FANDOM


I shall use w for the small omega (first transfinite ordinal) and W for the capital omega (first uncountable ordinal).

BAN separator BEAF structure FGH ordinal
[1] 1 3
[2] X w^w
[3] X2 w^w^2
[4] X3 w^w^3
[1,2] XX w^w^w
[2,2] XX+1 w^w^(w+1)
[1,3] X2X w^w^(w2)
[1,4] X3X w^w^(w3)
[1,1,2] XX2 w^w^(w^2)
[1,1,1,2] XX3 w^w^(w^3)
[1 [2] 2] XXX w^w^w^w
[2 [2] 2] XXX+1 w^w^(w^w+1)
[1,2 [2] 2] XXX+X w^w^(w^w+w)
[1 [2] 3] X2XX w^w^((w^w)2)
[1 [2] 4] X3XX w^w^((w^w)3)
[1 [2] 1,2] XXX+1 w^w^w^(w+1)
[1 [2] 1,1,2] XXX+2 w^w^w^(w+2)
[1 [2] 1 [2] 2] XX2X w^w^w^(w2)
[1 [3] 2] XXX2 w^w^w^(w^2)
[1 [4] 2] XXX3 w^w^w^(w^3)
[1 [1,2] 2] XXXX w^w^w^w^w
[1 [1 [2] 2] 2] XXXXX w^w^w^w^w^w
[1 [1 [1,2] 2] 2] X^^6 w^w^w^w^w^w^w
[1 \ 2] X^^X e_0
[2 \ 2] X(X^^X) e_0^w
[1,2 \ 2] X^X(X^^X) e_0^w^w
[1 [1 \ 2] 2 \ 2] (X^^X)^2 e_0^e_0
[1 [1 \ 2] 3 \ 2] (X^^X)^3 e_0^((e_0)2)
[1 [1 \ 2] 1,2 \ 2] (X^^X)^X e_0^((e_0)w)
[1 [1 \ 2] 1 [1 \ 2] 2 \ 2] (X^^X)^(X^^X) e_0^e_0^2
[1 [1 \ 2] 1 [1 \ 2] 3 \ 2] (X^^X)^(2(X^^X)) e_0^((e_0^2)2)
[1 [1 \ 2] 1 [1 \ 2] 1,2 \ 2] (X^^X)^(X(X^^X)) e_0^((e_0^2)w)
[1 [1 \ 2] 1 [1 \ 2] 1 [1 \ 2] 2 \ 2] (X^^X)^(X^^X)^2 e_0^e_0^2
[1 [1 \ 2] 1 [1 \ 2] 1 [1 \ 2] 1 [1 \ 2] 2 \ 2] (X^^X)^(X^^X)^3 e_0^e_0^3
[1 [2 \ 2] 2 \ 2] (X^^X)^(X^^X)^X e_0^e_0^w
[1 [3 \ 2] 2 \ 2] (X^^X)^(X^^X)^(X^2) e_0^e_0^w^2
[1 [1,2 \ 2] 2 \ 2] (X^^X)^(X^^X)^(X^X) e_0^e_0^w^w
[1 \ 3] X^^(2X) e_1
[1 \ 4] X^^(3X) e_2
[1 \ 1,2] X^^(X^2) e_w
[1 \ 2,2] X^^(X^2+X) e_(w+1)
[1 \ 3,2] X^^(X^2+2X) e_(w+2)
[1 \ 4,2] X^^(X^2+3X) e_(w+3)
[1 \ 1,3] X^^(2X^2) e_(w2)
[1 \ 1,4] X^^(3X^2) e_(w3)
[1 \ 1,1,2] X^^(X^3) e_(w^2)
[1 \ 1,1,1,2] X^^(X^4) e_(w^3)
[1 \ 1 [2] 2] X^^(X^X) e_(w^w)
[1 \ 1 [3] 2] X^^(X^X^2) e_(w^w^2)
[1 \ 1 [4] 2] X^^(X^X^3) e_(w^w^3)
[1 \ 1 [1,2] 2] X^^(X^X^X) e_(w^w^w)
[1 \ 1 [1 \ 2] 2] X^^(X^^X) e_(e_0)
[1 \ 1 [1 \ 1 [1 \ 2] 2] 2] X^^^4 e_(e_(e_0))
[1 \ 1 \ 2] X^^^X z_0
[1 \ 2 \ 2] (X^^^X)^^X e_(z_0+1)
[1 \ 3 \ 2] (X^^^X)^^(2X) e_(z_0+2)
[1 \ 1,2 \ 2] (X^^^X)^^(X^2) e_(z_0+w)
[1 \ 1 [2] 2 \ 2] (X^^^X)^^(X^X) e_(z_0+w^w)
[1 \ 1 [1 \ 2] 2 \ 2] (X^^^X)^^(X^^X) e_(z_0+e_0)
[1 \ 1 [1 \ 1 \ 2] 2 \ 2] (X^^^X)^^(X^^^X) e_((z_0)2)
[1 \ 1 [1 \ 1 \ 2] 3 \ 2] (X^^^X)^^(2(X^^^X)) e_((z_0)3)
