◊2} has recursion level C(C(Ω22+b+C(C(Ω22+C(Ω2+C(Ω22,0),a),0),a)2,0),0)
  • {1,,1{1,,1{1,,1,,2}2◊2}1{1,,1{1,,1,,2}2◊2}1 {1{1,,1{1,,1{1,,1,,2}2◊2}1{1,,1{1,,1,,2}2◊2}2{1,,1,,2}2◊2}1,2,,1◊2}2{1,,1,,2}2◊2} has recursion level C(C(Ω22+b+C(C(Ω22+C(Ω2+C(Ω22,0),a),0)+1,a),0),0)
  • {1,,1{1,,1{1,,1,,2}2◊2}1{1,,1{1,,1,,2}2◊2}1 {1{1,,2{1,,1{1,,1,,2}2◊2}1{1,,1{1,,1,,2}2◊2}2{1,,1,,2}2◊2}2,,1◊2}2{1,,1,,2}2◊2} has recursion level C(C(Ω22+b+C(C(Ω2,C(Ω22+C(Ω2+C(Ω22,0),a),0)),a),0),0)
  • {1,,1{1,,1{1,,1,,2}2◊2}1{1,,1{1,,1,,2}2◊2}1 {1{1,,1{1,,1{1,,1,,2}2◊2}1{1,,1{1,,1,,2}2◊2}1,2{1,,1,,2}2◊2}2,,1◊2}2{1,,1,,2}2◊2} has recursion level C(C(Ω22+b+C(C(Ω22+C(Ω2+C(Ω22,0),a)+1,0),a),0),0)
  • {1,,1{1,,1{1,,1,,2}2◊2}1{1,,1{1,,1,,2}2◊2}1 {1{1,,1{1,,1{1,,1,,2}2◊2}1{1,,1{1,,1,,2}2◊2}1{1,,1◊2}2{1,,1,,2}2◊2}2,,1◊2}2{1,,1,,2}2◊2} has recursion level C(C(Ω22+b+C(C(Ω22+C(Ω2+C(Ω22,0),a)+a,0),a),0),0)
  • {1,,1{1,,1{1,,1,,2}2◊2}1{1,,1{1,,1,,2}2◊2}1{1,,2◊2}2{1,,1,,2}2◊2} has recursion level C(C(Ω22+b+C(C(Ω22+C(Ω2+C(Ω22,0),a)+C(Ω2,a),0),a),0),0)
  • {1,,1{1,,1{1,,1,,2}2◊2}1{1,,1{1,,1,,2}2◊2}1{1,,1{1,,1,,2}2◊2}2{1,,1,,2}2◊2} has recursion level C(C(Ω22+b+C(C(Ω22+C(Ω2+C(Ω22,0),a)2,0),a),0),0)
  • {1,,1{2,,1{1,,1,,2}2◊2}2{1,,1,,2}2◊2} has recursion level C(C(Ω22+b+C(C(Ω22+ω^(C(Ω2+C(Ω22,0),a)+1),0),a),0),0)
  • {1,,1{1{1,,2{1,,1,,2}2◊2}2,,1{1,,1,,2}2◊2}2{1,,1,,2}2◊2} has recursion level C(C(Ω22+b+C(C(Ω22+C(C(Ω2,C(Ω2+C(Ω22,0),a)),C(Ω2+C(Ω22,0),a)),0),a),0),0)
  • {1,,1{1{1,,1{1,,1{1,,1,,2}2◊2}2{1,,1,,2}2◊2}2,,1{1,,1,,2}2◊2}2{1,,1,,2}2◊2} has recursion level C(C(Ω22+b+C(C(Ω22+C(C(Ω2+C(Ω2+C(Ω22,0),a),C(Ω2+C(Ω22,0),a)),C(Ω2+C(Ω22,0),a)),0),a),0),0)
  • {1,,1{1,,1{1,,1,,2}2◊2}2{1,,1,,2}2◊2} has recursion level C(C(Ω22+b+C(C(Ω22+C(C(Ω2+C(Ω2,C(Ω2+C(Ω22,0),a)),C(Ω2+C(Ω22,0),a)),C(Ω2+C(Ω22,0),a)),0),a),0),0)
  • {1,,1{1,,1,,2}3◊2} has recursion level C(C(Ω22+b+C(C(Ω22+C(C(Ω22,0),C(Ω2+C(Ω22,0),a)),0),a),0),0)
  • {1,,1{1,,1{1,,1,,2}3◊2}1{1,,2{1,,1,,2}2◊2}2{1,,1,,2}3◊2} has recursion level C(C(Ω22+b+C(C(Ω22+C(Ω2,C(Ω2+C(Ω22,0),a)),0),a),0),0)
  • {1,,1{1,,1{1,,1,,2}3◊2}1{1,,1{1,,1,,2}3◊2}2{1,,1,,2}3◊2} has recursion level C(C(Ω22+b+C(C(Ω22+C(Ω2+C(Ω22,0),C(Ω2+C(Ω22,0),a)),0),a),0),0)
  • {1,,1{1,,1,,2}1,2◊2} has recursion level C(C(Ω22+b+C(C(Ω22+C(Ω2+C(Ω22,0)+1,a),0),a),0),0)
  • {1,,1{1,,1,,2}1{1,,1◊2}2◊2} has recursion level C(C(Ω22+b+C(C(Ω22+C(Ω2+C(Ω22,0)+a,a),0),a),0),0)
  • {1,,1{1,,1,,2}1{1,,1,,2}2◊2} has recursion level C(C(Ω22+b+C(C(Ω22+C(Ω2+C(Ω22,0)2,a),0),a),0),0)
  • {1,,1{1{1,,2,,2}2,,1,,2}2◊2} has recursion level C(C(Ω22+b+C(C(Ω22+C(Ω2+C(C(Ω2,C(Ω22,0)),C(Ω22,0)),a),0),a),0),0)
  • {1,,1{1{1,,1{1,,1,,2}2,,2}2,,1,,2}2◊2} has recursion level C(C(Ω22+b+C(C(Ω22+C(Ω2+C(C(Ω2+C(Ω22,0),C(Ω22,0)),C(Ω22,0)),a),0),a),0),0)
  • {1,,1{1{1,,1{1,,2,,2}2,,2}2,,1,,2}2◊2} has recursion level C(C(Ω22+b+C(C(Ω22+C(Ω2+C(C(Ω22,C(Ω22,0)),C(Ω22,0)),a),0),a),0),0)
  • {1,,1{1{1,,1{1,,2,,2}1,2,,2}2,,1,,2}2◊2} has recursion level C(C(Ω22+b+C(C(Ω22+C(Ω2+C(C(Ω22+1,0),C(Ω22,0)),a),0),a),0),0)
  • {1,,1{1{1,,1{1,,2,,2}1{1,,1{1{1,,1{1,,2,,2}1,2,,2}2,,1,,2}2}2,,2}2,,1,,2}2◊2} has recursion level C(C(Ω22+b+C(C(Ω22+C(Ω2+C(C(Ω22+C(Ω2+C(C(Ω22+1,0),C(Ω22,0)),0),0),C(Ω22,0)),a),0),a),0),0)
  • {1,,1{1{1,,1{1,,2,,2}1{1,,1◊2}2,,2}2,,1,,2}2◊2} has recursion level C(C(Ω22+b+C(C(Ω22+C(Ω2+C(C(Ω22+a,0),C(Ω22,0)),a),0),a),0),0)
  • {1,,1{1{1,,1{1,,2,,2}1{1,,1{1,,1,,2}2◊2}2,,2}2,,1,,2}2◊2} has recursion level C(C(Ω22+b+C(C(Ω22+C(Ω2+C(C(Ω22+C(Ω2+C(Ω22,0),a),0),C(Ω22,0)),a),0),a),0),0)
  • {1,,1{1{1,,1{1,,2,,2}1{1,,1{1{1,,1{1,,2,,2}1{1,,1◊2}2,,2}2,,1,,2}2◊2}2,,2}2,,1,,2}2◊2} has recursion level C(C(Ω22+b+C(C(Ω22+C(Ω2+C(C(Ω22+C(Ω2+C(C(Ω22+a,0),C(Ω22,0)),a),0),C(Ω22,0)),a),0),a),0),0)
  • {1,,1◊3} has recursion level C(C(Ω22+b+C(C(Ω22+b,0),a),0),0)
  • {1,,1{1,,1◊3}1{1,,2{1,,1,,2}2}2◊3} has recursion level C(C(Ω22+b+C(C(Ω22+b,0),a)+C(Ω2,C(Ω2+C(Ω22,0),0)),0),0)
  • {1,,1{1,,1◊3}1{1,,1◊2}2◊3} has recursion level C(C(Ω22+b+C(C(Ω22+b,0),a)+a,0),0)
  • {1,,1{1,,1◊3}1{1{1,,1◊3}2,,1◊2}2◊3} has recursion level C(C(Ω22+b+C(C(Ω22+b,0),a)2,0),0)
  • {1,,1{1,,1◊3}1{1{1,,1◊3}1,2,,1◊2}2◊3} has recursion level C(C(Ω22+b+C(C(Ω22+b,0)+1,a),0),0)
  • {1,,1{1,,1◊3}1{1{1{1,,1◊3}2,,1◊3}2,,1◊2}2◊3} has recursion level C(C(Ω22+b+C(C(Ω22+b,0)2,a),0),0)
  • {1,,1{1,,1◊3}1{1{1,,2◊3}2,,1◊2}2◊3} has recursion level C(C(Ω22+b+C(C(Ω2,C(Ω22+b,0)),a),0),0)
  • {1,,1{1,,1◊3}1{1{1,,1{1,,1◊2}2◊3}2,,1◊2}2◊3} has recursion level C(C(Ω22+b+C(C(Ω2+a,C(Ω22+b,0)),a),0),0)
  • {1,,1{1,,1◊3}1{1{1,,1{1,,1{1,,1,,2}2◊2}2◊3}2,,1◊2}2◊3} has recursion level C(C(Ω22+b+C(C(Ω22,C(Ω22+b,0)),a),0),0)
  • {1,,1{1,,1◊3}1{1{1,,1{1,,1◊3}2◊3}2,,1◊2}2◊3} has recursion level C(C(Ω22+b+C(C(Ω22+b,C(Ω22+b,0)),a),0),0)
  • {1,,1{1,,1◊3}1{1{1,,1{1,,1◊3}1,2◊3}2,,1◊2}2◊3} has recursion level C(C(Ω22+b+C(C(Ω22+b+1,0),a),0),0)
  • {1,,1{1,,1◊3}1{1{1,,1{1,,1◊3}1{1,,1◊2}2◊3}2,,1◊2}2◊3} has recursion level C(C(Ω22+b+C(C(Ω22+b+a,0),a),0),0)
  • {1,,1{1,,1◊3}1{1{1,,1{1,,1◊3}1{1{1,,1{1,,1◊3}1,2◊3}2,,1◊2}2◊3}2,,1◊2}2◊3} has recursion level C(C(Ω22+b+C(C(Ω22+b+C(C(Ω22+b+1,0),a),0),a),0),0)
  • {1,,1{1,,1◊3}1{1,,2◊2}2◊3} has recursion level C(C(Ω22+b+C(Ω2,a),0),0)
  • {1,,1{1,,1◊3}1{1,,1{1,,1,,2}2◊2}2◊3} has recursion level C(C(Ω22+b+C(Ω2+C(Ω22,0),a),0),0)
  • {1,,1{1,,1◊3}1{1,,1{1{1,,2,,2}2,,1,,2}2◊2}2◊3} has recursion level C(C(Ω22+b+C(Ω2+C(C(Ω2,C(Ω22,0)),C(Ω22,0)),a),0),0)
  • {1,,1{1,,1◊3}1{1,,1{1{1,,1{1,,2,,2}1,2,,2}2,,1,,2}2◊2}2◊3} has recursion level C(C(Ω22+b+C(Ω2+C(C(Ω22+1,0),C(Ω22,0)),a),0),0)
  • {1,,1{1,,1◊3}1{1,,1{1{1,,1{1,,2,,2}1{1,,1◊2}2,,2}2,,1,,2}2◊2}2◊3} has recursion level C(C(Ω22+b+C(Ω2+C(C(Ω22+a,0),C(Ω22,0)),a),0),0)
  • {1,,1{1,,1◊3}1{1,,1◊3}2◊3} has recursion level C(C(Ω22+b+C(Ω2+C(C(Ω22+b,0),C(Ω22,0)),a),0),0)
  • {1,,1◊4} has recursion level C(C(Ω22+b+C(C(Ω22+b,0),C(Ω2+C(C(Ω22+b,0),C(Ω22,0)),a)),0),0)
  • {1,,1{1,,1◊4}1{1,,1◊4}2◊4} has recursion level C(C(Ω22+b+C(Ω2+C(C(Ω22+b,0),C(Ω22,0)),C(Ω2+C(C(Ω22+b,0),C(Ω22,0)),a)),0),0)
  • {1,,1◊5} has recursion level C(C(Ω22+b+C(C(Ω22+b,0),C(Ω2+C(C(Ω22+b,0),C(Ω22,0)),C(Ω2+C(C(Ω22+b,0),C(Ω22,0)),a))),0),0)
  • {1,,1◊1,2} has recursion level C(C(Ω22+b+C(Ω2+C(C(Ω22+b,0),C(Ω22,0))+1,0),0),0)
  • {1,,1◊1{1,,1◊1,2}2} has recursion level C(C(Ω22+b+C(Ω2+C(C(Ω22+b,0),C(Ω22,0))+C(Ω2+C(C(Ω22+b,0),C(Ω22,0))+1,0),0),0),0)
  • {1,,1◊1{1,,1,,2}2} has recursion level C(C(Ω22+b+C(Ω2+C(C(Ω22+b,0),C(Ω22,0))+C(Ω22,0),0),0),0)
  • {1,,1◊1{1{1,,2,,2}2,,1,,2}2} has recursion level C(C(Ω22+b+C(Ω2+C(C(Ω22+b,0),C(Ω22,0))+C(C(Ω2,C(Ω22,0)),C(Ω22,0)),0),0),0)
  • {1,,1◊1{1{1,,1{1,,2,,2}2,,2}2,,1,,2}2} has recursion level C(C(Ω22+b+C(Ω2+C(C(Ω22+b,0),C(Ω22,0))+C(C(Ω22,C(Ω22,0)),C(Ω22,0)),0),0),0)
  • {1,,1◊1{1{1,,1{1,,2,,2}1,2,,2}2,,1,,2}2} has recursion level C(C(Ω22+b+C(Ω2+C(C(Ω22+b,0),C(Ω22,0))+C(C(Ω22+1,0),C(Ω22,0)),0),0),0)
  • {1,,1◊1{1{1,,1{1,,2,,2}1{1,,1◊1{1{1,,1{1,,2,,2}1,2,,2}2,,1,,2}2}2,,2}2,,1,,2}2} has recursion level C(C(Ω22+b+C(Ω2+C(C(Ω22+b,0),C(Ω22,0))+C(C(Ω22+C(Ω2+C(C(Ω22+b,0),C(Ω22,0))+C(C(Ω22+1,0),C(Ω22,0)),0),0),C(Ω22,0)),0),0),0)
  • {1,,1◊1◊2} has recursion level C(C(Ω22+b+C(Ω2+C(C(Ω22+b,0),C(Ω22,0))2,0),0),0)

Now let ordinal b = C(Ω2+C(Ω2,C(Ω22,0)),0).

  • {1,,1{2,,1{1,,2,,2}1{1,,1,,2}2,,2}2} has recursion level C(C(Ω22+b+C(Ω2+ω^(C(C(Ω22+b,0),C(Ω22,0))+1),0),0),0)
  • {1{1{1,,1{1,,2,,2}1{1,,1,,2}2,,2}2,,1,,2}2{1,,1{1,,2,,2}1{1,,1,,2}2,,2}2,,1,,2} has recursion level C(C(Ω22+b+C(Ω2+ω^(C(C(Ω22+b,0),C(Ω22,0))2),0),0),0)
  • {1{1,,2,,2}2{1,,1{1,,2,,2}1{1,,1,,2}2,,2}2,,1,,2} has recursion level C(C(Ω22+b+C(Ω2+C(C(Ω2,C(Ω22,0)),C(C(Ω22+b,0),C(Ω22,0))),0),0),0)
  • {1{1,,1{1,,2,,2}1{1,,1,,2}2,,2}3,,1,,2} has recursion level C(C(Ω22+b+C(Ω2+C(C(Ω22+b,0),C(C(Ω22+b,0),C(Ω22,0))),0),0),0)
  • {1{1,,1{1,,2,,2}1{1,,1,,2}2,,2}1,2,,1,,2} has recursion level C(C(Ω22+b+C(Ω2+C(C(Ω22+b,0)+1,C(Ω22,0)),0),0),0)
  • {1{1,,1{1,,2,,2}1{1,,1,,2}2,,2}1{1,,2,,2}2,,1,,2} has recursion level C(C(Ω22+b+C(Ω2+C(C(Ω22+b,0)+C(Ω2,C(Ω22,0)),C(Ω22,0)),0),0),0)
  • {1{1,,1,2,,2}2,,1{1,,2,,2}1{1,,1,,2}2,,2} has recursion level C(C(Ω22+b+C(Ω2+C(C(Ω22+b,0)+C(Ω2+1,C(Ω22,0)),C(Ω22,0)),0),0),0)
  • {1{1,,1{1,,2,,2}1,2,,2}2,,1{1,,2,,2}1{1,,1,,2}2,,2} has recursion level C(C(Ω22+b+C(Ω2+C(C(Ω22+b,0)+C(Ω22+1,0),C(Ω22,0)),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,,2} has recursion level C(C(Ω22+b+C(Ω2+C(C(Ω22+b,0)2,C(Ω22,0)),0),0),0)
  • {1,,2{1,,2,,2}1{1,,1,,2}2,,2} has recursion level C(C(Ω22+b+C(Ω2+C(C(Ω2,C(Ω22+b,0)),C(Ω22,0)),0),0),0)
  • {1,,1{1,,1,,2}2{1,,2,,2}1{1,,1,,2}2,,2} has recursion level C(C(Ω22+b+C(Ω2+C(C(Ω2+C(Ω22,0),C(Ω22+b,0)),C(Ω22,0)),0),0),0)
  • {1,,1{1{1,,1,2{1,,2,,2}1{1,,1,,2}2,,2}2,,1,,2}2{1,,2,,2}1{1,,1,,2}2,,2} has recursion level C(C(Ω22+b+C(Ω2+C(C(Ω2+C(C(Ω2+1,C(Ω22+b,0)),C(Ω22,0)),C(Ω22+b,0)),C(Ω22,0)),0),0),0)
  • {1,,1{1,,2,,2}2{1,,1,,2}2,,2} has recursion level C(C(Ω22+b+C(Ω2+C(C(Ω22,C(Ω22+b,0)),C(Ω22,0)),0),0),0)
  • {1,,1{1,,2,,2}1,2{1,,1,,2}2,,2} has recursion level C(C(Ω22+b+C(Ω2+C(C(Ω22+1,C(Ω22+b,0)),C(Ω22,0)),0),0),0)
  • {1,,1{1,,2,,2}1{1,,1{1,,1{1,,2,,2}1,2{1,,1,,2}2,,2}2}2{1,,1,,2}2,,2} has recursion level C(C(Ω22+b+C(Ω2+C(C(Ω22+C(Ω2+C(C(Ω22+1,C(Ω22+b,0)),C(Ω22,0)),0),C(Ω22+b,0)),C(Ω22,0)),0),0),0)
  • {1,,1{1,,2,,2}1{1,,1,,2}3,,2} has recursion level C(C(Ω22+b+C(Ω2+C(C(Ω22+b,C(Ω22+b,0)),C(Ω22,0)),0),0),0)
  • {1,,1{1,,2,,2}1{1,,1,,2}1,2,,2} has recursion level C(C(Ω22+b+C(Ω2+C(C(Ω22+b+1,0),C(Ω22,0)),0),0),0)
  • {1,,1{1,,2,,2}1{1,,1,,2}1{1,,1{1,,1{1,,2,,2}1{1,,1,,2}1,2,,2}2}2,,2} has recursion level C(C(Ω22+b+C(Ω2+C(C(Ω22+b+C(Ω2+C(C(Ω22+b+1,0),C(Ω22,0)),0),0),C(Ω22,0)),0),0),0)
  • {1,,1{1,,2,,2}1{1,,1,,2}1{1,,1,,2}2,,2} has recursion level C(C(Ω22+b2,0),0)
  • {1,,1{1,,2,,2}1{2,,1,,2}2,,2} has recursion level C(C(Ω22+ω^(b+1),0),0)
  • {1,,1{1,,2,,2}1{1{1,,1{1,,1,,2}2}2,,1,,2}2,,2} has recursion level C(C(Ω22+ω^(b+C(Ω2+C(Ω22,0),0)),0),0)
  • {1,,1{1,,2,,2}1{1{1,,1{1,,1{1,,2,,2}1,2,,2}2}2,,1,,2}2,,2} has recursion level C(C(Ω22+ω^(b+C(Ω2+C(C(Ω22+1,0),C(Ω22,0)),0)),0),0)
  • {1,,1{1,,2,,2}1{1{1,,1{1,,1{1,,2,,2}1{2,,1,,2}2,,2}2}2,,1,,2}2,,2} has recursion level C(C(Ω22+ω^(b+C(Ω2+C(C(Ω22+ω^(b+1),0),C(Ω22,0)),0)),0),0)
  • {1,,1{1,,2,,2}1{1{1,,1,,2}2,,1,,2}2,,2} has recursion level C(C(Ω22+ω^(b2),0),0)
  • {1,,1{1,,2,,2}1{1{1,,2,,2}2,,1,,2}2,,2} has recursion level C(C(Ω22+C(C(Ω2,b),b),0),0)
  • {1,,1{1,,2,,2}1{1{1,,1{1,,1{1,,1,,2}2}2,,2}2,,1,,2}2,,2} has recursion level C(C(Ω22+C(C(Ω2+C(Ω2+C(Ω22,0),0),b),b),0),0)
  • {1,,1{1,,2,,2}1{1{1,,1{1,,1{1,,1{1,,2,,2}1{1,,1,,2}2,,2}2}2,,2}2,,1,,2}2,,2} has recursion level C(C(Ω22+C(C(Ω2+C(Ω2+C(C(Ω22+b,0),C(Ω22,0)),0),b),b),0),0)
  • {1,,1{1,,2,,2}1{1{1,,1{1,,1{1,,1{1,,2,,2}1{1{1,,2,,2}2,,1,,2}2,,2}2}2,,2}2,,1,,2}2,,2} has recursion level C(C(Ω22+C(C(Ω2+C(Ω2+C(C(Ω22+C(C(Ω2,b),b),0),C(Ω22,0)),0),b),b),0),0)
  • {1,,1{1,,2,,2}1{1{1,,1{1,,1,,2}2,,2}2,,1,,2}2,,2} has recursion level C(C(Ω22+C(C(Ω2+b,b),b),0),0)
  • {1,,1{1,,2,,2}1{1{1,,1{1,,2,,2}2,,2}2,,1,,2}2,,2} has recursion level C(C(Ω22+C(C(Ω2+C(Ω2,b),b),b),0),0)
  • {1,,1{1,,2,,2}1{1{1,,1{1,,2,,2}1,2,,2}2,,1,,2}2,,2} has recursion level C(C(Ω22+C(C(Ω2+C(Ω2,b)+1,b),b),0),0)
  • {1,,1{1,,2,,2}1{1,,2,,2}2,,2} has recursion level C(C(Ω22+C(C(Ω2+C(Ω2,b)2,b),b),0),0)
  • {1,,1{1,,1,2,,2}2,,2} has recursion level C(C(Ω22+C(C(Ω2+C(Ω2+1,b),b),b),0),0)
  • {1,,1{1,,1{1,,1,,2}2,,2}2,,2} has recursion level C(C(Ω22+C(C(Ω2+C(Ω2+b,b),b),b),0),0)
  • {1,,1,,3} has recursion level C(C(Ω22+C(C(Ω22,0),b),0),0)

So {1,,1,,3} has recursion level C(C(Ω22+C(C(Ω22,0),C(Ω2+C(Ω2,C(Ω22,0)),0)),0),0). Here’re some approximations, to make the comparisons above more simple:

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

However, C(Ω22,0) still works in an irregular way.

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

Now let ordinal b = C(Ω2+C(Ω2,C(Ω22,0)),0).

  • {1,,1{1{1,,1{1,,1,,3}2}2,,1{1,,1,,2}2}2{1,,1,,2}2} has recursion level C(C(Ω22+C(C(Ω22,0),b),0),0)
  • {1,,1{1{1,,1{1,,1,,3}2}2,,1{1,,1,,2}2}1{1{1,,1{1,,1,,2}3}2,,1{1,,1,,2}2}2{1,,1,,2}2} has recursion level C(C(Ω22+C(C(Ω22,0),b)+C(C(Ω22,0),C(Ω2+C(Ω22,0),0)),0),0)
  • {1,,1{1{1,,1{1,,1,,3}2}2,,1{1,,1,,2}2}1{1{1,,1{1,,1,,2}1,2}2,,1{1,,1,,2}2}2{1,,1,,2}2} has recursion level C(C(Ω22+C(C(Ω22,0),b)+C(C(Ω22+C(Ω2+C(Ω22,0)+1,0),0),C(Ω2+C(Ω22,0),0)),0),0)
  • {1,,1{1{1,,1{1,,1,,3}2}2,,1{1,,1,,2}2}1{1{1,,1{1,,1{1,,2,,2}1,2,,2}2}2,,1{1,,1,,2}2}2{1,,1,,2}2} has recursion level C(C(Ω22+C(C(Ω22,0),b)+C(C(Ω22+C(Ω2+C(C(Ω22+1,0),C(Ω22,0)),0),0),C(Ω2+C(Ω22,0),0)),0),0)
  • {1,,1{1{1,,1{1,,1,,3}2}2,,1{1,,1,,2}2}1{1{1,,1{1,,1{1,,2,,2}1{1,,1,,2}2,,2}2}2,,1{1,,1,,2}2}2{1,,1,,2}2} has recursion level C(C(Ω22+C(C(Ω22,0),b)+C(C(Ω22+b,0),C(Ω2+C(Ω22,0),0)),0),0)
  • {1,,1{1{1,,1{1,,1,,3}2}2,,1{1,,1,,2}2}1 {1{1,,1{1,,1{1,,2,,2}1{1,,1,,2}1{1,,1,,2}2,,2}2}2,,1{1,,1,,2}2}2{1,,1,,2}2} has recursion level C(C(Ω22+C(C(Ω22,0),b)+C(C(Ω22+b2,0),C(Ω2+C(Ω22,0),0)),0),0)
  • {1,,1{1{1,,1{1,,1,,3}2}2,,1{1,,1,,2}2}1{1{1,,1{1,,1,,3}2}2,,1{1,,1,,2}2}2{1,,1,,2}2} has recursion level C(C(Ω22+C(C(Ω22,0),b)+C(C(Ω22+C(C(Ω22,0),b),0),C(Ω2+C(Ω22,0),0)),0),0)
  • {1,,1{1{1,,1{1,,1,,3}2}1,2,,1{1,,1,,2}2}2{1,,1,,2}2} has recursion level C(C(Ω22+C(C(Ω22,0),b)+C(C(Ω22+C(C(Ω22,0),b),0)+1,C(Ω2+C(Ω22,0),0)),0),0)
  • {1{1,,1{1,,1,,2}3}2,,1{1,,1,,3}2} has recursion level C(C(Ω22+C(C(Ω22,0),b)+C(C(Ω22+C(C(Ω22,0),b),0)+C(Ω22,0),C(Ω2+C(Ω22,0),0)),0),0)
  • {1{1,,1{1,,1{1,,2,,2}1{1,,1,,2}2,,2}2}2,,1{1,,1,,3}2} has recursion level C(C(Ω22+C(C(Ω22,0),b)+C(C(Ω22+C(C(Ω22,0),b),0)+C(Ω22+b,0),C(Ω2+C(Ω22,0),0)),0),0)
  • {1{1,,1{1,,1{1,,2,,2}1{1,,1,,2}1{1,,1,,2}2,,2}2}2,,1{1,,1,,3}2} has recursion level C(C(Ω22+C(C(Ω22,0),b)+C(C(Ω22+C(C(Ω22,0),b),0)+C(Ω22+b2,0),C(Ω2+C(Ω22,0),0)),0),0)
  • {1{1,,1{1,,1{1,,2,,2}1{1,,2,,2}2,,2}2}2,,1{1,,1,,3}2} has recursion level C(C(Ω22+C(C(Ω22,0),b)+C(C(Ω22+C(C(Ω22,0),b),0)+C(Ω22+C(C(Ω2+C(Ω2,b)2,b),b),0),C(Ω2+C(Ω22,0),0)),0),0)
  • {1{1,,1{1,,1,,3}2}2,,1{1,,1,,3}2} has recursion level C(C(Ω22+C(C(Ω22,0),b)+C(C(Ω22+C(C(Ω22,0),b),0)2,C(Ω2+C(Ω22,0),0)),0),0)
  • {1,,2{1,,1,,3}2} has recursion level C(C(Ω22+C(C(Ω22,0),b)+C(C(Ω2,C(Ω22+C(C(Ω22,0),b),0)),C(Ω2+C(Ω22,0),0)),0),0)
  • {1,,1{1,,1{1,,1,,2}3}2{1,,1,,3}2} has recursion level C(C(Ω22+C(C(Ω22,0),b)+C(C(Ω22,C(Ω22+C(C(Ω22,0),b),0)),C(Ω2+C(Ω22,0),0)),0),0)
  • {1,,1{1,,1{1,,1{1,,2,,2}1{1,,1,,2}2,,2}2}2{1,,1,,3}2} has recursion level C(C(Ω22+C(C(Ω22,0),b)+C(C(Ω22+b,C(Ω22+C(C(Ω22,0),b),0)),C(Ω2+C(Ω22,0),0)),0),0)
  • {1,,1{1,,1{1,,1,,3}2}2{1,,1,,3}2} has recursion level C(C(Ω22+C(C(Ω22,0),b)+C(C(Ω22+C(C(Ω22,0),b),C(Ω22+C(C(Ω22,0),b),0)),C(Ω2+C(Ω22,0),0)),0),0)
  • {1,,1{1,,1{1,,1,,3}2}1,2{1,,1,,3}2} has recursion level C(C(Ω22+C(C(Ω22,0),b)+C(C(Ω22+C(C(Ω22,0),b)+1,0),C(Ω2+C(Ω22,0),0)),0),0)
  • {1,,1{1,,1{1,,1,,3}2}1{1,,1{1,,1,,2}2}2{1,,1,,3}2} has recursion level C(C(Ω22+C(C(Ω22,0),b)+C(C(Ω22+C(C(Ω22,0),b)+C(Ω2+C(Ω22,0),0),0),C(Ω2+C(Ω22,0),0)),0),0)
  • {1,,1{1,,1{1,,1,,3}2}1{1,,2{1,,1,,2}2}2{1,,1,,3}2} has recursion level C(C(Ω22+C(C(Ω22,0),b)+C(Ω2,C(Ω2+C(Ω22,0),0)),0),0)
  • {1,,1{1,,1{1,,1,,3}2}1{1,,1{1,,1{1,,2,,2}1,2,,2}2}2{1,,1,,3}2} has recursion level C(C(Ω22+C(C(Ω22,0),b)+C(Ω2+C(C(Ω22+1,0),C(Ω22,0)),0),0),0)
  • {1,,1{1,,1{1,,1,,3}2}1{1,,1{1,,1{1,,2,,2}1{1,,1,,2}2,,2}2}2{1,,1,,3}2} has recursion level C(C(Ω22+C(C(Ω22,0),b)+C(Ω2+C(C(Ω22+b,0),C(Ω22,0)),0),0),0)
  • {1,,1{1,,1{1,,1,,3}2}1{1,,1{1,,1{1,,2,,2}1{1,,1,,2}1{1,,1,,2}2,,2}2}2{1,,1,,3}2} has recursion level C(C(Ω22+C(C(Ω22,0),b)+C(Ω2+C(C(Ω22+b2,0),C(Ω22,0)),0),0),0)
  • {1,,1{1,,1{1,,1,,3}2}1{1,,1{1,,1,,3}2}2{1,,1,,3}2} has recursion level C(C(Ω22+C(C(Ω22,0),b)+C(Ω2+C(C(Ω22+C(C(Ω22,0),b),0),C(Ω22,0)),0),0),0)
  • {1,,1{1,,1,,2}2{1,,1,,3}2} has recursion level C(C(Ω22+C(C(Ω22,0),b)+C(C(Ω22,0),C(Ω2+C(C(Ω22+C(C(Ω22,0),b),0),C(Ω22,0)),0)),0),0)
  • {1,,1{1,,1,,2}1,2{1,,1,,3}2} has recursion level C(C(Ω22+C(C(Ω22,0),b)+C(Ω2+C(Ω22,0)+1,C(Ω2+C(C(Ω22+C(C(Ω22,0),b),0),C(Ω22,0)),0)),0),0)
  • {1,,1{1,,2,,2}2{1,,1,,3}2} has recursion level C(C(Ω22+C(C(Ω22,0),b)+C(Ω2+C(C(Ω2,C(Ω22,0)),C(Ω22,0)),C(Ω2+C(C(Ω22+C(C(Ω22,0),b),0),C(Ω22,0)),0)),0),0)
  • {1,,1{1,,1,,3}3} has recursion level C(C(Ω22+C(C(Ω22,0),b)+C(Ω2+C(C(Ω22+C(C(Ω22,0),b),0),C(Ω22,0)),C(Ω2+C(C(Ω22+C(C(Ω22,0),b),0),C(Ω22,0)),0)),0),0)
  • {1,,1{1,,1,,3}1,2} has recursion level C(C(Ω22+C(C(Ω22,0),b)+C(Ω2+C(C(Ω22+C(C(Ω22,0),b),0),C(Ω22,0))+1,0),0),0)
  • {1,,1{1,,1,,3}1{1,,1,,3}2} has recursion level C(C(Ω22+C(C(Ω22,0),b)+C(Ω2+C(C(Ω22+C(C(Ω22,0),b),0),C(Ω22,0))2,0),0),0)
  • {1,,1{2,,1{1,,1,,3}2,,2}2} has recursion level C(C(Ω22+C(C(Ω22,0),b)+C(Ω2+ω^(C(C(Ω22+C(C(Ω22,0),b),0),C(Ω22,0))+1),0),0),0)
  • {1{1,,2,,2}2{1,,1{1,,1,,3}2,,2}2,,1,,2} has recursion level C(C(Ω22+C(C(Ω22,0),b)+C(Ω2+C(C(Ω2,C(Ω22,0)),C(C(Ω22+C(C(Ω22,0),b),0),C(Ω22,0))),0),0),0)
  • {1{1,,1{1,,2,,2}1,2,,2}2{1,,1{1,,1,,3}2,,2}2,,1,,2} has recursion level C(C(Ω22+C(C(Ω22,0),b)+C(Ω2+C(C(Ω22+1,0),C(C(Ω22+C(C(Ω22,0),b),0),C(Ω22,0))),0),0),0)
  • {1{1,,1{1,,2,,2}1{1,,1,,2}2,,2}2{1,,1{1,,1,,3}2,,2}2,,1,,2} has recursion level C(C(Ω22+C(C(Ω22,0),b)+C(Ω2+C(C(Ω22+b,0),C(C(Ω22+C(C(Ω22,0),b),0),C(Ω22,0))),0),0),0)
  • {1{1,,1{1,,1,,3}2,,2}3,,1,,2} has recursion level C(C(Ω22+C(C(Ω22,0),b)+C(Ω2+C(C(Ω22+C(C(Ω22,0),b),0),C(C(Ω22+C(C(Ω22,0),b),0),C(Ω22,0))),0),0),0)
  • {1{1,,1{1,,1,,3}2,,2}1,2,,1,,2} has recursion level C(C(Ω22+C(C(Ω22,0),b)+C(Ω2+C(C(Ω22+C(C(Ω22,0),b),0)+1,C(Ω22,0)),0),0),0)
  • {1{1,,1{1,,1,,3}2,,2}1{1,,1,,2}2,,1,,2} has recursion level C(C(Ω22+C(C(Ω22,0),b)+C(Ω2+C(C(Ω22+C(C(Ω22,0),b),0)+C(Ω22,0),C(Ω22,0)),0),0),0)
  • {1{1,,1{1,,2,,2}1{1,,1,,2}2,,2}2,,1{1,,2,,2}1{1{1,,1{1,,1,,3}2,,2}2,,1,,2}2,,2} has recursion level C(C(Ω22+C(C(Ω22,0),b)+C(Ω2+C(C(Ω22+C(C(Ω22,0),b),0)+C(Ω22+b,0),C(Ω22,0)),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,,1{1,,1,,3}2,,2}2,,1,,2}2,,2} has recursion level C(C(Ω22+C(C(Ω22,0),b)+C(Ω2+C(C(Ω22+C(C(Ω22,0),b),0)2,C(Ω22,0)),0),0),0)
  • {1,,2{1,,2,,2}1{1{1,,1{1,,1,,3}2,,2}2,,1,,2}2,,2} has recursion level C(C(Ω22+C(C(Ω22,0),b)+C(Ω2+C(C(Ω2,C(Ω22+C(C(Ω22,0),b),0)),C(Ω22,0)),0),0),0)
  • {1,,1{1,,2,,2}2{1{1,,1{1,,1,,3}2,,2}2,,1,,2}2,,2} has recursion level C(C(Ω22+C(C(Ω22,0),b)+C(Ω2+C(C(Ω22,C(Ω22+C(C(Ω22,0),b),0)),C(Ω22,0)),0),0),0)
  • {1,,1{1,,2,,2}1{1,,1,,2}2{1{1,,1{1,,1,,3}2,,2}2,,1,,2}2,,2} has recursion level C(C(Ω22+C(C(Ω22,0),b)+C(Ω2+C(C(Ω22+b,C(Ω22+C(C(Ω22,0),b),0)),C(Ω22,0)),0),0),0)
  • {1,,1{1,,2,,2}1{1{1,,1{1,,1,,3}2,,2}2,,1,,2}3,,2} has recursion level C(C(Ω22+C(C(Ω22,0),b)+C(Ω2+C(C(Ω22+C(C(Ω22,0),b),C(Ω22+C(C(Ω22,0),b),0)),C(Ω22,0)),0),0),0)
  • {1,,1{1,,2,,2}1{1{1,,1{1,,1,,3}2,,2}2,,1,,2}1,2,,2} has recursion level C(C(Ω22+C(C(Ω22,0),b)+C(Ω2+C(C(Ω22+C(C(Ω22,0),b)+1,0),C(Ω22,0)),0),0),0)
  • {1,,1{1,,2,,2}1{1{1,,1{1,,1,,3}2,,2}2,,1,,2}1{1,,1,,2}2,,2} has recursion level C(C(Ω22+C(C(Ω22,0),b)+b,0),0)
  • {1,,1{1,,2,,2}1{1{1,,1{1,,1,,3}2,,2}2,,1,,2}1{1{1,,2,,2}2,,1,,2}2,,2} has recursion level C(C(Ω22+C(C(Ω22,0),b)+C(C(Ω2,b),b),0),0)
  • {1,,1{1,,2,,2}1{1{1,,1{1,,1,,3}2,,2}2,,1,,2}1{1{1,,1{1,,1,,3}2,,2}2,,1,,2}2,,2} has recursion level C(C(Ω22+C(C(Ω22,0),b)2,0),0)
  • {1,,1{1,,2,,2}1{1{1,,2,,2}2{1,,1{1,,1,,3}2,,2}2,,1,,2}2,,2} has recursion level C(C(Ω22+C(C(Ω2,b),C(C(Ω22,0),b)),0),0)
  • {1,,1{1,,2,,2}1{1{1,,1{1,,1,,3}2,,2}3,,1,,2}2,,2} has recursion level C(C(Ω22+C(C(Ω22,0),C(C(Ω22,0),b)),0),0)
  • {1,,1{1,,2,,2}1{1{1,,1{1,,1,,3}2,,2}1,2,,1,,2}2,,2} has recursion level C(C(Ω22+C(C(Ω22,0)+1,b),0),0)
  • {1,,1{1,,2,,2}1{1{1,,1{1,,1,,3}2,,2}1{1,,2,,2}2,,1,,2}2,,2} has recursion level C(C(Ω22+C(C(Ω22,0)+C(Ω2,b),b),0),0)
  • {1,,1{1,,2,,2}1{1{1{1,,1,2,,2}2,,1{1,,1,,3}2,,2}2,,1,,2}2,,2} has recursion level C(C(Ω22+C(C(Ω22,0)+C(Ω2+1,b),b),0),0)
  • {1{1,,1{1,,1,,3}2,,2}2,,1{1,,1,,3}2,,2} has recursion level C(C(Ω22+C(C(Ω22,0)2,b),0),0)
  • {1,,2{1,,1,,3}2,,2} has recursion level C(C(Ω22+C(C(Ω2,C(Ω22,0)),b),0),0)
  • {1,,1,2{1,,1,,3}2,,2} has recursion level C(C(Ω22+C(C(Ω2+1,C(Ω22,0)),b),0),0)
  • {1,,1{1,,1,,2}2{1,,1,,3}2,,2} has recursion level C(C(Ω22+C(C(Ω2+b,C(Ω22,0)),b),0),0)
  • {1,,1{1,,2,,2}2{1,,1,,3}2,,2} has recursion level C(C(Ω22+C(C(Ω2+C(Ω2,b),C(Ω22,0)),b),0),0)
  • {1,,1{1,,1{1,,1,,2}2,,2}2{1,,1,,3}2,,2} has recursion level C(C(Ω22+C(C(Ω2+C(Ω2+b,b),C(Ω22,0)),b),0),0)
  • {1,,1{1,,1{1,,1,,3}2,,2}2{1,,1,,3}2,,2} has recursion level C(C(Ω22+C(C(Ω22,C(Ω22,0)),b),0),0)
  • {1,,1{1,,1{1,,1,,3}2,,2}1,2{1,,1,,3}2,,2} has recursion level C(C(Ω22+C(C(Ω22+1,0),b),0),0)
  • {1,,1{1,,1{1,,1,,3}2,,2}1{1,,1,,2}2{1,,1,,3}2,,2} has recursion level C(C(Ω22+C(C(Ω22+b,0),b),0),0)
  • {1,,1{1,,1{1,,1,,3}2,,2}1{1{1,,1{1,,1{1,,1,,3}2,,2}1,2{1,,1,,3}2,,2}2,,1,,2}2{1,,1,,3}2,,2} has recursion level C(C(Ω22+C(C(Ω22+C(C(Ω22+1,0),b),0),b),0),0)
  • {1,,1{1,,1{1,,1,,3}2,,2}1{1,,2,,2}2{1,,1,,3}2,,2} has recursion level C(C(Ω22+C(Ω2,b),0),0)
  • {1,,1{1,,1{1,,1,,3}2,,2}1{1,,1{1,,2,,2}2,,2}2{1,,1,,3}2,,2} has recursion level C(C(Ω22+C(Ω2+C(Ω2,b),b),0),0)
  • {1,,1{1,,1{1,,1,,3}2,,2}1{1,,1{1,,1,,3}2,,2}2{1,,1,,3}2,,2} has recursion level C(C(Ω22+C(Ω2+C(Ω22,0),b),0),0)
  • {1,,1{1{1,,2{1,,1,,3}2,,2}2,,1{1,,1,,3}2,,2}2{1,,1,,3}2,,2} has recursion level C(C(Ω22+C(C(Ω2,C(Ω2+C(Ω22,0),b)),C(Ω2+C(Ω22,0),b)),0),0)
  • {1,,1{1,,2{1,,1,,3}2,,2}2{1,,1,,3}2,,2} has recursion level C(C(Ω22+C(C(Ω2+C(Ω2,C(Ω2+C(Ω22,0),b)),C(Ω2+C(Ω22,0),b)),C(Ω2+C(Ω22,0),b)),0),0)
  • {1,,1{1,,1,,3}3,,2} has recursion level C(C(Ω22+C(C(Ω22,0),C(Ω2+C(Ω22,0),b)),0),0)
  • {1,,1{1,,1{1,,1,,3}3,,2}1{1,,1{1,,1,,3}3,,2}2{1,,1,,3}3,,2} has recursion level C(C(Ω22+C(Ω2+C(Ω22,0),C(Ω2+C(Ω22,0),b)),0),0)
  • {1,,1{1,,1,,3}1,2,,2} has recursion level C(C(Ω22+C(Ω2+C(Ω22,0)+1,b),0),0)
  • {1,,1{1,,1,,3}1{1,,1{1,,1{1,,1,,3}1,2,,2}2}2,,2} has recursion level C(C(Ω22+C(Ω2+C(Ω22,0)+C(Ω2+C(C(Ω22+C(Ω2+C(Ω22,0)+1,b),0),C(Ω22,0)),0),b),0),0)
  • {1,,1{1,,1,,3}1{1,,1,,2}2,,2} has recursion level C(C(Ω22+C(Ω2+C(Ω22,0)+b,b),0),0)
  • {1,,1{1,,1,,3}1{1,,2,,2}2,,2} has recursion level C(C(Ω22+C(Ω2+C(Ω22,0)+C(Ω2,b),b),0),0)
  • {1,,1{1,,1,,3}1{1,,1{1,,1,,3}2,,2}2,,2} has recursion level C(C(Ω22+C(Ω2+C(Ω22,0)+C(Ω2+C(Ω22,0),b),b),0),0)
  • {1,,1{1,,1,,3}1{1,,1{1,,1,,3}1,2,,2}2,,2} has recursion level C(C(Ω22+C(Ω2+C(Ω22,0)+C(Ω2+C(Ω22,0)+1,b),b),0),0)
  • {1,,1{1,,1,,3}1{1,,1,,3}2,,2} has recursion level C(C(Ω22+C(Ω2+C(Ω22,0)2,b),0),0)
  • {1,,1{2,,1,,3}2,,2} has recursion level C(C(Ω22+C(Ω2+ω^(C(Ω22,0)+1),b),0),0)
  • {1{1,,1,,3}2,,1,,3} has recursion level C(C(Ω22+C(Ω2+ω^(C(Ω22,0)2),b),0),0)
  • {1,,2,,3} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2,C(Ω22,0)),C(Ω22,0)),b),0),0)
  • {1,,1,2,,3} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2+1,C(Ω22,0)),C(Ω22,0)),b),0),0)
  • {1,,1{1,,1,,2}2,,3} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2+b,C(Ω22,0)),C(Ω22,0)),b),0),0)
  • {1,,1{1,,1,,3}2,,3} has recursion level C(C(Ω22+C(Ω2+C(C(Ω2+C(Ω22,0),C(Ω22,0)),C(Ω22,0)),b),0),0)
  • {1,,1{1,,2,,3}2,,3} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22,C(Ω22,0)),C(Ω22,0)),b),0),0)
  • {1,,1{1,,2,,3}1,2,,3} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22+1,0),C(Ω22,0)),b),0),0)
  • {1,,1{1,,2,,3}1{1,,1,,2}2,,3} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22+b,0),C(Ω22,0)),b),0),0)
  • {1,,1{1,,2,,3}1{1,,1{1,,1{1,,2,,3}1{1,,1,,2}2,,3}2,,2}2,,3} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22+C(Ω2+C(C(Ω22+b,0),C(Ω22,0)),b),0),C(Ω22,0)),b),0),0)
  • {1,,1{1,,2,,3}1{1,,1,,3}2,,3} has recursion level C(C(Ω22+C(Ω2+C(Ω2,C(Ω22,0)),b),0),0)

