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) |