Well I had to do it at some point.
Part 2 - http://googology.wikia.com/wiki/User_blog:Boboris02/Measuring_the_strength_of_ABHAN_part_2
Part 3 - http://googology.wikia.com/wiki/User_blog:Boboris02/Measuring_the_strength_of_ABHAN_part_3
Linear arrays - up to \(\omega^\omega\)[]
|
\(b(a,a)\) |
\(a^2\) |
|---|---|
| \(b(a,a,1)\) | \(f_2(a)\) |
| \(b(a,a,n)\) | \(f_{n+1}(a)\) |
| \(b(a,a,0,1)\) | \(f_{\omega}(a)\) |
| \(b(a,a,n,1)\) | \(f_{\omega +n}(a)\) |
| \(b(a,a,n,k)\) | \(f_{\omega k + n}(a)\) |
| \(b(a,a,0,0,1)\) | \(f_{\omega^2}(a)\) |
| \(b(a,a,n,k,i)\) | \(f_{\omega^2 i + \omega k + n}(a)\) |
| \(b(a,a,0,0,0,1)\) | \(f_{\omega^3}(a)\) |
| \(b(a,a,n,k,l,m)\) | \(f_{\omega^3 m + \omega^2 l + \omega k + n}(a)\) |
| \(b(a,a,b,c,d,e,.....)\) | \(f_{...... + \omega^3 e + \omega^2 d + \omega c +b}(a)\) |
Limit of l-ABHAN is \(\omega^\omega\).
Two rows - up to \(\omega^{\omega 2}\)[]
| \(b(a,a\{1\}1)\) | \(f_{\omega^\omega}(a)\) |
|---|---|
| \(b(a,a,1\{1\}1)\) | \(f_{\omega^\omega +1}(a)\) |
| \(b(a,a,n\{1\}1)\) | \(f_{\omega^\omega +n}(a)\) |
| \(b(a,a,0,1\{1\}1)\) | \(f_{\omega^\omega + \omega}(a)\) |
| \(b(a,a,n,k\{1\}1)\) | \(f_{\omega^\omega + \omega k + n}(a)\) |
| \(b(a,a,b,c,d,.....\{1\}1)\) | \(f_{\omega^\omega + ..... +\omega^2 d + \omega c + b}(a)\) |
| \(b(a,a\{1\}2)\) | \(f_{\omega^\omega 2}(a)\) |
| \(b(a,a,b,c,d,e,....\{1\}n)\) | \(f_{\omega^\omega n + ...... \omega^3 e + \omega^2 d + \omega c +b}(a)\) |
| \(b(a,a\{1\}0,1)\) | \(f_{\omega^{\omega +1}}(a)\) |
| \(b(a,a,b,c,d,e,...\{1\}n,1)\) | \(f_{\omega^{\omega +1} + \omega^\omega n + ...... \omega^3 e + \omega^2 d + \omega c +b}(a)\) |
| \(b(a,a\{1\}0,2)\) | \(f_{\omega^{\omega +1} 2}(a)\) |
| \(b(a,a\{1\}0,0,1)\) | \(f_{\omega^{\omega +2}}(a)\) |
| \(b(a,a\{1\}n_1,n_2,.....,n_i,n_{i+1})\) | \(f_{\omega^{\omega + i}n_{i+1} + \omega^{\omega + i - 1}n_{i} + ...... + \omega^{\omega +1}n_2 + \omega^{\omega}n_1}(a)\) |
Planar - up to \(\omega^{\omega^2}\)[]
| \(b(a,a\{1\}0\{1\}1)\) | \(f_{\omega^{\omega 2}}(a)\) |
|---|---|
| \(b(a,a,1\{1\}0\{1\}1)\) | \(f_{\omega^{\omega 2} +1}(a)\) |
| \(b(a,a,b,c,d,e....\{1\}0\{1\}1)\) | \(f_{\omega^{\omega 2} +.....+\omega^3 e \omega^2 d + \omega c + b}(a)\) |
| \(b(a,a\{1\}1\{1\}1)\) | \(f_{\omega^{\omega 2} + \omega^\omega}(a)\) |