Now let separator ♦ = {1,,1{1,,2,,3}1{1,,1,,3}2,,3} and let ordinal a = C(Ω2+C(Ω2,C(Ω22,0)),0), and b = C(Ω2+C(Ω2,C(Ω22,0)),a) = C(Ω2+C(Ω2,C(Ω22,0)),C(Ω2+C(Ω2,C(Ω22,0)),0)).

  • {1,,1{1♦2,,1{1,,1,,2}2}2{1,,1,,2}2} has recursion level C(C(Ω22+b,0),0)
  • {1,,1{2♦2,,1{1,,1,,2}2}2{1,,1,,2}2} has recursion level C(C(Ω22+b+ω^(C(C(Ω22+b,0),C(Ω2+C(Ω22,0),0))+1),0),0)
  • {1,,1{1♦1,2,,1{1,,1,,2}2}2{1,,1,,2}2} has recursion level C(C(Ω22+b+C(C(Ω22+b,0)+1,C(Ω2+C(Ω22,0),0)),0),0)
  • {1{1,,1{1,,1{1,,2,,2}1{1,,1,,2}2,,2}2}2,,1♦2} has recursion level C(C(Ω22+b+C(C(Ω22+b,0)+C(Ω22+a,0),C(Ω2+C(Ω22,0),0)),0),0)
  • {1{1,,1{1,,1,,3}2}2,,1♦2} has recursion level C(C(Ω22+b+C(C(Ω22+b,0)+C(Ω22+C(C(Ω22,0),a),0),C(Ω2+C(Ω22,0),0)),0),0)
  • {1{1,,1{1,,1{1,,1{1,,1,,3}2,,2}1,2{1,,1,,3}2,,2}2}2,,1♦2} has recursion level C(C(Ω22+b+C(C(Ω22+b,0)+C(Ω22+C(C(Ω22+1,0),a),0),C(Ω2+C(Ω22,0),0)),0),0)
  • {1{1,,1{1,,1{1,,1{1,,1,,3}2,,2}1{1,,2,,2}2{1,,1,,3}2,,2}2}2,,1♦2} has recursion level C(C(Ω22+b+C(C(Ω22+b,0)+C(Ω22+C(Ω2,a),0),C(Ω2+C(Ω22,0),0)),0),0)
  • {1{1,,1{1,,1{1,,1,,3}1,2,,2}2}2,,1♦2} has recursion level C(C(Ω22+b+C(C(Ω22+b,0)+C(Ω22+C(Ω2+C(Ω22,0)+1,a),0),C(Ω2+C(Ω22,0),0)),0),0)
  • {1{1,,1♦2}2,,1♦2} has recursion level C(C(Ω22+b+C(C(Ω22+b,0)2,C(Ω2+C(Ω22,0),0)),0),0)
  • {1,,2♦2} has recursion level C(C(Ω22+b+C(C(Ω2,C(Ω22+b,0)),C(Ω2+C(Ω22,0),0)),0),0)
  • {1,,1{1,,1{1,,1,,2}3}2♦2} has recursion level C(C(Ω22+b+C(C(Ω22,C(Ω22+b,0)),C(Ω2+C(Ω22,0),0)),0),0)
  • {1,,1{1,,1{1,,1{1,,2,,2}1{1,,1,,2}2,,2}2}2♦2} has recursion level C(C(Ω22+b+C(C(Ω22+a,C(Ω22+b,0)),C(Ω2+C(Ω22,0),0)),0),0)
  • {1,,1{1,,1♦2}2♦2} has recursion level C(C(Ω22+b+C(C(Ω22+b,C(Ω22+b,0)),C(Ω2+C(Ω22,0),0)),0),0)
  • {1,,1{1,,1♦2}1,2♦2} has recursion level C(C(Ω22+b+C(C(Ω22+b+1,0),C(Ω2+C(Ω22,0),0)),0),0)
  • {1,,1{1,,1♦2}1{1,,2{1,,1,,2}2}2♦2} has recursion level C(C(Ω22+b+C(Ω2,C(Ω2+C(Ω22,0),0)),0),0)
  • {1,,1{1,,1♦2}1{1,,1{1,,1,,2}1,2}2♦2} has recursion level C(C(Ω22+b+C(Ω2+C(Ω22,0)+1,0),0),0)
  • {1,,1{1,,1♦2}1{1,,1{1,,1{1,,2,,2}1{1,,1,,2}2,,2}2}2♦2} has recursion level C(C(Ω22+b+C(Ω2+C(C(Ω22+a,0),C(Ω22,0)),0),0),0)
  • {1,,1{1,,1♦2}1{1,,1{1,,1,,3}2}2♦2} has recursion level C(C(Ω22+b+C(Ω2+C(C(Ω22+C(C(Ω22,0),a),0),C(Ω22,0)),0),0),0)
  • {1,,1{1,,1♦2}1{1,,1{1,,1{1,,1{1,,1,,3}2,,2}1{1,,2,,2}2{1,,1,,3}2,,2}2}2 ♦2} has recursion level C(C(Ω22+b+C(Ω2+C(C(Ω22+C(Ω2,a),0),C(Ω22,0)),0),0),0)
  • {1,,1{1,,1♦2}1{1,,1{1,,1{1,,1{1,,1,,3}2,,2}1{1,,1{1,,1,,3}2,,2}2{1,,1,,3}2,,2}2}2♦2} has recursion level C(C(Ω22+b+C(Ω2+C(C(Ω22+C(Ω2+C(Ω22,0),a),0),C(Ω22,0)),0),0),0)
  • {1,,1{1,,1♦2}1{1,,1♦2}2♦2} has recursion level C(C(Ω22+b+C(Ω2+C(C(Ω22+b,0),C(Ω22,0)),0),0),0)
  • {1,,1{1,,1,,2}2♦2} has recursion level C(C(Ω22+b+C(C(Ω22,0),C(Ω2+C(C(Ω22+b,0),C(Ω22,0)),0)),0),0)
  • {1,,1{1,,1,,2}1,2♦2} has recursion level C(C(Ω22+b+C(C(Ω22+C(Ω2+C(Ω22,0)+1,C(Ω2+C(C(Ω22+b,0),C(Ω22,0)),0)),0),C(Ω2+C(C(Ω22+b,0),C(Ω22,0)),0)),0),0)
  • {1,,1{1,,1{1,,2,,2}1,2,,2}2♦2} has recursion level C(C(Ω22+b+C(C(Ω22+C(Ω2+C(C(Ω22+1,0),C(Ω22,0)),C(Ω2+C(C(Ω22+b,0),C(Ω22,0)),0)),0),C(Ω2+C(C(Ω22+b,0),C(Ω22,0)),0)),0),0)
  • {1,,1{1,,1{1,,2,,2}1{1,,1,,2}2,,2}2♦2} has recursion level C(C(Ω22+b+C(C(Ω22+a,0),C(Ω2+C(C(Ω22+b,0),C(Ω22,0)),0)),0),0)
  • {1,,1{1,,1{1,,1{1,,2,,2}1{1,,1,,2}2,,2}2♦2}1{1,,1♦2}2{1,,1{1,,2,,2}1{1,,1,,2}2,,2}2♦2} has recursion level C(C(Ω22+b+C(C(Ω22+a,0),C(Ω2+C(C(Ω22+b,0),C(Ω22,0)),0))+C(Ω2+C(C(Ω22+b,0),C(Ω22,0)),0),0),0)
  • {1,,1{1,,1{1,,1{1,,2,,2}1{1,,1,,2}2,,2}2♦2}1{1{1,,1{1,,1{1,,2,,2}1{1,,1,,2}2,,2}2♦2}2,,1♦2} 2{1,,1{1,,2,,2}1{1,,1,,2}2,,2}2♦2} has recursion level C(C(Ω22+b+C(C(Ω22+a,0),C(Ω2+C(C(Ω22+b,0),C(Ω22,0)),0))2,0),0)
  • {1,,1{1,,1{1,,1{1,,2,,2}1{1,,1,,2}2,,2}2♦2}1{1{1,,1{1,,1{1,,2,,2}1{1,,1,,2}2,,2}2♦2}1,2,,1♦2} 2{1,,1{1,,2,,2}1{1,,1,,2}2,,2}2♦2} has recursion level C(C(Ω22+b+C(C(Ω22+a,0)+1,C(Ω2+C(C(Ω22+b,0),C(Ω22,0)),0)),0),0)
  • {1,,1{1,,1{1,,1{1,,2,,2}1{1,,1,,2}2,,2}2♦2}1 {1{1,,1{1,,1{1,,1{1,,2,,2}1{1,,1,,2}2,,2}2♦2}1,2{1,,1{1,,2,,2}1{1,,1,,2}2,,2}2♦2}2,,1♦2} 2{1,,1{1,,2,,2}1{1,,1,,2}2,,2}2♦2} has recursion level C(C(Ω22+b+C(C(Ω22+a+1,0),C(Ω2+C(C(Ω22+b,0),C(Ω22,0)),0)),0),0)
  • {1,,1{1,,1{1,,1{1,,2,,2}1{1,,1,,2}2,,2}2♦2}1{1,,2♦2}2{1,,1{1,,2,,2}1{1,,1,,2}2,,2}2♦2} has recursion level C(C(Ω22+b+C(C(Ω22+a+C(Ω2,C(Ω2+C(C(Ω22+b,0),C(Ω22,0)),0)),0),C(Ω2+C(C(Ω22+b,0),C(Ω22,0)),0)),0),0)
  • {1,,1{1,,1{1,,2,,2}1{1,,1,,2}2,,2}3♦2} has recursion level C(C(Ω22+b+C(C(Ω22+a2,0),C(Ω2+C(C(Ω22+b,0),C(Ω22,0)),0)),0),0)
  • {1,,1{1,,1,,3}2♦2} has recursion level C(C(Ω22+b+C(C(Ω22+C(C(Ω22,0),a),0),C(Ω2+C(C(Ω22+b,0),C(Ω22,0)),0)),0),0)
  • {1,,1♦3} has recursion level C(C(Ω22+b+C(C(Ω22+b,0),C(Ω2+C(C(Ω22+b,0),C(Ω22,0)),0)),0),0)
  • {1,,1{1,,1♦3}1{1,,1♦2}2♦3} has recursion level C(C(Ω22+b+C(C(Ω22+b,0),C(Ω2+C(C(Ω22+b,0),C(Ω22,0)),0))+C(Ω2+C(C(Ω22+b,0),C(Ω22,0)),0),0),0)
  • {1,,1{1,,1♦3}1{1{1,,1♦3}2,,1♦2}2♦3} has recursion level C(C(Ω22+b+C(C(Ω22+b,0),C(Ω2+C(C(Ω22+b,0),C(Ω22,0)),0))2,0),0)
  • {1,,1{1,,1♦3}1{1{1,,1♦3}1,2,,1♦2}2♦3} has recursion level C(C(Ω22+b+C(C(Ω22+b,0)+1,C(Ω2+C(C(Ω22+b,0),C(Ω22,0)),0)),0),0)
  • {1,,1{1,,1♦3}1{1{1,,2♦3}2,,1♦2}2♦3} has recursion level C(C(Ω22+b+C(C(Ω2,C(Ω22+b,0)),C(Ω2+C(C(Ω22+b,0),C(Ω22,0)),0)),0),0)
  • {1,,1{1,,1♦3}1{1{1,,1{1,,1♦3}2♦3}2,,1♦2}2♦3} has recursion level C(C(Ω22+b+C(C(Ω22+b,C(Ω22+b,0)),C(Ω2+C(C(Ω22+b,0),C(Ω22,0)),0)),0),0)
  • {1,,1{1,,1♦3}1{1{1,,1{1,,1♦3}1,2♦3}2,,1♦2}2♦3} has recursion level C(C(Ω22+b+C(C(Ω22+b+1,0),C(Ω2+C(C(Ω22+b,0),C(Ω22,0)),0)),0),0)
  • {1,,1{1,,1♦3}1{1,,2♦2}2♦3} has recursion level C(C(Ω22+b+C(Ω2,C(Ω2+C(C(Ω22+b,0),C(Ω22,0)),0)),0),0)
  • {1,,1{1,,1♦3}1{1,,1{1,,1,,2}2♦2}2♦3} has recursion level C(C(Ω22+b+C(Ω2+C(Ω22,0),C(Ω2+C(C(Ω22+b,0),C(Ω22,0)),0)),0),0)
  • {1,,1{1,,1♦3}1{1,,1♦3}2♦3} has recursion level C(C(Ω22+b+C(Ω2+C(C(Ω22+b,0),C(Ω22,0)),C(Ω2+C(C(Ω22+b,0),C(Ω22,0)),0)),0),0)
  • {1,,1♦1,2} has recursion level C(C(Ω22+b+C(Ω2+C(C(Ω22+b,0),C(Ω22,0))+1,0),0),0)
  • {1,,1♦1{1,,1{1,,2,,2}1{1,,1,,2}2,,2}2} has recursion level C(C(Ω22+b+C(Ω2+C(C(Ω22+b,0),C(Ω22,0))+C(C(Ω22+a,0),C(Ω22,0)),0),0),0)
  • {1,,1♦1♦2} has recursion level C(C(Ω22+b+C(Ω2+C(C(Ω22+b,0),C(Ω22,0))2,0),0),0)
  • {1,,1{2,,1♦2,,2}2} has recursion level C(C(Ω22+b+C(Ω2+ω^(C(C(Ω22+b,0),C(Ω22,0))+1),0),0),0)
  • {1{1,,1,,2}2,,1♦2,,2} has recursion level C(C(Ω22+b+C(Ω2+ω^(C(C(Ω22+b,0),C(Ω22,0))+C(Ω22,0)),0),0),0)
  • {1{1,,2,,2}2{1,,1♦2,,2}2,,1,,2} has recursion level C(C(Ω22+b+C(Ω2+C(C(Ω2,C(Ω22,0)),C(C(Ω22+b,0),C(Ω22,0))),0),0),0)
  • {1{1,,1{1,,2,,2}1{1,,1,,2}2,,2}2{1,,1♦2,,2}2,,1,,2} has recursion level C(C(Ω22+b+C(Ω2+C(C(Ω22+a,0),C(C(Ω22+b,0),C(Ω22,0))),0),0),0)
  • {1{1,,1♦2,,2}3,,1,,2} has recursion level C(C(Ω22+b+C(Ω2+C(C(Ω22+b,0),C(C(Ω22+b,0),C(Ω22,0))),0),0),0)
  • {1{1,,1♦2,,2}1,2,,1,,2} has recursion level C(C(Ω22+b+C(Ω2+C(C(Ω22+b,0)+1,C(Ω22,0)),0),0),0)
  • {1{1,,1{1,,2,,2}1{1,,1,,2}2,,2}2,,1{1,,2,,2}1{1{1,,1♦2,,2}2,,1,,2}2,,2} has recursion level C(C(Ω22+b+C(Ω2+C(C(Ω22+b,0)+C(Ω22+a,0),C(Ω22,0)),0),0),0)
  • {1{1,,1{1,,2,,2}1{1{1,,1♦2,,2}2,,1,,2}2,,2}2,,1{1,,2,,2}1{1{1,,1♦2,,2}2,,1,,2}2,,2} has recursion level C(C(Ω22+b+C(Ω2+C(C(Ω22+b,0)2,C(Ω22,0)),0),0),0)
  • {1,,2{1,,2,,2}1{1{1,,1♦2,,2}2,,1,,2}2,,2} has recursion level C(C(Ω22+b+C(Ω2+C(C(Ω2,C(Ω22+b,0)),C(Ω22,0)),0),0),0)
  • {1,,1{1,,2,,2}2{1{1,,1♦2,,2}2,,1,,2}2,,2} has recursion level C(C(Ω22+b+C(Ω2+C(C(Ω22,C(Ω22+b,0)),C(Ω22,0)),0),0),0)
  • {1,,1{1,,2,,2}1{1,,1,,2}2{1{1,,1♦2,,2}2,,1,,2}2,,2} has recursion level C(C(Ω22+b+C(Ω2+C(C(Ω22+a,C(Ω22+b,0)),C(Ω22,0)),0),0),0)
  • {1,,1{1,,2,,2}1{1{1,,2,,2}2,,1,,2}2{1{1,,1♦2,,2}2,,1,,2}2,,2} has recursion level C(C(Ω22+b+C(Ω2+C(C(Ω22+C(C(Ω2,a),a),C(Ω22+b,0)),C(Ω22,0)),0),0),0)
  • {1,,1{1,,2,,2}1{1{1,,1{1,,1,,3}2,,2}2,,1,,2}2{1{1,,1♦2,,2}2,,1,,2}2,,2} has recursion level C(C(Ω22+b+C(Ω2+C(C(Ω22+C(C(Ω22,0),a),C(Ω22+b,0)),C(Ω22,0)),0),0),0)
  • {1,,1{1,,2,,2}1{1{1,,1{1,,1{1,,1,,3}2,,2}1,2{1,,1,,3}2,,2}2,,1,,2}2{1{1,,1♦2,,2}2,,1,,2}2,,2} has recursion level C(C(Ω22+b+C(Ω2+C(C(Ω22+C(C(Ω22+1,0),a),C(Ω22+b,0)),C(Ω22,0)),0),0),0)
  • {1,,1{1,,2,,2}1{1{1,,1{1,,1{1,,1,,3}2,,2}1{1,,2,,2}2{1,,1,,3}2,,2}2,,1,,2}2{1{1,,1♦2,,2}2,,1,,2}2,,2} has recursion level C(C(Ω22+b+C(Ω2+C(C(Ω22+C(Ω2,a),C(Ω22+b,0)),C(Ω22,0)),0),0),0)
  • {1,,1{1,,2,,2}1{1{1,,1{1,,1{1,,1,,3}2,,2}1{1,,1{1,,1,,3}2,,2}2{1,,1,,3}2,,2}2,,1,,2} 2{1{1,,1♦2,,2}2,,1,,2}2,,2} has recursion level C(C(Ω22+b+C(Ω2+C(C(Ω22+C(Ω2+C(Ω22,0),a),C(Ω22+b,0)),C(Ω22,0)),0),0),0)
  • {1,,1{1,,2,,2}1{1{1,,1{1,,1,,3}1,2,,2}2,,1,,2}2{1{1,,1♦2,,2}2,,1,,2}2,,2} has recursion level C(C(Ω22+b+C(Ω2+C(C(Ω22+C(Ω2+C(Ω22,0)+1,a),C(Ω22+b,0)),C(Ω22,0)),0),0),0)
  • {1,,1{1,,2,,2}1{1{1,,1{1,,1{1,,2,,3}1,2,,3}2,,2}2,,1,,2}2{1{1,,1♦2,,2}2,,1,,2}2,,2} has recursion level C(C(Ω22+b+C(Ω2+C(C(Ω22+C(Ω2+C(C(Ω22+1,0),C(Ω22,0)),a),C(Ω22+b,0)),C(Ω22,0)),0),0),0)
  • {1,,1{1,,2,,2}1{1{1,,1♦2,,2}2,,1,,2}3,,2} has recursion level C(C(Ω22+b+C(Ω2+C(C(Ω22+b,C(Ω22+b,0)),C(Ω22,0)),0),0),0)
  • {1,,1{1,,2,,2}1{1{1,,1♦2,,2}2,,1,,2}1,2,,2} has recursion level C(C(Ω22+b+C(Ω2+C(C(Ω22+b+1,0),C(Ω22,0)),0),0),0)
  • {1,,1{1,,2,,2}1{1{1,,1♦2,,2}2,,1,,2}1{1,,1{1,,1{1,,2,,2}1{1{1,,1♦2,,2}2,,1,,2}1,2,,2}2}2,,2} has recursion level C(C(Ω22+b+C(Ω2+C(C(Ω22+b+C(Ω2+C(C(Ω22+b+1,0),C(Ω22,0)),0),0),C(Ω22,0)),0),0),0)
  • {1,,1{1,,2,,2}1{1{1,,1♦2,,2}2,,1,,2}1{1,,1,,2}2,,2} has recursion level C(C(Ω22+b+a,0),0)
  • {1,,1{1,,2,,2}1{1{1,,1♦2,,2}2,,1,,2}1{1{1,,1{1,,1,,3}2,,2}2,,1,,2}2,,2} has recursion level C(C(Ω22+b+C(C(Ω22,0),a),0),0)
  • {1,,1{1,,2,,2}1{1{1,,1♦2,,2}2,,1,,2}1{1{1,,1{1,,1{1,,1,,3}2,,2}1{1,,1,,2}2{1,,1,,3}2,,2}2,,1,,2}2,,2} has recursion level C(C(Ω22+b+C(C(Ω22+a,0),a),0),0)
  • {1,,1{1,,2,,2}1{1{1,,1♦2,,2}2,,1,,2}1 {1{1,,1{1,,1{1,,1,,3}2,,2}1{1,,1{1,,1,,3}2,,2}2{1,,1,,3}2,,2}2,,1,,2}2,,2} has recursion level C(C(Ω22+b+C(C(Ω22+C(Ω2+C(Ω22,0),a),0),a),0),0)
  • {1,,1{1,,2,,2}1{1{1,,1♦2,,2}2,,1,,2}1{1{1,,1{1,,1,,3}1,2,,2}2,,1,,2}2,,2} has recursion level C(C(Ω22+b+C(C(Ω22+C(Ω2+C(Ω22,0)+1,a),0),a),0),0)
  • {1,,1{1,,2,,2}1{1{1,,1♦2,,2}2,,1,,2}1{1{1,,1{1,,1{1,,2,,3}1,2,,3}2,,2}2,,1,,2}2,,2} has recursion level C(C(Ω22+b+C(C(Ω22+C(Ω2+C(C(Ω22+1,0),C(Ω22,0)),a),0),a),0),0)
  • {1,,1{1,,2,,2}1{1{1,,1♦2,,2}2,,1,,2}1{1{1,,1♦2,,2}2,,1,,2}2,,2} has recursion level C(C(Ω22+b+C(C(Ω22+b,0),a),0),0)
  • {1,,1{1,,2,,2}1{2{1,,1♦2,,2}2,,1,,2}2,,2} has recursion level C(C(Ω22+b+ω^(C(C(Ω22+b,0),a)+1),0),0)
  • {1,,1{1,,2,,2}1{1{1,,1♦2,,2}1,2,,1,,2}2,,2} has recursion level C(C(Ω22+b+C(C(Ω22+b,0)+1,a),0),0)
  • {1,,1{1,,2,,2}1{1{1{1,,1,2,,2}2,,1♦2,,2}2,,1,,2}2,,2} has recursion level C(C(Ω22+b+C(C(Ω22+b,0)+C(Ω2+1,a),a),0),0)
  • {1{1,,1{1,,1,,3}2,,2}2,,1♦2,,2} has recursion level C(C(Ω22+b+C(C(Ω22+b,0)+C(Ω22,0),a),0),0)
  • {1{1,,1♦2,,2}2,,1♦2,,2} has recursion level C(C(Ω22+b+C(C(Ω22+b,0)2,a),0),0)
  • {1,,2♦2,,2} has recursion level C(C(Ω22+b+C(C(Ω2,C(Ω22+b,0)),a),0),0)
  • {1,,1{1,,2,,2}2♦2,,2} has recursion level C(C(Ω22+b+C(C(Ω2+C(Ω2,a),C(Ω22+b,0)),a),0),0)
  • {1,,1{1,,1{1,,2,,2}2,,2}2♦2,,2} has recursion level C(C(Ω22+b+C(C(Ω2+C(Ω2+C(Ω2,a),a),C(Ω22+b,0)),a),0),0)
  • {1,,1{1,,1{1,,1,,3}2,,2}2♦2,,2} has recursion level C(C(Ω22+b+C(C(Ω22,C(Ω22+b,0)),a),0),0)
  • {1,,1{1,,1♦2,,2}2♦2,,2} has recursion level C(C(Ω22+b+C(C(Ω22+b,C(Ω22+b,0)),a),0),0)
  • {1,,1{1,,1♦2,,2}1,2♦2,,2} has recursion level C(C(Ω22+b+C(C(Ω22+b+1,0),a),0),0)
  • {1,,1{1,,1♦2,,2}1{1{1,,1{1,,1♦2,,2}1,2♦2,,2}2,,1,,2}2♦2,,2} has recursion level C(C(Ω22+b+C(C(Ω22+b+C(C(Ω22+b+1,0),a),0),a),0),0)
  • {1,,1{1,,1♦2,,2}1{1,,2,,2}2♦2,,2} has recursion level C(C(Ω22+b+C(Ω2,a),0),0)
  • {1,,1{1,,1♦2,,2}1{1,,1{1,,1,,3}2,,2}2♦2,,2} has recursion level C(C(Ω22+b+C(Ω2+C(Ω22,0),a),0),0)
  • {1,,1{1,,1♦2,,2}1{1,,1{1,,1,,3}1,2,,2}2♦2,,2} has recursion level C(C(Ω22+b+C(Ω2+C(Ω22,0)+1,a),0),0)
  • {1,,1{1,,1♦2,,2}1{1,,1{1,,1{1,,2,,3}1,2,,3}2,,2}2♦2,,2} has recursion level C(C(Ω22+b+C(Ω2+C(C(Ω22+1,0),C(Ω22,0)),a),0),0)
  • {1,,1{1,,1♦2,,2}1{1,,1♦2,,2}2♦2,,2} has recursion level C(C(Ω22+b+C(Ω2+C(C(Ω22+b,0),C(Ω22,0)),a),0),0)
  • {1,,1{1,,1,,3}1,2♦2,,2} has recursion level C(C(Ω22+b+C(Ω2+C(Ω22,0)+1,C(Ω2+C(C(Ω22+b,0),C(Ω22,0)),a)),0),0)
  • {1,,1{1,,1{1,,2,,3}1,2,,3}2♦2,,2} has recursion level C(C(Ω22+b+C(Ω2+C(C(Ω22+1,0),C(Ω22,0)),C(Ω2+C(C(Ω22+b,0),C(Ω22,0)),a)),0),0)
  • {1,,1♦3,,2} has recursion level C(C(Ω22+b+C(Ω2+C(C(Ω22+b,0),C(Ω22,0)),C(Ω2+C(C(Ω22+b,0),C(Ω22,0)),a)),0),0)
  • {1,,1♦1,2,,2} has recursion level C(C(Ω22+b+C(Ω2+C(C(Ω22+b,0),C(Ω22,0))+1,a),0),0)
  • {1,,1{2,,1{1,,2,,3}1{1,,1,,3}2,,3}2,,2} has recursion level C(C(Ω22+b+C(Ω2+ω^(C(C(Ω22+b,0),C(Ω22,0))+1),a),0),0)
  • {1{1,,2,,3}2♦2,,1,,3} has recursion level C(C(Ω22+b+C(Ω2+C(C(Ω2,C(Ω22,0)),C(C(Ω22+b,0),C(Ω22,0))),a),0),0)
  • {1♦1,2,,1,,3} has recursion level C(C(Ω22+b+C(Ω2+C(C(Ω22+b,0)+1,C(Ω22,0)),a),0),0)
  • {1♦2,,1{1,,2,,3}1{1,,1,,3}2,,3} has recursion level C(C(Ω22+b+C(Ω2+C(C(Ω22+b,0)2,C(Ω22,0)),a),0),0)
  • {1,,2{1,,2,,3}1{1,,1,,3}2,,3} has recursion level C(C(Ω22+b+C(Ω2+C(C(Ω2,C(Ω22+b,0)),C(Ω22,0)),a),0),0)
  • {1,,1{1,,2,,3}1,2{1,,1,,3}2,,3} has recursion level C(C(Ω22+b+C(Ω2+C(C(Ω22+1,C(Ω22+b,0)),C(Ω22,0)),a),0),0)
  • {1,,1{1,,2,,3}1{1,,1,,3}3,,3} has recursion level C(C(Ω22+b+C(Ω2+C(C(Ω22+b,C(Ω22+b,0)),C(Ω22,0)),a),0),0)
  • {1,,1{1,,2,,3}1{1,,1,,3}1,2,,3} has recursion level C(C(Ω22+b+C(Ω2+C(C(Ω22+b+1,0),C(Ω22,0)),a),0),0)
  • {1,,1{1,,2,,3}1{1,,1,,3}1{1,,1,,3}2,,3} has recursion level C(C(Ω22+b2,0),0)
  • {1,,1{1,,2,,3}1{1{1,,2,,3}2,,1,,3}2,,3} has recursion level C(C(Ω22+C(C(Ω2,b),b),0),0)
  • {1,,1{1,,2,,3}1{1{1,,1{1,,1,,3}2,,3}2,,1,,3}2,,3} has recursion level C(C(Ω22+C(C(Ω2+b,b),b),0),0)
  • {1,,1{1,,2,,3}1{1{1,,1{1,,2,,3}2,,3}2,,1,,3}2,,3} has recursion level C(C(Ω22+C(C(Ω2+C(Ω2,b),b),b),0),0)
  • {1,,1{1,,2,,3}1{1,,2,,3}2,,3} has recursion level C(C(Ω22+C(C(Ω2+C(Ω2,b)2,b),b),0),0)
  • {1,,1{1,,1{1,,1,,3}2,,3}2,,3} has recursion level C(C(Ω22+C(C(Ω2+C(Ω2+b,b),b),b),0),0)
  • {1,,1,,4} has recursion level C(C(Ω22+C(C(Ω22,0),b),0),0)

And so on, the recursion level of separator {1,,1,,k} (k > 2) is C(C(Ω22+C(C(Ω22,0),αk-2),0),0), where α0 = 0 and αk+1 = C(Ω2+C(Ω2,C(Ω22,0)),αk), between C(C(Ω22+αk-2,0),0) and C(C(Ω22+αk-1,0),0). So {1,,1,,1,2} has recursion level C(C(Ω22+C(Ω2+C(Ω2,C(Ω22,0))+1,0),0),0). Now let ordinal d = C(Ω2,C(Ω22,0)).

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

So {1,,1,,1,,2} has recursion level C(C(Ω22+C(Ω2+C(Ω2,C(Ω22,0))+C(Ω22,0),0),0),0). Here’re some approximations, to make the comparisons above more simple:

  • {1,,1,,2} approximately corresponds to C(Ω2+C(Ω2,C(Ω22,0)),0)
  • {1,,2,,2} approximately corresponds to C(Ω2,C(Ω2+C(Ω2,C(Ω22,0)),0))
  • {1,,1{1,,1,,2}2,,2} approximately corresponds to C(Ω2+C(Ω2+C(Ω2,C(Ω22,0)),0),C(Ω2+C(Ω2,C(Ω22,0)),0))
  • {1,,1{1,,1,,3}2,,2} approximately corresponds to C(Ω2+C(Ω22,0),C(Ω2+C(Ω2,C(Ω22,0)),0))
  • {1,,1{1,,1{1,,2,,3}1{1,,1,,3}2,,3}2,,2} approximately corresponds to C(Ω2+C(C(Ω22+C(Ω2+C(Ω2,C(Ω22,0)),C(Ω2+C(Ω2,C(Ω22,0)),0)),0),C(Ω22,0)),C(Ω2+C(Ω2,C(Ω22,0)),0))
  • {1,,1,,3} approximately corresponds to C(Ω2+C(Ω2,C(Ω22,0)),C(Ω2+C(Ω2,C(Ω22,0)),0))
  • {1,,2,,3} approximately corresponds to C(Ω2,C(Ω2+C(Ω2,C(Ω22,0)),C(Ω2+C(Ω2,C(Ω22,0)),0)))
  • {1,,1{1,,1,,4}2,,3} approximately corresponds to C(Ω2+C(Ω22,0),C(Ω2+C(Ω2,C(Ω22,0)),C(Ω2+C(Ω2,C(Ω22,0)),0)))
  • {1,,1,,4} approximately corresponds to C(Ω2+C(Ω2,C(Ω22,0)),C(Ω2+C(Ω2,C(Ω22,0)),C(Ω2+C(Ω2,C(Ω22,0)),0)))
  • {1,,1,,1,2} approximately corresponds to C(Ω2+C(Ω2,C(Ω22,0))+1,0)
  • {1,,1,,1{1,,1{1,,1,,2}2}2} approximately corresponds to C(Ω2+C(Ω2,C(Ω22,0))+C(Ω2+C(Ω22,0),0),0)
  • {1,,1,,1{1,,1,,2}2} approximately corresponds to C(Ω2+C(Ω2,C(Ω22,0))+C(Ω2+C(Ω2,C(Ω22,0)),0),0)

Beyond {1,,1,,1,,2}: Conclusion

Due to the outside layer of C(C(Ω22+____,0),0), the comparisons above seem erratic. But now we find the patterns on them. The story begins from C(Ω2+C(Ω22,0),0), corresponding to {1,,1{1,,1,,2}2}. Then εC(Ω2+C(Ω22,0),0)+1 = C(C(Ω2,C(Ω2+C(Ω22,0),0)),C(Ω2+C(Ω22,0),0)) corresponds to {1{1,,2{1,,1,,2}2}2,,1{1,,1,,2}2}. When the α in C(α,C(Ω2+C(Ω22,0),0)) get larger (but α < Ω2), it’ll correspond to {1 A 2,,1{1,,1,,2}2} with higher-leveled A (here A can be very high-leveled). The least ordinal larger than all those C(α,C(Ω2+C(Ω22,0),0)) is C(Ω2,C(Ω2+C(Ω22,0),0)), so C(Ω2,C(Ω2+C(Ω22,0),0)) works as the 1-separator in C(____,C(Ω2+C(Ω22,0),0)), just like that {1,,2{1,,1,,2}2} is the 1-separator in { ____ ,,1{1,,1,,2}2}. Next, C(Ω2,C(Ω2,C(Ω2+C(Ω22,0),0))) corresponds to {1,,3{1,,1,,2}2}, and so on. C(Ω2+C(Ω22,0),C(Ω2+C(Ω22,0),0)) corresponds to {1,,1{1,,1,,2}3}. Generally, for β < C(Ω22,0), the first fixed point of x -> C(Ω2+x,β) is C(Ω2+C(Ω22,0),β), and C(Ω22,0) works as the 1-separator in C(Ω2+____,β), just like that {1,,1,,2} works as the 1-separator in {1,, ____ }. But this correspondence fails at the next key point – When the α in C(Ω2+C(α,C(Ω22,0)),0) get larger (but α < Ω2), it’ll correspond to {1,,1 A 2} with higher-leveled A (here A can be very high-leveled). Note that without the limit from outside layer of C(C(Ω22+____,0),0), the α can be larger than, for instance, C(Ω23,0). The least ordinal larger than all those C(Ω2+C(α,C(Ω22,0)),0) is C(Ω2+C(Ω2,C(Ω22,0)),0), so actually C(Ω2+C(Ω2,C(Ω22,0)),0) corresponds to {1,,1,,2}, the 1-separator in {1,, ____ }. But the C(Ω22,0) still works as the 1-separator in α inside C(Ω2+α,β) (α < Ω2, β < C(Ω22,0)). This property is important even in further comparisons. Next, when the α in C(α,C(Ω2+C(Ω2,C(Ω22,0)),0)) get larger (but α < Ω2), it’ll correspond to {1 A 2,,1,,2} with higher-leveled A (here A can be very high-leveled). The least ordinal larger than all those C(α,C(Ω2+C(Ω2,C(Ω22,0)),0)) is C(Ω2,C(Ω2+C(Ω2,C(Ω22,0)),0)), so C(Ω2,C(Ω2+C(Ω2,C(Ω22,0)),0)) corresponds to {1,,2,,2}, the 1-separator in { ____ ,,1,,2}. Next, C(Ω2+C(Ω2,C(Ω22,0)),C(Ω2+C(Ω2,C(Ω22,0)),0)) corresponds to {1,,1,,3}, the 1-separator in {1,, ____ ,,2}. Generally, for β < C(Ω22,0), when the α in C(Ω2+C(α,C(Ω22,0)),β) get larger (but α < Ω2), it can traverse very high-leveled separator A in, for instance, {1,,1…,,1,,1 A 2…}. The least ordinal larger than all those C(Ω2+C(α,C(Ω22,0)),β) is C(Ω2+C(Ω2,C(Ω22,0)),β), so C(Ω2,C(Ω22,0)) works as the 2-separator in C(Ω2+____,β), just like the double comma. But this correspondence fails at some higher point (see the later post). Now let separator ◊ = {1,,1,,1{1,,1,,1,,2}2} and let ordinal d = C(Ω2,C(Ω22,0)), and a = C(Ω2+d+C(Ω22,0),0) = C(Ω2+C(Ω2,C(Ω22,0))+C(Ω22,0),0).

