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”.

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

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24 thoughts on “pDAN vs. Taranovsky’s ordinal notation

  1. Could you tell me:
    What ordinal corresponds to C(C(W_2*2+C(W_2*2,0),0),0)?
    What ordinal corresponds to C(C(W_2*2+C(W_2*2,1),0),0)?
    What ordinal corresponds to C(C(W_2*2+C(W,C(W_2*2,0)),0),0)?
    What ordinal corresponds to a {1,,1,,1,2} seperator?

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    • I haven’t analysed so far to C(C(W_2*2+C(W_2*2,0),0),0).
      Also, C(W_2*2,1) is not a valid term.

      {1,,1,,1,2} has recursion level ψ(I_ω) in normal ordinal notations. For {1,,1,,1,2} itself, it corresponds to I_ω.

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  2. Username5243 says:

    If I get it right then:

    W = C(W_2,0) corresponds to W in standard notation
    C(W_2,W) = C(W_2,C(W_2,0)) corresponds to W_2 in standard notation
    C(W_2,C(W_2,C(W_2,0))) corresponds to W_3 in standard notation
    C(W_2+1,0) corresponds to W_w
    C(W_2+2,0) corresponds to W_(w^2)
    C(W_2+W,0) corresponds to W_W
    And generally W_(w^a) correspodns to C(W_2+a,0).
    Then:

    C(W_2*2,0) corresponds to psi_I(0)
    C(W_2,C(W_2*2,0)) corresponds to W_(psi_I(0)+1)
    C(W_2*2,C(W_2*2,0)) corresponds to W_(psi_I(0)*2)
    C(W_2*2+1,0) corresponds to W_(psi_I(0)*w)
    C(W_2*2+W,0) corresponds to W_(psi_I(0)*W)
    C(W_2*2+C(W_2+C(W_2*2,0),0),0) corresponds to W_(psi_I(0)^2)

    But, beyond this point, I do not understand how it works fully.

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    • Aarex Tiaokhiao says:

      Hmmm….

      C(W_2+C(W_2+C(W_2*2,0),0),0) ~ psi_I(0)
      C(W,psi_I(0)) ~ psi_I(0)+W
      C(C(C(W_2,W),W),psi_I(0)) ~ psi_I(0)+e(W+1)
      C(C(W_2,W),psi_I(0)) ~ psi_I(0)+W_2
      C(C(W_2+1,0),psi_I(0)) ~ psi_I(0)+W_w
      C(C(W_2+C(W_2+1,0),0),psi_I(0)) ~ psi_I(0)+W_W_w
      C(psi_I(0),psi_I(0)) ~ psi_I(0)2
      C(W_2,psi_I(0)) ~ W_(psi_I(0)+1)

      You are right.

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    • No.

      C(W_2*2,0) is much stronger. You can see it as the 1-separator inside C(W_2+____,0). By Taranovsky’s words, it should be the least ordinal x such that Lx is a Σ1 elementary substructure of L.

      Then C(W_2,C(W_2*2,0)) works as the 2-separator inside C(W_2+____,0), also corresponding to the double comma in pDAN.

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    • I haven’t analyse so far to C(C(W_2*3,0),0).
      Also, C(C(W_2*2+C(W_2+C(W_2,C(W_2,C(W_2*2,0))),0),0),0) already surpasses the range of “standard ordinal notations”.

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  3. Username5243 says:

    Can’t you use a more standard ordinal notation. I don’t understand Taranovsky’s notation, and I’m not even sure all googologists that understand ordinals this high understand it. I certainly don’t.

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    • “Standard ordinal notation” can’t reach the limit of pDAN, but Taranovsky’s ordinal notation can.

      By the way, I only introduced Cantor’s normal form, binary φ function, binary θ function and Taranovsky’s notation in this website. If I show comparisons to higher “standard ordinal notations”, I must introduce them, but they’re too complex, such as The Pi-4 reflecting one, the Pi-n reflecting one, or even higher. Compared with them, Taranovsky states that:

      C(Ω+C(Ω,C(Ω2,0)),0) corresponds to the recursively inaccessible,
      C(Ω+C(Ω,C(Ω2,0))^2,0) corresponds to the recursively Mahlo,
      C(Ω+C(Ω,C(Ω2,0))^C(Ω,C(Ω2,0)),0) corresponds to the Pi-3 reflecting,
      C(Ω+C(Ω,C(Ω2,0))^C(Ω,C(Ω2,0))^(n-2),0) corresponds to the Pi-n reflecting,
      (using the 2nd system)

      So it’s a better choice to use Taranovsky’s ordinal notation.

      Like

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