| \(b(a,a\{1\}b,c,d,e,....\{1\}1)\) | \(f_{\omega^{\omega 2} + ..... + \omega^{\omega +3}e +\omega^{\omega +2}d + \omega^{\omega +1}c + \omega^\omega b}\) |
| \(b(a,a\{1\}0\{1\}2)\) | \(f_{\omega^{\omega 2} 2}(a)\) |
| \(b(a,a\{1\}0\{1\}b,c,d,e,...)\) | \(f_{..... + \omega^{\omega 2 + 3}e + \omega^{\omega 2 + 2}d + \omega^{\omega 2 + 1}c + \omega^{\omega 2}b}\) |
| \(b(a,a\{1\}0\{1\}0\{1\}1)\) | \(f_{\omega^{\omega 3}}(a)\) |
| \(b(a,a\{1\}n_1\{1\}n_2,......\{1\}n_i)\) | \(f_{\omega^{\omega i}n_i + ..... + \omega^{\omega 2}n_2 + \omega^{\omega}n_1}(a)\) |
| \(b(a,a\{1\}0\{1\}0\{1\}0\{1\}.....\{1\}0\{1\}1)\) w/ \(n\) \(\{1\}\)'s |
\(f_{\omega^{\omega n}}(a)\) |
Cubic - up to \(\omega^{\omega^3}\)[]
|
\(b(a,a\{2\}1)\) |
\(f_{\omega^{\omega^2}}(a)\) |
|---|---|
| \(b(a,a,1\{2\}1)\) | \(f_{\omega^{\omega^2}+1}(a)\) |
| \(b(a,a,0,1\{2\}1)\) | \(f_{\omega^{\omega^2}+\omega}(a)\) |
| \(b(a,a\{1\}0\{2\}1)\) | \(f_{\omega^{\omega^2} + \omega^\omega}(a)\) |
| \(b(a,a\{1\}1\{2\}1)\) | \(f_{\omega^{\omega^2}+\omega^\omega 2}(a)\) |
| \(b(a,a\{1\}0,1\{2\}1)\) | \(f_{\omega^{\omega^2}+\omega^{\omega +1}}(a)\) |
| \(b(a,a\{1\}0\{1\}0\{2\}1)\) | \(f_{\omega^{\omega^2}+\omega^{\omega 2}}(a)\) |
| \(b(a,a\{2\}2)\) | \(f_{\omega^{\omega^2}2}(a)\) |
| \(b(a,a\{2\}0,1)\) | \(f_{\omega^{\omega^2 +1}}(a)\) |
| \(b(a,a\{2\}0\{1\}1)\) | \(f_{\omega^{\omega^2 + \omega}}(a)\) |
| \(b(a,a\{2\}0\{1\}0\{1\}1)\) | \(f_{\omega^{\omega^2 + \omega 2}}\) |
| \(b(a,a\{2\}0\{2\}1)\) | \(f_{\omega^{\omega^2 2}}(a)\) |
| \(b(a,a\{2\}0\{2\}0\{2\}1)\) | \(f_{\omega^{\omega^2 3}}(a)\) |
|
\(b(a,a\{2\}0\{2\}0\{2\}.....\{2\}1)\) w/ \(n\) \(\{2\}\)'s |
\(f_{\omega^{\omega^2 n}}(a)\) |
Multi-dimentional - up to \(\omega^{\omega^\omega}\)[]
| \(b(a,a\{3\}1)\) | \(f_{\omega^{\omega^3}}(a)\) |
|---|---|
| \(b(a,a,1\{3\}1)\) | \(f_{\omega^{\omega^3}+1}(a)\) |
| \(b(a,a,0,1\{3\}1)\) | \(f_{\omega^{\omega^3}+\omega}(a)\) |
| \(b(a,a\{1\}0\{3\}1)\) | \(f_{\omega^{\omega^3}+\omega^\omega}(a)\) |
| \(b(a,a\{2\}0\{3\}1)\) | \(f_{\omega^{\omega^3}+\omega^{\omega^2}}(a)\) |
| \(b(a,a\{3\}2)\) | \(f_{\omega^{\omega^3}2}(a)\) |
| \(b(a,a\{3\}0,1)\) | \(f_{\omega^{\omega^3 +1}}\) |
| \(b(a,a\{3\}0\{1\}1)\) | \(f_{\omega^{\omega^3 +\omega}}(a)\) |
| \(b(a,a\{3\}0\{3\}1)\) | \(f_{\omega^{\omega^3 2}}(a)\) |