  • {1,,1{1{1,,1◊2}2,,1{1,,1,,2}2}2{1,,1,,2}2} has recursion level C(C(Ω22+a,0),0)
  • {1,,1{2{1,,1◊2}2,,1{1,,1,,2}2}2{1,,1,,2}2} has recursion level C(C(Ω22+a+ω^(C(C(Ω22+a,0),C(Ω2+C(Ω22,0),0))+1),0),0)
  • {1,,1{1{1,,1◊2}1,2,,1{1,,1,,2}2}2{1,,1,,2}2} has recursion level C(C(Ω22+a+C(C(Ω22+a,0)+1,C(Ω2+C(Ω22,0),0)),0),0)
  • {1{1,,1{1,,1,,2}1,2}2,,1◊2} has recursion level C(C(Ω22+a+C(C(Ω22+a,0)+C(Ω22+C(Ω2+C(Ω22,0)+1,0),0),C(Ω2+C(Ω22,0),0)),0),0)
  • {1{1,,1{1,,1{1,,2,,2}1{1,,1,,2}2,,2}2}2,,1◊2} has recursion level C(C(Ω22+a+C(C(Ω22+a,0)+C(Ω22+C(Ω2+d,0),0),C(Ω2+C(Ω22,0),0)),0),0)
  • {1{1,,1◊2}2,,1◊2} has recursion level C(C(Ω22+a+C(C(Ω22+a,0)2,C(Ω2+C(Ω22,0),0)),0),0)
  • {1,,2◊2} has recursion level C(C(Ω22+a+C(C(Ω2,C(Ω22+a,0)),C(Ω2+C(Ω22,0),0)),0),0)
  • {1,,1{1,,1◊2}2◊2} has recursion level C(C(Ω22+a+C(C(Ω22+a,C(Ω22+a,0)),C(Ω2+C(Ω22,0),0)),0),0)
  • {1,,1{1,,1◊2}1,2◊2} has recursion level C(C(Ω22+a+C(C(Ω22+a+1,0),C(Ω2+C(Ω22,0),0)),0),0)
  • {1,,1{1,,1◊2}1{1,,2{1,,1,,2}2}2◊2} has recursion level C(C(Ω22+a+C(Ω2,C(Ω2+C(Ω22,0),0)),0),0)
  • {1,,1{1,,1◊2}1{1,,1{1,,1,,2}1,2}2◊2} has recursion level C(C(Ω22+a+C(Ω2+C(Ω22,0)+1,0),0),0)
  • {1,,1{1,,1◊2}1{1,,1{1,,1{1,,2,,2}1{1,,1,,2}2,,2}2}2◊2} has recursion level C(C(Ω22+a+C(Ω2+C(C(Ω22+C(Ω2+d,0),0),C(Ω22,0)),0),0),0)
  • {1,,1{1,,1◊2}1{1,,1◊2}2◊2} has recursion level C(C(Ω22+a+C(Ω2+C(C(Ω22+a,0),C(Ω22,0)),0),0),0)
  • {1,,1{1,,1,,2}2◊2} has recursion level C(C(Ω22+a+C(C(Ω22,0),C(Ω2+C(C(Ω22+a,0),C(Ω22,0)),0)),0),0)
  • {1,,1{1,,1{1,,1,,2}2◊2}1{1,,1◊2}2{1,,1,,2}2◊2} has recursion level C(C(Ω22+a+C(C(Ω22,0),C(Ω2+C(C(Ω22+a,0),C(Ω22,0)),0))+C(Ω2+C(C(Ω22+a,0),C(Ω22,0)),0),0),0)
  • {1,,1{1,,1{1,,1,,2}2◊2}1{1{1,,1{1,,1,,2}2◊2}2,,1◊2}2{1,,1,,2}2◊2} has recursion level C(C(Ω22+a+C(C(Ω22,0),C(Ω2+C(C(Ω22+a,0),C(Ω22,0)),0))2,0),0)
  • {1,,1{1,,1{1,,1,,2}2◊2}1{1{1,,2{1,,1,,2}2◊2}2,,1◊2}2{1,,1,,2}2◊2} has recursion level C(C(Ω22+a+C(C(Ω2,C(Ω22,0)),C(Ω2+C(C(Ω22+a,0),C(Ω22,0)),0)),0),0)
  • {1,,1{1,,1{1,,1,,2}2◊2}1{1{1,,1{1,,1{1,,1,,2}2◊2}2{1,,1,,2}2◊2}2,,1◊2}2{1,,1,,2}2◊2} has recursion level C(C(Ω22+a+C(C(Ω22,C(Ω22,0)),C(Ω2+C(C(Ω22+a,0),C(Ω22,0)),0)),0),0)
  • {1,,1{1,,1{1,,1,,2}2◊2}1{1{1,,1{1,,1{1,,1,,2}2◊2}1,2{1,,1,,2}2◊2}2,,1◊2}2{1,,1,,2}2◊2} has recursion level C(C(Ω22+a+C(C(Ω22+1,0),C(Ω2+C(C(Ω22+a,0),C(Ω22,0)),0)),0),0)
  • {1,,1{1,,1{1,,1,,2}2◊2}1{1,,2◊2}2{1,,1,,2}2◊2} has recursion level C(C(Ω22+a+C(C(Ω22+C(Ω2,C(Ω2+C(C(Ω22+a,0),C(Ω22,0)),0)),0),C(Ω2+C(C(Ω22+a,0),C(Ω22,0)),0)),0),0)
  • {1,,1{1,,1{1,,1,,2}2◊2}1{1,,1{1,,1,,2}2◊2}2{1,,1,,2}2◊2} has recursion level C(C(Ω22+a+C(C(Ω22+C(Ω2+C(Ω22,0),C(Ω2+C(C(Ω22+a,0),C(Ω22,0)),0)),0),C(Ω2+C(C(Ω22+a,0),C(Ω22,0)),0)),0),0)
  • {1,,1{1,,1,,2}1,2◊2} has recursion level C(C(Ω22+a+C(C(Ω22+C(Ω2+C(Ω22,0)+1,C(Ω2+C(C(Ω22+a,0),C(Ω22,0)),0)),0),C(Ω2+C(C(Ω22+a,0),C(Ω22,0)),0)),0),0)
  • {1,,1{1,,1{1,,2,,2}1,2,,2}2◊2} has recursion level C(C(Ω22+a+C(C(Ω22+C(Ω2+C(C(Ω22+1,0),C(Ω22,0)),C(Ω2+C(C(Ω22+a,0),C(Ω22,0)),0)),0),C(Ω2+C(C(Ω22+a,0),C(Ω22,0)),0)),0),0)
  • {1,,1{1,,1{1,,2,,2}1{1,,1,,2}2,,2}2◊2} has recursion level C(C(Ω22+a+C(C(Ω22+C(Ω2+d,0),0),C(Ω2+C(C(Ω22+a,0),C(Ω22,0)),0)),0),0)
  • {1,,1{1,,1,,1,2}2◊2} has recursion level C(C(Ω22+a+C(C(Ω22+C(Ω2+d+1,0),0),C(Ω2+C(C(Ω22+a,0),C(Ω22,0)),0)),0),0)
  • {1,,1◊3} has recursion level C(C(Ω22+a+C(C(Ω22+a,0),C(Ω2+C(C(Ω22+a,0),C(Ω22,0)),0)),0),0)
  • {1,,1{1,,1◊3}1{1,,1◊2}2◊3} has recursion level C(C(Ω22+a+C(C(Ω22+a,0),C(Ω2+C(C(Ω22+a,0),C(Ω22,0)),0))+C(Ω2+C(C(Ω22+a,0),C(Ω22,0)),0),0),0)
  • {1,,1{1,,1◊3}1{1{1,,1◊3}2,,1◊2}2◊3} has recursion level C(C(Ω22+a+C(C(Ω22+a,0),C(Ω2+C(C(Ω22+a,0),C(Ω22,0)),0))2,0),0)
  • {1,,1{1,,1◊3}1{1{1,,1◊3}1,2,,1◊2}2◊3} has recursion level C(C(Ω22+a+C(C(Ω22+a,0)+1,C(Ω2+C(C(Ω22+a,0),C(Ω22,0)),0)),0),0)
  • {1,,1{1,,1◊3}1{1{1{1,,1{1,,1,,2}2◊2}2,,1◊3}2,,1◊2}2◊3} has recursion level C(C(Ω22+a+C(C(Ω22+a,0)+C(Ω22,0),C(Ω2+C(C(Ω22+a,0),C(Ω22,0)),0)),0),0)
  • {1,,1{1,,1◊3}1{1{1{1,,1◊3}2,,1◊3}2,,1◊2}2◊3} has recursion level C(C(Ω22+a+C(C(Ω22+a,0)2,C(Ω2+C(C(Ω22+a,0),C(Ω22,0)),0)),0),0)
  • {1,,1{1,,1◊3}1{1{1,,2◊3}2,,1◊2}2◊3} has recursion level C(C(Ω22+a+C(C(Ω2,C(Ω22+a,0)),C(Ω2+C(C(Ω22+a,0),C(Ω22,0)),0)),0),0)
  • {1,,1{1,,1◊3}1{1{1,,1{1,,1◊3}1,2◊3}2,,1◊2}2◊3} has recursion level C(C(Ω22+a+C(C(Ω22+a+1,0),C(Ω2+C(C(Ω22+a,0),C(Ω22,0)),0)),0),0)
  • {1,,1{1,,1◊3}1{1{1,,1{1,,1◊3}1{1,,1◊2}2◊3}2,,1◊2}2◊3} has recursion level C(C(Ω22+a+C(C(Ω22+a+C(Ω2+C(C(Ω22+a,0),C(Ω22,0)),0),0),C(Ω2+C(C(Ω22+a,0),C(Ω22,0)),0)),0),0)
  • {1,,1{1,,1◊3}1{1,,2◊2}2◊3} has recursion level C(C(Ω22+a+C(Ω2,C(Ω2+C(C(Ω22+a,0),C(Ω22,0)),0)),0),0)
  • {1,,1{1,,1◊3}1{1,,1{1,,1,,2}2◊2}2◊3} has recursion level C(C(Ω22+a+C(Ω2+C(Ω22,0),C(Ω2+C(C(Ω22+a,0),C(Ω22,0)),0)),0),0)
  • {1,,1{1,,1◊3}1{1,,1◊3}2◊3} has recursion level C(C(Ω22+a+C(Ω2+C(C(Ω22+a,0),C(Ω22,0)),C(Ω2+C(C(Ω22+a,0),C(Ω22,0)),0)),0),0)
  • {1,,1◊1,2} has recursion level C(C(Ω22+a+C(Ω2+C(C(Ω22+a,0),C(Ω22,0))+1,0),0),0)
  • {1,,1◊1◊2} has recursion level C(C(Ω22+a+C(Ω2+C(C(Ω22+a,0),C(Ω22,0))2,0),0),0)
  • {1,,1{2,,1◊2,,2}2} has recursion level C(C(Ω22+a+C(Ω2+ω^(C(C(Ω22+a,0),C(Ω22,0))+1),0),0),0)
  • {1{1,,1,,2}2,,1,,1{1,,1,,1,,2}2} has recursion level C(C(Ω22+a+C(Ω2+ω^(C(C(Ω22+a,0),C(Ω22,0))+C(Ω22,0)),0),0),0)
  • {1{1,,1{1,,2,,2}1,2,,2}2{1,,1◊2,,2}2,,1,,2} has recursion level C(C(Ω22+a+C(Ω2+C(C(Ω22+1,0),C(C(Ω22+a,0),C(Ω22,0))),0),0),0)
  • {1{1,,1◊2,,2}1,2,,1,,2} has recursion level C(C(Ω22+a+C(Ω2+C(C(Ω22+a,0)+1,C(Ω22,0)),0),0),0)
  • {1{1,,1{1,,2,,2}1{1{1,,1◊2,,2}2,,1,,2}2,,2}2,,1{1,,2,,2}1{1{1,,1◊2,,2}2,,1,,2}2,,2} has recursion level C(C(Ω22+a+C(Ω2+C(C(Ω22+a,0)2,C(Ω22,0)),0),0),0)
  • {1,,2{1,,2,,2}1{1{1,,1◊2,,2}2,,1,,2}2,,2} has recursion level C(C(Ω22+a+C(Ω2+C(C(Ω2,C(Ω22+a,0)),C(Ω22,0)),0),0),0)
  • {1,,1{1,,2,,2}2{1{1,,1◊2,,2}2,,1,,2}2,,2} has recursion level C(C(Ω22+a+C(Ω2+C(C(Ω22,C(Ω22+a,0)),C(Ω22,0)),0),0),0)
  • {1,,1{1,,2,,2}1{1,,1,,2}2{1{1,,1◊2,,2}2,,1,,2}2,,2} has recursion level C(C(Ω22+a+C(Ω2+C(C(Ω22+C(Ω2+d,0),C(Ω22+a,0)),C(Ω22,0)),0),0),0)
  • {1,,1{1,,2,,2}1{1{1,,1{1,,1,,3}2,,2}2,,1,,2}2{1{1,,1◊2,,2}2,,1,,2}2,,2} has recursion level C(C(Ω22+a+C(Ω2+C(C(Ω22+C(C(Ω22,0),C(Ω2+d,0)),C(Ω22+a,0)),C(Ω22,0)),0),0),0)
  • {1,,1{1,,2,,2}1{1{1,,1{1,,1,,1,2}2,,2}2,,1,,2}2{1{1,,1◊2,,2}2,,1,,2}2,,2} has recursion level C(C(Ω22+a+C(Ω2+C(C(Ω22+C(Ω2+d+1,0),C(Ω22+a,0)),C(Ω22,0)),0),0),0)
  • {1,,1{1,,2,,2}1{1{1,,1◊2,,2}2,,1,,2}3,,2} has recursion level C(C(Ω22+a+C(Ω2+C(C(Ω22+a,C(Ω22+a,0)),C(Ω22,0)),0),0),0)
  • {1,,1{1,,2,,2}1{1{1,,1◊2,,2}2,,1,,2}1,2,,2} has recursion level C(C(Ω22+a+C(Ω2+C(C(Ω22+a+1,0),C(Ω22,0)),0),0),0)
  • {1,,1{1,,2,,2}1{1{1,,1◊2,,2}2,,1,,2}1{1,,1,,2}2,,2} has recursion level C(C(Ω22+a+C(Ω2+d,0),0),0)
  • {1,,1{1,,2,,2}1{1{1,,1◊2,,2}2,,1,,2}1{1{1,,1{1,,1,,3}2,,2}2,,1,,2}2,,2} has recursion level C(C(Ω22+a+C(C(Ω22,0),C(Ω2+d,0)),0),0)
  • {1,,1{1,,2,,2}1{1{1,,1◊2,,2}2,,1,,2}1{1{1,,1◊2,,2}2,,1,,2}2,,2} has recursion level C(C(Ω22+a+C(C(Ω22+a,0),C(Ω2+d,0)),0),0)
  • {1,,1{1,,2,,2}1{1{1,,1◊2,,2}1,2,,1,,2}2,,2} has recursion level C(C(Ω22+a+C(C(Ω22+a,0)+1,C(Ω2+d,0)),0),0)
  • {1,,2◊2,,2} has recursion level C(C(Ω22+a+C(C(Ω2,C(Ω22+a,0)),C(Ω2+d,0)),0),0)
  • {1,,1{1,,2,,2}2◊2,,2} has recursion level C(C(Ω22+a+C(C(Ω2+C(Ω2,C(Ω2+d,0)),C(Ω22+a,0)),C(Ω2+d,0)),0),0)
  • {1,,1{1,,1{1,,1,,3}2,,2}2◊2,,2} has recursion level C(C(Ω22+a+C(C(Ω22,C(Ω22+a,0)),C(Ω2+d,0)),0),0)
  • {1,,1{1,,1◊2,,2}2◊2,,2} has recursion level C(C(Ω22+a+C(C(Ω22+a,C(Ω22+a,0)),C(Ω2+d,0)),0),0)
  • {1,,1{1,,1◊2,,2}1,2◊2,,2} has recursion level C(C(Ω22+a+C(C(Ω22+a+1,0),C(Ω2+d,0)),0),0)
  • {1,,1{1,,1◊2,,2}1{1,,2,,2}2◊2,,2} has recursion level C(C(Ω22+a+C(Ω2,C(Ω2+d,0)),0),0)
  • {1,,1{1,,1◊2,,2}1{1,,1{1,,1,,3}2,,2}2◊2,,2} has recursion level C(C(Ω22+a+C(Ω2+C(Ω22,0),C(Ω2+d,0)),0),0)
  • {1,,1{1,,1◊2,,2}1{1,,1◊2,,2}2◊2,,2} has recursion level C(C(Ω22+a+C(Ω2+C(C(Ω22+a,0),C(Ω22,0)),C(Ω2+d,0)),0),0)
  • {1,,1◊1,2,,2} has recursion level C(C(Ω22+a+C(Ω2+C(C(Ω22+a,0),C(Ω22,0))+1,C(Ω2+d,0)),0),0)
  • {1{1,,1,,3}2,,1,,1{1,,1,,1,,2}2} has recursion level C(C(Ω22+a+C(Ω2+ω^(C(C(Ω22+a,0),C(Ω22,0))+C(Ω22,0)),C(Ω2+d,0)),0),0)
  • {1{1,,1,,4}2,,1,,1{1,,1,,1,,2}2} has recursion level C(C(Ω22+a+C(Ω2+ω^(C(C(Ω22+a,0),C(Ω22,0))+C(Ω22,0)),C(Ω2+d,C(Ω2+d,0))),0),0)
  • {1{1,,1,,1,2}2,,1,,1{1,,1,,1,,2}2} has recursion level C(C(Ω22+a+C(Ω2+d+1,0),0),0)
  • {1{1,,1,,1{1,,1,,2}2}2,,1,,1{1,,1,,1,,2}2} has recursion level C(C(Ω22+a+C(Ω2+d+C(Ω2+d,0),0),0),0)
  • {1◊2,,1,,1{1,,1,,1,,2}2} has recursion level C(C(Ω22+a2,0),0)
  • {1,,2,,1{1,,1,,1,,2}2} has recursion level C(C(Ω22+C(C(Ω2,a),a),0),0)
  • {1,,1,,2{1,,1,,1,,2}2} has recursion level C(C(Ω22+C(C(Ω22,0),a),0),0)
  • {1,,1{1,,1,,2{1,,1,,1,,2}2}1,2,,1{1,,1,,1,,2}2} has recursion level C(C(Ω22+C(Ω2+C(Ω22,0)+1,a),0),0)
  • {1,,1{1,,2,,2{1,,1,,1,,2}2}1,2,,2{1,,1,,1,,2}2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22+1,0),C(Ω22,0)),a),0),0)
  • {1,,1{1,,2,,2{1,,1,,1,,2}2}1{1,,1,,2}2,,2{1,,1,,1,,2}2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22+C(Ω2+d,0),0),C(Ω22,0)),a),0),0)
  • {1,,1{1,,2,,2{1,,1,,1,,2}2}1◊2,,2{1,,1,,1,,2}2} has recursion level C(C(Ω22+C(Ω2+C(C(Ω22+a,0),C(Ω22,0)),a),0),0)
  • {1,,1{1,,2,,2{1,,1,,1,,2}2}1{1,,2,,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,a),0),C(Ω22,0)),a),0),0)
  • {1,,1{1,,2,,2{1,,1,,1,,2}2}1{1,,1{1,,1,,2{1,,1,,1,,2}2}1,2,,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(Ω22,0)+1,a),0),C(Ω22,0)),a),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+d,a),0),0)
  • {1,,1,,1,2{1,,1,,1,,2}2} has recursion level C(C(Ω22+C(Ω2+d+1,a),0),0)
  • {1,,1,,1◊2{1,,1,,1,,2}2} has recursion level C(C(Ω22+C(Ω2+d+a,a),0),0)
  • {1,,1,,1{1,,1,,1,,2}3} has recursion level C(C(Ω22+C(Ω2+d+C(Ω22,0),C(Ω2+d+C(Ω22,0),0)),0),0)
  • {1,,1,,1{1,,1,,1,,2}4} has recursion level C(C(Ω22+C(Ω2+d+C(Ω22,0),C(Ω2+d+C(Ω22,0),C(Ω2+d+C(Ω22,0),0))),0),0)
  • {1,,1,,1{1,,1,,1,,2}1,2} has recursion level C(C(Ω22+C(Ω2+d+C(Ω22,0)+1,0),0),0)

Now let ordinal d = C(Ω2,C(Ω22,0)).

  • {1,,1,,1{1,,1,,1,,2}1{1,,1,,1,,2}2} has recursion level C(C(Ω22+C(Ω2+d+C(Ω22,0)2,0),0),0)
  • {1,,1,,1{2,,1,,1,,2}2} has recursion level C(C(Ω22+C(Ω2+d+ω^(C(Ω22,0)+1),0),0),0)
  • {1,,1,,1{1{1,,1,,1,,2}2,,1,,1,,2}2} has recursion level C(C(Ω22+C(Ω2+d+ω^(C(Ω22,0)2),0),0),0)
  • {1,,2,,1,,2} has recursion level C(C(Ω22+C(Ω2+d+C(d,C(Ω22,0)),0),0),0)
  • {1,,1,2,,1,,2} has recursion level C(C(Ω22+C(Ω2+d+C(C(Ω2+1,C(Ω22,0)),C(Ω22,0)),0),0),0)
  • {1,,1{1,,1,,1,,2}2,,1,,2} has recursion level C(C(Ω22+C(Ω2+d+C(C(Ω2+C(Ω22,0),C(Ω22,0)),C(Ω22,0)),0),0),0)
  • {1,,1{1{1,,2,,1,,2}2,,1,,1,,2}2,,1,,2} has recursion level C(C(Ω22+C(Ω2+d+C(C(Ω2+C(d,C(Ω22,0)),C(Ω22,0)),C(Ω22,0)),0),0),0)
  • {1,,1{1{1,,1{1,,1,,1,,2}2,,1,,2}2,,1,,1,,2}2,,1,,2} has recursion level C(C(Ω22+C(Ω2+d+C(C(Ω2+C(C(Ω2+C(Ω22,0),C(Ω22,0)),C(Ω22,0)),C(Ω22,0)),C(Ω22,0)),0),0),0)
  • {1,,1{1,,2,,1,,2}2,,1,,2} has recursion level C(C(Ω22+C(Ω2+d+C(C(Ω2+d,C(Ω22,0)),C(Ω22,0)),0),0),0) (here’s a new irregular thing)
  • {1,,1{1,,2,,1,,2}1,2,,1,,2} has recursion level C(C(Ω22+C(Ω2+d+C(C(Ω2+d+1,C(Ω22,0)),C(Ω22,0)),0),0),0)
  • {1,,1{1,,2,,1,,2}1{1,,1,,1,,2}2,,1,,2} has recursion level C(C(Ω22+C(Ω2+d+C(C(Ω2+d+C(Ω22,0),C(Ω22,0)),C(Ω22,0)),0),0),0)
  • {1,,1{1,,2,,1,,2}1{1,,2,,1,,2}2,,1,,2} has recursion level C(C(Ω22+C(Ω2+d+C(C(Ω22,C(Ω22,0)),C(Ω22,0)),0),0),0)
  • {1,,1{1,,2,,1,,2}1{1,,2,,1,,2}1,2,,1,,2} has recursion level C(C(Ω22+C(Ω2+d+C(C(Ω22+1,0),C(Ω22,0)),0),0),0)
  • {1,,1{1,,2,,1,,2}1{1,,2,,1,,2}1{1,,1,,1{1,,1,,1,,2}2}2,,1,,2} has recursion level C(C(Ω22+C(Ω2+d+C(C(Ω22+C(Ω2+d+C(Ω22,0),0),0),C(Ω22,0)),0),0),0)
  • {1,,1{1,,2,,1,,2}1{1,,2,,1,,2}1{1,,1,,1,,2}2,,1,,2} has recursion level C(C(Ω22+C(Ω2+d2,0),0),0)
  • {1,,1{1,,2,,1,,2}1{1,,2,,1,,2}1{1,,1,,1,,2}1{1,,1,,1,,2}2,,1,,2} has recursion level C(C(Ω22+C(Ω2+d2,0)2,0),0)
  • {1,,1{1,,2,,1,,2}1{1,,2,,1,,2}1{1,,2,,1,,2}2,,1,,2} has recursion level C(C(Ω22+C(C(Ω2+C(Ω2,C(Ω2+d2,0))3,C(Ω2+d2,0)),C(Ω2+d2,0)),0),0)
  • {1,,1,,2,,2} has recursion level C(C(Ω22+C(C(Ω22,0),C(Ω2+d2,0)),0),0)
  • {1,,1{1,,1{1,,1,,2,,2}2,,1,,2}1,2{1,,1,,2,,2}2,,1,,2} has recursion level C(C(Ω22+C(C(Ω22+1,0),C(Ω2+d2,0)),0),0)
  • {1,,1{1,,1,,2,,2}1,2,,1,,2} has recursion level C(C(Ω22+C(Ω2+C(Ω22,0)+1,C(Ω2+d2,0)),0),0)
  • {1,,1{1,,2,,2,,2}1{1,,1,,2,,2}2,,2,,2} has recursion level C(C(Ω22+C(Ω2+d,C(Ω2+d2,0)),0),0)
  • {1,,1{1,,2,,3,,2}1{1,,1,,3,,2}2,,3,,2} has recursion level C(C(Ω22+C(Ω2+d,C(Ω2+d,C(Ω2+d2,0))),0),0)
  • {1,,1,,1,2,,2} has recursion level C(C(Ω22+C(Ω2+d+1,C(Ω2+d2,0)),0),0)
  • {1,,1,,1{1,,1,,2}2,,2} has recursion level C(C(Ω22+C(Ω2+d+C(Ω2+d,0),C(Ω2+d2,0)),0),0)
  • {1,,1,,1{1,,1,,1,,2}2,,2} has recursion level C(C(Ω22+C(Ω2+d+C(Ω2+d2,0),C(Ω2+d2,0)),0),0)
  • {1,,1,,1{1,,1,,1,2,,2}2,,2} has recursion level C(C(Ω22+C(Ω2+d+C(Ω2+d+1,C(Ω2+d2,0)),C(Ω2+d2,0)),0),0)
  • {1,,1,,1,,3} has recursion level C(C(Ω22+C(Ω2+d+C(Ω22,0),C(Ω2+d2,0)),0),0)
  • {1,,1,,1{1,,1,,1,,3}1,2,,2} has recursion level C(C(Ω22+C(Ω2+d+C(Ω22,0)+1,C(Ω2+d2,0)),0),0)
  • {1,,2,,1,,3} has recursion level C(C(Ω22+C(Ω2+d+C(d,C(Ω22,0)),C(Ω2+d2,0)),0),0)
  • {1,,1{1,,2,,1,,3}2,,1,,3} has recursion level C(C(Ω22+C(Ω2+d+C(C(Ω2+d,C(Ω22,0)),C(Ω22,0)),C(Ω2+d2,0)),0),0)
  • {1,,1{1,,2,,1,,3}1{1,,2,,1,,3}2,,1,,3} has recursion level C(C(Ω22+C(Ω2+d+C(C(Ω22,C(Ω22,0)),C(Ω22,0)),C(Ω2+d2,0)),0),0)
  • {1,,1{1,,2,,1,,3}1{1,,2,,1,,3}1{1,,1,,1,,3}2,,1,,3} has recursion level C(C(Ω22+C(Ω2+d2,C(Ω2+d2,0)),0),0)
  • {1,,1,,1,,4} has recursion level C(C(Ω22+C(Ω2+d+C(Ω22,0),C(Ω2+d2,C(Ω2+d2,0))),0),0)
  • {1,,1{1,,2,,1,,4}1{1,,2,,1,,4}1{1,,1,,1,,4}2,,1,,4} has recursion level C(C(Ω22+C(Ω2+d2,C(Ω2+d2,C(Ω2+d2,0))),0),0)
  • {1,,1,,1,,1,2} has recursion level C(C(Ω22+C(Ω2+d2+1,0),0),0)
  • {1,,1,,1,,1{1,,1,,1,,2}2} has recursion level C(C(Ω22+C(Ω2+d2+C(Ω2+d2,0),0),0),0)
  • {1,,1,,1,,1,,2} has recursion level C(C(Ω22+C(Ω2+d2+C(Ω22,0),0),0),0)
  • {1,,1,,1,,1{1,,1,,1,,1,,2}1,2} has recursion level C(C(Ω22+C(Ω2+d2+C(Ω22,0)+1,0),0),0)
  • {1,,2,,1,,1,,2} has recursion level C(C(Ω22+C(Ω2+d2+C(d,C(Ω22,0)),0),0),0)
  • {1,,1{1,,2,,1,,1,,2}2,,1,,1,,2} has recursion level C(C(Ω22+C(Ω2+d2+C(C(Ω2+d,C(Ω22,0)),C(Ω22,0)),0),0),0)
  • {1,,1{1,,2,,1,,1,,2}1{1,,2,,1,,1,,2}2,,1,,1,,2} has recursion level C(C(Ω22+C(Ω2+d2+C(C(Ω2+d2,C(Ω22,0)),C(Ω22,0)),0),0),0)
  • {1,,1{1,,2,,1,,1,,2}1{1,,2,,1,,1,,2}1{1,,2,,1,,1,,2}2,,1,,1,,2} has recursion level C(C(Ω22+C(Ω2+d2+C(C(Ω22,C(Ω22,0)),C(Ω22,0)),0),0),0)
  • {1,,1{1,,2,,1,,1,,2}1{1,,2,,1,,1,,2}1{1,,2,,1,,1,,2}1,2,,1,,1,,2} has recursion level C(C(Ω22+C(Ω2+d2+C(C(Ω22+1,0),C(Ω22,0)),0),0),0)
  • {1,,1{1,,2,,1,,1,,2}1{1,,2,,1,,1,,2}1{1,,2,,1,,1,,2}1{1,,1,,1,,1,,2}2,,1,,1,,2} has recursion level C(C(Ω22+C(Ω2+d3,0),0),0)
  • {1,,1{1,,1,,2,,1,,2}1,2,,1,,1,,2} has recursion level C(C(Ω22+C(Ω2+C(Ω22,0)+1,C(Ω2+d3,0)),0),0)
  • {1,,1,,1,2,,1,,2} has recursion level C(C(Ω22+C(Ω2+d+1,C(Ω2+d3,0)),0),0)
  • {1,,1,,1,,1,2,,2} has recursion level C(C(Ω22+C(Ω2+d2+1,C(Ω2+d3,0)),0),0)
  • {1,,1,,1,,1,,3} has recursion level C(C(Ω22+C(Ω2+d2+C(Ω22,0),C(Ω2+d3,0)),0),0)
  • {1,,1{1,,2,,1,,1,,3}1{1,,2,,1,,1,,3}1{1,,2,,1,,1,,3}1{1,,1,,1,,1,,3}2,,1,,1,,3} has recursion level C(C(Ω22+C(Ω2+d3,C(Ω2+d3,0)),0),0)
  • {1,,1,,1,,1,,1,2} has recursion level C(C(Ω22+C(Ω2+d3+1,0),0),0)
  • {1,,1,,1,,1,,1,,2} has recursion level C(C(Ω22+C(Ω2+d3+C(Ω22,0),0),0),0)
  • {1,,1,,1,,1,,1,,1,,2} has recursion level C(C(Ω22+C(Ω2+d4+C(Ω22,0),0),0),0)
  • {1{2,,}2} has recursion level C(C(Ω22+C(Ω2+ω^(d+1),0),0),0)

Something strange appears here. The recursion level of {1{1{2,,}2}2} is only equal to that of {1{1,,2}2}, because s(n,n{1{1{2,,}2}2}2) = s(n,n{1{1,,1,,…1,,1,,2}2}2) = (case B2, step 6) s(n,n{1{1…{1,2}…2}2}2) with n-1 1′s. Then {1{1{1{2,,}2}2{2,,}2}2} has equal recursion level to {1{1,,2}2,,2}. Also, {1,,1{1{1{2,,}2}2{2,,}2}2} has equal recursion level to {1{1,,1,,2}2,,1,,2}, {1,,1,,1{1{1{2,,}2}2{2,,}2}2} has equal recursion level to {1{1,,1,,1,,2}2,,1,,1,,2}, and so on – they’re all less than the recursion level of {1{2,,}2}. However, {1{1,,2}2{2,,}2} has equal recursion level to {1{1,,2}2,,2}, {1{1,,1,,2}2{2,,}2} has equal recursion level to {1{1,,1,,2}2,,1,,2}, {1{1,,1,,1,,2}2{2,,}2} has equal recursion level to {1{1,,1,,1,,2}2,,1,,1,,2}, and so on, so {1{1{2,,}2}2{2,,}2} has equal recursion level to {1{2,,}2}. What’s more, this “recursion level platform” happens on higher level of separators, such as {1{1{2,,}2}3{2,,}2}, {1{1{2,,}2}1{1{2,,}2}2{2,,}2}, {1{1{1{2,,}2}2{2,,}2}2{2,,}2}, {1,,2{2,,}2}, {1,,3{2,,}2}, {1,,1,,2{2,,}2}, {1,,1,,1,,2{2,,}2}, and so on – until {1{2,,}3}. Inside {1 ____ 2}, the {1{2,,}2} actually works in the same way as {1,,2}. Further, {1,,2{2,,}2} actually works as {1,,3}, {1,,1,,2{2,,}2} actually works as {1,,1,,2}, and {1{2,,}3} actually works as the real {1{2,,}2} – {1{1{2,,}3}2} has equal recursion level to {1{2,,}2}. Also, {1,,1{1{2,,}3}2}, {1,,1,,1{1{2,,}3}2}, and even the single {1{2,,}3}, all have equal recursion level to {1{2,,}2}. The main reason of the “recursion level platform” described above is at the step 5 in case B2. For separator {1,,1,,…1,,1,,2} (which is resulted from {1{2,,}2}), the “X ,, n-1 Y” is {1,,1,,…1,,1,,1}, then lv(“X ,, n-1 Y”) = lv({1}), so we always turn into step 6 (expanding rule), with no adding between the separator {1,,1,,…1,,1,,2} and the outside {1 ____ 2}. However, for separator {1,,1,,…1,,1,,2{2,,}2} (which is resulted from {1{2,,}3}), the “X ,, n-1 Y” is {1,,1,,…1,,1,,1{2,,}2}, then lv(“X ,, n-1 Y”) = lv({1{2,,}2}), so for an outside layer lower-leveled than {1{2,,}2} (e.g. the {1 ____ 2} layer), we can turn into step 7 (adding rule), then we leave the “recursion level platform”. “Recursion level platform” also happens on higher {1{2,,}1{2,,}2}, {1{3,,}2}, {1{1,2,,}2}, {1{1`2,,}2}, {1{1{1{1,2,,}2}2,,}2}, {1{1{1{1,,2,,}2}2,,}2} and even {1{1{1,,2{1,,2,,}2}2,,}2}. But {1{2,,2,,}2} reduces to {1{1,,2,,}…1{1,,2,,}1{1,,2,,}2}, and D({1{1,,2,,}…1{1,,2,,}1{1,,2,,}2}) = {1{1,,2,,}…1{1,,2,,}1{1,,1,,}2} = (rule 2) {1{1,,2,,}…1{1,,2,,}1,,2}, so there’s no “recursion level platform” here. Now let ordinal d = C(Ω2,C(Ω22,0)).