[1 \ 1 [1 \ 1 \ 2] 1,2 \ 2] (X^^^X)^^(X(X^^^X)) e_((z_0)w)
[1 \ 1 [1 \ 1 \ 2] 1 [1 \ 1 \ 2] 2 \ 2] (X^^^X)^^(X^^^X)^2 e_(z_0^2)
[1 \ 1 [2 \ 1 \ 2] 2 \ 2] (X^^^X)^^(X^^^X)^X e_(z_0^w)
[1 \ 1 [1 [1 \ 1 \ 2] 2 \ 1 \ 2] 2 \ 2] (X^^^X)^^(X^^^X)^(X^^^X) e_(z_0^z_0)
[1 \ 1 [1 \ 2 \ 2] 2 \ 2] (X^^^X)^^(X^^^X)^^X e_(e_(z_0+1))
[1 \ 1 \ 3] X^^^(2X) z_1
[1 \ 1 \ 4] X^^^(3X) z_2
[1 \ 1 \ 1,2] X^^^(X^2) z_w
[1 \ 1 \ 1 [2] 2] X^^^(X^X) z_(w^w)
[1 \ 1 \ 1 [1,2] 2] X^^^(X^X^X) z_(w^w^w)
[1 \ 1 \ 1 [1 \ 2] 2] X^^^(X^^X) z_(e_0)
[1 \ 1 \ 1 [1 \ 1 \ 2] 2] X^^^(X^^^X) z_(z_0)
[1 \ 1 \ 1 \ 2] X^^^^X n_0
[1 \ 1 \ 1 \ 1 \ 2] X^^^^^X theta(4,0)
[1 \\ 2] {X,X,X} theta(w,0)
[1 \ 2 \\ 2] {X,X,X}^^X e_(theta(w,0)+1)
[1 \ 1 \ 2 \\ 2] {X,X,X}^^^X z_(theta(w,0)+1)
[1 \\ 3] {X,2X,X} theta(w,1)
[1 \\ 4] {X,3X,X} theta(w,2)
[1 \\ 1,2] {X,X^2,X} theta(w,w)
[1 \\ 1 [2] 2] {X,X^X,X} theta(w,w^w)
[1 \\ 1 [1,2] 2] {X,X^X^X,X} theta(w,w^w^w)
[1 \\ 1 [1 \ 2] 2] {X,X^^X,X} theta(w,e_0)
[1 \\ 1 [1 \ 1 \ 2] 2] {X,X^^^X,X} theta(w,z_0)
[1 \\ 1 [1 \\ 2] 2] {X,{X,X,X},X} theta(w,theta(w,0))
[1 \\ 1 \ 2] {X,X,X+1} theta(w+1)
[1 \\ 1 \ 1 \ 2] {X,X,X+2} theta(w+2)
[1 \\ 1 \\ 2] {X,X,2X} theta(w2)
[1 \\ 1 \\ 1 \\ 2] {X,X,3X} theta(w3)
[1 \\\ 2] {X,X,X^2} theta(w^2)
[1 \\\\ 2] {X,X,X^3} theta(w^3)
[1 [1,2]\ 2] {X,X,X^X} theta(w^w)
[1 [1,2]\ 1 \ 2] {X,X,X^X+1} theta(w^w+1)
[1 [1,2]\ 1 \\ 2] {X,X,X^X+X} theta(w^w+w)
[1 [1,2]\ 1 [1,2]\ 2] {X,X,2X^X} theta((w^w)2)
[1 [2,2]\ 2] {X,X,X^(X+1)} theta(w^(w+1))
[1 [3,2]\ 2] {X,X,X^(X+2)} theta(w^(w+2))
[1 [1,3]\ 2] {X,X,X^(2X)} theta(w^(w2))
[1 [1,4]\ 2] {X,X,X^(3X)} theta(w^(w3))
[1 [1,1,2]\ 2] {X,X,X^(X^2)} theta(w^w^2)
[1 [1,1,1,2]\ 2] {X,X,X^(X^3)} theta(w^w^3)
[1 [1 [2] 2]\ 2] {X,X,X^(X^X)} theta(w^w^w)
[1 [1 [1,2] 2]\ 2] {X,X,X^(X^(X^X))} theta(w^w^w^w)
[1 [1 \ 2]\ 2] {X,X,X^^X} theta(e_0)
[1 [1 \ 1 \ 2]\ 2] {X,X,X^^^X} theta(z_0)
[1 [1 \\ 2]\ 2] {X,X,{X,X,X}} theta(theta(w))
[1 [1 \\\ 2]\ 2] {X,X,{X,X,X^2}} theta(theta(w^2))
[1 [1 [1,2]\ 2]\ 2] {X,X,{X,X,X^X}} theta(theta(w^w))
[1 [1 [1 \ 2]\ 2]\ 2] {X,X,{X,X,X^^X}} theta(theta(e_0))
[1 [1 ¬ 3] 2] {X,X,1,2} theta(W)

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