| \(b(a,a\{4\}1)\) | \(f_{\omega^{\omega^4}}(a)\) |
| \(b(a,a\{4\}0,1)\) | \(f_{\omega^{\omega^4 +1}}(a)\) |
| \(b(a,a\{4\}0\{4\}1)\) | \(f_{\omega^{\omega^4 2}}(a)\) |
| \(b(a,a\{4\}0\{4\}0\{4\}......\{4\}1)\) w/ \(n\) \(\{4\}\)'s | \(f_{\omega^{\omega^4 n}}(a)\) |
| \(b(a,a{5}1)\) | \(f_{\omega^{\omega^5}}(a)\) |
| \(b(a,a\{n\}1)\) | \(f_{\omega^{\omega^n}}(a)\) |
Limit of md-ABHAN is \(\omega^{\omega^\omega}\)
Hyperdimentional - up to \(\varepsilon_0\)[]
| \(b(a,a\{0,1\}1)\) | \(f_{\omega^{\omega^\omega}}(a)\) |
|---|---|
| \(b(a,a,1\{0,1\}1)\) | \(f_{\omega^{\omega^\omega}+1}(a)\) |
| \(b(a,a\{1\}0\{0,1\}1)\) | \(f_{\omega^{\omega^\omega}+\omega^\omega}(a)\) |
| \(b(a,a\{n\}0\{0,1\}1)\) | \(f_{\omega^{\omega^\omega}+\omega^{\omega^n}}(a)\) |
| \(b(a,a\{0,1\}2)\) | \(f_{\omega^{\omega^\omega}2}(a)\) |
| \(b(a,a\{0,1\}0,1)\) | \(f_{\omega^{\omega^\omega +1}}(a)\) |
| \(b(a,a\{0,1\}0\{1\}1)\) | \(f_{\omega^{\omega^\omega +\omega}}(a)\) |
| \(b(a,a\{0,1\}0\{0,1\}1)\) | \(f_{\omega^{\omega^\omega 2}}(a)\) |
| \(b(a,a\{1,1\}1)\) | \(f_{\omega^{\omega^{\omega +1}}}(a)\) |
| \(b(a,a\{n,1\}1)\) | \(f_{\omega^{\omega^{\omega +n}}}(a)\) |
| \(b(a,a\{0,2\}1)\) | \(f_{\omega^{\omega^{\omega 2}}}(a)\) |
| \(b(a,a\{b,c,d,e,....\}1)\) | \(f_{\omega^{\omega^{.... +\omega^3 e + \omega^2 d + \omega c +b}}}(a)\) |
| \(b(a,a\{0\{1\}1\}1)\) | \(f_{\omega^{\omega^{\omega^\omega}}}(a)\) |
| \(b(a,a\{0\{n\}1\}1)\) | \(f_{\omega^{\omega^{\omega^{\omega^n}}}}(a)\) |
| \(b(a,a\{0\{b,c,d,e,...\}1\}1)\) | \(f_{\omega^{\omega^{\omega^{\omega^{.....+\omega^3 e + \omega^2 d + \omega c +b}}}}}(a)\) |
| \(b(a,a\{0\{0\{1\}1\}1\}1)\) | \(f_{\omega^{\omega^{\omega^{\omega^{\omega^\omega}}}}}(a)\) |
| \(b(a,a\{0\{0\{b,c,d,e,...\}1\}1\}1)\) | \(f_{\omega^{\omega^{\omega^{\omega^{\omega^{.....+\omega^3 e + \omega^2 d + \omega c +b}}}}}}(a)\) |
| \(b(a,a\{0\{0\{0\{1\}1\}1\}1\}1)\) | \(f_{\omega^{\omega^{\omega^{\omega^{\omega^{\omega^\omega}}}}}}(a)\) |
| \(b(a,a\{0\{0\{0\{0,1\}1\}1\}1\}1)\) | \(f_{\omega^{\omega^{\omega^{\omega^{\omega^{\omega^{\omega^\omega}}}}}}}(a)\) |
| \(b(a,a\{0\{0\{0\{..\{1\}..\}1\}1\}1\}1)\) w/ \(n\) ones from the center out. | \(f_{\omega\uparrow\uparrow (n*2)}(a)\) |
Limit of hd-ABHAN is \(\varepsilon_0\).