  • {1,,1{1{2,,}3}2} has recursion level C(C(Ω22+C(Ω2+ω^(d+1),0),0),0)
  • {1,,1{1,,1{1{2,,}3}2}1{1,,2{1,,1,,2}2}2{1{2,,}3}2} has recursion level C(C(Ω22+C(Ω2+ω^(d+1),0)+C(Ω2,C(Ω2+C(Ω22,0),0)),0),0)
  • {1,,1{1,,1{1{2,,}3}2}1{1,,1{1{2,,}3}2}2{1{2,,}3}2} has recursion level C(C(Ω22+C(Ω2+ω^(d+1),0)+C(Ω2+C(C(Ω22+C(Ω2+ω^(d+1),0),0),C(Ω22,0)),0),0),0)
  • {1,,1{1{2,,}3}1,2} has recursion level C(C(Ω22+C(Ω2+ω^(d+1),0)+C(Ω2+C(C(Ω22+C(Ω2+ω^(d+1),0),0),C(Ω22,0))+1,0),0),0)
  • {1,,1{2{2,,}3}2} has recursion level C(C(Ω22+C(Ω2+ω^(d+1),0)+C(Ω2+ω^(C(C(Ω22+C(Ω2+ω^(d+1),0),0),C(Ω22,0))+1),0),0),0)
  • {1{1,,1,,2}2{2,,}3} has recursion level C(C(Ω22+C(Ω2+ω^(d+1),0)+C(Ω2+ω^(C(C(Ω22+C(Ω2+ω^(d+1),0),0),C(Ω22,0))+C(Ω22,0)),0),0),0)
  • {1,,1{1,,1{1,,1,,1{1{2,,}3}2}2,,2}1{1,,2,,2}2{1,,1,,1{1{2,,}3}2}2,,2} has recursion level C(C(Ω22+C(Ω2+ω^(d+1),0)+C(Ω2,C(Ω2+d,0)),0),0)
  • {1,,1{1,,1{1,,1,,1{1{2,,}3}2}2,,2}1{1,,1{1,,1,,1{1{2,,}3}2}2,,2}2{1,,1,,1{1{2,,}3}2}2,,2} has recursion level C(C(Ω22+C(Ω2+ω^(d+1),0)+C(Ω2+C(C(Ω22+C(Ω2+ω^(d+1),0),0),C(Ω22,0)),C(Ω2+d,0)),0),0)
  • {1,,1{1,,1,,1{1{2,,}3}2}1,2,,2} has recursion level C(C(Ω22+C(Ω2+ω^(d+1),0)+C(Ω2+C(C(Ω22+C(Ω2+ω^(d+1),0),0),C(Ω22,0))+1,C(Ω2+d,0)),0),0)
  • {1{1,,1,,3}2,,1,,1{1{2,,}3}2} has recursion level C(C(Ω22+C(Ω2+ω^(d+1),0)+C(Ω2+ω^(C(C(Ω22+C(Ω2+ω^(d+1),0),0),C(Ω22,0))+C(Ω22,0)),C(Ω2+d,0)),0),0)
  • {1{1,,1,,1,2}2,,1,,1{1{2,,}3}2} has recursion level C(C(Ω22+C(Ω2+ω^(d+1),0)+C(Ω2+d+1,0),0),0)
  • {1{1,,1,,1{1{2,,}2}2}2,,1,,1{1{2,,}3}2} has recursion level C(C(Ω22+C(Ω2+ω^(d+1),0)+C(Ω2+d+C(Ω22,0),0),0),0)
  • {1{1,,1,,1{1{2,,}3}2}2,,1,,1{1{2,,}3}2} has recursion level C(C(Ω22+C(Ω2+ω^(d+1),0)+C(Ω2+d+C(C(Ω22+C(Ω2+ω^(d+1),0),0),C(Ω22,0)),0),0),0)
  • {1,,1{1,,1,,2{1{2,,}3}2}1,2,,1{1{2,,}3}2} has recursion level C(C(Ω22+C(Ω2+ω^(d+1),0)+C(Ω2+C(Ω22,0)+1,C(Ω2+d+C(C(Ω22+C(Ω2+ω^(d+1),0),0),C(Ω22,0)),0)),0),0)
  • {1,,1{1,,2,,2{1{2,,}3}2}1{1,,1,,2{1{2,,}3}2}2,,2{1{2,,}3}2} has recursion level C(C(Ω22+C(Ω2+ω^(d+1),0)+C(Ω2+d,C(Ω2+d+C(C(Ω22+C(Ω2+ω^(d+1),0),0),C(Ω22,0)),0)),0),0)
  • {1,,1,,1,2{1{2,,}3}2} has recursion level C(C(Ω22+C(Ω2+ω^(d+1),0)+C(Ω2+d+1,C(Ω2+d+C(C(Ω22+C(Ω2+ω^(d+1),0),0),C(Ω22,0)),0)),0),0)
  • {1,,1,,1{1{2,,}3}1,2} has recursion level C(C(Ω22+C(Ω2+ω^(d+1),0)+C(Ω2+d+C(C(Ω22+C(Ω2+ω^(d+1),0),0),C(Ω22,0))+1,0),0),0)
  • {1,,1,,1{2{2,,}3}2} has recursion level C(C(Ω22+C(Ω2+ω^(d+1),0)+C(Ω2+d+ω^(C(C(Ω22+C(Ω2+ω^(d+1),0),0),C(Ω22,0))+1),0),0),0)
  • {1{1,,1,,1,,2}2{2,,}3} has recursion level C(C(Ω22+C(Ω2+ω^(d+1),0)+C(Ω2+d+ω^(C(C(Ω22+C(Ω2+ω^(d+1),0),0),C(Ω22,0))+C(Ω22,0)),0),0),0)
  • {1{1,,1,,1,,1,,2}2{2,,}3} has recursion level C(C(Ω22+C(Ω2+ω^(d+1),0)+C(Ω2+d2+ω^(C(C(Ω22+C(Ω2+ω^(d+1),0),0),C(Ω22,0))+C(Ω22,0)),0),0),0)
  • {1{1{2,,}2}2{2,,}3} has recursion level C(C(Ω22+C(Ω2+ω^(d+1),0)2,0),0)
  • {1,,1{1{1{2,,}3}2{2,,}3}2} has recursion level C(C(Ω22+C(Ω2+ω^(d+1),0)2,0),0)
  • {1{1{2,,}3}2{2,,}3} has recursion level C(C(Ω22+C(Ω2+ω^(d+1),0)2,0),0)
  • {1{1{2,,}3}1,2{2,,}3} has recursion level C(C(Ω22+ω^(C(Ω2+ω^(d+1),0)+1),0),0)
  • {1,,2{2,,}3} has recursion level C(C(Ω22+C(C(Ω2,C(Ω2+ω^(d+1),0)),C(Ω2+ω^(d+1),0)),0),0)
  • {1,,1,,2{2,,}3} has recursion level C(C(Ω22+C(C(Ω22,0),C(Ω2+ω^(d+1),0)),0),0)
  • {1,,1{1,,1,,2{2,,}3}1,2,,1{2,,}3} has recursion level C(C(Ω22+C(Ω2+C(Ω22,0)+1,C(Ω2+ω^(d+1),0)),0),0)
  • {1,,1{1,,2,,2{2,,}3}1{1,,1,,2{2,,}3}2,,2{2,,}3} has recursion level C(C(Ω22+C(Ω2+d,C(Ω2+ω^(d+1),0)),0),0)
  • {1,,1,,1,2{2,,}3} has recursion level C(C(Ω22+C(Ω2+d+1,C(Ω2+ω^(d+1),0)),0),0)
  • {1,,1,,1,,2{2,,}3} has recursion level C(C(Ω22+C(Ω2+d+C(Ω22,0),C(Ω2+ω^(d+1),0)),0),0)
  • {1,,1,,1,,1,,2{2,,}3} has recursion level C(C(Ω22+C(Ω2+d2+C(Ω22,0),C(Ω2+ω^(d+1),0)),0),0)
  • {1{2,,}5} has recursion level C(C(Ω22+C(Ω2+ω^(d+1),C(Ω2+ω^(d+1),0)),0),0)
  • {1{2,,}7} has recursion level C(C(Ω22+C(Ω2+ω^(d+1),C(Ω2+ω^(d+1),C(Ω2+ω^(d+1),0))),0),0)
  • {1{2,,}1,2} has recursion level C(C(Ω22+C(Ω2+ω^(d+1)+1,0),0),0)
  • {1{2,,}1,,2} has recursion level C(C(Ω22+C(Ω2+ω^(d+1)+C(Ω22,0),0),0),0)
  • {1{2,,}1{1{2,,}1,,2}1,2} has recursion level C(C(Ω22+C(Ω2+ω^(d+1)+C(Ω22,0)+1,0),0),0)
  • {1,,1{1{2,,}1,,2}2{2,,}1,,2} has recursion level C(C(Ω22+C(Ω2+ω^(d+1)+C(C(Ω2+C(Ω22,0),C(Ω22,0)),C(Ω22,0)),0),0),0)
  • {1,,1{1,,2{2,,}1,,2}2{2,,}1,,2} has recursion level C(C(Ω22+C(Ω2+ω^(d+1)+C(C(Ω2+d,C(Ω22,0)),C(Ω22,0)),0),0),0)
  • {1,,1{1,,2{2,,}1,,2}1,2{2,,}1,,2} has recursion level C(C(Ω22+C(Ω2+ω^(d+1)+C(C(Ω2+d+1,C(Ω22,0)),C(Ω22,0)),0),0),0)
  • {1,,1{1,,2{2,,}1,,2}1{1,,2{2,,}1,,2}2{2,,}1,,2} has recursion level C(C(Ω22+C(Ω2+ω^(d+1)+C(C(Ω2+d2,C(Ω22,0)),C(Ω22,0)),0),0),0)
  • {1,,1{2,,2{2,,}1,,2}2{2,,}1,,2} has recursion level C(C(Ω22+C(Ω2+ω^(d+1)+C(C(Ω2+ω^(d+1),C(Ω22,0)),C(Ω22,0)),0),0),0)
  • {1,,1{2,,2{2,,}1,,2}1,2{2,,}1,,2} has recursion level C(C(Ω22+C(Ω2+ω^(d+1)+C(C(Ω2+ω^(d+1)+1,C(Ω22,0)),C(Ω22,0)),0),0),0)
  • {1,,1{2,,2{2,,}1,,2}1{1,,2{2,,}1,,2}2{2,,}1,,2} has recursion level C(C(Ω22+C(Ω2+ω^(d+1)+C(C(Ω22,C(Ω22,0)),C(Ω22,0)),0),0),0)
  • {1,,1{2,,2{2,,}1,,2}1{1,,2{2,,}1,,2}1,2{2,,}1,,2} has recursion level C(C(Ω22+C(Ω2+ω^(d+1)+C(C(Ω22+1,0),C(Ω22,0)),0),0),0)
  • {1,,1{2,,2{2,,}1,,2}1{1,,2{2,,}1,,2}1{1{2,,}1,,2}2{2,,}1,,2} has recursion level C(C(Ω22+C(Ω2+ω^(d+1)+d,0),0),0)
  • {1{2,,}1,,1,2} has recursion level C(C(Ω22+C(Ω2+ω^(d+1)+d+1,0),0),0)
  • {1{2,,}1,,1,,2} has recursion level C(C(Ω22+C(Ω2+ω^(d+1)+d+C(Ω22,0),0),0),0)
  • {1{2,,}1,,1,,1,,2} has recursion level C(C(Ω22+C(Ω2+ω^(d+1)+d2+C(Ω22,0),0),0),0)
  • {1{2,,}1{2,,}2} has recursion level C(C(Ω22+C(Ω2+ω^(d+1)2,0),0),0)
  • {1{3,,}2} has recursion level C(C(Ω22+C(Ω2+ω^(d+2),0),0),0)
  • {1{1{1{2,,}2}2,,}2} has recursion level C(C(Ω22+C(Ω2+ω^(d+C(Ω2+ω^(d+1),0)),0),0),0)
  • {1{1,,2,,}2} has recursion level C(C(Ω22+C(Ω2+ω^(d+C(Ω22,0)),0),0),0)

Now let separator ■ = {1{1,,2,,}2} and let ordinal d = C(Ω2,C(Ω22,0)), and ordinal a = C(Ω2+ω^(d+C(Ω22,0)),0) = C(Ω2+ω^(C(Ω2,C(Ω22,0))+C(Ω22,0)),0).

  • {1{1■2,,}3} has recursion level C(C(Ω22+a,0),0)
  • {1,,1{1,,1{1{1■2,,}3}2}1{1,,1{1{1■2,,}3}2}2{1{1■2,,}3}2} has recursion level C(C(Ω22+a+C(Ω2+C(C(Ω22+a,0),C(Ω22,0)),0),0),0)
  • {1,,1{1{1■2,,}3}1,2} has recursion level C(C(Ω22+a+C(Ω2+C(C(Ω22+a,0),C(Ω22,0))+1,0),0),0)
  • {1{1,,1,,2}2{1■2,,}3} has recursion level C(C(Ω22+a+C(Ω2+ω^(C(C(Ω22+a,0),C(Ω22,0))+C(Ω22,0)),0),0),0)
  • {1{1,,1,,1,,2}2{1■2,,}3} has recursion level C(C(Ω22+a+C(Ω2+d+ω^(C(C(Ω22+a,0),C(Ω22,0))+C(Ω22,0)),0),0),0)
  • {1{1,,1,,1,,1,,2}2{1■2,,}3} has recursion level C(C(Ω22+a+C(Ω2+d2+ω^(C(C(Ω22+a,0),C(Ω22,0))+C(Ω22,0)),0),0),0)
  • {1{1{2,,}2}2{1■2,,}3} has recursion level C(C(Ω22+a+C(Ω2+ω^(d+1),0),0),0)
  • {1{1{1■2,,}3}2{1■2,,}3} has recursion level C(C(Ω22+a2,0),0)
  • {1,,2{1■2,,}3} has recursion level C(C(Ω22+C(C(Ω2,a),a),0),0)
  • {1,,1,,2{1■2,,}3} has recursion level C(C(Ω22+C(C(Ω22,0),a),0),0)
  • {1,,1,,1,,2{1■2,,}3} has recursion level C(C(Ω22+C(Ω2+d+C(Ω22,0),a),0),0)
  • {1,,1,,1,,1,,2{1■2,,}3} has recursion level C(C(Ω22+C(Ω2+d2+C(Ω22,0),a),0),0)
  • {1{2,,}2{1■2,,}3} has recursion level C(C(Ω22+C(Ω2+ω^(d+1),a),0),0)
  • {1{1{1{1■2,,}3}2,,}2{1■2,,}3} has recursion level C(C(Ω22+C(Ω2+ω^(d+a),a),0),0)
  • {1{1{1{1■2,,}3}2,,}1{2,,}2{1■2,,}3} has recursion level C(C(Ω22+C(Ω2+ω^(d+a)+ω^(d+1),a),0),0)
  • {1{2{1{1■2,,}3}2,,}2{1■2,,}3} has recursion level C(C(Ω22+C(Ω2+ω^(d+a+1),a),0),0)
  • {1{1{1{1,,2{1■2,,}3}2{1■2,,}3}2,,}2{1■2,,}3} has recursion level C(C(Ω22+C(Ω2+ω^(d+C(C(Ω2,a),a)),a),0),0)
  • {1{1{1{1{1{1{1■2,,}3}2,,}2{1■2,,}3}2{1■2,,}3}2,,}2{1■2,,}3} has recursion level C(C(Ω22+C(Ω2+ω^(d+C(C(Ω22+C(Ω2+ω^(d+a),a),0),a)),a),0),0)
  • {1{1{1,,2{1■2,,}3}2,,}2{1■2,,}3} has recursion level C(C(Ω22+C(Ω2+ω^(d+C(Ω2,a)),a),0),0)
  • {1{1{1{2,,}2{1■2,,}3}2,,}2{1■2,,}3} has recursion level C(C(Ω22+C(Ω2+ω^(d+C(Ω2+ω^(d+1),a)),a),0),0)
  • {1{1{1{1{1{1■2,,}3}2,,}2{1■2,,}3}2,,}2{1■2,,}3} has recursion level C(C(Ω22+C(Ω2+ω^(d+C(Ω2+ω^(d+a),a)),a),0),0)
  • {1{1■2,,}5} has recursion level C(C(Ω22+C(Ω2+ω^(d+C(Ω22,0)),a),0),0)
  • {1{1■2,,}1,2} has recursion level C(C(Ω22+C(Ω2+ω^(d+C(Ω22,0))+1,0),0),0)
  • {1{1■2,,}1,,2} has recursion level C(C(Ω22+C(Ω2+ω^(d+C(Ω22,0))+C(Ω22,0),0),0),0)
  • {1{1■2,,}1{2,,}2} has recursion level C(C(Ω22+C(Ω2+ω^(d+C(Ω22,0))+ω^(d+1),0),0),0)
  • {1{2■2,,}2} has recursion level C(C(Ω22+C(Ω2+ω^(d+C(Ω22,0)+1),0),0),0)
  • {1{1■3,,}2} has recursion level C(C(Ω22+C(Ω2+ω^(d+C(Ω22,0)2),0),0),0)
  • {1{1■1,2,,}2} has recursion level C(C(Ω22+C(Ω2+ω^(d+ω^(C(Ω22,0)+1)),0),0),0)
  • {1{1{2{1,,2,,}2}2,,}2} has recursion level C(C(Ω22+C(Ω2+ω^(d+ω^ω^(C(Ω22,0)+1)),0),0),0)
  • {1,,2{1,,2,,}2} has recursion level C(C(Ω22+C(Ω2+ω^(d+C(d,C(Ω22,0))),0),0),0)
  • {1,,1{1,,2{1,,2,,}2}2{1,,2,,}2} has recursion level C(C(Ω22+C(Ω2+ω^(d+C(C(Ω2+d,C(Ω22,0)),C(Ω22,0))),0),0),0)
  • {1,,1{2,,2{1,,2,,}2}2{1,,2,,}2} has recursion level C(C(Ω22+C(Ω2+ω^(d+C(C(Ω2+ω^(d+1),C(Ω22,0)),C(Ω22,0))),0),0),0)
  • {1,,1{1■2,,2{1,,2,,}2}2{1,,2,,}2} has recursion level C(C(Ω22+C(Ω2+ω^(d+C(C(Ω2+ω^(d+C(Ω22,0)),C(Ω22,0)),C(Ω22,0))),0),0),0)
  • {1,,1{1{1,,2{1,,2,,}2}2,,2{1,,2,,}2}2{1,,2,,}2} has recursion level C(C(Ω22+C(Ω2+ω^(d+C(C(Ω22,C(Ω22,0)),C(Ω22,0))),0),0),0)
  • {1,,1{1{1,,2{1,,2,,}2}2,,2{1,,2,,}2}1,2{1,,2,,}2} has recursion level C(C(Ω22+C(Ω2+ω^(d+C(C(Ω22+1,0),C(Ω22,0))),0),0),0)
  • {1,,1{1{1,,2{1,,2,,}2}2,,2{1,,2,,}2}1{1{1■2,,}2}2{1,,2,,}2} has recursion level C(C(Ω22+C(Ω2+ω^(d+C(C(Ω22+a,0),C(Ω22,0))),0),0),0)
  • {1,,1{1{1,,2{1,,2,,}2}2,,2{1,,2,,}2}1■2{1,,2,,}2} has recursion level C(C(Ω22+C(Ω2+ω^(d2),0),0),0)
  • {1,,1{1{1,,2{1,,2,,}2}2,,2{1,,2,,}2}1■1■2{1,,2,,}2} has recursion level C(C(Ω22+C(Ω2+ω^(d2),0)2,0),0)
  • {1,,1,,2{1,,2,,}2} has recursion level C(C(Ω22+C(C(Ω22,0),C(Ω2+ω^(d2),0)),0),0)
  • {1,,1,,1,,2{1,,2,,}2} has recursion level C(C(Ω22+C(Ω2+d+C(Ω22,0),C(Ω2+ω^(d2),0)),0),0)
  • {1,,1,,1,,1,,2{1,,2,,}2} has recursion level C(C(Ω22+C(Ω2+d2+C(Ω22,0),C(Ω2+ω^(d2),0)),0),0)
  • {1{2,,}2{1,,2,,}2} has recursion level C(C(Ω22+C(Ω2+ω^(d+1),C(Ω2+ω^(d2),0)),0),0)
  • {1{1■2,,}2{1,,2,,}2} has recursion level C(C(Ω22+C(Ω2+ω^(d+C(Ω2+ω^(d2),0)),C(Ω2+ω^(d2),0)),0),0)
  • {1{1,,2,,}3} has recursion level C(C(Ω22+C(Ω2+ω^(d+C(Ω22,0)),C(Ω2+ω^(d2),0)),0),0)
  • {1{1,,2,,}4} has recursion level C(C(Ω22+C(Ω2+ω^(d+C(Ω22,0)),C(Ω2+ω^(d2),C(Ω2+ω^(d2),0))),0),0)
  • {1{1,,2,,}1,2} has recursion level C(C(Ω22+C(Ω2+ω^(d2)+1,0),0),0)

Now let ordinal d = C(Ω2,C(Ω22,0)).

  • {1{1,,2,,}1{2,,}2} has recursion level C(C(Ω22+C(Ω2+ω^(d2)+ω^(d+1),0),0),0)
  • {1{1,,2,,}1{1,,2,,}2} has recursion level C(C(Ω22+C(Ω2+ω^(d2)+ω^(d+C(Ω22,0)),0),0),0)
  • {1{1,,2,,}1{1,,2,,}1{1,,2,,}2} has recursion level C(C(Ω22+C(Ω2+ω^(d2)2+ω^(d+C(Ω22,0)),0),0),0)
  • {1{2,,2,,}2} has recursion level C(C(Ω22+C(Ω2+ω^(d2+1),0),0),0)
  • {1{1,,3,,}2} has recursion level C(C(Ω22+C(Ω2+ω^(d2+C(Ω22,0)),0),0),0)
  • {1,,1,,2{1,,3,,}2} has recursion level C(C(Ω22+C(C(Ω22,0),C(Ω2+ω^(d3),0)),0),0)
  • {1{1,,3,,}1,2} has recursion level C(C(Ω22+C(Ω2+ω^(d3)+1,0),0),0)
  • {1{2,,3,,}2} has recursion level C(C(Ω22+C(Ω2+ω^(d3+1),0),0),0)
  • {1{1,,4,,}2} has recursion level C(C(Ω22+C(Ω2+ω^(d3+C(Ω22,0)),0),0),0)
  • {1{1,,1,2,,}2} has recursion level C(C(Ω22+C(Ω2+ω^ω^(d+1),0),0),0)
  • {1{1,,1,,2,,}2} has recursion level C(C(Ω22+C(Ω2+ω^ω^(d+C(Ω22,0)),0),0),0)
  • {1,,1,,2{1,,1,,2,,}2} has recursion level C(C(Ω22+C(C(Ω22,0),C(Ω2+ω^ω^(d2),0)),0),0)
  • {1{1,,1,,2,,}1,2} has recursion level C(C(Ω22+C(Ω2+ω^ω^(d2)+1,0),0),0)
  • {1{2,,1,,2,,}2} has recursion level C(C(Ω22+C(Ω2+ω^(ω^(d2)+1),0),0),0)
  • {1{1,,1,,3,,}2} has recursion level C(C(Ω22+C(Ω2+ω^(ω^(d2)+ω^(d+C(Ω22,0))),0),0),0)
  • {1{1,,1,,1,2,,}2} has recursion level C(C(Ω22+C(Ω2+ω^ω^(d2+1),0),0),0)
  • {1{1,,1,,1,,2,,}2} has recursion level C(C(Ω22+C(Ω2+ω^ω^(d2+C(Ω22,0)),0),0),0)
  • {1{1{2,,}2,,}2} has recursion level C(C(Ω22+C(Ω2+ω^ω^ω^(d+1),0),0),0)
  • {1{1{1,,2,,}2,,}2} has recursion level C(C(Ω22+C(Ω2+ω^ω^ω^(d+C(Ω22,0)),0),0),0)
  • {1{1{1,,1,,2,,}2,,}2} has recursion level C(C(Ω22+C(Ω2+ω^ω^ω^ω^(d+C(Ω22,0)),0),0),0)
  • {1{1{1{1,,2,,}2,,}2,,}2} has recursion level C(C(Ω22+C(Ω2+ω^ω^ω^ω^ω^(d+C(Ω22,0)),0),0),0)

Let f(n) = s(n,n{1{1{1…{1,2,,}…2,,}2,,}2}2) where there’re n separators with a double comma. So f(n) is the limit of pDAN. It eventually outgrows all above, and has growth rate C(C(Ω22+C(Ω2+C(C(Ω2,C(Ω2,C(Ω22,0))),C(Ω2,C(Ω22,0))),0),0),0) = C(C(Ω22+C(Ω2C(Ω2,C(Ω22,0))+1,0),0),0).

Numbers from extended array notation

Here are some typical numbers from extended array notation. They’re also examples to help you understand exAN.

Two row series

Those numbers are defined using expression s(a1,a2,…ak-1,ak{2}b1,b2,…bk-1,bk). The simplest ones are already defined in LAN numbers page, e.g.

In s(3,2{2}2), we start the process. We land at the 3rd entry – the 2 in s(3,2{2}2), and we meet case B2, so change s(3,2{2}2) into s(3,2{2}2{2}1). Then we focus on the first entry of the former {2}, and we meet case B3, so change s(3,2{2}2{2}1) into s(3,2{1}1{1}2{2}1). Since {1} is the comma, we get s(3,2{2}2) = s(3,2,1,2{2}1). That’s how exAN rules work. As a result,

s(3,2{2}2) = s(3,2,1,2{2}1) = s(3,2,1,2) = s(3,3,2,1) = s(3,3,2) = tribo
s(3,3{2}2) = s(3,3,1,1,2{2}1) = s(3,3,1,1,2) = s(3,3,3,3,1) = s(3,3,3,3) = tetentri
s(3,4{2}2) = s(3,4,1,1,1,2{2}1) = s(3,4,1,1,1,2) = s(3,3,3,3,4,1) = s(3,3,3,3,4) = truchaindupribolplex
s(4,2{2}2) = s(4,2,1,2{2}1) = s(4,2,1,2) = s(4,4,2,1) = s(4,4,2) = tetbo
s(4,4{2}2) = s(4,4,1,1,1,2{2}1) = s(4,4,1,1,1,2) = s(4,4,4,4,4,1) = s(4,4,4,4,4) = pententet
s(5,2{2}2) = s(5,2,1,2{2}1) = s(5,2,1,2) = s(5,5,2,1) = s(5,5,2) = pentbo
s(5,5{2}2) = s(5,5,1,1,1,1,2{2}1) = s(5,5,1,1,1,1,2) = s(5,5,5,5,5,5,1) = s(5,5,5,5,5,5) = hexenpent
s(6,2{2}2) = s(6,2,1,2{2}1) = s(6,2,1,2) = s(6,6,2,1) = s(6,6,2) = hexabo

Some larger numbers are listed below.

Linel group

This group includes linel, linelplex, lintrienel, lintetenel and linpentenel in order.

Linel = s(3,2,2{2}2) = s(3,s(3,1,2{2}2),1{2}2) = s(3,s(3,1,1{2}2),1{2}2) = s(3,s(3,1,1,2{2}1),1{2}2) = s(3,s(3,1,1,2),1{2}2) = s(3,s(3,3,1,1),1{2}2) = s(3,s(3,3),1{2}2) = s(3,27,1{2}2) = s(3,27,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2{2}1) = s(3,27,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2) = s(3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,27,1) = s(3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,27) (with 28 3′s).

Linelplex = s(3,3,2{2}2) = s(3,s(3,2,2{2}2),1{2}2) = s(3,s(3,2,2{2}2),1,1…,1,2{2}1) = s(3,s(3,2,2{2}2),1,1…,1,2) (with s(3,2,2{2}2) 1′s) = s(3,3,…3,3,s(3,2,2{2}2),1) = s(3,3,…3,3,s(3,2,2{2}2)) (with s(3,2,2{2}2) + 1 3′s). It’s so hard to express in LAN! Note: though s(3,2{2}2) = s(3,3,2), s(3,2{2}3) ≠ s(3,3,2{2}2). In fact, s(3,2{2}3) = s(3,2,1,2{2}2) = s(3,3,2,1{2}2) = s(3,3,…3,3,s(3,2,2{2}2)) (with s(3,2,2{2}2) + 2 3′s) – that’s because rule 2 only applies to “the 1 in the last entry”.

Lintrienel = s(3,3,3{2}2) = s(3,s(3,2,3{2}2),2{2}2) = s(3,s(3,s(3,1,3{2}2),2{2}2),2{2}2) = s(3,s(3,s(3,1,1{2}2),2{2}2),2{2}2) = s(3,s(3,27,2{2}2),2{2}2). The name “lintrienel” bases on “linel” and adds an “-trien-“, which means an “3,3,3” string in the “first row”.

Lintetenel = s(3,3,3,3{2}2) and linpentenel = s(3,3,3,3,3{2}2). And, for the same reason, s(3,3{2}3) ≠ s(3,3,3,3{2}2).

Dulinel group

This group includes dulinel, dulinelplex, dulintrienel and dulintetenel in order.

Dulinel = s(3,2,2{2}3) = s(3,s(3,1,2{2}3),1{2}3) = s(3,s(3,1,1{2}3),1{2}3) = s(3,s(3,1,1,2{2}2),1{2}3) = s(3,s(3,3,1,1{2}2),1{2}3) = s(3,s(3,3,1,1,1,1,2{2}1),1{2}3) = s(3,s(3,3,1,1,1,1,2),1{2}3) = s(3,hexentri,1{2}3) = s(3,hexentri,1,1…,1,2{2}2) (with hexentri 1′s) = s(3,3,…3,3,hexentri,1{2}2) (with hexentri + 1 3′s).

Dulinelplex = s(3,3,2{2}3) = s(3,s(3,2,2{2}3),1{2}3) = s(3,s(3,2,2{2}3),1{2}3) = s(3,dulinel,1{2}3) = s(3,dulinel,1,1…,1,2{2}2) (with dulinel 1′s) = s(3,3,…3,3,hexentri,1{2}2) (with dulinel + 1 3′s).

Dulintrienel = s(3,3,3{2}3) and dulintetenel = s(3,3,3,3{2}3).

Dienlinel group

This group includes dienlinel, dienlintrienel, trienlinel, tetenlinel and pentenlinel in order.

Dienlinel = s(3,2,2{2}3,3), dienlintrienel = s(3,3,3{2}3,3), trienlinel = s(3,2,2{2}3,3,3), tetenlinel = s(3,2,2{2}3,3,3,3) and pentenlinel = s(3,2,2{2}3,3,3,3,3). Here the “dien-“, “trien-“, “teten-” and “penten-” mean 2, 3, 4 or 5 entries of 3 respectively.

Dimensional series

Those numbers are defined using arrays with separator {n} only (where n is a positive integer). A special result is that s(a,1,1…,1{n}2 #) = s(a,1,1…,1{n-1}2 #), so s(a,1,1…,1{n}2 #) = s(a,1,1…,1,2 #).

Linbel group

This group includes linbel, lintrienbel, linblinel, linbdulinel, linbdienlinel, dulinbel, dienlinbel and trienlinbel in order.

Linbel = s(3,2,2{2}1{2}2) = s(3,s(3,1,2{2}1{2}2),1{2}1{2}2) = s(3,s(3,1,1{2}1{2}2),1{2}1{2}2) = s(3,s(3,1,1{2}1,2),1{2}1{2}2) = s(3,s(3,3,3{2}1,1),1{2}1{2}2) = s(3,trientri,1{2}1{2}2) = s(3,trientri,1{2}1,1,…1,1,2) with trientri 1′s. It marks the end of 2-row arrays (arrays with only one {2} and no separator higher than {2})!

Lintrienbel = s(3,3,3{2}1{2}2), linblinel = s(3,2,2{2}2{2}2), linbdulinel = s(3,2,2{2}3{2}2), linbdienlinel = s(3,2,2{2}3,3{2}2), dulinbel = s(3,2,2{2}1{2}3), dienlinbel = s(3,2,2{2}1{2}3,3) and trienlinbel = s(3,2,2{2}1{2}3,3,3).

Lintrel group

This group includes lintrel, lintetel, linpenel and linhexel in order.

Lintrel = s(3,2,2{2}1{2}1{2}2), lintetel = s(3,2,2{2}1{2}1{2}1{2}2), linpenel = s(3,2,2{2}1{2}1{2}1{2}1{2}2) and linhexel = s(3,2,2{2}1{2}1{2}1{2}1{2}1{2}2). They mark the end of 3-row, 4-row, 5-row and 6-row arrays respectively.

Planel group

This group includes planel, planelplex, plantrienel, planlinel and planlinbel in order.

Planel = s(3,2,2{3}2) = s(3,s(3,1,2{3}2),1{3}2) = s(3,s(3,1,1{3}2),1{3}2) = s(3,s(3,1,1{2}2),1{3}2) = s(3,s(3,1,1,2),1{3}2) = s(3,s(3,3,1,1),1{3}2) = s(3,27,1{3}2) = s(3,27,1{2}1{2}1{2}1{2}1{2}1{2}1{2}1{2}1{2}1{2}1{2}1{2}1{2}1{2}1{2}1{2}1{2}1{2}1{2}1{2}1{2}1{2}1{2}1{2}1{2}1{2}1{2}2). And planelplex = s(3,3,2{3}2) = s(3,planel,1{2}1{2}…1{2}1{2}2) with planel {2}′s – that marks the end of planar arrays (arrays with commas and {2}′s only)!

Plantrienel = s(3,3,3{3}2), planlinel = s(3,2,2{2}2{3}2) and planlinbel = s(3,2,2{2}1{2}2{3}2).

planbel group

This group includes duplanel, dienplanel, trienplanel, planbel and plantrel in order.

Duplanel = s(3,2,2{3}3), dienplanel = s(3,2,2{3}3,3), trienplanel = s(3,2,2{3}3,3,3), planbel = s(3,2,2{3}1{3}2) and plantrel = s(3,2,2{3}1{3}1{3}2).

Cubel group

This group includes cubel, tesseral and penteral in order.

Cubel = s(3,2,2{4}2) = s(3,s(3,1,2{4}2),1{4}2) = s(3,s(3,1,1{4}2),1{4}2) = s(3,s(3,1,1,2),1{4}2) = s(3,s(3,3,1,1),1{4}2) = s(3,27,1{4}2) = s(3,27,1{3}1{3}1{3}1{3}1{3}1{3}1{3}1{3}1{3}1{3}1{3}1{3}1{3}1{3}1{3}1{3}1{3}1{3}1{3}1{3}1{3}1{3}1{3}1{3}1{3}1{3}1{3}2).

Tesseral = s(3,2,2{5}2) = s(3,27,1{5}2) = s(3,27,1{4}1{4}1{4}1{4}1{4}1{4}1{4}1{4}1{4}1{4}1{4}1{4}1{4}1{4}1{4}1{4}1{4}1{4}1{4}1{4}1{4}1{4}1{4}1{4}1{4}1{4}1{4}2) – now you see how much tesseral is larger than cubel, but we only change the {4} into the {5}.

Finally penteral = s(3,2,2{6}2).

Post-dimensional series

In definition of those numbers, there’re at least layer-2 entries. Though rules for them are the same as before, the reducing process seems more complex. So let me explain more to you.

Dimentril group

This group includes dimensol, dimentriensol, dimenlinsol, dimenlinbsol, dimentril, diendimensol, lindimensol, and dimenbol in order.

Dimentril = s(3,3{1,2}3). To solve it, we start the process, and meet the 3 in s(3,3{1,2}3). It’s case B2, so change the “{1,2}3” into “{1,2}2{1,2}2“, and jump to the 1 in s(3,3{1,2}2{1,2}2). Then we meet the 2 in s(3,3{1,2}2{1,2}2). It’s case B1, so s(3,3{1,2}3) = s(3,3{3,1}2{1,2}2) = s(3,3{3}2{1,2}2). The next time we start the process, we first meet case B2 then case B3, and s(3,3{3}2{1,2}2) = s(3,3{2}1{2}1{2}2{1,2}2) = s(3,3{2}1{2}1,1,1,2{1,2}2) = s(3,3{2}3{2}3,3,3,1{1,2}2). Also s(3,1{1,2}1,1,2) = s(3,3{1,2}3,1,1) = dimentril, and s(3,3{1,2}1,2) = s(3,3{1,2}3,1) = dimentril. The name “dimentril” combines “dimension” and “-tril”, because it’s above dimensional arrays, and all entries at base layer are 3.

However, dimentril is too far above dimensional arrays, so let’s take some more steps to see its hugeness.

Dimensol = s(3,3{1,2}2) = s(3,3{3,1}2{1,2}1) = s(3,3{3}2) = s(3,3{2}1{2}1{2}2) = s(3,3{2}1{2}1,1,1,2) = s(3,3{2}3{2}3,3,3) – it’s not so large, even smaller than lintrel.

But dimentriensol is larger than all dimensional numbers. It’s s(3,3,3{1,2}2) = s(3,s(3,2,3{1,2}2),2{1,2}2) = s(3,s(3,s(3,1,3{1,2}2),2{1,2}2),2{1,2}2) = s(3,s(3,s(3,1,1{1,2}2),2{1,2}2),2{1,2}2) = s(3,s(3,s(3,3,3,2),2{1,2}2),2{1,2}2), where s(3,n,2{1,2}2) = s(3,s(3,n-1,2{1,2}2),1{1,2}2) = s(3,3,3{s(3,n-1,2{1,2}2)}2) – an s(3,n-1,2{1,2}2)-1 dimension array!

Dimenlinsol = s(3,3{2}2{1,2}2) = s(3,3,1,1,2{1,2}2) = s(3,3,3,3,1{1,2}2), and dimenlinbsol = s(3,3{2}1{2}2{1,2}2) = s(3,3{2}1,1,1,2{1,2}2) = s(3,3{2}3,3,3,1{1,2}2). And dimentril is the 3rd term beginning with dimenlinsol and dimenlinbsol.

Going beyond dimentril, diendimensol = s(3,3{1,2}3,3) = s(3,3{3}2{1,2}2,3), lindimensol = s(3,3{1,2}1{2}2) = s(3,3{1,2}1,1,1,2) = s(3,3{1,2}3,3,3), and dimenbol = s(3,3{1,2}1{1,2}2) = s(3,3{1,2}3{3,1}2{1,2}1) = s(3,3{1,2}3{3}2).

Dimensolplex group (New Naming System)

From now on, we start another naming system. In the system, we use 4 “namebases” to name numbers result from some typical expressions. They’re

  • “ol” – to name expressions s(3,3A2)
  • “trien” – to name expressions s(3,3,3A2)
  • “tril” – to name expressions s(3,3A3)
  • “bol” – to name expressions s(3,3A1A2)

where A is a separator. The first group beyond dimenbol is dimensolplex group. This group contains 19 numbers as follows. Though I use the same suffix “-plex” as before, they have fully different meaning.

  • dimensolplex = s(3,3{2,2}2) = s(3,3{1,2}1{1,2}1{1,2}2) = s(3,3{1,2}3{1,2}3{3}2). It’s similar to dimenbol, but with 3 “dimension rows” instead of 2.
  • dimentrienplex = s(3,3,3{2,2}2). Notice the change of the namebase from dimentril group. This time we have not just 3 dimension rows, because the {2,2} reduces only when you meet it in the process. Actually dimentrienplex = s(3,s(3,s(3,3,3{1,2}2),2{2,2}2),2{2,2}2), where s(3,n,2{2,2}2) reduces to an array with s(3,n-1,2{2,2}2) dimension rows!
  • dimentrilplex = s(3,3{2,2}3) = s(3,3{1,2}1{1,2}1{1,2}2{2,2}2) = s(3,3{1,2}3{1,2}3{3}2{2,2}2).
  • dimenbolplex = s(3,3{2,2}1{2,2}2) = s(3,3{2,2}1{1,2}1{1,2}1{1,2}2) = s(3,3{2,2}3{1,2}3{1,2}3{3}2).
  • dimentrienbiplex = s(3,3,3{3,2}2)
  • dimentrilbiplex = s(3,3{3,2}3)
  • dimenbolbiplex = s(3,3{3,2}1{3,2}2)
  • dimensoltriplex = s(3,3{4,2}2)
  • dimentrientriplex = s(3,3,3{4,2}2)
  • dimentriltriplex = s(3,3{4,2}3)
  • dimenboltriplex = s(3,3{4,2}1{4,2}2)
  • dimensolquadriplex = s(3,3{5,2}2)
  • dimentrienquadriplex = s(3,3,3{5,2}2)
  • dimentrilquadriplex = s(3,3{5,2}3)
  • dimenbolquadriplex = s(3,3{5,2}1{5,2}2)
  • dimensolquintiplex = s(3,3{6,2}2)
  • dimentrienquintiplex = s(3,3,3{6,2}2)
  • dimentrilquintiplex = s(3,3{6,2}3)
  • dimenbolquintiplex = s(3,3{6,2}1{6,2}2)

So the suffix n-plex means that the separator A becomes {n+1,2}. However, there’s one missing number…

-dex group

Now we pick up the missing number – dimensoldex, but you may think that the missing number is dimensolbiplex. Actually they’re the same thing, because

Dimensoldex = s(3,3{1,3}2) = s(3,3{3,2}2) = dimensolbiplex. The reducing process is similar to dimensol or dimentril. First we start the process, and meet the “2” in s(3,3{1,3}2) – that’s case B2, so change the “{1,3}2” into “{1,3}2{1,3}1″, and jump to the “1“. Then we meet the “3” in s(3,3{1,3}2{1,3}1) – that’s case B3, so s(3,3{1,3}2) = s(3,3{3,2}2{1,3}1) = s(3,3{3,2}2).

The suffix “-dex” change the {1,2} into {1,3}, so we get dimentriendex = s(3,3,3{1,3}2), dimentrildex = s(3,3{1,3}3) and dimenboldex = s(3,3{1,3}1{1,3}2).

The {1,3} is not just {3,2} – it’s {b,2} where b is the iterator only when you meet it in the process. Dimensoldex = dimensolbiplex – that’s a bit weak. But could you imagine what’s the n such that dimentriendex ≈ dimentrien-n-plex? Try to reduce it, and you’ll get similar answer to what I said about dimentriensol and dimentrienplex.

Then we can add “n-plex” suffixes to those numbers, such as

  • dimensoldexplex = s(3,3{2,3}2)
  • dimentriendexplex = s(3,3,3{2,3}2)
  • dimentrildexplex = s(3,3{2,3}3)
  • dimenboldexplex = s(3,3{2,3}1{2,3}2)
  • dimentriendexbiplex = s(3,3,3{3,3}2)
  • dimentrildexbiplex = s(3,3{3,3}3)
  • dimenboldexbiplex = s(3,3{3,3}1{3,3}2)

(still with one missing stuff…) So the suffix “n-plex” means the first entry of the separator A changed from 1 to n+1.

The next step is adding “-plex” prior to the “-dex”. And we get some larger numbers –

  • dimensolplexdex = s(3,3{1,4}2) = s(3,3{3,3}2), a.k.a. the missing stuff – dimensoldexbiplex.
  • dimentrienplexdex = s(3,3,3{1,4}2)
  • dimentrilplexdex = s(3,3{1,4}3)
  • dimenbolplexdex = s(3,3{1,4}1{1,4}2)
  • dimensolplexdexplex = s(3,3{2,4}2)
  • dimentrienplexdexplex = s(3,3,3{2,4}2)
  • dimentrilplexdexplex = s(3,3{2,4}3)
  • dimenbolplexdexplex = s(3,3{2,4}1{2,4}2)

And with “-biplex” prior to the “-dex”:

  • dimensolbiplexdex = s(3,3{1,5}2) = s(3,3{3,4}2)
  • dimentrienbiplexdex = s(3,3,3{1,5}2)
  • dimentrilbiplexdex = s(3,3{1,5}3)
  • dimenbolbiplexdex = s(3,3{1,5}1{1,5}2)
  • dimensolbiplexdexplex = s(3,3{2,5}2)
  • dimentrienbiplexdexplex = s(3,3,3{2,5}2)
  • dimentrilbiplexdexplex = s(3,3{2,5}3)
  • dimenbolbiplexdexplex = s(3,3{2,5}1{2,5}2)

With “-triplex” prior to the “-dex”:

  • dimensoltriplexdex = s(3,3{1,6}2) = s(3,3{3,5}2)
  • dimentrientriplexdex = s(3,3,3{1,6}2)
  • dimentriltriplexdex = s(3,3{1,6}3)
  • dimenboltriplexdex = s(3,3{1,6}1{1,6}2)

And so on. The “n-plex” prior to the “-dex” means adding n to the 2nd entry of the separator A.

-bidex group

Here’s how the suffix “-bidex” works.

Dimensolbidex = s(3,3{1,1,2}2). To solve it, first we start the process, and meet the “2” in s(3,3{1,1,2}2) – that’s case B2, so change the “{1,1,2}2” into “{1,1,2}2{1,1,2}1″, and jump to the “1“. Then we meet the “2” in s(3,3{1,1,2}2{1,1,2}1) – that’s case B3, so s(3,3{1,1,2}2) = s(3,3{1,3,1}2{1,1,2}1) = s(3,3{1,3}2) – it’s still dimensoldex (or dimensolbiplex).

  • dimentrienbidex = s(3,3,3{1,1,2}2) – that’s much larger.
  • dimentrilbidex = s(3,3{1,1,2}3)
  • dimenbolbidex = s(3,3{1,1,2}1{1,1,2}2)

And adding “-plex” after “-biplex”:

  • dimensolbidexplex = s(3,3{2,1,2}2)
  • dimentrienbidexplex = s(3,3,3{2,1,2}2)
  • dimentrilbidexplex = s(3,3{2,1,2}3)
  • dimenbolbidexplex = s(3,3{2,1,2}1{2,1,2}2)

We can also add “-plex” prior to the “-biplex”, but this step is too big. The “-bidex” means two “-dex”es, so we can insert “-plex” between them, such as

  • dimensoldexplexdex = s(3,3{1,2,2}2) = s(3,3{3,1,2}2)
  • dimentriendexplexdex = s(3,3,3{1,2,2}2)
  • dimentrildexplexdex = s(3,3{1,2,2}3)
  • dimenboldexplexdex = s(3,3{1,2,2}1{1,2,2}2)
  • dimensoldexplexdexplex = s(3,3{2,2,2}2)
  • dimentriendexplexdexplex = s(3,3,3{2,2,2}2)
  • dimentrildexplexdexplex = s(3,3{2,2,2}3)
  • dimenboldexplexdexplex = s(3,3{2,2,2}1{2,2,2}2)
  • dimentriendexbiplexdex = s(3,3,3{1,3,2}2)
  • dimentrildexbiplexdex = s(3,3{1,3,2}3)
  • dimenboldexbiplexdex = s(3,3{1,3,2}1{1,3,2}2)

Then we reach “-plex” prior to “-bidex”.

  • dimensolthrex = s(3,3{1{2}2}2) = s(3,3{1,1,1,2}2) = s(3,3{1,1,3}2) = s(3,3{1,3,2}2) = s(3,3{3,2,2}2), a.k.a. dimensoltridex, dimensolplexbidex, dimensoldexbiplexdex and dimensoldexplexdexbiplex.
  • dimentrienplexbidex = s(3,3,3{1,1,3}2)
  • dimentrilplexbidex = s(3,3{1,1,3}3)
  • dimenbolplexbidex = s(3,3{1,1,3}1{1,1,3}2)
  • dimensolplexbidexplex = s(3,3{2,1,3}2)
  • dimentrienplexbidexplex = s(3,3,3{2,1,3}2)
  • dimentrilplexbidexplex = s(3,3{2,1,3}3)
  • dimenbolplexbidexplex = s(3,3{2,1,3}1{2,1,3}2)
  • dimensolplexdexplexdex = s(3,3{1,2,3}2) = s(3,3{3,1,3}2)
  • dimentrienplexdexplexdex = s(3,3,3{1,2,3}2)
  • dimentrilplexdexplexdex = s(3,3{1,2,3}3)
  • dimenbolplexdexplexdex = s(3,3{1,2,3}1{1,2,3}2)
  • dimentrienplexdexbiplexdex = s(3,3,3{1,3,3}2)
  • dimentrilplexdexbiplexdex = s(3,3{1,3,3}3)
  • dimenbolplexdexbiplexdex = s(3,3{1,3,3}1{1,3,3}2)

And with “-biplex”:

  • dimensolbiplexbidex = s(3,3{1,1,4}2) = s(3,3{1,3,3}2) = s(3,3{3,2,3}2), a.k.a. dimensolplexdexbiplexdex.
  • dimentrienbiplexbidex = s(3,3,3{1,1,4}2)
  • dimentrilbiplexbidex = s(3,3{1,1,4}3)
  • dimenbolbiplexbidex = s(3,3{1,1,4}1{1,1,4}2)

And so on. The “n-plex” prior to the “-bidex” means adding n to the 3rd entry of the separator A.

Multi-dex group

The suffix “-tridex” means 3 “-dex”es, so we can insert “-plex”es before them, after one, after two, or after them, such as

  • dimentrientridex = s(3,3,3{1,1,1,2}2)
  • dimentriltridex = s(3,3{1,1,1,2}3)
  • dimenboltridex = s(3,3{1,1,1,2}1{1,1,1,2}2)
  • dimensoltridexplex = s(3,3{2,1,1,2}2)
  • dimensolbidexplexdex = s(3,3{1,2,1,2}2) = s(3,3{3,1,1,2}2)
  • dimensolbidexplexdexplex = s(3,3{2,2,1,2}2)
  • dimensoldexplexbidex = s(3,3{1,1,2,2}2) = s(3,3{1,3,1,2}2) = s(3,3{3,2,1,2}2)
  • dimensoldexplexbidexplex = s(3,3{2,1,2,2}2)
  • dimensoldexplexdexplexdex = s(3,3{1,2,2,2}2) = s(3,3{3,1,2,2}2)
  • dimensoldexplexdexplexdexplex = s(3,3{2,2,2,2}2)
  • dimensolquadridex = s(3,3{1,1,1,1,2}2) = s(3,3{1,1,1,3}2) = s(3,3{1,1,3,2}2) = s(3,3{1,3,2,2}2) = s(3,3{3,2,2,2}2), a.k.a. dimensolplextridex.
  • dimensolbiplextridex = s(3,3{1,1,1,4}2) = s(3,3{1,1,3,3}2) = s(3,3{1,3,2,3}2) = s(3,3{3,2,2,3}2)

The suffix “-quadridex” means 4 “-dex”es, so we can also insert “-plex”es.

  • dimentrienquadridex = s(3,3,3{1,1,1,1,2}2)
  • dimentrilquadridex = s(3,3{1,1,1,1,2}3)
  • dimenbolquadridex = s(3,3{1,1,1,1,2}1{1,1,1,1,2}2)
  • dimensolquadridexplex = s(3,3{2,1,1,1,2}2)
  • dimensoltridexplexdex = s(3,3{1,2,1,1,2}2) = s(3,3{3,1,1,1,2}2)
  • dimensoltridexplexdexplex = s(3,3{2,2,1,1,2}2)
  • dimensolbidexplexbidex = s(3,3{1,1,2,1,2}2) = s(3,3{1,3,1,1,2}2) = s(3,3{3,2,1,1,2}2)
  • dimensolbidexplexbidexplex = s(3,3{2,1,2,1,2}2)
  • dimensolbidexplexdexplexdex = s(3,3{1,2,2,1,2}2) = s(3,3{3,1,2,1,2}2)
  • dimensolbidexplexdexplexdexplex = s(3,3{2,2,2,1,2}2)
  • dimensoldexplextridex = s(3,3{1,1,1,2,2}2) = s(3,3{1,1,3,1,2}2) = s(3,3{1,3,2,1,2}2) = s(3,3{3,2,2,1,2}2)
  • dimensoldexplextridexplex = s(3,3{2,1,1,2,2}2)
  • dimensoldexplexbidexplexdex = s(3,3{1,2,1,2,2}2) = s(3,3{3,1,1,2,2}2)
  • dimensoldexplexbidexplexdexplex = s(3,3{2,2,1,2,2}2)
  • dimensoldexplexdexplexbidex = s(3,3{1,1,2,2,2}2) = s(3,3{1,3,1,2,2}2) = s(3,3{3,2,1,2,2}2)
  • dimensoldexplexdexplexbidexplex = s(3,3{2,1,2,2,2}2)
  • dimensoldexplexdexplexdexplexdex = s(3,3{1,2,2,2,2}2) = s(3,3{3,1,2,2,2}2)
  • dimensoldexplexdexplexdexplexdexplex = s(3,3{2,2,2,2,2}2)
  • dimensolquintidex = s(3,3{1,1,1,1,1,2}2) = s(3,3{1,1,1,1,3}2) = s(3,3{1,1,1,3,2}2) = s(3,3{1,1,3,2,2}2) = s(3,3{1,3,2,2,2}2) = s(3,3{3,2,2,2,2}2), a.k.a. dimensolplexquadridex.
  • dimensolbiplexquadridex = s(3,3{1,1,1,1,4}2) = s(3,3{1,1,1,3,3}2) = s(3,3{1,1,3,2,3}2) = s(3,3{1,3,2,2,3}2) = s(3,3{3,2,2,2,3}2)

And the suffix “-quintidex” means 5 “-dex”es.

  • dimentrienquintidex = s(3,3,3{1,1,1,1,1,2}2)
  • dimentrilquintidex = s(3,3{1,1,1,1,1,2}3)
  • dimenbolquintidex = s(3,3{1,1,1,1,1,2}1{1,1,1,1,1,2}2)
  • dimensolquintidexplex = s(3,3{2,1,1,1,1,2}2)
  • dimensolquadridexplexdex = s(3,3{1,2,1,1,1,2}2) = s(3,3{3,1,1,1,1,2}2)
  • dimensoltridexplexbidex = s(3,3{1,1,2,1,1,2}2) = s(3,3{1,3,1,1,1,2}2) = s(3,3{3,2,1,1,1,2}2)
  • dimensolbidexplextridex = s(3,3{1,1,1,2,1,2}2) = s(3,3{1,1,3,1,1,2}2) = s(3,3{1,3,2,1,1,2}2) = s(3,3{3,2,2,1,1,2}2)
  • dimensoldexplexquadridex = s(3,3{1,1,1,1,2,2}2) = s(3,3{1,1,1,3,1,2}2) = s(3,3{1,1,3,2,1,2}2) = s(3,3{1,3,2,2,1,2}2) = s(3,3{3,2,2,2,1,2}2)
  • dimensolplexquintidex = s(3,3{1,1,1,1,1,3}2) = s(3,3{1,1,1,1,3,2}2) = s(3,3{1,1,1,3,2,2}2) = s(3,3{1,1,3,2,2,2}2) = s(3,3{1,3,2,2,2,2}2) = s(3,3{3,2,2,2,2,2}2). Also s(3,3{1,1,1,1,1,1,2}2) = s(3,3{1,1,1,1,1,3}2) = dimensolplexquintidex.

And so on. Now have a look at effects of suffix “-dex” and “-plex”. First, both modify the “separator A” in the definition of namebases. The “n-dex” increase the (n+1)-th entry of A from 1 to 2, then the “n-plex” prior to m “-dex”es increase the (m+1)-th entry of A by n. However, since we start on A = {1,2}, names below dimensolbidex have an offset on the 2nd entry of A.

Naming space

Now I introduce an idea about naming numbers – the naming space. The size of naming space of a naming system shows how many regular names the system can produce.

Here’s how “-plex” and “-dex” works in my naming system. We can have arbitrary many “-plex”es, and shorten as “n-plex”, so this forms a naming space of {\omega}. Notice that I don’t write {\aleph_0} because the levels of “-plex”, “-biplex”, “-triplex”, etc. are ordered. We want “-biplex” to be a higher-leveled modifier than “-plex”, and “-triplex” is even higher-leveled, and so on.

Next, a “-dex” is a stronger modifier than all the “n-plex”es, and “-dex” followed by “n-plex” form another naming space of {\omega}, and of {\omega2} in total. Then “-plexdex” followed by “n-plex” form another {\omega}, and so on. So “n-plexdex-m-plex” form a naming space of {\omega^2}. We still want them to be ordered, e.g. “-plexdex” is higher-leveled than all the “-dex-n-plex”es.

Next, 2 “-dex”es separate “-plex”es into 3 groups, so they form a naming space of {\omega^3}; 3 “-dex”es separate “-plex”es into 4 groups, so they form a naming space of {\omega^4}, and so on. Multi-dex and “-plex”es then form a naming space of {\omega^\omega}.

What about the notation system? With “-plex”es and “-dex”es I named numbers up to separator {1{2}2}, which has recursion level {\omega^\omega} – that’s equal to the size of the naming space, so we can have a name for numbers at every separator recursion level. However, function f(n) = s(n,n{1{2}2}2) has growth rate {\omega^{\omega^{\omega^\omega}}>\omega^\omega}, so the naming space is not enough if I must name for every growth rate (in FGH scale). I indeed named numbers up to separator {1{2}2}, but you can see what’s wrong with the naming system. The problem is that there’re many naming holes. e.g. between two adjacent terms in the naming system, dimentrienquintidex (where s(n,n,3{1,1,1,1,1,2}2) has growth rate {\omega^{\omega^{\omega^5}}+2}) and dimentrilquintidex (where s(n,n{1,1,1,1,1,2}3) has growth rate {\omega^{\omega^{\omega^5}}2}), there’re many “unnamed growth rate” (i.e. without any named numbers), so they form a naming hole.

So how to deal with naming holes? One way is to leave it empty, but we can’t feel the hugeness of the step from before the hole to after the hole – it’s not good. A second way is to change the order of the modifiers we use, to form a larger naming space, then it fit our need. But, the new order may be more chaotic, and the result names may be longer and unclear. That’s a “notational way”, with more and more rules, not suit for naming. Another way is to add new modifiers to the naming system, which can fill the naming holes. Then the types of modifiers will be greater as we go on. That’s a “naming way”, with more and more new modifiers, and we get more and more colorful names – that’s good. So I choose the last one for the next steps.

-threx group

Now I introduce a new modifier – the suffix “-threx”. It means the 3rd single modifier after “-plex” and “-dex”. Here’s the basic framework of the naming system. First, we have “n-threx”, meaning n “-threx”es, and this form a naming space of {\omega}. Second, we insert “-dex”es between “-threx”es, and this step multiply the naming space by {\omega^\omega}. Finally, we insert “-plex”es between suffixes already existing, and this step multiply the naming space by {\omega^\omega} again. So the size of the naming space is {\omega^{\omega2}}. However, I still want the system can go up to separator recursion level {\omega^{\omega^\omega}}! So I must need more modifiers…

Let’s begin with a single “-threx”.

  • dimentrienthrex = s(3,3,3{1{2}2}2)
  • dimentrilthrex = s(3,3{1{2}2}3)
  • dimenbolthrex = s(3,3{1{2}2}1{1{2}2}2)

Then add “-plex”es after the “-threx”.

  • dimensolthrexplex = s(3,3{2{2}2}2)
  • dimentrienthrexplex = s(3,3,3{2{2}2}2)
  • dimentrilthrexplex = s(3,3{2{2}2}3)
  • dimenbolthrexplex = s(3,3{2{2}2}1{2{2}2}2)
  • dimensolthrexbiplex = s(3,3{3{2}2}2)
  • dimentrienthrexbiplex = s(3,3,3{3{2}2}2)
  • dimentrilthrexbiplex = s(3,3{3{2}2}3)
  • dimenbolthrexbiplex = s(3,3{3{2}2}1{3{2}2}2)

Then we can add “-plex”es prior to the “-threx”, such as

  • dimentrienplexthrex = s(3,3,3{1{2}3}2)
  • dimentrilplexthrex = s(3,3{1{2}3}3)
  • dimenbolplexthrex = s(3,3{1{2}3}1{1{2}3}2)

But wait! There’s a naming hole between {1,2{2}2} and {1{2}3}. To fill the naming hole, I use a new modifier.

  • dimensolthrexdexiplex = s(3,3{1,2{2}2}2) = s(3,3{3,1{2}2}2), which is slightly larger than dimensolthrexbiplex.
  • dimentrienthrexdexiplex = s(3,3,3{1,2{2}2}2)
  • dimentrilthrexdexiplex = s(3,3{1,2{2}2}3)
  • dimenbolthrexdexiplex = s(3,3{1,2{2}2}1{1,2{2}2}2)
  • dimensolthrexdexibiplex = s(3,3{2,2{2}2}2)
  • dimensolthrexbidexiplex = s(3,3{1,1,2{2}2}2) = s(3,3{1,3,1{2}2}2)
  • dimensolthrexbidexibiplex = s(3,3{2,1,2{2}2}2)
  • dimensolthrexdexiplexdexiplex = s(3,3{1,2,2{2}2}2) = s(3,3{3,1,2{2}2}2), a.k.a. dimensolthrexbidexitriplex.
  • dimensolthrexdexiplexdexibiplex = s(3,3{2,2,2{2}2}2)
  • dimensolthrexdex = s(3,3{1{2}1,2}2) = s(3,3{1{2}3}2) = s(3,3{1,1,1,2{2}2}2), a.k.a. dimensolplexthrex and dimensolthrextridexiplex.

Notice that I use “-dexi” instead of “-dex”, and I always leave one more “-plex” at the end of the name. That’s my way to fill the naming hole. Now continue upward…

  • dimensolplexthrexplex = s(3,3{2{2}3}2)
  • dimensolbiplexthrex = s(3,3{1{2}4}2) = s(3,3{1,1,1,2{2}3}2) = s(3,3{1,1,3,1{2}3}2) – and we have the same way to fill the naming hole between the first two numbers.
  • dimensolthrexdexplex = s(3,3{2{2}1,2}2)
  • dimensolthrexdexdexiplex = s(3,3{1,2{2}1,2}2)
  • dimensolthrexplexdex = s(3,3{1{2}2,2}2)
  • dimensolthrexplexdexplex = s(3,3{2{2}2,2}2)
  • dimensolplexthrexdex = s(3,3{1{2}1,3}2). Also dimensolthrexbidex = s(3,3{1{2}1,1,2}2) = s(3,3{1{2}1,3}2) = dimensolplexthrexdex.
  • dimensolplexthrexdexplex = s(3,3{2{2}1,3}2)
  • dimensolplexthrexplexdex = s(3,3{1{2}2,3}2)
  • dimensolplexthrexplexdexplex = s(3,3{2{2}2,3}2)

A “-threx” and a “-dex” can form 3 places where “n-plex” can insert. The last place corresponds to increasing the 1st entry of {1{2}1,2}, the 2nd place for the 2nd entry of {1{2}1,2}, and the 1st place for the last entry. For more “-threx”es and “-dex”es, this still holds.

  • dimensolthrexbidexplex = s(3,3{2{2}1,1,2}2)
  • dimensolthrexdexplexdex = s(3,3{1{2}2,1,2}2)
  • dimensolthrexplexbidex = s(3,3{1{2}1,2,2}2)
  • dimensoldexthrex = s(3,3{1{2}1{2}2}2) = s(3,3{1{2}1,1,1,2}2) = s(3,3{1{2}1,1,3}2), a.k.a. dimensolthrextridex and dimensolplexthrexbidex.
  • dimensoldexthrexplex = s(3,3{2{2}1{2}2}2)
  • dimensoldexplexthrex = s(3,3{1{2}2{2}2}2)
  • dimensoldexthrexdex = s(3,3{1{2}1{2}1,2}2) = s(3,3{1{2}1{2}3}2), a.k.a. dimensolplexdexthrex .
  • dimensoldexthrexdexplex = s(3,3{2{2}1{2}1,2}2)
  • dimensoldexthrexplexdex = s(3,3{1{2}2{2}1,2}2)
  • dimensoldexplexthrexdex = s(3,3{1{2}1{2}2,2}2)
  • dimensoldexthrexbidex = s(3,3{1{2}1{2}1,1,2}2) = s(3,3{1{2}1{2}1,3}2), a.k.a. dimensolplexdexthrexdex.
  • dimensoldexthrexbidexplex = s(3,3{2{2}1{2}1,1,2}2)
  • dimensoldexthrexdexplexdex = s(3,3{1{2}2{2}1,1,2}2)
  • dimensoldexthrexplexbidex = s(3,3{1{2}1{2}2,1,2}2)
  • dimensoldexplexthrexbidex = s(3,3{1{2}1{2}1,2,2}2)
  • extendol = s(3,3{1`2}2) = s(3,3{1{1,2}2}2) = s(3,3{1{3}2}2) = s(3,3{1{2}1{2}1{2}2}2) = s(3,3{1{2}1{2}1,1,1,2}2) = s(3,3{1{2}1{2}1,1,3}2), a.k.a. dimensoltetrex, dimensolbithrex, dimensolbidexthrex, dimensoldexthrextridex and dimensolplexdexthrexbidex. For more information about this number, see a later page “numbers from expanding array notation”.

Generally, a “-dex” add an entry at the end of the layer-1 separator (i.e. the “separator A”). If you want some expressions with extra entry not at the end of the separator A, then you go into a naming hole, and you need “-dexi”es. e.g. dimensoldexthrexdexiplexdex = s(3,3{1{2}1,2{2}1,2}2) = s(3,3{1{2}3,1{2}1,2}2).

  • dimensolbithrexplex = s(3,3{2{3}2}2)
  • dimensolthrexplexthrex = s(3,3{1{2}2{3}2}2). Hey! There’s an extra entry! This entry corresponds to {1{2}2}, or the strength of a “-threx”.
  • dimensolthrexdexiplexthrex = s(3,3{1{2}1,2{3}2}2) – for other extra entries we need “-dexi”es.
  • dimensolthrexthrexiplexthrex = s(3,3{1{2}1{2}2{3}2}2) – now “-threxi” appears.
  • dimensolthrexthrexidexiiplexthrex = s(3,3{1{2}1,2{2}2{3}2}2) – now “-dexii” appears. That’s a way to fill an “2nd-order” naming hole, where even with “-dexi” and “-threxi” we still can’t cover. If there’re still naming holes, we’ll use more Roman numerals such as “-dexiii”, “-dexiv”, “-dexv”, “-dexvi”, etc.
  • dimensolbithrexdex = s(3,3{1{3}1,2}2) = s(3,3{1{3}3}2), a.k.a. dimensolplexbithrex.
  • dimensolbithrexdexplex = s(3,3{2{3}1,2}2)
  • dimensolbithrexplexdex = s(3,3{1{2}2{3}1,2}2)
  • dimensolthrexplexthrexdex = s(3,3{1{3}2,2}2)
  • dimensolbithrexbidex = s(3,3{1{3}1,1,2}2) = s(3,3{1{3}1,3}2), a.k.a. dimensolplexbithrexdex.
  • dimensolthrexdexthrex = s(3,3{1{3}1{2}2}2) = s(3,3{1{3}1,1,1,2}2) = s(3,3{1{3}1,1,3}2), a.k.a. dimensolbithrextridex and dimensolplexbithrexbidex.
  • dimensolthrexdexthrexdex = s(3,3{1{3}1{2}1,2}2)
  • dimensolthrexbidexthrex = s(3,3{1{3}1{2}1{2}2}2)
  • dimensoldexbithrex = s(3,3{1{3}1{3}2}2) = s(3,3{1{3}1{2}1{2}1{2}2}2), a.k.a. dimensolthrextridexthrex.
  • dimensoltrithrex = s(3,3{1{4}2}2) = s(3,3{1{3}1{3}1{3}2}2), a.k.a. dimensolbidexbithrex.
  • dimensoltrithrexplex = s(3,3{2{4}2}2)
  • dimensolbithrexplexthrex = s(3,3{1{2}2{4}2}2)
  • dimensolthrexplexbithrex = s(3,3{1{3}2{4}2}2)
  • dimensoltrithrexdex = s(3,3{1{4}1,2}2) = s(3,3{1{4}3}2), a.k.a. dimensolplextrithrex.
  • dimensolbithrexdexthrex = s(3,3{1{4}1{2}2}2)
  • dimensolthrexdexbithrex = s(3,3{1{4}1{3}2}2)
  • dimensoldextrithrex = s(3,3{1{4}1{4}2}2)
  • dimensolquadrithrex = s(3,3{1{5}2}2) = s(3,3{1{4}1{4}1{4}2}2), a.k.a. dimensolbidextrithrex.
  • dimensolquintithrex = s(3,3{1{6}2}2)

Now I show you how the main naming system (excluding the filling naming hole part) works for the “-threx” level.

  1. At the beginning, 4 namebases are mapped into expressions. Separator A is initialized as {1,2}.
  2. Append “-threx”es to the name. The entry inside the separator inside A increases 1 every appending.
  3. Insert “-dex”es to the name. For every insertion, look at the position from the end of the name. If it’s the k-th position, the A will add an entry at the end, where the separator corresponds to the result of “appending k ‘-threx’es at step 2”.
  4. Insert “-plex”es to the name. For every insertion, look at the position from the end of the name. If it’s the k-th position, the k-th entry of A will increases 1.
  5. Replace multiple same suffixes with “bi-“, “tri-“, “quadri-” etc. ones.

Using ordinals, the naming process can be more clear. A “-threx” means {\omega^{\omega^\alpha}\mapsto\omega^{\omega^{\alpha+\beta}}}, a “-dex” means {\omega^\alpha\mapsto\omega^{\alpha+\beta}}, and a “-plex” means {\alpha\mapsto\alpha+\beta} in separator recursion level, where the β is defined by the result of adding k suffixes in previous steps, where k is the position of the suffix we focus from the end of the name. What’s more important, this way works even beyond the “-threx” group!

-tetrex and beyond

Using suffix “-tetrex”, we have a naming space of {\omega^{\omega3}}.

  • dimentrientetrex = s(3,3,3{1{1,2}2}2)
  • dimentriltetrex = s(3,3{1{1,2}2}3)
  • dimenboltetrex = s(3,3{1{1,2}2}1{1{1,2}2}2)
  • dimensoltetrexplex = s(3,3{2{1,2}2}2)
  • dimensoltetrexdexiplex = s(3,3{1,2{1,2}2}2) = s(3,3{3,1{1,2}2}2)
  • dimensoltetrexplexdexiplex = s(3,3{1,3{1,2}2}2)
  • dimensoltetrexbidexiplex = s(3,3{1,1,2{1,2}2}2) = s(3,3{1,3,1{1,2}2}2)
  • dimensoltetrexthrexiplex = s(3,3{1{2}2{1,2}2}2) = s(3,3{1,1,1,2{2}1{1,2}2}2)
  • dimensoltetrexthrexidexiiplex = s(3,3{1,2{2}2{1,2}2}2) = s(3,3{3,1{2}2{1,2}2}2)
  • dimensoltetrexthrexidexiplex = s(3,3{1{2}1,2{1,2}2}2) = s(3,3{1{2}3,1{1,2}2}2)
  • dimensoltetrexdexithrexiplex = s(3,3{1{2}1{2}2{1,2}2}2) = s(3,3{1{2}1,1,1,2{2}1{1,2}2}2)
  • dimensoltetrexdex = s(3,3{1{1,2}1,2}2) = s(3,3{1{1,2}3}2) = s(3,3{1{3}2{1,2}2}2), a.k.a. dimensolplextetrex and dimensoltetrexbithrexiplex.
  • dimensoltetrexdexplex = s(3,3{2{1,2}1,2}2)
  • dimensoltetrexplexdex = s(3,3{1{1,2}2,2}2)
  • dimensoltetrexbidex = s(3,3{1{1,2}1,1,2}2) = s(3,3{1{1,2}1,3}2), a.k.a. dimensolplextetrexdex.
  • dimensoltetrexthrexidex = s(3,3{1{1,2}1{2}2}2) – Now a “-threxi” appears prior to a “-dex”.
  • dimensoldextetrex = s(3,3{1{1,2}1{1,2}2}2) = s(3,3{1{1,2}1{3}2}2), a.k.a. dimensoltetrexbithrexidex.
  • dimensoldextetrexplex = s(3,3{2{1,2}1{1,2}2}2)
  • dimensoldexplextetrex = s(3,3{1{1,2}2{1,2}2}2)
  • dimensoldextetrexdex = s(3,3{1{1,2}1{1,2}1,2}2) = s(3,3{1{1,2}1{1,2}3}2), a.k.a. dimensolplexdextetrex.
  • dimensoltetrexthrex = s(3,3{1{2,2}2}2) = s(3,3{1{1,2}1{1,2}1{1,2}2}2), a.k.a. dimensolbidextetrex.
  • dimensoltetrexthrexplex = s(3,3{2{2,2}2}2)
  • dimensoltetrexplexthrex = s(3,3{1{1,2}2{2,2}2}2)
  • dimensoltetrexthrexdex = s(3,3{1{2,2}1,2}2) = s(3,3{1{2,2}3}2), a.k.a. dimensolplextetrexthrex.
  • dimensoltetrexthrexdexplex = s(3,3{2{2,2}1,2}2)
  • dimensoltetrexthrexplexdex = s(3,3{1{1,2}2{2,2}1,2}2)
  • dimensoltetrexplexthrexdex = s(3,3{1{2,2}2,2}2)
  • dimensoltetrexthrexbidex = s(3,3{1{2,2}1,1,2}2) = s(3,3{1{2,2}1,3}2), a.k.a. dimensolplextetrexthrexdex.
  • dimensoltetrexdexthrex = s(3,3{1{2,2}1{1,2}2}2)
  • dimensoltetrexdexthrexplex = s(3,3{2{2,2}1{1,2}2}2)
  • dimensoltetrexdexplexthrex = s(3,3{1{1,2}2{2,2}1{1,2}2}2)
  • dimensoltetrexplexdexthrex = s(3,3{1{2,2}2{1,2}2}2)
  • dimensoltetrexdexthrexdex = s(3,3{1{2,2}1{1,2}1,2}2) = s(3,3{1{2,2}1{1,2}3}2), a.k.a. dimensolplextetrexdexthrex.
  • dimensoltetrexdexthrexbidex = s(3,3{1{2,2}1{1,2}1,1,2}2)
  • dimensoltetrexbidexthrex = s(3,3{1{2,2}1{1,2}1{1,2}2}2)
  • dimensoldextetrexthrex = s(3,3{1{2,2}1{2,2}2}2)
  • dimensoldextetrexthrexdex = s(3,3{1{2,2}1{2,2}1,2}2)
  • dimensoldextetrexthrexbidex = s(3,3{1{2,2}1{2,2}1,1,2}2)
  • dimensoldextetrexdexthrex = s(3,3{1{2,2}1{2,2}1{1,2}2}2)
  • dimensoldextetrexdexthrexdex = s(3,3{1{2,2}1{2,2}1{1,2}1,2}2)
  • dimensoldextetrexdexthrexbidex = s(3,3{1{2,2}1{2,2}1{1,2}1,1,2}2)
  • dimensoldextetrexbidexthrex = s(3,3{1{2,2}1{2,2}1{1,2}1{1,2}2}2)
  • dimensolbitetrex = s(3,3{1{1,1,2}2}2) = s(3,3{1{1,3}2}2) = s(3,3{1{3,2}2}2) = s(3,3{1{2,2}1{2,2}1{2,2}2}2), a.k.a. dimensolthrextetrex, dimensoltetrexbithrex and dimensolbidextetrexthrex.
  • dimensolbitetrexplex = s(3,3{2{1,1,2}2}2)
  • dimensoltetrexplextetrex = s(3,3{1{1,2}2{1,1,2}2}2)
  • dimensolbitetrexdex = s(3,3{1{1,1,2}1,2}2) = s(3,3{1{1,1,2}3}2), a.k.a. dimensolplexbitetrex.
  • dimensolbitetrexdexplex = s(3,3{2{1,1,2}1,2}2)
  • dimensolbitetrexplexdex = s(3,3{1{1,2}2{1,1,2}1,2}2)
  • dimensoltetrexplextetrexdex = s(3,3{1{1,1,2}2,2}2)
  • dimensolbitetrexbidex = s(3,3{1{1,1,2}1,1,2}2) = s(3,3{1{1,1,2}1,3}2), a.k.a. dimensolplexbitetrexdex.
  • dimensoltetrexdextetrex = s(3,3{1{1,1,2}1{1,2}2}2)
  • dimensoltetrexdextetrexplex = s(3,3{2{1,1,2}1{1,2}2}2)
  • dimensoltetrexdexplextetrex = s(3,3{1{1,2}2{1,1,2}1{1,2}2}2)
  • dimensoltetrexplexdextetrex = s(3,3{1{1,1,2}2{1,2}2}2)
  • dimensoltetrexdextetrexdex = s(3,3{1{1,1,2}1{1,2}1,2}2) = s(3,3{1{1,1,2}1{1,2}3}2), a.k.a. dimensolplextetrexdextetrex.
  • dimensoltetrexdextetrexbidex = s(3,3{1{1,1,2}1{1,2}1,1,2}2)
  • dimensoltetrexbidextetrex = s(3,3{1{1,1,2}1{1,2}1{1,2}2}2)
  • dimensoltetrexbidextetrexdex = s(3,3{1{1,1,2}1{1,2}1{1,2}1,2}2)
  • dimensoltetrexbidextetrexbidex = s(3,3{1{1,1,2}1{1,2}1{1,2}1,1,2}2)
  • dimensoldexbitetrex = s(3,3{1{1,1,2}1{1,1,2}2}2)
  • dimensoldexbitetrexplex = s(3,3{2{1,1,2}1{1,1,2}2}2)
  • dimensoldextetrexplextetrex = s(3,3{1{1,2}2{1,1,2}1{1,1,2}2}2)
  • dimensoldexplexbitetrex = s(3,3{1{1,1,2}2{1,1,2}2}2)
  • dimensoldexbitetrexdex = s(3,3{1{1,1,2}1{1,1,2}1,2}2) = s(3,3{1{1,1,2}1{1,1,2}3}2), a.k.a. dimensolplexdexbitetrex.
  • dimensoldexbitetrexbidex = s(3,3{1{1,1,2}1{1,1,2}1,1,2}2)
  • dimensoldextetrexdextetrex = s(3,3{1{1,1,2}1{1,1,2}1{1,2}2}2)
  • dimensoldextetrexbidextetrex = s(3,3{1{1,1,2}1{1,1,2}1{1,2}1{1,2}2}2)
  • dimensolbitetrexthrex = s(3,3{1{2,1,2}2}2) = s(3,3{1{1,1,2}1{1,1,2}1{1,1,2}2}2), a.k.a. dimensolbidexbitetrex.
  • dimensolbitetrexthrexplex = s(3,3{2{2,1,2}2}2)
  • dimensolbitetrexplexthrex = s(3,3{1{1,2}2{2,1,2}2}2)
  • dimensoltetrexplextetrexthrex = s(3,3{1{1,1,2}2{2,1,2}2}2)
  • dimensolbitetrexthrexdex = s(3,3{1{2,1,2}1,2}2) = s(3,3{1{2,1,2}3}2), a.k.a. dimensolplexbitetrexthrex.
  • dimensolbitetrexthrexbidex = s(3,3{1{2,1,2}1,1,2}2)
  • dimensolbitetrexdexthrex = s(3,3{1{2,1,2}1{1,2}2}2)
  • dimensolbitetrexdexthrexdex = s(3,3{1{2,1,2}1{1,2}1,2}2)
  • dimensolbitetrexdexthrexbidex = s(3,3{1{2,1,2}1{1,2}1,1,2}2)
  • dimensolbitetrexbidexthrex = s(3,3{1{2,1,2}1{1,2}1{1,2}2}2)
  • dimensoltetrexdextetrexthrex = s(3,3{1{2,1,2}1{1,1,2}2}2)
  • dimensoltetrexdextetrexdexthrex = s(3,3{1{2,1,2}1{1,1,2}1{1,2}2}2)
  • dimensoltetrexdextetrexbidexthrex = s(3,3{1{2,1,2}1{1,1,2}1{1,2}1{1,2}2}2)
  • dimensoltetrexbidextetrexthrex = s(3,3{1{2,1,2}1{1,1,2}1{1,1,2}2}2)
  • dimensoldexbitetrexthrex = s(3,3{1{2,1,2}1{2,1,2}2}2)
  • dimensoldexbitetrexthrexdex = s(3,3{1{2,1,2}1{2,1,2}1,2}2)
  • dimensoldexbitetrexdexthrex = s(3,3{1{2,1,2}1{2,1,2}1{1,2}2}2)
  • dimensoldextetrexdextetrexthrex = s(3,3{1{2,1,2}1{2,1,2}1{1,1,2}2}2)
  • dimensoltetrexthrextetrex = s(3,3{1{1,2,2}2}2) = s(3,3{1{3,1,2}2}2) = s(3,3{1{2,1,2}1{2,1,2}1{2,1,2}2}2), a.k.a. dimensolbitetrexbithrex and dimensolbidexbitetrexthrex.
  • dimensoltetrexthrextetrexdex = s(3,3{1{1,2,2}1,2}2)
  • dimensoltetrexthrexdextetrex = s(3,3{1{1,2,2}1{1,2}2}2)
  • dimensoltetrexdexthrextetrex = s(3,3{1{1,2,2}1{1,1,2}2}2)
  • dimensoldextetrexthrextetrex = s(3,3{1{1,2,2}1{1,2,2}2}2)
  • dimensoltetrexthrextetrexthrex = s(3,3{1{2,2,2}2}2)
  • dimensolpentex = s(3,3{1{1{2}2}2}2) = s(3,3{1{1,1,1,2}2}2) = s(3,3{1{1,1,3}2}2) = s(3,3{1{1,3,2}2}2) = s(3,3{1{3,2,2}2}2), a.k.a. dimensoltritetrex, dimensolthrexbitetrex, dimensoltetrexbithrextetrex and dimensoltetrexthrextetrexbithrex.
  • dimensoltritetrexplex = s(3,3{2{1,1,1,2}2}2)
  • dimensolbitetrexplextetrex = s(3,3{1{1,2}2{1,1,1,2}2}2)
  • dimensoltetrexplexbitetrex = s(3,3{1{1,1,2}2{1,1,1,2}2}2)
  • dimensoltritetrexdex = s(3,3{1{1,1,1,2}1,2}2) = s(3,3{1{1,1,1,2}3}2), a.k.a. dimensolplextritetrex.
  • dimensolbitetrexdextetrex = s(3,3{1{1,1,1,2}1{1,2}2}2)
  • dimensoltetrexdexbitetrex = s(3,3{1{1,1,1,2}1{1,1,2}2}2)
  • dimensoldextritetrex = s(3,3{1{1,1,1,2}1{1,1,1,2}2}2)
  • dimensoltritetrexthrex = s(3,3{1{2,1,1,2}2}2)
  • dimensolbitetrexthrextetrex = s(3,3{1{1,2,1,2}2}2)
  • dimensoltetrexthrexbitetrex = s(3,3{1{1,1,2,2}2}2)
  • dimensolquadritetrex = s(3,3{1{1,1,1,1,2}2}2) = s(3,3{1{1,1,1,3}2}2), a.k.a. dimensolthrextritetrex.
  • dimensolquadritetrexplex = s(3,3{2{1,1,1,1,2}2}2)
  • dimensolquadritetrexdex = s(3,3{1{1,1,1,1,2}1,2}2) = s(3,3{1{1,1,1,1,2}3}2), a.k.a. dimensolplexquadritetrex.
  • dimensoldexquadritetrex = s(3,3{1{1,1,1,1,2}1{1,1,1,1,2}2}2)
  • dimensolquadritetrexthrex = s(3,3{1{2,1,1,1,2}2}2)
  • dimensolquintitetrex = s(3,3{1{1,1,1,1,1,2}2}2) = s(3,3{1{1,1,1,1,3}2}2), a.k.a. dimensolthrexquadritetrex.