Nested (only linear) - up to \(\varphi(\omega,0)\)[]
Up to \(\varepsilon_1\)[]
| \(b(a,a\{0\backslash 1\}1)\) | \(f_{\varepsilon_0}(a)\) |
|---|---|
| \(b(a,a,1\{0\backslash 1\}1)\) | \(f_{\varepsilon_0 +1}(a)\) |
| \(b(a,a,0,1\{0\backslash 1\}1)\) | \(f_{\varepsilon_0 +\omega}(a)\) |
| \(b(a,a\{1\}0\{0\backslash 1\}1)\) | \(f_{\varepsilon_0 +\omega^\omega}(a)\) |
| \(b(a,a\{0,1\}0\{0\backslash 1\}1)\) | \(f_{\varepsilon_0 +\omega^{\omega^\omega}}(a)\) |
| \(b(a,a\{0\{1\}1\}0\{0\backslash 1\}1)\) | \(f_{\varepsilon_0 +\omega^{\omega^{\omega^\omega}}}(a)\) |
| \(b(a,a\{0\{0,1\}1\}0\{0\backslash 1\}1)\) | \(f_{\varepsilon_0 +\omega^{\omega^{\omega^{\omega^\omega}}}}(a)\) |
| \(b(a,a\{0\backslash 1\}2)\) | \(f_{\varepsilon_0 2}(a)\) |
| \(b(a,a\{0\backslash 1\}0,1)\) | \(f_{\varepsilon_0 \omega}(a)\) |
| \(b(a,a\{0\backslash 1\}0\{1\}1)\) | \(f_{\varepsilon_0 \omega^\omega}(a)\) |
| \(b(a,a\{0\backslash 1\}0\{0,1\}1)\) | \(f_{\varepsilon_0 \omega^{\omega^\omega}}(a)\) |
| \(b(a,a\{0\backslash 1\}0\{0\backslash 1\}1)\) | \(f_{\varepsilon_0^2}(a)\) |
| \(b(a,a\{0\backslash 1\}0\{0\backslash 1\}0\{0\backslash 1\}1)\) | \(f_{\varepsilon_0^3}(a)\) |
| \(b(a,a\{1\backslash 1\}1)\) | \(f_{\varepsilon_0^\omega}(a)\) |
| \(b(a,a\{1\backslash 1\}n)\) | \(f_{\varepsilon_0^\omega n}(a)\) |
| \(b(a,a\{1\backslash 1\}0\{1\backslash 1\}1)\) | \(f_{\varepsilon_0^{\omega 2}}(a)\) |
| \(b(a,a\{2\backslash 1\}1)\) | \(f_{\varepsilon_0^{\omega^2}}(a)\) |
| \(b(a,a\{n\backslash 1\}1)\) | \(f_{\varepsilon_0^{\omega^n}}(a)\) |
| \(b(a,a\{0,1\backslash 1\}1)\) | \(f_{\varepsilon_0^{\omega^\omega}}(a)\) |
| \(b(a,a\{0\{1\}1\backslash 1\}1)\) | \(f_{\varepsilon_0^{\omega^{\omega^\omega}}}(a)\) |
| \(b(a,a\{0\{0,1\}1\backslash 1\}1)\) | \(f_{\varepsilon_0^{\omega^{\omega^{\omega^\omega}}}}(a)\) |
| \(b(a,a\{0\{0\backslash1\}1\backslash 1\}1)\) | \(f_{\varepsilon_0^{\varepsilon_0}}(a)\) |
| \(b(a,a\{0\{0\{0\backslash 1\}1\backslash 1\}1\backslash 1\}1)\) | \(f_{\varepsilon_0^{\varepsilon_0^{\varepsilon_0}}}(a)\) |
| \(b(a,a\{0\{0\{0\{0\backslash 1\}1\backslash 1\}1\backslash 1\}1\backslash 1\}1)\) | \(f_{\varepsilon_0^{\varepsilon_0^{\varepsilon_0^{\varepsilon_0}}}}(a)\) |
| \(b(a,a\{0\{0\{.....\}1\backslash 1\}1\backslash 1\}1)\) \W \(n\) \(\backslash\)'s | \(f_{\varepsilon_0^{\varepsilon_0^{.....}}}(a)\) \W \(n\) \(\varepsilon_0\)'s |
Up to \(\zeta_0\)[]
| \(b(a,a\{0\backslash 2\}1)\) | \(f_{\varepsilon_1}(a)\) |
|---|---|
| \(b(a,a,n\{0\backslash 2\}1)\) | \(f_{\varepsilon_1 + n}(a)\) |
| \(b(a,a,0,1\{0\backslash 2\}1)\) | \(f_{\varepsilon_1 + \omega}(a)\) |
| \(b(a,a\{1\}0\{0\backslash 2\}1)\) | \(f_{\varepsilon_1 + \omega^\omega}(a)\) |
| \(b(a,a\{0,1\}0\{0\backslash 2\}1)\) | \(f_{\varepsilon_1 + \omega^{\omega^\omega}}(a)\) |
| \(b(a,a\{0\{1\}1\}0\{0\backslash 2\}1)\) | \(f_{\varepsilon_1 + \omega^{\omega^{\omega^\omega}}}(a)\) |
| \(b(a,a\{0\backslash 1\}0\{0\backslash 2\}1)\) | \(f_{\varepsilon_1 + \varepsilon_0}(a)\) |
| \(b(a,a\{0\backslash 2\}2)\) | \(f_{\varepsilon_1 2}(a)\) |
| \(b(a,a\{0 \backslash 2\}n)\) | \(f_{\varepsilon_1 n}(a)\) |
| \(b(a,a\{0\backslash 2\}0,1)\) | \(f_{\varepsilon_1 \omega}(a)\) |
| \(b(a,a\{0\backslash 1\}0\{1\}1)\) | \(f_{\varepsilon_1 \omega^\omega}(a)\) |
| \(b(a,a\{0\backslash 2\}0\{0\backslash 1\}1)\) | \(f_{\varepsilon_1 \varepsilon_0}(a)\) |
| \(b(a,a\{0\backslash 2\}0\{0\backslash 2\}1)\) | \(f_{\varepsilon^{2}_1}(a)\) |
| \(b(a,n\{1\backslash 2\}1)\) | \(f_{\varepsilon^{n}_1}(a)\) |
| \(b(a,a\{1\backslash 2\}1)\) | \(f_{\varepsilon^{\omega}_1}(a)\) |
| \(b(a,a\{0\backslash 2\}0\{1\backslash 2\}1)\) | \(f_{\varepsilon^{\omega}_1 + \varepsilon_1}(a)\) |
| \(b(a,a\{1\backslash 2\}2)\) | \(f_{\varepsilon^{\omega}_1 2}(a)\) |
| \(b(a,a\{1\backslash 2\}0\{0\backslash 2\}1)\) | \(f_{\varepsilon^{\omega + 1}_1}(a)\) |
| \(b(a,a\{1\backslash 2\}0\{1\backslash 2\}1)\) | \(f_{\varepsilon^{\omega 2}}(a)\) |
| \(b(a,n\{2\backslash 2\}1)\) | \(f_{\varepsilon^{\omega n}_1}\) |
From now on,we are going to focus mainly on the seperators.