Using suffix “-pentex”, we have a naming space of {\omega^{\omega4}}.

  • dimentrienpentex = s(3,3,3{1{1{2}2}2}2)
  • dimentrilpentex = s(3,3{1{1{2}2}2}3)
  • dimenbolpentex = s(3,3{1{1{2}2}2}1{1{1{2}2}2}2)
  • dimensolpentexplex = s(3,3{2{1{2}2}2}2)
  • dimensolpentexdex = s(3,3{1{1{2}2}1,2}2) = s(3,3{1{1{2}2}3}2), a.k.a. dimensolplexpentex.
  • dimensoldexpentex = s(3,3{1{1{2}2}1{1{2}2}2}2)
  • dimensolpentexthrex = s(3,3{1{2{2}2}2}2)
  • dimensolpentextetrex = s(3,3{1{1{2}1,2}2}2) = s(3,3{1{1{2}3}2}2), a.k.a. dimensolthrexpentex.
  • dimensoltetrexpentex = s(3,3{1{1{2}1{2}2}2}2)
  • dimensolhex = s(3,3{1{1{1,2}2}2}2) = s(3,3{1{1{3}2}2}2), a.k.a. dimensolbipentex.
  • dimensolbipentexplex = s(3,3{2{1{3}2}2}2)
  • dimensolbipentexdex = s(3,3{1{1{3}2}1,2}2) = s(3,3{1{1{3}2}3}2), a.k.a. dimensolplexbipentex.
  • dimensoldexbipentex = s(3,3{1{1{3}2}1{1{3}2}2}2)
  • dimensolbipentexthrex = s(3,3{1{2{3}2}2}2)
  • dimensolbipentextetrex = s(3,3{1{1{3}1,2}2}2) = s(3,3{1{1{3}3}2}2), a.k.a. dimensolthrexbipentex.
  • dimensoltetrexbipentex = s(3,3{1{1{3}1{3}2}2}2)
  • dimensoltripentex = s(3,3{1{1{4}2}2}2)
  • dimensolquadripentex = s(3,3{1{1{5}2}2}2)
  • dimensolquintipentex = s(3,3{1{1{6}2}2}2)

Using suffix “-hex”, we have a naming space of {\omega^{\omega5}}.

  • dimentrienhex = s(3,3,3{1{1{1,2}2}2}2)
  • dimentrilhex = s(3,3{1{1{1,2}2}2}3)
  • dimenbolhex = s(3,3{1{1{1,2}2}2}1{1{1{1,2}2}2}2)
  • dimensolhexplex = s(3,3{2{1{1,2}2}2}2)
  • dimensolhexdex = s(3,3{1{1{1,2}2}1,2}2) = s(3,3{1{1{1,2}2}3}2), a.k.a. dimensolplexhex.
  • dimensoldexhex = s(3,3{1{1{1,2}2}1{1{1,2}2}2}2)
  • dimensolhexthrex = s(3,3{1{2{1,2}2}2}2)
  • dimensolhextetrex = s(3,3{1{1{1,2}1,2}2}2) = s(3,3{1{1{1,2}3}2}2), a.k.a. dimensolthrexhex.
  • dimensoltetrexhex = s(3,3{1{1{1,2}1{1,2}2}2}2)
  • dimensolhexpentex = s(3,3{1{1{2,2}2}2}2)
  • dimensolbihex = s(3,3{1{1{1,1,2}2}2}2) = s(3,3{1{1{1,3}2}2}2), a.k.a. dimensolpentexhex.
  • dimensolheptex = s(3,3{1{1{1{2}2}2}2}2) = s(3,3{1{1{1,1,1,2}2}2}2), a.k.a. dimensoltrihex.
  • dimensolquadrihex = s(3,3{1{1{1,1,1,1,2}2}2}2)
  • dimensolquintihex = s(3,3{1{1{1,1,1,1,1,2}2}2}2)

And further…

  • dimentrienheptex = s(3,3,3{1{1{1{2}2}2}2}2)
  • dimentrilheptex = s(3,3{1{1{1{2}2}2}2}3)
  • dimenbolheptex = s(3,3{1{1{1{2}2}2}2}1{1{1{1{2}2}2}2}2)
  • dimensoloctex = s(3,3{1{1{1{1,2}2}2}2}2) = s(3,3{1{1{1{3}2}2}2}2), a.k.a. dimensolbiheptex.
  • dimensolbioctex = s(3,3{1{1{1{1,1,2}2}2}2}2)
  • dimensolennex = s(3,3{1{1{1{1{2}2}2}2}2}2) = s(3,3{1{1{1{1,1,1,2}2}2}2}2), a.k.a. dimensoltrioctex.
  • dimensoldecex = s(3,3{1{1{1{1{1,2}2}2}2}2}2) = s(3,3{1{1{1{1{3}2}2}2}2}2), a.k.a. dimensolbiennex.

So an “n-ex” means {\omega^{\omega^{.^{.^{.^{\omega^\alpha}}}}}\mapsto\omega^{\omega^{.^{.^{.^{\omega^{\alpha+\beta}}}}}}} (where the α is at the n-th level of the power tower), where β is defined by the result of adding k suffixes in previous steps, where k is the position of the suffix we focus from the end of the name. Extending the suffixes to “n-ex” for any n, we have a naming space of {\omega^{\omega^2}}, which is certainly not enough for every separator recursion level in exAN. But we can add modifiers “n-ex-r” (where r is in Roman numerals), then we have a naming space of {\varepsilon_0} – that’s enough for every separator recursion level in exAN.

What’s more, ε0 is the first ordinal that separator recursion level catches up with growth rate (in FGH scale). If we build a “junior” naming system, where the only difference from the normal one is that the “n-ex” rule is for growth rate instead of separator recursion level, then the “n-ex” is analogous to the “(n+2)-ex” in the “junior” naming system. e.g.

dimensolennex jr. = s(3,3{1{1{1{2}2}2}2}2) = dimensolheptex

So we can name every growth rate if we use the “junior” naming system. But here I don’t use that because I don’t want to left “-plex” and “-dex” below separators (which happens in the “junior” naming system), and separators are important in parts beyond exAN.

Numbers from linear array notation

Here are some typical numbers from linear array notation. They’re also examples to help you understand LAN.

3-entry series

Those numbers are defined using s(a,b,c) with a, b, c > 1.

Tribo group

This group includes tribo, tetbo, pentbo and hexabo in order.

Tribo = s(3,3,2) = 3↑↑3 = 3^(3↑↑2) = 3^(3^(3↑↑1)) = 3^(3^3) = 3^27 = 7625597484987. It also equals those expressions: s(3,2,3) =(by rule 3) s(3,s(3,1,3),2) = s(3,s(3,1,2),2) = s(3,s(3,1,1),2) =(by rule 2) s(3,s(3,1),2) = s(3,3^1,2) = s(3,3,2). Using exAN, s(3,2{2}2) = s(3,2{1}1{1}2{2}1) = s(3,2,1,2) = s(3,3,2,1) = s(3,3,2). This numbers is quite small, even in real life. The amount of quarks in something weighing 32.2 grams, is about tribo squared. If we line tribo cubed quarks up, they just form a line about 47 thousand light years long, which is less than the radius of the milky way galaxy.

Tetbo = s(4,4,2) = 4↑↑4 = 4^(4↑↑3) = 4^(4^(4↑↑2)) = 4^(4^(4^(4↑↑1))) = 4^(4^(4^4)) = 4^(4^256) ≈ 10^(8.0723×10^153). It’s also equals those expressions: s(4,2,3) =(by rule 3) s(4,s(4,1,3),2) = s(4,s(4,1,1),2) =(by rule 2) s(4,s(4,1),2) = s(4,4^1,2) = s(4,4,2). Using exAN, s(4,2{2}2) = s(4,2{1}1{1}2{2}1) = s(4,2,1,2) = s(4,4,2,1) = s(4,4,2). This number gets large in real life. Tetbo Planck times is already longer than the Poincare recurrence time of the observed universe.

Pentbo = s(5,5,2) = 5↑↑5 = 5^(5↑↑4) = 5^(5^(5↑↑3)) = 5^(5^(5^(5↑↑2))) = 5^(5^(5^(5^(5↑↑1)))) = 5^(5^(5^(5^5))) = 5^(5^(5^3125)) ≈ 10^(10^(1.33574×10^2184)). It’s also equals those expressions: s(5,2,3) = s(5,s(5,1,3),2) = s(5,s(5,1,1),2) = s(5,s(5,1),2) = s(5,5^1,2) = s(5,5,2). Using exAN, s(5,2{2}2) = s(5,2{1}1{1}2{2}1) = s(5,2,1,2) = s(5,5,2,1) = s(5,5,2). Pentbo Planck times is longer than the Poincare recurrence time of a black hole with the same weight as the observed universe.

Hexabo = s(6,6,2) = 6↑↑6 = 6^(6↑↑5) = 6^(6^(6↑↑4)) = 6^(6^(6^(6↑↑3))) = 6^(6^(6^(6^(6↑↑2)))) = 6^(6^(6^(6^(6^(6↑↑1))))) = 6^(6^(6^(6^(6^6)))) = 6^(6^(6^(6^46656))) ≈ 10^(10^(10^(2.0692×10^36305))). It’s also equals those expressions: s(6,2,3) = s(6,s(6,1,3),2) = s(6,s(6,1,1),2) = s(6,s(6,1),2) = s(6,6^1,2) = s(6,6,2). Using exAN, s(6,2{2}2) = s(6,2{1}1{1}2{2}1) = s(6,2,1,2) = s(6,6,2,1) = s(6,6,2). This number stands above all the numbers used in real life science.

Trientri group

This group includes trientri, tettro, pentro and hextro in order.

Trientri = s(3,3,3) = 3↑↑↑3 = 3↑↑(3↑↑↑2) = 3↑↑(3↑↑(3↑↑↑1)) = 3↑↑(3↑↑3) = 3↑↑tribo = 3↑↑7625597484987. Also s(3,2,4) = s(3,s(3,1,4),3) = s(3,s(3,1,1),3) = s(3,s(3,1),3) = s(3,3^1,3) = s(3,3,3) = trientri, s(3,1,1,3) = s(3,3,1,2) = s(3,3,3,1) = s(3,3,3) = trientri, and s(3,1,1,1,2) = s(3,3,3,1,1) = s(3,3,3,1) = s(3,3,3) = trientri (so s(3,1,c,1,2) = trientri for any c). The name “trientri” comes from “3 entries of 3”. It’s also called tritri by Jonathan Bowers. Trientri can be expressed in exponentiation as follows:
1#trientri

Tettro = s(4,4,3) = 4↑↑↑4 = 4↑↑(4↑↑↑3) = 4↑↑(4↑↑(4↑↑↑2)) = 4↑↑(4↑↑(4↑↑(4↑↑↑1))) = 4↑↑(4↑↑(4↑↑4)) = 4↑↑(4↑↑tetbo). Also s(4,2,4) = s(4,s(4,1,4),3) = s(4,4,3) = tettro, and s(4,3,1,2) = s(4,4,3,1) = s(4,4,3) = tettro. Tettro can be expressed in exponentiation as follows:
2#tettro

Pentro = s(5,5,3) = 5↑↑↑5 = 5↑↑(5↑↑(5↑↑(5↑↑5))) = 5↑↑(5↑↑(5↑↑pentbo)). Also s(5,2,4) = s(5,s(5,1,4),3) = s(5,5,3) = pentro, and s(5,3,1,2) = s(5,5,3,1) = s(5,5,3) = pentro. Pentro can be expressed in exponentiation as follows:
3#pentro

Hextro = s(6,6,3) = 6↑↑↑6 = 6↑↑(6↑↑(6↑↑(6↑↑(6↑↑6)))) = 6↑↑(6↑↑(6↑↑(6↑↑hexabo))). Also s(6,2,4) = s(6,s(6,1,4),3) = s(6,6,3) = hextro, and s(6,3,1,2) = s(6,6,3,1) = s(6,6,3) = hextro. Hextro can be expressed in exponentiation as follows:
4#hextro

Trientet group

This group includes triteto, trientet, penteto and hexteto in order.

Triteto = s(3,3,4) = 3↑↑↑↑3 = 3↑↑↑(3↑↑↑3) = 3↑↑↑trientri. Also s(3,2,5) = s(3,s(3,1,5),4) = s(3,3,4) = triteto, and s(3,4,1,2) = s(3,3,4,1) = s(3,3,4) = triteto. It’s also called grahal by Aarex Tiaokhiao. Triteto can be expressed in exponentiation as follows:
5#triteto

Trientet = s(4,4,4) = 4↑↑↑↑4 = 4↑↑↑(4↑↑↑(4↑↑↑4)) = 4↑↑↑(4↑↑↑tettro). Also s(4,2,5) = s(4,s(4,1,5),4) = s(4,4,4) = trientet, s(4,1,1,3) = s(4,4,1,2) = s(4,4,4,1) = s(4,4,4) = trientet, and s(4,1,1,1,2) = s(4,4,4,1,1) = s(4,4,4,1) = s(4,4,4) = trientet. The name “trientet” comes from “3 entries of 4”. It’s also called tritet by Jonathan Bowers. Trientet can be expressed in exponentiation as follows:
6#trientet

Penteto = s(5,5,4) = 5↑↑↑↑5 = 5↑↑↑(5↑↑↑(5↑↑↑(5↑↑↑5))) = 5↑↑↑(5↑↑↑(5↑↑↑pentro)). Also s(5,2,5) = s(5,s(5,1,5),4) = s(5,5,4) = penteto, and s(5,4,1,2) = s(5,5,4,1) = s(5,5,4) = penteto. Penteto can be expressed in exponentiation as follows:
7#penteto

Hexteto = s(6,6,4) = 6↑↑↑↑6 = 6↑↑↑(6↑↑↑(6↑↑↑(6↑↑↑(6↑↑↑6)))) = 6↑↑↑(6↑↑↑(6↑↑↑(6↑↑↑hextro))). Also s(6,2,5) = s(6,s(6,1,5),4) = s(6,6,4) = hexteto, and s(6,4,1,2) = s(6,6,4,1) = s(6,6,4) = hexteto. Hexteto can be expressed in exponentiation as follows:
8#hexteto

Trienpent group

This group includes tripeno, tetpeno, trienpent and hexpeno in order.

Tripeno = s(3,3,5) = 3↑↑↑↑↑3 = 3↑↑↑↑(3↑↑↑↑3) = 3↑↑↑↑triteto. Also s(3,2,6) = s(3,s(3,1,6),5) = s(3,3,5) = tripeno, and s(3,5,1,2) = s(3,3,5,1) = s(3,3,5) = tripeno. Tripeno can be expressed in exponentiation as follows:
9#tripeno

Tetpeno = s(4,4,5) = 4↑↑↑↑↑4 = 4↑↑↑↑(4↑↑↑↑(4↑↑↑↑4)) = 4↑↑↑↑(4↑↑↑↑trientet). Also s(4,2,6) = s(4,s(4,1,6),5) = s(4,4,5) = tetpeno, and s(4,5,1,2) = s(4,4,5,1) = s(4,4,5) = tetpeno. Tetpeno can be expressed in exponentiation as follows:
10#tetpeno

Trienpent = s(5,5,5) = 5↑↑↑↑↑5 = 5↑↑↑↑(5↑↑↑↑(5↑↑↑↑(5↑↑↑↑5))) = 5↑↑↑↑(5↑↑↑↑(5↑↑↑↑penteto)). Also s(5,2,6) = s(5,s(5,1,6),5) = s(5,5,5) = trienpent, s(5,1,1,3) = s(5,5,1,2) = s(5,5,5,1) = s(5,5,5) = trienpent, and s(5,1,1,1,2) = s(5,5,5,1,1) = s(5,5,5,1) = s(5,5,5) = trienpent. It’s also called tripent by Jonathan Bowers. Trienpent can be expressed in exponentiation as follows:
11#trienpent

Hexpeno = s(6,6,5) = 6↑↑↑↑↑6 = 6↑↑↑↑(6↑↑↑↑(6↑↑↑↑(6↑↑↑↑(6↑↑↑↑6)))) = 6↑↑↑↑(6↑↑↑↑(6↑↑↑↑(6↑↑↑↑hexteto))). Also s(6,2,6) = s(6,s(6,1,6),5) = s(6,6,5) = hexpeno, and s(6,5,1,2) = s(6,6,5,1) = s(6,6,5) = hexpeno. Hexpeno can be expressed in exponentiation as follows:
12#hexpeno

Trienhex group

This group includes trihexo, tethexo, penhexo and trienhex in order.

Trihexo = s(3,3,6) = 3↑↑↑↑↑↑3 = 3↑↑↑↑↑(3↑↑↑↑↑3) = 3↑↑↑↑↑tripeno. Also s(3,2,7) = s(3,s(3,1,7),6) = s(3,3,6) = trihexo, and s(3,6,1,2) = s(3,3,6,1) = s(3,3,6) = trihexo. Trihexo can be expressed in exponentiation as follows:
13#trihexo

Tethexo, penhexo and trienhex are s(4,4,6), s(5,5,6) and s(6,6,6) respectively.

4-entry series

Those numbers are defined using s(a,b,c,d) with a, b, d > 1.

Primitol group

This group includes primitol, primitolplex, primitolbiplex, primitoltriplex, primitolquadriplex, primibol, primibolplex, primitrol, primitrolplex, primitetol, primitetolplex, primipenol and primipenolplex in order.

Primitol = s(3,2,2,2) = s(3,s(3,1,2,2),1,2) = s(3,s(3,1,1,2),1,2) = s(3,s(3,3,1,1),1,2) = s(3,s(3,3,1),1,2) = s(3,s(3,3),1,2) = s(3,3^3,1,2) = s(3,27,1,2) = s(3,3,27,1) = s(3,3,27) = 3↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑3. Let “stage 1” be 3^3 = 27, stage 2 = tribo = 3↑↑3 = 7625597484987, then stage 3 = trientri = 3↑↑↑3, stage 4 = triteto = 3↑↑↑↑3, stage 5 = tripeno = 3↑↑↑↑↑3 and stage 6 = trihexo = 3↑↑↑↑↑↑3. We have some pictures for stage 3 ~ 6, and you can image how large the stage 27 is – that’s primitol. It’s hard to define it in primitive recursive arithmetic, so it’s named “primitol”.

Primitolplex = s(3,3,2,2) = s(3,s(3,2,2,2),1,2) = s(3,3,s(3,2,2,2),1) = s(3,3,s(3,2,2,2)) = s(3,3,s(3,3,27)) = s(3,3,primitol). Also s(3,2,1,3) = s(3,3,2,2) = primitolplex. It’s the “stage primitol” number described above. Primitolplex can be expressed in up-arrow notation as follows:
14#s(3,3,2,2)

Then, primitolbiplex, primitoltriplex and primitolquadriplex are s(3,4,2,2), s(3,5,2,2) and s(3,6,2,2) respectively. In up-arrow notation, they can be expressed as15#s(3,4,2,2),16#s(3,5,2,2)and17#s(3,6,2,2)respectively.

Primibol = s(3,2,3,2) = s(3,s(3,1,3,2),2,2) = s(3,s(3,1,2,2),2,2) = s(3,s(3,1,1,2),2,2) = s(3,s(3,3,1,1),2,2) = s(3,27,2,2). We can say it’s “primitol-25-plex”. Primibol can be expressed in up-arrow notation as follows:
18#s(3,2,3,2)

Primibolplex = s(3,3,3,2) = s(3,s(3,2,3,2),2,2) = s(3,primibol,2,2). Also s(3,1,1,4) = s(3,3,1,3) = s(3,3,3,2) = primibolplex, and in exAN, s(3,2,1{2}2) = s(3,2,1,1,2{2}1) = s(3,2,1,1,2) = s(3,3,3,2,1) = s(3,3,3,2) = primibolplex. It also equals 3→3→3→3 in Conway’s chained arrow notation. Primibolplex can be expressed in up-arrow notation as follows:
19#s(3,3,3,2)

Then, primitrol, primitetol and primipenol are s(3,2,4,2), s(3,2,5,2) and s(3,2,6,2) respectively. Primitrolplex, primitetolplex and primipenolplex are s(3,n,1,3) = s(3,3,n,2) with n = 4, 5 and 6 respectively.

Primitrol, primitrolplex, primitetol and primitetolplex can be expressed in up-arrow notation as20#s(3,2,4,2),21#s(3,3,4,2),22#s(3,2,5,2)and22#s(3,3,5,2)respectively. (The pictures for primipenol and primipenolplex are too large and complex)

Tetentri group

This group includes duprimitol, duprimitolplex, duprimibol and tetentri in order.

Duprimitol = s(3,2,2,3) = s(3,s(3,1,2,3),1,3) = s(3,s(3,1,1,3),1,3) = s(3,s(3,3,1,2),1,3) = s(3,s(3,3,3,1),1,3) = s(3,s(3,3,3),1,3) = s(3,3,s(3,3,3),2) = s(3,3,trientri,2). We can say it’s “primi-(trientri-1)-olplex”. Then, duprimitolplex = s(3,3,2,3) = s(3,s(3,2,2,3),1,3) = s(3,3,s(3,2,2,3),2) = s(3,3,duprimitol,2), also s(3,2,1,4) = s(3,3,2,3) = duprimitolplex. And duprimibol = s(3,2,3,3) = s(3,s(3,1,3,3),2,3) = s(3,s(3,1,1,3),2,3) = s(3,trientri,2,3).

Tetentri = s(3,3,3,3) = s(3,s(3,2,3,3),2,3) = s(3,duprimibol,2,3). Also s(3,1,1,5) = s(3,3,1,4) = s(3,3,3,3) = tetentri, s(3,1,1,2,2) = s(3,3,1,1,2) = s(3,3,3,3,1) = s(3,3,3,3) = tetentri, s(3,1,1,1,1,2) = s(3,3,3,3,1,1) = s(3,3,3,3,1) = s(3,3,3,3) = tetentri, and in exAN s(3,3{2}2) = s(3,3,1,1,2{2}1) = s(3,3,1,1,2) = s(3,3,3,3,1) = s(3,3,3,3) = tetentri. It also equals 3→3→3→3→3 in Conway’s chained arrow notation. The name “tetentri” comes from “4 entries of 3”, but we can also name it duprimibolplex.

Tetentet group

This group includes truprimitol, truprimitolplex, truprimibol, truprimibolplex and tetentet in order.

Truprimitol = s(3,2,2,4) = s(3,s(3,1,2,4),1,4) = s(3,s(3,1,1,4),1,4) = s(3,s(3,3,1,3),1,4) = s(3,s(3,3,3,2),1,4) = s(3,3,s(3,3,3,2),3) = s(3,3,primibolplex,3). Truprimitolplex = s(3,3,2,4) = s(3,s(3,2,2,4),1,4) = s(3,3,s(3,2,2,4),3) = s(3,3,truprimitol,3), also s(3,2,1,5) = s(3,3,2,4) = truprimitolplex.

Truprimibol = s(3,2,3,4) = s(3,s(3,1,3,4),2,4) = s(3,s(3,1,1,4),2,4) = s(3,s(3,3,3,2),2,4) = s(3,primibolplex,2,4). Truprimibolplex = s(3,3,3,4) = s(3,s(3,2,3,4),2,4) = s(3,truprimibol,2,4), also s(3,1,1,6) = s(3,3,1,5) = s(3,3,3,4) = truprimibolplex, and s(3,4,1,1,2) = s(3,3,3,4,1) = s(3,3,3,4) = truprimibolplex.

Tetentet = s(4,4,4,4) = s(4,s(4,3,4,4),3,4) = s(4,s(4,s(4,2,4,4),3,4),3,4) = s(4,s(4,s(4,s(4,1,4,4),3,4),3,4),3,4) = s(4,s(4,s(4,s(4,1,1,4),3,4),3,4),3,4) = s(4,s(4,s(4,s(4,4,1,3),3,4),3,4),3,4) = s(4,s(4,s(4,s(4,4,4,2),3,4),3,4),3,4) = s(4,s(4,s(4,s(4,s(4,s(4,s(4,1,4,2),3,2),3,2),3,2),3,4),3,4),3,4) = s(4,s(4,s(4,s(4,s(4,s(4,s(4,1,1,2),3,2),3,2),3,2),3,4),3,4),3,4) = s(4,s(4,s(4,s(4,s(4,s(4,s(4,4,1,1),3,2),3,2),3,2),3,4),3,4),3,4) = s(4,s(4,s(4,s(4,s(4,s(4,256,3,2),3,2),3,2),3,4),3,4),3,4). Also s(4,1,1,6) = s(4,4,1,5) = s(4,4,4,4) = tetentet, s(4,1,1,2,2) = s(4,4,1,1,2) = s(4,4,4,4,1) = s(4,4,4,4) = tetentet, and s(4,1,1,1,1,2) = s(4,4,4,4,1,1) = s(4,4,4,4,1) = s(4,4,4,4) = tetentet. It also equals 4→4→4→4→4→4 in Conway’s chained arrow notation.

Tetenpent group

This group includes quadprimitol, quadprimitolplex, quadprimibol, quadprimibolplex and tetenpent in order.

Quadprimitol = s(3,2,2,5) = s(3,s(3,1,2,5),1,5) = s(3,s(3,1,1,5),1,5) = s(3,s(3,3,1,4),1,5) = s(3,s(3,3,3,3),1,5) = s(3,3,s(3,3,3,3),4) = s(3,3,tetentri,4). Quadprimitolplex = s(3,3,2,5) = s(3,s(3,2,2,5),1,5) = s(3,3,s(3,2,2,5),4) = s(3,3,quadprimitol,4), also s(3,2,1,6) = s(3,3,2,5) = quadprimitolplex.

Quadprimibol = s(3,2,3,5) = s(3,s(3,1,3,5),2,5) = s(3,s(3,1,1,5),2,5) = s(3,tetentri,2,5). Quadprimibolplex = s(3,3,3,5) = s(3,s(3,2,3,5),2,5) = s(3,quadprimibol,2,5), also s(3,1,1,7) = s(3,3,1,6) = s(3,3,3,5) = quadprimibolplex, and s(3,5,1,1,2) = s(3,3,3,5,1) = s(3,3,3,5) = quadprimibolplex.

Tetenpent = s(5,5,5,5) = s(5,s(5,s(5,s(5,s(5,1,5,5),4,5),4,5),4,5),4,5) = s(5,s(5,s(5,s(5,s(5,1,1,5),4,5),4,5),4,5),4,5) = s(5,s(5,s(5,s(5,s(5,5,1,4),4,5),4,5),4,5),4,5) = s(5,s(5,s(5,s(5,s(5,5,5,3),4,5),4,5),4,5),4,5) = s(5,s(5,s(5,s(5,s(5,s(5,s(5,s(5,s(5,1,5,3),4,3),4,3),4,3),4,3),4,5),4,5),4,5),4,5) = s(5,s(5,s(5,s(5,s(5,s(5,s(5,s(5,s(5,1,1,3),4,3),4,3),4,3),4,3),4,5),4,5),4,5),4,5) = s(5,s(5,s(5,s(5,s(5,s(5,s(5,s(5,s(5,5,1,2),4,3),4,3),4,3),4,3),4,5),4,5),4,5),4,5) = s(5,s(5,s(5,s(5,s(5,s(5,s(5,s(5,s(5,5,5,1),4,3),4,3),4,3),4,3),4,5),4,5),4,5),4,5) = s(5,s(5,s(5,s(5,s(5,s(5,s(5,s(5,s(5,5,5),4,3),4,3),4,3),4,3),4,5),4,5),4,5),4,5) = s(5,s(5,s(5,s(5,s(5,s(5,s(5,s(5,trienpent,4,3),4,3),4,3),4,3),4,5),4,5),4,5),4,5). Also s(5,1,1,7) = s(5,5,1,6) = s(5,5,5,5) = tetenpent, s(5,1,1,2,2) = s(5,5,1,1,2) = s(5,5,5,5,1) = s(5,5,5,5) = tetenpent, and s(5,1,1,1,1,2) = s(5,5,5,5,1,1) = s(5,5,5,5,1) = s(5,5,5,5) = tetenpent. It also equals 5→5→5→5→5→5→5 in Conway’s chained arrow notation.

Tetenhex group

This group includes quinprimitol, quinprimitolplex, quinprimibol, quinprimibolplex and tetenhex in order.

Quinprimitol = s(3,2,2,6) = s(3,s(3,1,2,6),1,6) = s(3,s(3,1,1,6),1,6) = s(3,s(3,3,1,5),1,6) = s(3,s(3,3,3,4),1,6) = s(3,3,s(3,3,3,4),5) = s(3,3,truprimibolplex,5). Quinprimitolplex = s(3,3,2,6) = s(3,s(3,2,2,6),1,6) = s(3,3,s(3,2,2,6),5) = s(3,3,quinprimitol,5), also s(3,2,1,7) = s(3,3,2,6) = quinprimitolplex.

Quinprimibol = s(3,2,3,6) = s(3,s(3,1,3,6),2,6) = s(3,s(3,1,1,6),2,6) = s(3,truprimibolplex,2,6). Quinprimibolplex = s(3,3,3,6) = s(3,s(3,2,3,6),2,6) = s(3,quinprimibol,2,6), also s(3,1,1,8) = s(3,3,1,7) = s(3,3,3,6) = quinprimibolplex, and s(3,6,1,1,2) = s(3,3,3,6,1) = s(3,3,3,6) = quinprimibolplex.

Tetenhex = s(6,6,6,6) = s(6,s(6,s(6,s(6,s(6,s(6,1,6,6),5,6),5,6),5,6),5,6),5,6), where s(6,1,6,6) = s(6,1,1,6) = s(6,6,1,5) = s(6,6,6,4) = s(6,s(6,s(6,s(6,s(6,s(6,1,6,4),5,4),5,4),5,4),5,4),5,4), where s(6,1,6,4) = s(6,1,1,4) = s(6,6,1,3) = s(6,6,6,2) = s(6,s(6,s(6,s(6,s(6,s(6,1,6,2),5,2),5,2),5,2),5,2),5,2), where s(6,1,6,2) = s(6,1,1,2) = s(6,6,1,1) = s(6,6,1) = s(6,6) = 46656. So tetenhex = s(6,s(6,s(6,s(6,s(6,s(6,s(6,s(6,s(6,s(6,s(6,s(6,s(6,s(6,s(6,46656,5,2),5,2),5,2),5,2),5,2),5,4),5,4),5,4),5,4),5,4),5,6),5,6),5,6),5,6),5,6). Also s(6,1,1,8) = s(6,6,1,7) = s(6,6,6,6) = tetenhex, s(6,1,1,2,2) = s(6,6,1,1,2) = s(6,6,6,6,1) = s(6,6,6,6) = tetenhex, and s(6,1,1,1,1,2) = s(6,6,6,6,1,1) = s(6,6,6,6,1) = s(6,6,6,6) = tetenhex. It also equals 6→6→6→6→6→6→6→6 in Conway’s chained arrow notation.

5+ entry series

Those numbers are defined using LAN of 5 or more entries.

Chainol group

This group includes chainol, chainolplex, chainolbiplex, chainoltriplex, chainolquadriplex, chainbol, chainbolplex, chaintrol and chaintrolplex in order.

Chainol = s(3,2,2,1,2) = s(3,s(3,1,2,1,2),1,1,2) = s(3,s(3,1,1,1,2),1,1,2) = s(3,s(3,3,3,1,1),1,1,2) = s(3,s(3,3,3,1),1,1,2) = s(3,s(3,3,3),1,1,2) = s(3,3,3,s(3,3,3),1) = s(3,3,3,s(3,3,3)) = s(3,3,3,trientri) = 3→3→3→…3→3→3 with trientri+2 3′s. Let stage 1 = trientri, stage 2 = primibolplex, stage 3 = tetentri, stage 4 = truprimibolplex, stage 5 = quadprimibolplex and stage 6 = quinprimibolplex, etc. then chainol is the stage trientri number. It’s very hard to define (or bound) it in Conway’s chained arrow notation, so it’s named “chainol”.

Chainolplex = s(3,3,2,1,2) = s(3,s(3,2,2,1,2),1,1,2) = s(3,3,3,s(3,2,2,1,2),1) = s(3,3,3,s(3,2,2,1,2)) = s(3,3,3,chainol). Also s(3,2,1,2,2) = s(3,3,2,1,2) = chainolplex. Then chainolbiplex, chainoltriplex and chainolquadriplex are s(3,4,2,1,2), s(3,5,2,1,2) and s(3,6,2,1,2) respectively.

Chainbol = s(3,2,3,1,2) = s(3,s(3,1,3,1,2),2,1,2) = s(3,s(3,1,1,1,2),2,1,2) = s(3,trientri,2,1,2). Chainbolplex = s(3,3,3,1,2) = s(3,s(3,2,3,1,2),2,1,2) = s(3,chainbol,2,1,2), also s(3,1,1,3,2) = s(3,3,1,2,2) = s(3,3,3,1,2) = chainbolplex, and s(3,1,1,1,3) = s(3,3,3,1,2) = chainbolplex.

Chaintrol = s(3,2,4,1,2) = s(3,s(3,1,4,1,2),3,1,2) = s(3,s(3,1,1,1,2),3,1,2) = s(3,trientri,3,1,2). Chaintrolplex = s(3,3,4,1,2) = s(3,s(3,2,4,1,2),3,1,2) = s(3,chaintrol,3,1,2), also s(3,4,1,2,2) = s(3,3,4,1,2) = chaintrolplex.

Chainprimol group

This group includes chainprimol, chainprimolplex, chainpribol, chainpribolplex, chainduprimol, chainduprimolplex, chaindupribol, chaindupribolplex, chaintruprimol, chaintruprimolplex, chaintrupribol and chaintrupribolplex in order.

Chainprimol = s(3,2,2,2,2) = s(3,s(3,1,2,2,2),1,2,2) = s(3,s(3,1,1,2,2),1,2,2) = s(3,s(3,3,1,1,2),1,2,2) = s(3,s(3,3,3,3,1),1,2,2) = s(3,s(3,3,3,3),1,2,2) = s(3,3,s(3,3,3,3),1,2) = s(3,3,tetentri,1,2), and we can say it’s chain-(tetentri-1)-olplex. Chainprimolplex = s(3,3,2,2,2) = s(3,s(3,2,2,2,2),1,2,2) = s(3,3,s(3,2,2,2,2),1,2) = s(3,3,chainprimol,1,2), also s(3,2,1,3,2) = s(3,3,2,2,2) = chainprimolplex.

Chainpribol = s(3,2,3,2,2) = s(3,s(3,1,3,2,2),2,2,2) = s(3,s(3,1,1,2,2),2,2,2) = s(3,tetentri,2,2,2), and we can say it’s chainprimol-(tetentri-2)-plex. Chainpribolplex = s(3,3,3,2,2) = s(3,s(3,2,3,2,2),2,2,2) = s(3,chainpribol,2,2,2), also s(3,2,1,1,3) = s(3,3,3,2,2) = chainpribolplex, and s(3,1,1,4,2) = s(3,3,1,3,2) = s(3,3,3,2,2) = chainpribolplex.

Chainduprimol = s(3,2,2,3,2) = s(3,s(3,1,2,3,2),1,3,2) = s(3,s(3,1,1,3,2),1,3,2) = s(3,s(3,3,1,2,2),1,3,2) = s(3,s(3,3,3,1,2),1,3,2) = s(3,3,s(3,3,3,1,2),2,2) = s(3,3,chainbolplex,2,2), and we can say it’s chainpri-(chainbolplex-1)-olplex. Chainduprimolplex = s(3,3,2,3,2) = s(3,s(3,2,2,3,2),1,3,2) = s(3,3,s(3,2,2,3,2),2,2) = s(3,3,chainduprimol,2,2), also s(3,2,1,4,2) = s(3,3,2,3,2) = chainduprimolplex.

Chaindupribol = s(3,2,3,3,2) = s(3,s(3,1,3,3,2),2,3,2) = s(3,s(3,1,1,3,2),2,3,2) = s(3,chainbolplex,2,3,2). Chaindupribolplex = s(3,3,3,3,2) = s(3,s(3,2,3,3,2),2,3,2) = s(3,chaindupribol,2,3,2), also s(3,1,1,5,2) = s(3,3,1,4,2) = s(3,3,3,3,2) = chaindupribolplex, s(3,1,1,2,3) = s(3,3,1,1,3) = s(3,3,3,3,2) = chaindupribolplex, and s(3,2,1,1,1,2) = s(3,3,3,3,2,1) = s(3,3,3,3,2) = chaindupribolplex.

Chaintruprimol = s(3,2,2,4,2) = s(3,s(3,1,2,4,2),1,4,2) = s(3,s(3,1,1,4,2),1,4,2) = s(3,s(3,3,1,3,2),1,4,2) = s(3,s(3,3,3,2,2),1,4,2) = s(3,3,s(3,3,3,2,2),3,2) = s(3,3,chainpribolplex,3,2). Chaintruprimolplex = s(3,3,2,4,2) = s(3,s(3,2,2,4,2),1,4,2) = s(3,3,s(3,2,2,4,2),3,2) = s(3,3,chaintruprimol,3,2). Chaintrupribol = s(3,2,3,4,2) = s(3,s(3,1,3,4,2),2,4,2) = s(3,s(3,1,1,4,2),2,4,2) = s(3,chainpribolplex,2,4,2). Chaintrupribolplex = s(3,3,3,4,2) = s(3,s(3,2,3,4,2),2,4,2) = s(3,chaintrupribol,2,4,2), also s(3,4,1,1,3) = s(3,3,3,4,2) = chaintrupribolplex, and s(3,1,1,6,2) = s(3,3,1,5,2) = s(3,3,3,4,2) = chaintrupribolplex.

Pententri group

This group includes duchainol, duchainolplex, duchainbol, duchainbolplex, duchainprimol, duchainprimolplex, duchainpribol, duchainpribolplex, duchainduprimol, duchainduprimolplex, duchaindupribol and pententri in order.

Duchainol = s(3,2,2,1,3) = s(3,s(3,1,2,1,3),1,1,3) = s(3,s(3,1,1,1,3),1,1,3) = s(3,s(3,3,3,1,2),1,1,3) = s(3,3,3,s(3,3,3,1,2),2) = s(3,3,3,chainbolplex,2), and we can say it’s chain-(chainbolplex-1)-upribolplex. Duchainolplex = s(3,3,2,1,3) = s(3,s(3,2,2,1,3),1,1,3) = s(3,3,3,s(3,2,2,1,3),2) = s(3,3,3,duchainol,2), also s(3,2,1,2,3) = s(3,3,2,1,3) = duchainolplex. Duchainbol = s(3,2,3,1,3) = s(3,s(3,1,3,1,3),2,1,3) = s(3,s(3,1,1,1,3),2,1,3) = s(3,s(3,3,3,1,2),2,1,3) = s(3,chainbolplex,2,1,3). Duchainbolplex = s(3,3,3,1,3) = s(3,s(3,2,3,1,3),2,1,3) = s(3,duchainbol,2,1,3), also s(3,1,1,3,3) = s(3,3,1,2,3) = s(3,3,3,1,3) = duchainbolplex, and s(3,1,1,1,4) = s(3,3,3,1,3) = duchainbolplex.