| \(\{2\backslash 2\}\) | \(f_{\varepsilon^{\omega^2}_1}\) |
|---|---|
| \(\{n\backslash 2\}\) | \(f_{\varepsilon^{\omega^n}_1}\) |
| \(\{0,1\backslash 2\}\) | \(f_{\varepsilon^{\omega^\omega}_1}\) |
| \(\{0\{1\}1\backslash 2\}\) | \(f_{\varepsilon^{\omega^{\omega^\omega}}_1}\) |
| \(\{0\{0,1\}1\backslash 2\}\) | \(f_{\varepsilon^{\omega^{\omega^{\omega^\omega}}}_1}\) |
| \(\{0\{0\backslash 1\}1\backslash 2\}\) | \(f_{\varepsilon^{\varepsilon_0}_1}\) |
| \(\{0\{0\{0\backslash 1\}1\backslash 1\}1\backslash 2\}\) | \(f_{\varepsilon^{\varepsilon^{\varepsilon_0}_0}_1}\) |
| \(\{0\{0\backslash 2\}1\backslash 2\}\) | \(f_{\varepsilon^{\varepsilon_1}_1}\) |
| \(\{0\backslash 3\}\) | \(f_{\varepsilon_2}\) |
| \(\{0\backslash n\}\) | \(f_{\varepsilon_{n-1}}\) |
| \(\{0\backslash 0,1\}\) | \(f_{\varepsilon_{\omega}}\) |
| \(\{0\backslash 0\{0\backslash 1\}1\}\) | \(f_{\varepsilon_{\varepsilon_0}}\) |
| \(\{0\backslash 0\{0\backslash 0\{0\backslash 1\}1\}1\}\) | \(f_{\varepsilon_{\varepsilon_{\varepsilon_0}}}\) |
| \(\{0\backslash 0\{0\backslash 0\{.....\}1\}1\}\) | \(f_{\varepsilon_{\varepsilon_{.....}}}\) |
Up to \(\eta_0\)[]
| \(\{0\backslash 0\backslash 1\}\) | \(f_{\zeta_0}\) |
|---|---|
| \(\{1\backslash 0\backslash 1\}\) | \(f_{\zeta^{\omega}_0}\) |
| \(\{2\backslash 0\backslash 1\}\) | \(f_{\zeta^{\omega^2}_0}\) |
| \(\{n\backslash 0\backslash 1\}\) | \(f_{\zeta^{\omega^n}_0}\) |
| \(\{0,1\backslash 0\backslash 1\}\) | \(f_{\zeta^{\omega^\omega}_0}\) |
| \(\{0\{1\}1\backslash 0\backslash 1\}\) | \(f_{\zeta^{\omega^{\omega^\omega}}_0}\) |
| \(\{0\{0,1\}1\backslash 0\backslash 1\}\) | \(f_{\zeta^{\omega^{\omega^{\omega^\omega}}}_0}\) |
| \(\{0\{0\{....\}1\}1\backslash 0\backslash 1\}\) | \(f_{\zeta^{\omega^{\omega^{\omega^{\omega^{....}}}}}_0}\) |
| \(\{0\{0\backslash 1\}1\backslash 0\backslash 1\}\) | \(f_{\zeta^{\epsilon_0}_0}\) |
| \(\{A\backslash 0\backslash 1\}\) for \(A\) is an array in the seperator | \(f_{\zeta^{\omega^{\lambda}}_0}\) for \(\lambda\) is the ordinal corresponding to \(A\) |