Duchainprimol = s(3,2,2,2,3) = s(3,s(3,1,2,2,3),1,2,3) = s(3,s(3,1,1,2,3),1,2,3) = s(3,s(3,3,1,1,3),1,2,3) = s(3,s(3,3,3,3,2),1,2,3) = s(3,3,s(3,3,3,3,2),1,3) = s(3,3,chaindupribolplex,1,3), and we can say it’s duchain-(chaindupribolplex-1)-olplex. Duchainprimolplex = s(3,3,2,2,3) = s(3,s(3,2,2,2,3),1,2,3) = s(3,3,s(3,2,2,2,3),1,3) = s(3,3,duchainprimol,1,3). Duchainpribol = s(3,2,3,2,3) = s(3,s(3,1,3,2,3),2,2,3) = s(3,s(3,1,1,2,3),2,2,3) = s(3,chaindupribolplex,2,2,3). Duchainpribolplex = s(3,3,3,2,3) = s(3,s(3,2,3,2,3),2,2,3) = s(3,duchainpribol,2,2,3), also s(3,1,1,4,3) = s(3,3,1,3,3) = s(3,3,3,2,3) = duchainpribolplex, and s(3,2,1,1,4) = s(3,3,3,2,3) = duchainpribolplex.

Duchainduprimol = s(3,2,2,3,3) = s(3,s(3,1,2,3,3),1,3,3) = s(3,s(3,1,1,3,3),1,3,3) = s(3,s(3,3,1,2,3),1,3,3) = s(3,s(3,3,3,1,3),1,3,3) = s(3,3,duchainbolplex,2,3). Duchainduprimolplex = s(3,3,2,3,3) = s(3,s(3,2,2,3,3),1,3,3) = s(3,3,s(3,2,2,3,3),2,3) = s(3,3,duchainduprimol,2,3). Duchaindupribol = s(3,2,3,3,3) = s(3,s(3,1,3,3,3),2,3,3) = s(3,s(3,1,1,3,3),2,3,3) = s(3,duchainbolplex,2,3,3).

Pententri = s(3,3,3,3,3) = s(3,s(3,2,3,3,3),2,3,3) = s(3,duchaindupribol,2,3,3). Also s(3,1,1,5,3) = s(3,3,1,4,3) = s(3,3,3,3,3) = pententri, s(3,1,1,2,4) = s(3,3,1,1,4) = s(3,3,3,3,3) = pententri, s(3,1,1,2,1,2) = s(3,3,1,1,1,2) = s(3,3,3,3,3,1) = s(3,3,3,3,3) = pententri, s(3,1,1,1,1,1,2) = s(3,3,3,3,3,1,1) = s(3,3,3,3,3,1) = s(3,3,3,3,3) = pententri, and in exAN, s(3,3,1{2}2) = s(3,3,1,1,1,2{2}1) = s(3,3,1,1,1,2) = s(3,3,3,3,3,1) = s(3,3,3,3,3) = pententri. We can also name it duchaindupribolplex.

Pententet group

This group includes truchainol, truchainolplex, truchainbol, truchainbolplex, truchainprimol, truchainprimolplex, truchainpribol, truchainpribolplex, truchainduprimol, truchainduprimolplex, truchaindupribol, truchaindupribolplex, pententet, pentenpent and pentenhex in order.

Truchainol = s(3,2,2,1,4) = s(3,s(3,1,2,1,4),1,1,4) = s(3,s(3,1,1,1,4),1,1,4) = s(3,s(3,3,3,1,3),1,1,4) = s(3,3,3,s(3,3,3,1,3),3) = s(3,3,3,duchainbolplex,3). Truchainolplex = s(3,3,2,1,4) = s(3,s(3,2,2,1,4),1,1,4) = s(3,3,3,s(3,2,2,1,4),3) = s(3,3,3,truchainol,3). Truchainbol = s(3,2,3,1,4) = s(3,s(3,1,3,1,4),2,1,4) = s(3,s(3,1,1,1,4),2,1,4) = s(3,duchainbolplex,2,1,4).Truchainbolplex = s(3,3,3,1,4) = s(3,s(3,2,3,1,4),2,1,4) = s(3,truchainbol,2,1,4).

Truchainprimol = s(3,2,2,2,4) = s(3,s(3,1,2,2,4),1,2,4) = s(3,s(3,1,1,2,4),1,2,4) = s(3,s(3,3,1,1,4),1,2,4) = s(3,s(3,3,3,3,3),1,2,4) = s(3,3,s(3,3,3,3,3),1,4) = s(3,3,pententri,1,4). Truchainprimolplex = s(3,3,2,2,4) = s(3,s(3,2,2,2,4),1,2,4) = s(3,3,s(3,2,2,2,4),1,4) = s(3,3,truchainprimol,1,4). Truchainpribol = s(3,2,3,2,4) = s(3,s(3,1,3,2,4),2,2,4) = s(3,s(3,1,1,2,4),2,2,4) = s(3,pententri,2,2,4). Truchainpribolplex = s(3,3,3,2,4) = s(3,s(3,2,3,2,4),2,2,4) = s(3,truchainpribol,2,2,4).

Truchainduprimol = s(3,2,2,3,4) = s(3,s(3,1,2,3,4),1,3,4) = s(3,s(3,1,1,3,4),1,3,4) = s(3,s(3,3,1,2,4),1,3,4) = s(3,s(3,3,3,1,4),1,3,4) = s(3,3,s(3,3,3,1,4),2,4) = s(3,3,truchainbolplex,2,4). Truchainduprimolplex = s(3,3,2,3,4) = s(3,s(3,2,2,3,4),1,3,4) = s(3,3,s(3,2,2,3,4),2,4) = s(3,3,truchainduprimol,2,4). Truchaindupribol = s(3,2,3,3,4) = s(3,s(3,1,3,3,4),2,3,4) = s(3,s(3,1,1,3,4),2,3,4) = s(3,truchainbolplex,2,3,4). Truchaindupribolplex = s(3,3,3,3,4) = s(3,s(3,2,3,3,4),2,3,4) = s(3,truchaindupribol,2,3,4), also s(3,1,1,5,4) = s(3,3,1,4,4) = s(3,3,3,3,4) = truchaindupribolplex, s(3,1,1,2,5) = s(3,3,1,1,5) = s(3,3,3,3,4) = truchaindupribolplex, and in exAN, s(3,4{2}2) = s(3,4,1,1,1,2{2}1) = s(3,4,1,1,1,2) = s(3,3,3,3,4,1) = s(3,3,3,3,4) = truchaindupribolplex.

And, pententet, pentenpent and pentenhex are s(4,4,4,4,4), s(5,5,5,5,5) and s(6,6,6,6,6) respectively. Especially, in exAN, s(4,4{2}2) = s(4,4,1,1,1,2{2}1) = s(4,4,1,1,1,2) = s(4,4,4,4,4,1) = s(4,4,4,4,4) = pententet.

Choinol group

This group includes choinol, choinalplex, choinalbiplex, choinbol, choinbolplex, choinprimol, choinprimolplex, choinpribol, choinpribolplex, choinduprimol, choinduprimolplex, choindupribol and choindupribolplex in order.

Choinol = s(3,2,2,1,1,2) = s(3,s(3,1,2,1,1,2),1,1,1,2) = s(3,s(3,1,1,1,1,2),1,1,1,2) = s(3,s(3,3,3,3,1,1),1,1,1,2) = s(3,s(3,3,3,3),1,1,1,2) = s(3,3,3,3,s(3,3,3,3),1) = s(3,3,3,3,s(3,3,3,3)) = s(3,3,3,3,tetentri), and we can say it’s (tetentri-1)-uchaindupribolplex.

Choinolplex = s(3,3,2,1,1,2) = s(3,s(3,2,2,1,1,2),1,1,1,2) = s(3,3,3,3,s(3,2,2,1,1,2),1) = s(3,3,3,3,s(3,2,2,1,1,2)) = s(3,3,3,3,choinol), and choinalbiplex = s(3,4,2,1,1,2) = s(3,3,3,3,choinalplex).

Choinbol = s(3,2,3,1,1,2) = s(3,s(3,1,3,1,1,2),2,1,1,2) = s(3,s(3,1,1,1,1,2),2,1,1,2) = s(3,tetentri,2,1,1,2). Choinbolplex = s(3,3,3,1,1,2) = s(3,s(3,2,3,1,1,2),2,1,1,2) = s(3,choinbol,2,1,1,2), also s(3,1,1,3,1,2) = s(3,3,1,2,1,2) = s(3,3,3,1,1,2) = choinbolplex, and s(3,1,1,1,2,2) = s(3,3,3,1,1,2) = choinbolplex.

Choinprimol = s(3,2,2,2,1,2) = s(3,s(3,1,2,2,1,2),1,2,1,2) = s(3,s(3,1,1,2,1,2),1,2,1,2) = s(3,s(3,3,1,1,1,2),1,2,1,2) = s(3,s(3,3,3,3,3,1),1,2,1,2) = s(3,s(3,3,3,3,3),1,2,1,2) = s(3,3,s(3,3,3,3,3),1,1,2) = s(3,3,pententri,1,1,2). Choinprimolplex = s(3,3,2,2,1,2) = s(3,s(3,2,2,2,1,2),1,2,1,2) = s(3,3,s(3,2,2,2,1,2),1,1,2) = s(3,3,choinprimol,1,1,2). Choinpribol = s(3,2,3,2,1,2) = s(3,s(3,1,3,2,1,2),2,2,1,2) = s(3,s(3,1,1,2,1,2),2,2,1,2) = s(3,pententri,2,2,1,2). Choinpribolplex = s(3,3,3,2,1,2) = s(3,s(3,2,3,2,1,2),2,2,1,2) = s(3,choinpribol,2,2,1,2).

Choinduprimol = s(3,2,2,3,1,2), choinduprimolplex = s(3,3,2,3,1,2), choindupribol = s(3,2,3,3,1,2), and choindupribolplex = s(3,3,3,3,1,2). Also s(3,1,1,5,1,2) = s(3,3,1,4,1,2) = s(3,3,3,3,1,2) = choindupribolplex, and s(3,1,1,2,2,2) = s(3,3,1,1,2,2) = s(3,3,3,3,1,2) = choindupribolplex.

Choichainol group

This group includes choichainol, choichainolplex, choichainbolplex, choichainpribolplex, choichaindupribolplex and choiduchaindupribolplex in order.

Choichainol = s(3,2,2,1,2,2) = s(3,s(3,1,2,1,2,2),1,1,2,2) = s(3,s(3,1,1,1,2,2),1,1,2,2) = s(3,s(3,3,3,1,1,2),1,1,2,2) = s(3,3,3,s(3,3,3,1,1,2),1,2) = s(3,3,3,choinbolplex,1,2). Choichainolplex = s(3,3,2,1,2,2) = s(3,s(3,2,2,1,2,2),1,1,2,2) = s(3,3,3,s(3,2,2,1,2,2),1,2) = s(3,3,3,choichainol,1,2). Choichainbolplex = s(3,3,3,1,2,2) = s(3,s(3,2,3,1,2,2),2,1,2,2) = s(3,s(3,s(3,1,3,1,2,2),2,1,2,2),2,1,2,2) = s(3,s(3,s(3,1,1,1,2,2),2,1,2,2),2,1,2,2) = s(3,s(3,choinbolplex,2,1,2,2),2,1,2,2), also s(3,1,1,3,2,2) = s(3,3,1,2,2,2) = s(3,3,3,1,2,2) = choichainbolplex.

Choichainpribolplex = s(3,3,3,2,2,2) = s(3,s(3,s(3,1,3,2,2,2),2,2,2,2),2,2,2,2) = s(3,s(3,s(3,1,1,2,2,2),2,2,2,2),2,2,2,2) = s(3,s(3,choindupribolplex,2,2,2,2),2,2,2,2). Choichaindupribolplex = s(3,3,3,3,2,2) = s(3,s(3,s(3,1,3,3,2,2),2,3,2,2),2,3,2,2) = s(3,s(3,s(3,1,1,3,2,2),2,3,2,2),2,3,2,2) = s(3,s(3,s(3,3,1,2,2,2),2,3,2,2),2,3,2,2) = s(3,s(3,s(3,3,3,1,2,2),2,3,2,2),2,3,2,2) = s(3,s(3,choichainbolplex,2,3,2,2),2,3,2,2). Also s(3,1,1,5,2,2) = s(3,3,1,4,2,2) = s(3,3,3,3,2,2) = choichaindupribolplex, and s(3,1,1,2,3,2) = s(3,3,1,1,3,2) = s(3,3,3,3,2,2) = choichaindupribolplex.

Choiduchaindupribolplex = s(3,3,3,3,3,2) = s(3,s(3,s(3,1,3,3,3,2),2,3,3,2),2,3,3,2) = s(3,s(3,s(3,1,1,3,3,2),2,3,3,2),2,3,3,2) = s(3,s(3,s(3,3,1,2,3,2),2,3,3,2),2,3,3,2) = s(3,s(3,s(3,3,3,1,3,2),2,3,3,2),2,3,3,2) = s(3,s(3,s(3,s(3,s(3,1,3,1,3,2),2,1,3,2),2,1,3,2),2,3,3,2),2,3,3,2) = s(3,s(3,s(3,s(3,s(3,1,1,1,3,2),2,1,3,2),2,1,3,2),2,3,3,2),2,3,3,2) = s(3,s(3,s(3,s(3,s(3,3,3,1,2,2),2,1,3,2),2,1,3,2),2,3,3,2),2,3,3,2) = s(3,s(3,s(3,s(3,choichainbolplex,2,1,3,2),2,1,3,2),2,3,3,2),2,3,3,2). Also, s(3,1,1,5,3,2) = s(3,3,1,4,3,2) = s(3,3,3,3,3,2) = choiduchaindupribolplex, s(3,1,1,2,4,2) = s(3,3,1,1,4,2) = s(3,3,3,3,3,2) = choiduchaindupribolplex, and s(3,1,1,2,1,3) = s(3,3,1,1,1,3) = s(3,3,3,3,3,2) = choiduchaindupribolplex.

Up to here, the naming system works approximately as follows: e-uchoi-d-uchain-c-upri-b-ol-a-plex = s(3,a+2,b+1,c+1,d+1,e+1), where a, c, d, e ≥ 0 and b ≥ 1.

Hexentri group

Here’s the final group of LAN numbers. This group includes hexentri, hexentet, hexenpent and hexenhex.

Hexentri, hexentet, hexenpent and hexenhex are s(n,n,n,n,n,n) = s(n,n,1,1,1,1,2) = s(n,1,1,1,1,1,1,2) with n = 3, 4, 5 and 6 respectively. Especially, s(4,4,1{2}2) = s(4,4,1,1,1,1,2{2}1) = s(4,4,1,1,1,1,2) = hexentet, and s(5,5{2}2) = s(5,5,1,1,1,1,2{2}1) = s(5,5,1,1,1,1,2) = hexenpent.

Ordinal notations (part 3) – Taranovsky’s notation

Taranovsky’s notation is a simple yet strong ordinal notation, introduced by Dmytro Taranovsky.

Definition

First, Taranovsky’s notation is actually made up of many systems, called 1st system, 2nd system, 3rd system, and so on.

In the n-th system, we uses a binary function: C(α,β), and two constants: 0 and Ωn. Here the 0 is not an ordinal or a number – it’s just a notation for “something”. And the same, Ωn is not an ordinal – it’s just “something”.

Here’s the definition of standard form. First, 0 and Ωn are in standard form. Then C(α,β) is in standard form iff it fits all those shown below:

  1. α and β are in standard form
  2. β = 0, or β = Ωn, or β = C(γ,δ) with α ≤ γ
  3. α is n-built from below from <C(α,β)

But what’s “≤” and what’s “n-built from below from”? To answer this question, we need to define some more things.

First we need to define “n-built from below from” as follows.

  • α is 0-built from below from <β iff α < β.
  • α is (k+1)-built from below from β iff for all subterm γ of α, γ ≤ α or there is such a subterm δ of α that γ is subterm of δ and δ is k-built from below from β.
  • The “subterm” in α can be defined as follows.
    • In any part of expression of α, η is subterm of η itself.
    • In any part of expression of α, if η = C(γ,δ), and x is a subterm of γ or δ, then x is a subterm of η.

To define binary relation “<” and “≤”, we need another form of the notation – postfix form. We can make a postfix form of every normal expression in such a way. First delete all the (′s, )′s and commas, which means we’ll get a string only containing 0, Ωn and C’s. Then reverse the string. Now we get the postfix form. For expressions in standard form, we make the postfix form of them, then the comparisons (“≤”, “≥”, “<” and “>”) are done in the lexicographical order where C < 0 < Ωn.

Property: In n-th system, for b < Ωn, Ωn is 1-built from below from b but not 0-built from below from b.

So why is it an ordinal notation? Because the “<” relation of expressions in standard form fits the ordinal axioms.

Now, we know that 0 is truly the ordinal 0. But unlike in θ function, Ωn is not the n-th uncountable cardinal – it’s just some very large ordinal, but they’re also such common fixed point that {\omega^{\Omega_n}=\Omega_n,\;\varphi(\Omega_n,0)=\Omega_n,\;\Gamma_{\Omega_n}=\Omega_n}, etc.

One variable function

Any ordinal α can be expressed as {\alpha=C(\beta_1,C(\beta_2,\cdots C(\beta_k,0)))} or {\alpha=C(\beta_1,C(\beta_2,\cdots C(\beta_k,\Omega_n)))} in standard form (k can also be 0). The former one is less than Ωn (we call it low-ordinal here) while the latter one is greaterequal to Ωn (we call it high-ordinal here).

Define the one variable function C1 as follows, it’s actually another representation of ordinals: for low-ordinal α, {\alpha=C(\beta_1,C(\beta_2,\cdots C(\beta_{k-1},C(\beta_k,0))))=C_1(\omega^{\beta_k}+\omega^{\beta_{k-1}}+\cdots+\omega^{\beta_2}+\omega^{\beta_1})}.

For high-ordinal α, {\alpha=C(\beta_1,C(\beta_2,\cdots C(\beta_{k-1},C(\beta_k,\Omega_n))))=\Omega_n+\omega^{\beta_k}+\omega^{\beta_{k-1}}+\cdots+\omega^{\beta_2}+\omega^{\beta_1}}, which is the same as ordinal addition and exponentiation.

It turns out that, if {\alpha<\varepsilon_0}, then C1(α) = α, and C1(β+α) = C1(β)+α.

All the 1st system, 2nd system, 3rd system, n-th systems can be combined into one notation as follows: the constants are 0 and Ωn (for every positive integer n), and binary function C. Ωi = C(Ωi+1, 0) and the standard form always uses Ωi instead of C(Ωi+1, 0). And, to check for standard form and compare ordinals, use Ωi = C(Ωi+1, 0) to convert each Ω to Ωn for a single positive integer n (it does not matter which n) and then use the n-th ordinal notation system.

Here‘s a calculator in Python program, which convert ordinals into Cantor’s normal form, binary φ function, and C1-representation. Just note that it uses “C” instead of “C1” for the one variable function, and “C2” instead of “C” for the binary function.

Analysis

Below {\varepsilon_0}, doesn’t appear. For expressions without any Ωn′s, α is 1-built from below from β, so “α is n-built from below from β” is always true.

So C(0,0) = 1, C(0,C(0,0)) = 2, C(0,C(0,C(0,0))) = 3, and C(0,α) = α+1.
Then {C(C(0,0),0)=C(1,0)=\omega} and {C(1,\alpha)=\alpha+\omega}.
Then {C(C(0,C(0,0)),0)=C(2,0)=\omega^2} and {C(2,\alpha)=\alpha+\omega^2}.
{C(C(C(0,0),0),0)=C(\omega,0)=\omega^\omega} and {C(\omega,\alpha)=\alpha+\omega^\omega}.
{C(C(0,C(C(0,0),0)),0)=C(\omega+1,0)=\omega^{\omega+1}} and {C(\omega+1,\alpha)=\alpha+\omega^{\omega+1}}.
{C(C(C(0,0),C(C(0,0),0)),0)=C(\omega2,0)=\omega^{\omega2}} and {C(\omega2,\alpha)=\alpha+\omega^{\omega2}}.
{C(C(C(0,C(0,0)),0),0)=C(\omega^2,0)=\omega^{\omega^2}} and {C(\omega^2,\alpha)=\alpha+\omega^{\omega^2}}.
{C(C(C(C(0,0),0),0),0)=C(\omega^\omega,0)=\omega^{\omega^\omega}} and {C(\omega^\omega,\alpha)=\alpha+\omega^{\omega^\omega}}.
It turns out that, for {\alpha<\Omega_1,\;C(\alpha,\beta)=\beta+\omega^\alpha}, if C(α,β) is standard.

Here’re more results in 1st system.

{C(\Omega_1,0)=C_1(\Omega_1)=\varepsilon_0}
{C(\varepsilon_0,\varepsilon_0)=\varepsilon_02}
{C(\varepsilon_0,C(\varepsilon_0,\varepsilon_0))=\varepsilon_03}
{C(\varepsilon_0+1,\varepsilon_0)=\omega^{\varepsilon_0+1}}
{C(\varepsilon_02,\varepsilon_0)=\omega^{\varepsilon_02}}
{C(\omega^{\varepsilon_0+1},\varepsilon_0)=\omega^{\omega^{\varepsilon_0+1}}}
{C(\omega^{\varepsilon_02},\varepsilon_0)=\omega^{\omega^{\varepsilon_02}}}
{C(\omega^{\omega^{\varepsilon_0+1}},\varepsilon_0)=\omega^{\omega^{\omega^{\varepsilon_0+1}}}}
{C(\Omega_1,C(\Omega_1,0))=C_1(\Omega_12)=\varepsilon_1}
{C(\Omega_1,C(\Omega_1,C(\Omega_1,0)))=C_1(\Omega_13)=\varepsilon_2}
{C(C(0,\Omega_1),0)=C(\Omega_1+1,0)=C_1(\Omega_1\omega)=\varepsilon_\omega}. Note that ordinals above Ω1 are high-ordinals and C(0,Ω1) = Ω1+1 in 1st system.
{C(\Omega_1,C(\Omega_1+1,0))=C_1(\Omega_1\omega+\Omega_1)=\varepsilon_{\omega+1}}
{C(\Omega_1+1,C(\Omega_1+1,0))=C_1(\Omega_1\omega2)=\varepsilon_{\omega2}}
{C(\Omega_1+2,0)=C_1(\Omega_1\omega^2)=\varepsilon_{\omega^2}}
{C(\Omega_1+3,0)=C_1(\Omega_1\omega^3)=\varepsilon_{\omega^3}}
{C(\Omega_1+\omega,0)=C_1(\Omega_1\omega^\omega)=\varepsilon_{\omega^\omega}}
{C(\Omega_1+\omega+1,0)=C_1(\Omega_1\omega^{\omega+1})=\varepsilon_{\omega^{\omega+1}}}
{C(\Omega_1+\omega^2,0)=C_1(\Omega_1\omega^{\omega^2})=\varepsilon_{\omega^{\omega^2}}}
{C(\Omega_1+\omega^\omega,0)=C_1(\Omega_1\omega^{\omega^\omega})=\varepsilon_{\omega^{\omega^\omega}}}
{C(\Omega_1+\varepsilon_0,0)=C_1(\Omega_1\varepsilon_0)=\varepsilon_{\varepsilon_0}}
{C(\Omega_1+\varepsilon_{\varepsilon_0},0)=C_1(\Omega_1\varepsilon_{\varepsilon_0})=\varepsilon_{\varepsilon_{\varepsilon_0}}}
{C(\Omega_12,0)=C_1(\Omega_1^2)=\varphi(2,0)}
{C(\Omega_1,\varphi(2,0))=C_1(\Omega_1^2+\Omega_1)=\varepsilon_{\varphi(2,0)+1}}
{C(\Omega_1+1,\varphi(2,0))=\varepsilon_{\varphi(2,0)+\omega}}
{C(\Omega_1+\varepsilon_0,\varphi(2,0))=\varepsilon_{\varphi(2,0)+\varepsilon_0}}
{C(\Omega_1+\varepsilon_{\varepsilon_0},\varphi(2,0))=\varepsilon_{\varphi(2,0)+\varepsilon_{\varepsilon_0}}}
{C(\Omega_1+\varphi(2,0),\varphi(2,0))=\varepsilon_{\varphi(2,0)2}}
{C(\Omega_1+\varphi(2,0)+1,\varphi(2,0))=\varepsilon_{\omega^{\varphi(2,0)+1}}}
{C(\Omega_1+\varphi(2,0)2,\varphi(2,0))=\varepsilon_{\omega^{\varphi(2,0)2}}}
{C(\Omega_1+C(\Omega_1,\varphi(2,0)),\varphi(2,0))=\varepsilon_{\varepsilon_{\varphi(2,0)+1}}}
{C(\Omega_1+C(\Omega_1+C(\Omega_1,\varphi(2,0)),\varphi(2,0)),\varphi(2,0))=\varepsilon_{\varepsilon_{\varepsilon_{\varphi(2,0)+1}}}}
{C(\Omega_12,C(\Omega_12,0))=C_1(\Omega_1^22)=\varphi(2,1)}
{C(\Omega_12+1,0)=\varphi(2,\omega)}
{C(\Omega_12+C(\Omega_12,0),0)=\varphi(2,\varphi(2,0))}
{C(\Omega_13,0)=C_1(\Omega_1^3)=\varphi(3,0)}
{C(\Omega_1\omega,0)=C_1(\Omega_1^\omega)=\varphi(\omega,0)}
{C(\Omega_1,C(\Omega_1\omega,0))=C_1(\Omega_1^\omega+\Omega_1)=\varepsilon_{\varphi(\omega,0)+1}}
{C(\Omega_12,C(\Omega_1\omega,0))=C_1(\Omega_1^\omega+\Omega_1^2)=\varphi(2,\varphi(\omega,0)+1)}
{C(\Omega_1\omega,C(\Omega_1\omega,0))=C_1(\Omega_1^\omega2)=\varphi(\omega,1)}
{C(\Omega_1\omega+1,0)=\varphi(\omega,\omega)}
{C(\Omega_1\omega+\Omega_1,0)=C_1(\Omega_1^{\omega+1})=\varphi(\omega+1,0)}
{C(\Omega_1\omega2,0)=C_1(\Omega_1^{\omega2})=\varphi(\omega2,0)}
{C(\Omega_1\omega^2,0)=C_1(\Omega_1^{\omega^2})=\varphi(\omega^2,0)}
{C(\Omega_1C(\Omega_1,0),0)=\varphi(\varepsilon_0,0)}
{C(\Omega_1C(\Omega_12,0),0)=\varphi(\varphi(2,0),0)}
{C(\Omega_1C(\Omega_1\omega,0),0)=\varphi(\varphi(\omega,0),0)}
{C(\Omega_1C(\Omega_1C(\Omega_1,0),0),0)=\varphi(\varphi(\varepsilon_0,0),0)}
{C(\Omega_1^2,0)=C_1(\Omega_1^{\Omega_1})=\Gamma_0}
{C(\Omega_1,\Gamma_0)=\varepsilon_{\Gamma_0+1}}
{C(\Omega_1+\Gamma_0,\Gamma_0)=\varepsilon_{\Gamma_02}}
{C(\Omega_12,\Gamma_0)=\varphi(2,\Gamma_0+1)}
{C(\Omega_1\Gamma_0,\Gamma_0)=\varphi(\Gamma_0,1)}
{C(\Omega_1C(\Omega_1\Gamma_0,\Gamma_0),\Gamma_0)=\varphi(\varphi(\Gamma_0,1),0)}
{C(\Omega_1^2,C(\Omega_1^2,0))=C_1(\Omega_1^{\Omega_1}2)=\Gamma_1}
{C(\Omega_1^2+1,0)=\Gamma_\omega}
{C(\Omega_1^2+\Omega_1,0)=\theta(\Omega+1,0)}
{C(\Omega_1^2+\Omega_12,0)=\theta(\Omega+2,0)}
{C(\Omega_1^2+\Omega_1C(\Omega_1^2,0),0)=\theta(\Omega+\theta(\Omega,0),0)}
{C(\Omega_1^2+\Omega_1C(\Omega_1^2+1,0),0)=\theta(\Omega+\theta(\Omega,\omega),0)}
{C(\Omega_1^2+\Omega_1C(\Omega_1^2+\Omega_1C(\Omega_1^2,0),0),0)=\theta(\Omega+\theta(\Omega+\theta(\Omega,0),0),0)}
{C(\Omega_1^22,0)=C_1(\Omega_1^{\Omega_12})=\theta(\Omega2,0)}
{C(\Omega_1^23,0)=\theta(\Omega3,0)}
{C(\Omega_1^2\omega,0)=\theta(\Omega\omega,0)}
{C(\Omega_1^2C(\Omega_1^2,0),0)=\theta(\Omega\theta(\Omega,0),0)}
{C(\Omega_1^2C(\Omega_1^2C(\Omega_1^2,0),0),0)=\theta(\Omega\theta(\Omega\theta(\Omega,0),0),0)}
{C(\Omega_1^3,0)=C_1(\Omega_1^{\Omega_1^2})=\theta(\Omega^2,0)}
{C(\Omega_1^4,0)=C_1(\Omega_1^{\Omega_1^3})=\theta(\Omega^3,0)}
{C(\Omega_1^\omega,0)=C_1(\Omega_1^{\Omega_1^\omega})=\theta(\Omega^\omega,0)}
{C(\Omega_1^\omega,C(\Omega_1^\omega,0))=\theta(\Omega^\omega,1)}
{C(\Omega_1^\omega+1,0)=\theta(\Omega^\omega,\omega)}
{C(\Omega_1^\omega+\Omega_1,0)=\theta(\Omega^\omega+1,0)}
{C(\Omega_1^\omega+\Omega_12,0)=\theta(\Omega^\omega+2,0)}
{C(\Omega_1^\omega+\Omega_1^2,0)=\theta(\Omega^\omega+\Omega,0)}
{C(\Omega_1^\omega+\Omega_1^3,0)=\theta(\Omega^\omega+\Omega^2,0)}
{C(\Omega_1^\omega2,0)=\theta(\Omega^\omega2,0)}
{C(\Omega_1^{\omega+1},0)=\theta(\Omega^{\omega+1},0)}
{C(\Omega_1^{C(\Omega_1^\omega,0)},0)=\theta(\Omega^{\theta(\Omega^\omega,0)},0)}
{C(\Omega_1^{\Omega_1},0)=\theta(\Omega^{\Omega},0)}
{C(\Omega_1^{C(\Omega_1^{\Omega_1},0)},C(\Omega_1^{\Omega_1},0))=\theta(\Omega^{\theta(\Omega^\Omega,0)},1)}
{C(\Omega_1^{\Omega_1},C(\Omega_1^{\Omega_1},0))=\theta(\Omega^\Omega,1)}
{C(\Omega_1^{\Omega_1}+1,0)=\theta(\Omega^\Omega,\omega)}
{C(\Omega_1^{\Omega_1}+\Omega_1,0)=\theta(\Omega^\Omega+1,0)}
{C(\Omega_1^{\Omega_1}2,0)=\theta(\Omega^\Omega2,0)}
{C(\Omega_1^{\Omega_1+1},0)=\theta(\Omega^{\Omega+1},0)}
{C(\Omega_1^{\Omega_12},0)=\theta(\Omega^{\Omega2},0)}
{C(\Omega_1^{\Omega_1^2},0)=\theta(\Omega^{\Omega^2},0)}
{C(\Omega_1^{\Omega_1^{\Omega_1}},0)=\theta(\Omega^{\Omega^\Omega},0)}
{C(\Omega_1^{\Omega_1^{\Omega_1^{\Omega_1}}},0)=\theta(\Omega^{\Omega^{\Omega^\Omega}},0)}

It seems that the Ω1 in C function works very similar to the Ω in θ function. And they are also very similar to my EAN: Ω1 or Ω expands into “ω layers of” the C or θ where they are, and the results are the supremum of any finite layers of the C or θ where the Ω1 or Ω is.

The 1st system only handles ordinals up to the Bachmann-Howard ordinal. So we need 2nd system now. In 2nd system, Ω1 = C(Ω2,0), and the Bachmann-Howard ordinal is the supremum of all the C(C(Ω2,0),0), C(C(C(Ω2,0),C(Ω2,0)),0), C(C(C(C(Ω2,0),C(Ω2,0)),C(Ω2,0)),0), C(C(C(C(C(Ω2,0),C(Ω2,0)),C(Ω2,0)),C(Ω2,0)),0), etc. Thus the BHO equals {C(C(\Omega_2,C(\Omega_2,0)),0)=C(C(\Omega_2,\Omega_1),0)}.