| \(\{0\{0\backslash 0\backslash 1\}1\backslash 0\backslash 1\}\) | \(f_{\zeta^{\zeta_0}_0}\) |
| \(\{0\{0\{0\backslash 0\backslash 1\}1\backslash 0\backslash 1\}1\backslash 0\backslash 1\}\) | \(f_{\zeta^{\zeta^{\zeta_0}_0}_0}\) |
| \(\{0\backslash 2\backslash 1\}\) | \(f_{\varepsilon_{\zeta_0 +1}}\) |
| \(\{A\backslash 2\backslash 1\}\) | \(f_{\varepsilon^{\omega^\lambda}_{\zeta_0 +1}}\) |
| \(\{0\backslash n+1\backslash 1\}\) | \(f_{\varepsilon_{\zeta_0 +n}}\) |
| \(\{0\backslash 0,1\backslash 1\}\) | \(f_{\varepsilon_{\zeta_0 +\omega}}\) |
| \(\{0\backslash 0\{1\}1\backslash 1\}\) | \(f_{\varepsilon_{\zeta_0 +\omega^\omega}}\) |
| \(\{0\backslash 0\{0\backslash 1\}1\backslash 1\}\) | \(f_{\varepsilon_{\zeta_0 +\varepsilon_0}}\) |
| \(\{0\backslash 0\{0\backslash 0\backslash 1\}1\backslash 1\}\) | \(f_{\varepsilon_{\zeta_0 2}}\) |
| \(\{0\backslash 0\{0\backslash 0\backslash 1\}0,1\backslash 1\}\) | \(f_{\varepsilon_{\zeta_0 \omega}}\) |
| \(\{0\backslash 0\{1\backslash 0\backslash 1\}1\backslash 1\}\) | \(f_{\varepsilon_{\zeta^{\omega}_0}}\) |
| \(\{0\backslash 0\{0\backslash 1\backslash 1\}1\backslash 1\}\) | \(f_{\varepsilon_{\varepsilon_{\zeta_0 +1}}}\) |
| \(\{0\backslash 0\{0\backslash 0\{0\backslash 1\backslash 1\}1\backslash 1\}1\backslash 1\}\) | \(f_{\varepsilon_{\varepsilon_{\varepsilon_{\zeta_0}}}}\) |
| \(\{0\backslash 0\backslash 2\}\) | \(f_{\zeta_1}\) |
| \(\{0\backslash 0\backslash n+1\}\) | \(f_{\zeta_n}\) |
| \(\{0\backslash 0\backslash 0,1\}\) | \(f_{\zeta_\omega}\) |
| \(\{0\backslash 0\backslash 0\{0\backslash 1\}1\}\) | \(f_{\zeta_{\varepsilon_0}}\) |
| \(\{0\backslash 0\backslash A\}\) | \(f_{\zeta_\lambda}\) |
| \(\{0\backslash 0\backslash 0\{0\backslash 0\backslash 1\}1\}\) | \(f_{\zeta_{\zeta_0}}\) |
| \(\{0\backslash 0\backslash 0\{0\backslash 0\backslash 0\{0\backslash 0\backslash 1\}1\}1\}\) | \(f_{\zeta_{\zeta_{\zeta_0}}}\) |
For the sake of time and simplicity,I will,from now on,use \(\alpha\) as a shorthand for \(f_{\alpha}\).