{C(\Omega_1,C(C(\Omega_2,\Omega_1),0))=\varepsilon_{\theta(\varepsilon_{\Omega+1},0)+1}}
{C(\Omega_1^{\Omega_1},C(C(\Omega_2,\Omega_1),0))=\theta(\Omega^\Omega,\theta(\varepsilon_{\Omega+1},0)+1)}
{C(C(\Omega_2,\Omega_1),C(C(\Omega_2,\Omega_1),0))=\theta(\varepsilon_{\Omega+1},1)}
{C(C(\Omega_2,\Omega_1)+1,0)=\theta(\varepsilon_{\Omega+1},\omega)}
{C(C(\Omega_2,\Omega_1)+\Omega_1,0)=\theta(\varepsilon_{\Omega+1}+1,0)}
{C(C(\Omega_2,\Omega_1)+\Omega_1^2,0)=\theta(\varepsilon_{\Omega+1}+\Omega,0)}
{C(C(\Omega_2,\Omega_1)+\Omega_1^{\Omega_1},0)=\theta(\varepsilon_{\Omega+1}+\Omega^\Omega,0)}
{C(C(\Omega_2,\Omega_1)+C(C(\Omega_2,\Omega_1),\Omega_1),0)=\theta(\varepsilon_{\Omega+1}2,0)}. This ordinal (C(C(C(C(Ω21),Ω1),C(Ω21)),0)) is the supremum of all the C(C(C(Ω11),C(Ω21)),0), C(C(C(C(Ω11),Ω1),C(Ω21)),0), C(C(C(C(C(Ω11),Ω1),Ω1),C(Ω21)),0), etc. It’s not C(C(Ω21)2,0)=C(C(C(Ω2,C(Ω2,0)),C(Ω2,C(Ω2,0))),0)! And {C(C(\Omega_2,\Omega_1),\Omega_1)=\varepsilon_{\Omega_1+1}} is standard in 2nd system.
{C(C(\Omega_2,\Omega_1)+C(C(\Omega_2,\Omega_1),\Omega_1)+\Omega_1,0)=\theta(\varepsilon_{\Omega+1}2+1,0)}
{C(C(\Omega_2,\Omega_1)+C(C(\Omega_2,\Omega_1),\Omega_1)2,0)=\theta(\varepsilon_{\Omega+1}3,0)}
{C(C(\Omega_2,\Omega_1)+\omega^{C(C(\Omega_2,\Omega_1),\Omega_1)+1},0)=\theta(\omega^{\varepsilon_{\Omega+1}+1},0)}
{C(C(\Omega_2,\Omega_1)+\omega^{C(C(\Omega_2,\Omega_1),\Omega_1)2},0)=\theta(\omega^{\varepsilon_{\Omega+1}2},0)}
{C(C(\Omega_2,\Omega_1)+\omega^{\omega^{C(C(\Omega_2,\Omega_1),\Omega_1)+1}},0)=\theta(\omega^{\omega^{\varepsilon_{\Omega+1}+1}},0)}
{C(C(\Omega_2,\Omega_1)+C(C(\Omega_2,\Omega_1),C(C(\Omega_2,\Omega_1),\Omega_1)),0)=\theta(\varepsilon_{\Omega+2},0)}
{C(C(\Omega_2,\Omega_1)+C(C(\Omega_2,\Omega_1)+1,\Omega_1),0)=\theta(\varepsilon_{\Omega+\omega},0)}. Note again that C(0,C(Ω21)) = C(Ω21)+1.
{C(C(\Omega_2,\Omega_1)+C(C(\Omega_2,\Omega_1)+\Omega_1,\Omega_1),0)=\theta(\varepsilon_{\Omega2},0)}
{C(C(\Omega_2,\Omega_1)+C(C(\Omega_2,\Omega_1)+\Omega_1^{\Omega_1},\Omega_1),0)=\theta(\varepsilon_{\Omega^\Omega},0)}
{C(C(\Omega_2,\Omega_1)+C(C(\Omega_2,\Omega_1)+C(C(\Omega_2,\Omega_1),\Omega_1),\Omega_1),0)=\theta(\varepsilon_{\varepsilon_{\Omega+1}},0)}
{C(C(\Omega_2,\Omega_1)+C(C(\Omega_2,\Omega_1)+C(C(\Omega_2,\Omega_1)+1,\Omega_1),\Omega_1),0)=\theta(\varepsilon_{\varepsilon_{\Omega+\omega}},0)}
{C(C(\Omega_2,\Omega_1)+C(C(\Omega_2,\Omega_1)+C(C(\Omega_2,\Omega_1)+C(C(\Omega_2,\Omega_1),\Omega_1),\Omega_1),\Omega_1),0)}\\ {=\theta(\varepsilon_{\varepsilon_{\varepsilon_{\Omega+1}}},0)}
{C(C(\Omega_2,\Omega_1)2,0)=\theta(\varphi(2,\Omega+1),0)}. This ordinal (C(C(C(Ω21),C(Ω21)),0)) is the supremum of all the C(C(Ω21),0), C(C(C(C(Ω21),Ω1),C(Ω21)),0), C(C(C(C(C(C(Ω21),Ω1),C(Ω21)),Ω1),C(Ω21)),0), etc.
{C(C(\Omega_2,\Omega_1),C(C(\Omega_2,\Omega_1)2,0))=\theta(\varepsilon_{\Omega+1},\theta(\varphi(2,\Omega+1),0)+1)}
{C(C(\Omega_2,\Omega_1)+C(C(\Omega_2,\Omega_1),\Omega_1),C(C(\Omega_2,\Omega_1)2,0))=\theta(\varepsilon_{\Omega+1}2,\theta(\varphi(2,\Omega+1),0)+1)}
{C(C(\Omega_2,\Omega_1)+C(C(\Omega_2,\Omega_1)+C(C(\Omega_2,\Omega_1),\Omega_1),\Omega_1),C(C(\Omega_2,\Omega_1)2,0))}\\ {=\theta(\varepsilon_{\varepsilon_{\Omega+1}},\theta(\varphi(2,\Omega+1),0)+1)}
{C(C(\Omega_2,\Omega_1)2,C(C(\Omega_2,\Omega_1)2,0))=\theta(\varphi(2,\Omega+1),1)}
{C(C(\Omega_2,\Omega_1)2+1,0)=\theta(\varphi(2,\Omega+1),\omega)}
{C(C(\Omega_2,\Omega_1)2+\Omega_1,0)=\theta(\varphi(2,\Omega+1)+1,0)}
{C(C(\Omega_2,\Omega_1)2+C(C(\Omega_2,\Omega_1),\Omega_1),0)=\theta(\varphi(2,\Omega+1)+\varepsilon_{\Omega+1},0)}
{C(C(\Omega_2,\Omega_1)2+C(C(\Omega_2,\Omega_1)+C(C(\Omega_2,\Omega_1),\Omega_1),\Omega_1),0)}\\ {=\theta(\varphi(2,\Omega+1)+\varepsilon_{\varepsilon_{\Omega+1}},0)}
{C(C(\Omega_2,\Omega_1)2+C(C(\Omega_2,\Omega_1)2,\Omega_1),0)=\theta(\varphi(2,\Omega+1)2,0)}.
{C(C(\Omega_2,\Omega_1)2+C(C(\Omega_2,\Omega_1)2,\Omega_1)2,0)=\theta(\varphi(2,\Omega+1)3,0)}
{C(C(\Omega_2,\Omega_1)2+C(C(\Omega_2,\Omega_1),C(C(\Omega_2,\Omega_1)2,\Omega_1)),0)=\theta(\varepsilon_{\varphi(2,\Omega+1)+1},0)}
{C(C(\Omega_2,\Omega_1)2+C(C(\Omega_2,\Omega_1)+C(C(\Omega_2,\Omega_1)2,\Omega_1),C(C(\Omega_2,\Omega_1)2,\Omega_1)),0)}\\ {=\theta(\varepsilon_{\varphi(2,\Omega+1)2},0)}
{C(C(\Omega_2,\Omega_1)2+C(C(\Omega_2,\Omega_1)2,C(C(\Omega_2,\Omega_1)2,\Omega_1)),0)=\theta(\varphi(2,\Omega+2),0)}
{C(C(\Omega_2,\Omega_1)2+C(C(\Omega_2,\Omega_1)2+1,\Omega_1),0)=\theta(\varphi(2,\Omega+\omega),0)}
{C(C(\Omega_2,\Omega_1)2+C(C(\Omega_2,\Omega_1)2+\Omega_1,\Omega_1),0)=\theta(\varphi(2,\Omega2),0)}
{C(C(\Omega_2,\Omega_1)2+C(C(\Omega_2,\Omega_1)2+C(C(\Omega_2,\Omega_1)2,\Omega_1),\Omega_1),0)}\\ {=\theta(\varphi(2,\varphi(2,\Omega+1)),0)}
{C(C(\Omega_2,\Omega_1)3,0)=\theta(\varphi(3,\Omega+1),0)}
{C(\omega^{C(\Omega_2,\Omega_1)+1},0)=\theta(\varphi(\omega,\Omega+1),0)}
{C(\omega^{C(\Omega_2,\Omega_1)+1}+C(\Omega_2,\Omega_1),0)=\theta(\varphi(\omega+1,\Omega+1),0)}
{C(\omega^{C(\Omega_2,\Omega_1)+2},0)=\theta(\theta(\omega^2,\Omega),0)}
{C(\omega^{C(\Omega_2,\Omega_1)+\Omega_1},0)=\theta(\theta(\Omega,\Omega),0)}
{C(\omega^{C(\Omega_2,\Omega_1)+C(C(\Omega_2,\Omega_1),\Omega_1)},0)=\theta(\theta(\varepsilon_{\Omega+1},\Omega),0)}
{C(\omega^{C(\Omega_2,\Omega_1)+C(C(\Omega_2,\Omega_1)+1,\Omega_1)},0)=\theta(\theta(\varepsilon_{\Omega+\omega},\Omega),0)}
{C(\omega^{C(\Omega_2,\Omega_1)+C(C(\Omega_2,\Omega_1)+C(C(\Omega_2,\Omega_1),\Omega_1),\Omega_1)},0)=\theta(\theta(\varepsilon_{\varepsilon_{\Omega+1}},\Omega),0)}
{C(\omega^{C(\Omega_2,\Omega_1)+C(C(\Omega_2,\Omega_1)2,\Omega_1)},0)=\theta(\theta(\theta(2,\Omega),\Omega),0)}
{C(\omega^{C(\Omega_2,\Omega_1)+C(\omega^{C(\Omega_2,\Omega_1)+1},\Omega_1)},0)=\theta(\theta(\theta(\omega,\Omega),\Omega),0)}
{C(\omega^{C(\Omega_2,\Omega_1)+C(\omega^{C(\Omega_2,\Omega_1)+\Omega_1},\Omega_1)},0)=\theta(\theta(\theta(\Omega,\Omega),\Omega),0)}
{C(\omega^{C(\Omega_2,\Omega_1)2},0)=\theta(\Omega_2,0)}. This ordinal (C(C(C(C(Ω21),C(Ω21)),C(Ω21)),0)) is the supremum of all the C(C(C(Ω21),C(Ω21)),0), C(C(C(C(C(C(Ω21),C(Ω21)),Ω1),C(Ω21)),C(Ω21)),0), C(C(C(C(C(C(C(C(C(Ω21),C(Ω21)),Ω1),C(Ω21)),C(Ω21)),Ω1),C(Ω21)),C(Ω21)),0), etc.
{C(\omega^{C(\Omega_2,\Omega_1)2}+1,0)=\theta(\Omega_2,\omega)}
{C(\omega^{C(\Omega_2,\Omega_1)2}+\Omega_1,0)=\theta(\Omega_2+1,0)}
{C(\omega^{C(\Omega_2,\Omega_1)2}+C(C(\Omega_2,\Omega_1),\Omega_1),0)=\theta(\Omega_2+\varepsilon_{\Omega+1},0)}
{C(\omega^{C(\Omega_2,\Omega_1)2}+C(C(\Omega_2,\Omega_1)2,\Omega_1),0)=\theta(\Omega_2+\theta(2,\Omega),0)}
{C(\omega^{C(\Omega_2,\Omega_1)2}+C(\omega^{C(\Omega_2,\Omega_1)+\Omega_1},\Omega_1),0)=\theta(\Omega_2+\theta(\Omega,\Omega),0)}
{C(\omega^{C(\Omega_2,\Omega_1)2}+C(\omega^{C(\Omega_2,\Omega_1)+C(C(\Omega_2,\Omega_1),\Omega_1)},\Omega_1),0)=\theta(\Omega_2+\theta(\varepsilon_{\Omega+1},\Omega),0)}
{C(\omega^{C(\Omega_2,\Omega_1)2}+C(\omega^{C(\Omega_2,\Omega_1)2},\Omega_1),0)=\theta(\Omega_2+\theta(\Omega_2,\Omega),0)}
{C(\omega^{C(\Omega_2,\Omega_1)2}+C(\omega^{C(\Omega_2,\Omega_1)2},\Omega_1)2,0)=\theta(\Omega_2+\theta(\Omega_2,\Omega)2,0)}
{C(\omega^{C(\Omega_2,\Omega_1)2}+C(C(\Omega_2,\Omega_1),C(\omega^{C(\Omega_2,\Omega_1)2},\Omega_1)),0)=\theta(\Omega_2+\varepsilon_{\theta(\Omega_2,\Omega)+1},0)}
{C(\omega^{C(\Omega_2,\Omega_1)2}+C(\omega^{C(\Omega_2,\Omega_1)+\Omega_1},C(\omega^{C(\Omega_2,\Omega_1)2},\Omega_1)),0)=\theta(\Omega_2+\theta(\Omega,\theta(\Omega_2,\Omega)+1),0)}
{C(\omega^{C(\Omega_2,\Omega_1)2}+C(\omega^{C(\Omega_2,\Omega_1)+C(\omega^{C(\Omega_2,\Omega_1)2},\Omega_1)},C(\omega^{C(\Omega_2,\Omega_1)2},\Omega_1)),0)}\\ {=\theta(\Omega_2+\theta(\theta(\Omega_2,\Omega),\Omega+1),0)}
{C(\omega^{C(\Omega_2,\Omega_1)2}+C(\omega^{C(\Omega_2,\Omega_1)2},C(\omega^{C(\Omega_2,\Omega_1)2},\Omega_1)),0)=\theta(\Omega_2+\theta(\Omega_2,\Omega+1),0)}
{C(\omega^{C(\Omega_2,\Omega_1)2}+C(\omega^{C(\Omega_2,\Omega_1)2}+1,\Omega_1),0)=\theta(\Omega_2+\theta(\Omega_2,\Omega+\omega),0)}
{C(\omega^{C(\Omega_2,\Omega_1)2}+C(\omega^{C(\Omega_2,\Omega_1)2}+\Omega_1,\Omega_1),0)=\theta(\Omega_2+\theta(\Omega_2,\Omega2),0)}
{C(\omega^{C(\Omega_2,\Omega_1)2}+C(\omega^{C(\Omega_2,\Omega_1)2}+C(C(\Omega_2,\Omega_1),\Omega_1),\Omega_1),0)=\theta(\Omega_2+\theta(\Omega_2,\varepsilon_{\Omega+1}),0)}
{C(\omega^{C(\Omega_2,\Omega_1)2}+C(\omega^{C(\Omega_2,\Omega_1)2}+C(\omega^{C(\Omega_2,\Omega_1)2},\Omega_1),\Omega_1),0)=\theta(\Omega_2+\theta(\Omega_2,\theta(\Omega_2,\Omega)),0)}
{C(\omega^{C(\Omega_2,\Omega_1)2}+C(\Omega_2,\Omega_1),0)=\theta(\Omega_2+\theta(\Omega_2+1,\Omega),0)}
{C(\omega^{C(\Omega_2,\Omega_1)2}+C(\Omega_2,\Omega_1)2,0)=\theta(\Omega_2+\theta(\Omega_2+2,\Omega),0)}
{C(\omega^{C(\Omega_2,\Omega_1)2}+\omega^{C(\Omega_2,\Omega_1)+\Omega_1},0)=\theta(\Omega_2+\theta(\Omega_2+\Omega,\Omega),0)}
{C(\omega^{C(\Omega_2,\Omega_1)2}+\omega^{C(\Omega_2,\Omega_1)+C(\omega^{C(\Omega_2,\Omega_1)2},\Omega_1)},0)=\theta(\Omega_2+\theta(\Omega_2+\theta(\Omega_2,\Omega),\Omega),0)}
{C(\omega^{C(\Omega_2,\Omega_1)2}+\omega^{C(\Omega_2,\Omega_1)+C(\omega^{C(\Omega_2,\Omega_1)2}+C(\Omega_2,\Omega_1),\Omega_1)},0)}\\ {=\theta(\Omega_2+\theta(\Omega_2+\theta(\Omega_2+1,\Omega),\Omega),0)}
{C(\omega^{C(\Omega_2,\Omega_1)2}+\omega^{C(\Omega_2,\Omega_1)+C(\omega^{C(\Omega_2,\Omega_1)2}+\omega^{C(\Omega_2,\Omega_1)+C(\omega^{C(\Omega_2,\Omega_1)2},\Omega_1)},\Omega_1)},0)}\\ {=\theta(\Omega_2+\theta(\Omega_2+\theta(\Omega_2+\theta(\Omega_2,\Omega),\Omega),\Omega),0)}
{C(\omega^{C(\Omega_2,\Omega_1)2}2,0)=\theta(\Omega_22,0)}
{C(\omega^{C(\Omega_2,\Omega_1)2}3,0)=\theta(\Omega_23,0)}
{C(\omega^{C(\Omega_2,\Omega_1)2+1},0)=\theta(\Omega_2\omega,0)}
{C(\omega^{C(\Omega_2,\Omega_1)2+\Omega_1},0)=\theta(\Omega_2\Omega,0)}
{C(\omega^{C(\Omega_2,\Omega_1)3},0)=\theta(\Omega_2^2,0)}
{C(\omega^{\omega^{C(\Omega_2,\Omega_1)+1}},0)=\theta(\Omega_2^\omega,0)}
{C(\omega^{\omega^{C(\Omega_2,\Omega_1)2}},0)=\theta(\Omega_2^{\Omega_2},0)}
{C(\omega^{\omega^{\omega^{C(\Omega_2,\Omega_1)2}}},0)=\theta(\Omega_2^{\Omega_2^{\Omega_2}},0)}
{C(C(\Omega_2,C(\Omega_2,\Omega_1)),0)=\theta(\varepsilon_{\Omega_2+1},0)}
{C(C(\Omega_2,C(\Omega_2,\Omega_1))+\Omega_1,0)=\theta(\varepsilon_{\Omega_2+1}+1,0)}
{C(C(\Omega_2,C(\Omega_2,\Omega_1))+\omega^{C(\Omega_2,\Omega_1)2},0)=\theta(\varepsilon_{\Omega_2+1}+\Omega_2,0)}
{C(C(\Omega_2,C(\Omega_2,\Omega_1))+\omega^{\omega^{C(\Omega_2,\Omega_1)2}},0)=\theta(\varepsilon_{\Omega_2+1}+\Omega_2^{\Omega_2},0)}
{C(C(\Omega_2,C(\Omega_2,\Omega_1))+C(C(\Omega_2,C(\Omega_2,\Omega_1)),C(\Omega_2,\Omega_1)),0)=\theta(\varepsilon_{\Omega_2+1}2,0)}
{C(C(\Omega_2,C(\Omega_2,\Omega_1))+C(C(\Omega_2,C(\Omega_2,\Omega_1)),C(\Omega_2,\Omega_1))\omega,0)=\theta(\varepsilon_{\Omega_2+1}\omega,0)}
{C(C(\Omega_2,C(\Omega_2,\Omega_1))+C(C(\Omega_2,C(\Omega_2,\Omega_1)),C(C(\Omega_2,C(\Omega_2,\Omega_1)),C(\Omega_2,\Omega_1))),0)}\\ {=\theta(\varepsilon_{\Omega_2+2},0)}
{C(C(\Omega_2,C(\Omega_2,\Omega_1))+C(C(\Omega_2,C(\Omega_2,\Omega_1))+1,C(\Omega_2,\Omega_1)),0)=\theta(\varepsilon_{\Omega_2+\omega},0)}
{C(C(\Omega_2,C(\Omega_2,\Omega_1))+C(C(\Omega_2,C(\Omega_2,\Omega_1))+C(\Omega_2,\Omega_1),C(\Omega_2,\Omega_1)),0)=\theta(\varepsilon_{\Omega_22},0)}
{C(C(\Omega_2,C(\Omega_2,\Omega_1))+C(C(\Omega_2,C(\Omega_2,\Omega_1))+C(C(\Omega_2,C(\Omega_2,\Omega_1)),C(\Omega_2,\Omega_1)),0),C(\Omega_2,\Omega_1)),0)}\\ {=\theta(\varepsilon_{\varepsilon_{\Omega_2+1}},0)}
{C(C(\Omega_2,C(\Omega_2,\Omega_1))2,0)=\theta(\varphi(2,\Omega_2+1),0)}
{C(C(\Omega_2,C(\Omega_2,\Omega_1))3,0)=\theta(\varphi(3,\Omega_2+1),0)}
{C(\omega^{C(\Omega_2,C(\Omega_2,\Omega_1))+1},0)=\theta(\theta(\omega,\Omega_2),0)}
{C(\omega^{C(\Omega_2,C(\Omega_2,\Omega_1))+C(\Omega_2,\Omega_1)},0)=\theta(\theta(\Omega_2,\Omega_2),0)}
{\omega^{C(C(\Omega_2,C(\Omega_2,\Omega_1))+C(C(\Omega_2,C(\Omega_2,\Omega_1)),C(\Omega_2,\Omega_1))},0)=\theta(\theta(\varepsilon_{\Omega_2+1},\Omega_2),0)}
{C(\omega^{C(\Omega_2,C(\Omega_2,\Omega_1))2},0)=\theta(\Omega_3,0)}
{C(\omega^{C(\Omega_2,C(\Omega_2,\Omega_1))2}2,0)=\theta(\Omega_32,0)}
{C(\omega^{C(\Omega_2,C(\Omega_2,\Omega_1))3},0)=\theta(\Omega_3^2,0)}
{C(\omega^{\omega^{C(\Omega_2,C(\Omega_2,\Omega_1))+1}},0)=\theta(\Omega_3^\omega,0)}
{C(\omega^{\omega^{\omega^{C(\Omega_2,C(\Omega_2,\Omega_1))+1}}},0)=\theta(\Omega_3^{\Omega_3^\omega},0)}
{C(C(\Omega_2,C(\Omega_2,C(\Omega_2,\Omega_1))),0)=\theta(\varepsilon_{\Omega_3+1},0)}
{C(C(\Omega_2,C(\Omega_2,C(\Omega_2,C(\Omega_2,\Omega_1)))),0)=\theta(\varepsilon_{\Omega_4+1},0)}
{C(C(\Omega_2+1,0),0)=\theta(\Omega_\omega,0)}
{C(C(\Omega_2+1,0)+\Omega_1,0)=\theta(\Omega_\omega+1,0)}
{C(C(\Omega_2+1,0)+\omega^{C(\Omega_2,\Omega_1)2},0)=\theta(\Omega_\omega+\Omega_2,0)}
{C(C(\Omega_2+1,0)+\omega^{C(\Omega_2,C(\Omega_2,\Omega_1))2},0)=\theta(\Omega_\omega+\Omega_3,0)}
{C(C(\Omega_2+1,0)2,0)=\theta(\Omega_\omega2,0)}
{C(\omega^{C(\Omega_2+1,0)+1},0)=\theta(\Omega_\omega\omega,0)}
{C(\omega^{C(\Omega_2+1,0)2},0)=\theta(\Omega_\omega^2,0)}
{C(\omega^{\omega^{C(\Omega_2+1,0)+1}},0)=\theta(\Omega_\omega^\omega,0)}
{C(\omega^{\omega^{\omega^{C(\Omega_2+1,0)+1}}},0)=\theta(\Omega_\omega^{\Omega_\omega^\omega},0)}
{C(C(\Omega_2,C(\Omega_2+1,0)),0)=\theta(\varepsilon_{\Omega_\omega+1},0)}
{C(C(\Omega_2,C(\Omega_2+1,0))+C(\Omega_2+1,0),0)=\theta(\varepsilon_{\Omega_\omega+1}+\Omega_\omega,0)}
{C(C(\Omega_2,C(\Omega_2+1,0))+\omega^{C(\Omega_2+1,0)2},0)=\theta(\varepsilon_{\Omega_\omega+1}+\Omega_\omega^2,0)}
{C(C(\Omega_2,C(\Omega_2+1,0))+C(C(\Omega_2,C(\Omega_2+1,0)),C(\Omega_2+1,0)),0)=\theta(\varepsilon_{\Omega_\omega+1}2,0)}
{C(C(\Omega_2,C(\Omega_2+1,0))+C(C(\Omega_2,C(\Omega_2+1,0))+1,C(\Omega_2+1,0)),0)=\theta(\varepsilon_{\Omega_\omega+\omega},0)}
{C(C(\Omega_2,C(\Omega_2+1,0))2,0)=\theta(\varphi(2,\Omega_\omega+1),0)}
{C(\omega^{C(\Omega_2,C(\Omega_2+1,0))+1},0)=\theta(\theta(\omega,\Omega_\omega),0)}
{C(\omega^{C(\Omega_2,C(\Omega_2+1,0))+C(\Omega_2+1,0)},0)=\theta(\theta(\Omega_\omega,\Omega_\omega),0)}
{C(\omega^{C(\Omega_2,C(\Omega_2+1,0))+C(C(\Omega_2,C(\Omega_2+1,0)),C(\Omega_2+1,0))},0)=\theta(\theta(\varepsilon_{\Omega_\omega+1},\Omega_\omega),0)}
{C(\omega^{C(\Omega_2,C(\Omega_2+1,0))2},0)=\theta(\Omega_{\omega+1},0)}
{C(C(\Omega_2,C(\Omega_2,C(\Omega_2+1,0))),0)=\theta(\varepsilon_{\Omega_{\omega+1}+1},0)}
{C(C(\Omega_2,C(\Omega_2,C(\Omega_2,C(\Omega_2+1,0)))),0)=\theta(\varepsilon_{\Omega_{\omega+2}+1},0)}
{C(C(\Omega_2+1,C(\Omega_2+1,0)),0)=\theta(\Omega_{\omega2},0)}
{C(C(\Omega_2+1,C(\Omega_2+1,C(\Omega_2+1,0))),0)=\theta(\Omega_{\omega3},0)}
{C(C(\Omega_2+2,0),0)=\theta(\Omega_{\omega^2},0)}
{C(C(\Omega_2+\omega,0),0)=\theta(\Omega_{\omega^\omega},0)}
{C(C(\Omega_2+C(\Omega_1,0),0),0)=\theta(\Omega_{\varepsilon_0},0)}
{C(C(\Omega_2+C(C(\Omega_2,\Omega_1),0),0),0)=\theta(\Omega_{\theta(\varepsilon_{\Omega+1},0)},0)}
{C(C(\Omega_2+C(C(\Omega_2+1,0),0),0),0)=\theta(\Omega_{\theta(\Omega_\omega,0)},0)}
{C(C(\Omega_2+C(C(\Omega_2+C(C(\Omega_2+1,0),0),0),0),0),0)=\theta(\Omega_{\theta(\Omega_{\theta(\Omega_\omega,0)},0)},0)}
{C(C(\Omega_2+\Omega_1,0),0)=\theta(\Omega_\Omega,0)}
{C(C(\Omega_2,C(\Omega_2+\Omega_1,0)),0)=\theta(\varepsilon_{\Omega_\Omega+1},0)}
{C(C(\Omega_2+1,C(\Omega_2+\Omega_1,0)),0)=\theta(\Omega_{\Omega+\omega},0)}
{C(C(\Omega_2+C(C(\Omega_2+\Omega_1,0),0),C(\Omega_2+\Omega_1,0)),0)=\theta(\Omega_{\Omega+\theta(\Omega_\Omega,0)},0)}
{C(C(\Omega_2+\Omega_1,C(\Omega_2+\Omega_1,0)),0)=\theta(\Omega_{\Omega2},0)}
{C(C(\Omega_2+\Omega_1+1,0),0)=\theta(\Omega_{\Omega\omega},0)}
{C(C(\Omega_2+\Omega_12,0),0)=\theta(\Omega_{\Omega^2},0)}
{C(C(\Omega_2+\Omega_1^2,0),0)=\theta(\Omega_{\Omega^\Omega},0)}
{C(C(\Omega_2+\Omega_1^{\Omega_1},0),0)=\theta(\Omega_{\Omega^{\Omega^\Omega}},0)}
{C(C(\Omega_2+C(C(\Omega_2,\Omega_1),\Omega_1),0),0)=\theta(\Omega_{\varepsilon_{\Omega+1}},0)}
{C(C(\Omega_2+C(\omega^{C(\Omega_2,\Omega_1)2},\Omega_1),0),0)=\theta(\Omega_{\theta(\Omega_2,\Omega)},0)}
{C(C(\Omega_2+C(C(\Omega_2+1,0),\Omega_1),0),0)=\theta(\Omega_{\theta(\Omega_\omega,\Omega)},0)}
{C(C(\Omega_2+C(C(\Omega_2+\Omega_1,0),\Omega_1),0),0)=\theta(\Omega_{\theta(\Omega_\Omega,\Omega)},0)}
{C(C(\Omega_2+C(C(\Omega_2+C(C(\Omega_2+1,0),\Omega_1),0),\Omega_1),0),0)=\theta(\Omega_{\theta(\Omega_{\theta(\Omega_\omega,\Omega)},\Omega)},0)}
{C(C(\Omega_2+C(\Omega_2,\Omega_1),0),0)=\theta(\Omega_{\Omega_2},0)}
{C(C(\Omega_2+C(\Omega_2,C(\Omega_2,\Omega_1)),0),0)=\theta(\Omega_{\Omega_3},0)}
{C(C(\Omega_2+C(\Omega_2+1,0),0),0)=\theta(\Omega_{\Omega_\omega},0)}
{C(C(\Omega_2+C(\Omega_2+2,0),0),0)=\theta(\Omega_{\Omega_{\omega^2}},0)}
{C(C(\Omega_2+C(\Omega_2+\Omega_1,0),0),0)=\theta(\Omega_{\Omega_{\Omega}},0)}
{C(C(\Omega_2+C(\Omega_2+C(C(\Omega_2,\Omega_1),\Omega_1),0),0),0)=\theta(\Omega_{\Omega_{\varepsilon_{\Omega+1}}},0)}
{C(C(\Omega_2+C(\Omega_2+C(\Omega_2,\Omega_1),0),0),0)=\theta(\Omega_{\Omega_{\Omega_2}},0)}
{C(C(\Omega_2+C(\Omega_2+C(\Omega_2+1,0),0),0),0)=\theta(\Omega_{\Omega_{\Omega_\omega}},0)}
{C(C(\Omega_2+C(\Omega_2+C(\Omega_2+C(\Omega_2+1,0),0),0),0),0)=\theta(\Omega_{\Omega_{\Omega_{\Omega_\omega}}},0)}

The supremum of all the C(C(C(0,Ω2),0),0), C(C(C(C(C(0,Ω2),0),Ω2),0),0), C(C(C(C(C(C(C(0,Ω2),0),Ω2),0),Ω2),0),0), etc. is C(C(C(Ω22),0),0) = C(C(Ω22,0),0). And that’s the limit of θ function.

The fundamental sequences in Taranovsky’s notation can be simply defined. First, we define a function: L(α), it’s the amount of C’s in the standard form of ordinal α. Then we define:

{\alpha[n]=max\{\beta|\beta<\alpha\land L(\beta)\le L(\alpha)+n\}}

That’s it! So we get a very fast-growing function: {f_{C(C(\cdots C(\Omega_n2,0)\cdots,0),0)}(n)} with n C’s. It marks the strength of Taranovsky’s notation.

The full strength of Taranovsky’s notation is unknown yet. It might be as weak as second-order arithmetic (Z2), and might be as strong as second-order arithmetic with projective determinacy (Z2+PD), according to Taranovsky’s page.

Ordinal notations (part 2) – θ function

   θ function is a binary function. It’s defined as follows:

  • {C_0(\alpha,\beta)=\{\gamma|\gamma<\beta\}\cup\{0\}}
  • {C_{n+1}(\alpha,\beta)=\{\gamma+\delta|\gamma,\delta\in C_n(\alpha,\beta)\}\cup\{\theta(\gamma,\delta)|\gamma<\alpha\land\gamma,\delta\in C_n(\alpha,\beta)\}\cup\{\Omega_\gamma|\gamma\in C_n(\alpha,\beta)\}}
  • {C(\alpha,\beta)=\bigcup_{n<\omega}C_n(\alpha,\beta)}
  • {\theta(\alpha,\beta)=min\{\gamma|\gamma\notin C(\alpha,\gamma)\land(\forall\delta<\beta:\gamma>\theta(\alpha,\delta))\}}

Where {\Omega_0=0} and {\Omega_\alpha} means the αth uncountable cardinal.
   {\Omega_1=\Omega} is the first uncountable ordinal and has cardinality {\aleph_1}{\Omega_2} is the smallest ordinal that has cardinality {\aleph_2}. Generally {\Omega_\alpha} is the smallest ordinal that has cardinality {\aleph_\alpha}. They’re so large that {\omega^{\Omega_\alpha}=\Omega_\alpha,\ \varepsilon_{\Omega_\alpha}=\Omega_\alpha,\ \varphi(\Omega_\alpha,0)=\Omega_\alpha}, and for all {\beta<\Omega_\alpha,\ \varphi(\beta,\Omega_\alpha)=\Omega_\alpha}.

   It means that, θ(α,β) is the (1+β)-th ordinal such that it cannot be built from ordinals less than it by addition, applying θ(δ,_) where δ < α and getting an uncountable cardinal.

Explanation

   To get {\theta(0,\beta)} we need to deal with C(0,α). C0(0,0) already contains 0, so let’s begin from 1. For C(0,1), C0(0,1) only contains 0, adding and getting uncountable cardinals cannot get 1, so {\theta(0,0)=1}. For C(0,2), 1 is in it, so 2=1+1 is in it. And so on, 3 is in C(0,3), 4 is in C(0,4) ,etc. Until ω, it cannot be built from natural numbers by addition and getting uncountable cardinals, so {\theta(0,1)=\omega}. Further, {\theta(0,\beta)=\omega^\beta}.

   {\theta(1,0)} is the least ordinal that can’t be built from ordinals less than it by addition and applying {\theta(0,\beta)=\omega^\beta} (and getting uncountable cardinals) – that’s where the Cantor’s normal form cannot get. So {\theta(1,0)=\varepsilon_0}. Further, {\theta(1,\beta)=\varepsilon_\beta}.

   It seems that {\theta(\alpha,\beta)=\varphi(\alpha,\beta)} below {\Gamma_0}, making θ function an extension of φ function. Even {\theta(\Gamma_0,\beta)=\varphi(\Gamma_0,\beta)} is true.

Beyond Feferman–Schütte ordinal

   However, things suddenly go wrong. In {C(\Gamma_0+1,\Gamma_0)}, we cannot use {\theta(\Gamma_0,\beta)}, we can just use {\theta(\alpha,\beta)} for {\alpha<\Gamma_0}, so {\theta(\Gamma_0+1,0)=\Gamma_0}.
   From {\Gamma_0+1} on, we can use {\theta(\Gamma_0,\beta)} in {C(\Gamma_0+1,\alpha)}, so {\theta(\Gamma_0+1,1)=\varphi(\Gamma_0+1,0)}. And further, {\theta(\Gamma_0+1,1+\beta)=\varphi(\Gamma_0+1,\beta)}.

   Then {\theta(\alpha,0)=\Gamma_0} for {\Gamma_0<\alpha<\Omega}, and {\theta(\alpha,1+\beta)=\varphi(\alpha,\beta)} for {\Gamma_0<\alpha<\Gamma_1} – It meets another problem at {\Gamma_1}.

   {\theta(\Gamma_1,0)=\Gamma_0,\theta(\Gamma_1,1)=\Gamma_1}, and {\theta(\Gamma_1,1+\beta)=\varphi(\Gamma_1,\beta)}. Next, {\theta(\Gamma_1+1,0)=\Gamma_0,\theta(\Gamma_1+1,1)=\Gamma_1,\theta(\Gamma_1+1,2)=\varphi(\Gamma_1+1,0)}, and {\theta(\Gamma_1+1,2+\beta)=\varphi(\Gamma_1+1,\beta)}. Now this pattern {\theta(\alpha,1+\gamma+\beta)=\varphi(\alpha,\beta)} where {\Gamma_\gamma<\alpha\leq\Gamma_{\gamma+1}} will continue up to Ω, so it’s enough to define larger ordinals. But we cannot name, say, what’s the first fixed point of {\alpha\mapsto\Gamma_\alpha}, what’s LVO, what’s BHO, etc.

   To express large ordinals only using θ function (and Cantor’s normal form), we need to deal with something beyond {\theta(\Omega,0)}. Using the result above, {\theta(\Omega,\alpha)=\Gamma_\alpha} for all α < Ω. Now notice that C(Ω+1,α) contains Ω and can apply θ(Ω,α) in it, so {\Gamma_0} is in {C(\Omega+1,\Gamma_0)}. In fact, {\theta(\Omega+1,\beta)} is the (1+β)-th fixed point of {\alpha\mapsto\Gamma_\alpha}.

   It seems that, if we write θ function in this way: {\alpha=\theta(\gamma_1,\beta)}, where {\gamma_i=\omega^{\gamma_{i,1}}+\omega^{\gamma_{i,2}}+\cdots+\omega^{\gamma_{i,n_i}}+\omega^{\gamma_{i+1}}} with {\gamma_{i,1}\geq\gamma_{i,2}\geq\cdots\gamma_{i,n_i}\geq\gamma_{i+1}} and {\gamma_k=\Omega}, then it’s the (1+β)-th fixed point of {\alpha\mapsto\theta(\delta_1,0)} where {\delta_i=\omega^{\gamma_{i,1}}+\omega^{\gamma_{i,2}}+\cdots+\omega^{\gamma_{i,n_i}}+\omega^{\delta_{i+1}}} and {\delta_k=\alpha} itself. This pattern holds up to {\theta(\Gamma_{\Omega+1},0)} – even beyond Bachmann-Howard ordinal. For example, {\theta(\Omega^\Omega,0)=\theta(\omega^{\omega^{\Omega2}},0)} is the first fixed point of {\alpha\mapsto\theta(\omega^{\omega^{\Omega+\alpha}},0)}, which is called large Veblen ordinal. The small Veblen ordinal is {\theta(\Omega^\omega,0)=\theta(\omega^{\omega^{\Omega+1}},0)}, and the Bachmann-Howard ordinal is {\theta(\varepsilon_{\Omega+1},0)}.

Beyond Bachmann-Howard ordinal

   The (1+β)-th fixed point of {\alpha\mapsto\theta(\varphi(\alpha,\Omega+1),0)} is {\theta(\varphi(\Omega,1),\beta)}. For {\alpha<\Omega,\ \theta(\Omega,\alpha)=\Gamma_\alpha}, but {\theta(\Omega,\Omega+\alpha)=\varphi(\Omega,1+\alpha)} – The same thing at {\Gamma_0} happens at Ω now.

   Another important point is {\theta(\Omega_2,0)}. What is it? {\theta(\Gamma_{\Omega+1},0)} cannot be built from ordinal less than it and Ω by addition and {\theta(\alpha,\beta)} where {\alpha<\Gamma_{\Omega+1}}, so it’s {\theta(\Gamma_{\Omega+1},0)}. What’s more, for {\Gamma_{\Omega+1}\leq\alpha<\Omega_2,\ \theta(\alpha,0)=\theta(\Omega_2,0)}.
   Then {\theta(\Gamma_{\Omega+1},1)} is the 2nd ordinal that cannot be built from ordinal less than it and Ω by addition and {\theta(\alpha,\beta)} where {\alpha<\Gamma_{\Omega+1}} – it’s {\theta(\Omega_2,1)}. What’s more, for β < Ω and {\Gamma_{\Omega+1}\leq\alpha<\Omega_2,\ \theta(\alpha,\beta)=\theta(\Omega_2,\beta)}.
   {\theta(\Omega_2,\Omega)=\Gamma_{\Omega+1}}, and {\theta(\Omega_2,\Omega+1)=\Gamma_{\Omega+2}}. And for {\alpha<\Omega_2,\ \theta(\Omega_2,\Omega+\alpha)=\Gamma_{\Omega+1+\alpha}}. Note that {\Omega\in C(\alpha,\Omega)} for any α!

   The next step is {\theta(\Omega_2+\Gamma_{\Omega+1},0)}. Since {\Gamma_{\Omega+1}=\theta(\Omega_2,\Omega)}, it can be built from ordinals below it and Ω and Ω2 by addition and {\theta(\alpha,\beta)} where {\alpha<\Omega_22}, so it’s not {\theta(\Omega_22,0)} – there’re many things between them, such as {\theta(\Omega_2+\Gamma_{\Omega+2},0)}, {\theta(\Omega_2+\theta(\Omega_2+1,\Omega),0)}, {\theta(\Omega_2+\theta(\Omega_2+\Gamma_{\Omega+1},\Omega),0)}, {\theta(\Omega_2+\theta(\Omega_2+\theta(\Omega_2+\Gamma_{\Omega+1},\Omega),\Omega),0)}, etc.

   Then we turn to {\theta(\Gamma_{\Omega_2+1},0)}. For β < Ω2 and {\Gamma_{\Omega_2+1}\leq\alpha<\Omega_3,\ \theta(\alpha,\beta)=\theta(\Omega_3,\beta)}, but there’re many things between {\theta(\Omega_3+\Gamma_{\Omega_2+1},0)} and {\theta(\Omega_32,0)}.

Beyond θ(Ωω,0)

   The “getting an uncountable cardinal” in definition start to works now. {\theta(\Omega_\omega,0)} is the supremum of all the {\theta(\Omega_n,0)} for natural number n.

   {\theta(\Omega_\Omega,0)} is not {\theta(\Omega_{\Gamma_0},0)} because we have {\Gamma_0}, then {\Omega_{\Gamma_0}}, then {\theta(\Omega_{\Gamma_0},0)} in {C(\Omega_{\Omega},\theta(\Omega_{\Gamma_0},0))}. In fact, {\theta(\Omega_\Omega,\beta)} is the (1+β)-th fixed point of {\alpha\mapsto\theta(\Omega_\alpha,0)}.

   Also, {\theta(\Omega_{\Omega_2},0)} is not {\theta(\Omega_{\Gamma_{\Omega+1}},0)}; it’s {\theta(\Omega_{\alpha\mapsto\theta(\Omega_\alpha,\Omega)},0)}.

   The limit of θ function is the supremum of {\theta(\Omega,0),\ \theta(\Omega_\Omega,0),\ \theta(\Omega_{\Omega_\Omega},0)}, etc.

Ordinal notations

   Ordinals are important things in googology. Most of them are infinite, so they’re not just numbers. But we can use them to present some “levels”. Here I bring 4 ordinal notations I would use in this site – Cantor’s normal form, binary φ function, θ function, and Taranovsky’s notation.

Definition of ordinals

   Ordinals are something that fits these properties. First we need a binary relation “<” – that’s “less than” – on ordinals.

  1. Any ordinal a and b must be and only be one of those 3 relations: a < b, a = b, or b < a.
  2. a = a – that means a < a is always false.
  3. {a<b\land b<c\Rightarrow a<c}
  4. Any sets or classes of ordinals must have a minimum.

   We also write a < b as b > a. And {a\leq b} means a < b or a = b.

Basic notations

   Here’s a very basic notation. The minimum of all ordinals is 0, and {min\{\beta|\beta>\alpha\}=\alpha+1} is the successor of {\alpha}.

   We call an ordinal x successor ordinal iff it’s the successor of some ordinal. x is a limit ordinal iff it’s neither 0 nor successor ordinal. We call the least limit ordinal ω – it’s the supremum of all natural numbers.

   Then there’re some arithmetic on ordinals. First is the addition.

  • {\alpha+0=\alpha}
  • {\alpha+(\beta+1)=(\alpha+\beta)+1}
  • {\alpha+\beta=sup\{\alpha+\gamma|\gamma<\beta\}} for limit ordinal β

Then multiplication.

  • {\alpha0=0}
  • {\alpha(\beta+1)=\alpha\beta+\alpha}
  • {\alpha\beta=sup\{\alpha\gamma|\gamma<\beta\}} for limit ordinal β

Then exponentiation.

  • {\alpha^0=1}
  • {\alpha^{\beta+1}=\alpha^\beta\alpha}
  • {\alpha^\beta=sup\{\alpha^\gamma|\gamma<\beta\}} for limit ordinal β

   Then ordinals have such properties:

  1. {\alpha+(\beta+\gamma)=(\alpha+\beta)+\gamma}
  2. {\alpha(\beta\gamma)=(\alpha\beta)\gamma}
  3. {\alpha(\beta+\gamma)=\alpha\beta+\alpha\gamma}
  4. {\alpha^{\beta+\gamma}=\alpha^\beta\alpha^\gamma}
  5. {\alpha^{\beta\gamma}=(\alpha^\beta)^\gamma}

   Finally, we have a notation called Cantor’s normal form.

  • Ordinal {\alpha\leq\omega} is in Cantor’s normal form.
  • Ordinal {\alpha=\omega^{\beta_1}k_1+\omega^{\beta_2}k_2+\cdots+\omega^{\beta_n}k_n} where n and all ki‘s are positive integers and {\beta_1>\beta_2>\cdots>\beta_n} are all in Cantor’s normal form is in Cantor’s normal form.

   If we just use Cantor’s normal form built from 0, 1 and ω, we have such a limit – {\omega^{\omega^{\omega^{\cdots}}}=\varepsilon_0}.

Binary φ function

   Binary φ function is defined as follows:

  • {\varphi(0,\beta)=\omega^\beta}
  • {\varphi(\alpha,\beta)=min\{\gamma |(\forall\delta<\alpha:\gamma=\varphi(\delta,\gamma))\land(\forall\delta<\beta:\gamma>\varphi(\alpha,\delta))\}}

The second line means, {\varphi(\alpha,\beta)} is the (1+β)-th common fixed point that {\gamma\mapsto\varphi(\delta,\gamma)} for all δ < α.

Explanation

   {\varphi(0,\beta)=\omega^\beta} – that’s just a simple Cantor’s normal form.

   {\varepsilon_0=\varphi(1,0)} is the least ordinal such that {\omega^{\varepsilon_0}=\varepsilon_0}. We say it’s the first fixed point of {\alpha\mapsto\omega^\alpha}. The second fixed point is {\varepsilon_1=\varphi(1,1)}, the third is {\varepsilon_2=\varphi(1,2)}, and so on. The ω-th is {\varepsilon_\omega=\varphi(1,\omega)}.

   There’re also some ordinals such that {\alpha=\varepsilon_\alpha} – the fixed points of {\alpha\mapsto\varphi(1,\alpha)}. The first fixed point is {\varphi(2,0)}, the second is {\varphi(2,1)}, and so on.

   {\alpha\mapsto\varphi(2,\alpha)} also has fixed points. They’re {\varphi(3,\beta)}.

   Generally speaking, {\varphi(\alpha+1,\beta)} is the (1+β)-th fixed point of {\gamma\mapsto\varphi(\alpha,\gamma)}. This works for successor ordinal α.

   For limit case, such as ω, {\varphi(\omega,\beta)} is a fixed point of {\gamma\mapsto\varphi(0,\gamma)}, also {\gamma\mapsto\varphi(1,\gamma)}, also {\gamma\mapsto\varphi(2,\gamma)}, also {\gamma\mapsto\varphi(3,\gamma)}, and so on. It’s the common fixed point of {\gamma\mapsto\varphi(\delta,\gamma)} for all δ < ω, and it’s the (1+β)-th ordinal have such property. Notice that this “common fixed point” also works for successor case, so we get the definition.

   Using 0, 1, ω, by addition and φ function, we have such a limit – it’s the first fixed point of {\gamma\mapsto\varphi(\gamma,0)} – that’s {\Gamma_0}. Generally, we use {\Gamma_\beta} to present the (1+β)-th fixed point of {\gamma\mapsto\varphi(\gamma,0)}.