Up to \(\varphi(\omega,0)\)[]
| \(\{0\backslash 0\backslash 0\backslash 1\}\) | \(\eta_0\) |
|---|---|
| \(\{1\backslash 0\backslash 0\backslash 1\}\) | \(\eta^{\omega}_0\) |
| \(\{A\backslash 0\backslash 0\backslash 1\}\) | \(\eta^{\omega^\lambda}_0\) |
| \(\{0\{0\backslash 0\backslash 0\backslash 1\}1\backslash 0\backslash 0\backslash 1\}\) | \(\eta^{\eta_0}_0\) |
| \(\{0\{0\{0\backslash 0\backslash 0\backslash 1\}1\backslash 0\backslash 0\backslash 1\}1\backslash 0\backslash 0\backslash 1\}\) | \(\eta^{\eta^{\eta_0}_0}_0\) |
| \(\{0\backslash 1\backslash 0\backslash 1\}\) | \(\varepsilon_{\eta_0 +1}\) |
| \(\{0\backslash 0,1\backslash 0\backslash 1\}\) | \(\varepsilon_{\eta_0 +\omega}\) |
| \(\{0\backslash 0\{0\backslash 0\backslash 0\backslash 1\}1\backslash 0\backslash 1\}\) | \(\varepsilon_{\eta_0 2}\) |
| \(\{0\backslash A\backslash 0\backslash 1\}\) | \(\varepsilon_{\eta_0 +\lambda}\) |
| \(\{0\backslash 0\{0\backslash 1\backslash 0\backslash 1\}1\backslash 0\backslash 1\}\) | \(\varepsilon_{\varepsilon_{\eta_0 +1}}\) |
| \(\{0\backslash 0\backslash 1\backslash 1\}\) | \(\zeta_{\eta_0 +1}\) |
| \(\{0\backslash 0\backslash 0,1\backslash 1\}\) | \(\zeta_{\eta_0 +\omega}\) |
| \(\{0\backslash 0\backslash 0\{0\backslash 1\}1\backslash 1\}\) | \(\zeta_{\eta_0 +\varepsilon_0}\) |
| \(\{0\backslash 0\backslash 0\{0\backslash 0\backslash 0\backslash 1\}1\backslash 1\}\) | \(\zeta_{\eta_0 2}\) |
| \(\{0\backslash 0\backslash A\backslash 1\}\) | \(\zeta_{\eta_0 + \lambda}\) |
| \(\{0\backslash 0\backslash 0\{0\backslash 0\backslash 0\backslash 1\}1\backslash 1\}\) | \(\zeta_{\zeta_{\eta_0 +1}}\) |
| \(\{0\backslash 0\backslash 0\backslash 2\}\) | \(\eta_1\) |
| \(\{0\backslash 0\backslash 0\backslash n+1\}\) | \(\eta_n\) |
| \(\{0\backslash 0\backslash 0\backslash 0,1\}\) | \(\eta_\omega\) |
| \(\{0\backslash 0\backslash 0\backslash 0\{1\}1\}\) | \(\eta_{\omega^\omega}\) |
| \(\{0\backslash 0\backslash 0\backslash 0\{0\backslash 1\}1\}\) | \(\eta_{\varepsilon_0}\) |
| \(\{0\backslash 0\backslash 0\backslash A\}\) | \(\eta_\lambda\) |
| \(\{0\backslash 0\backslash 0\backslash 0\{0\backslash 0\backslash 0\backslash 1\}1\}\) | \(\eta_{\eta_0}\) |
| \(\{0\backslash 0\backslash 0\backslash 0\{0\backslash 0\backslash 0\backslash 0\{......\}1\}1\}\) | \(\eta_{\eta_{......}}\) |
| \(\{0\backslash 0\backslash 0\backslash 0\backslash 1\}\) | \(\varphi(4,0)\) |
| \(\{0\backslash 1\backslash 0\backslash 0\backslash 1\}\) | \(\varepsilon_{\varphi(4,0)+1}\) |
| \(\{0\backslash 0\backslash 1\backslash 0\backslash 1\}\) | \(\varphi(2,\varphi(4,0)+1)\) |
| \(\{0\backslash 0\backslash 0\backslash 1\backslash 1\}\) | \(\varphi(3,\varphi(4,0)+1)\) |
| \(\{0\backslash 0\backslash 0\backslash 0\backslash 2\}\) | \(\varphi(4,1)\) |
| \(\{0\backslash 0\backslash 0\backslash 0\backslash 0,1\}\) | \(\varphi(4,\omega)\) |
| \(\{0\backslash 0\backslash 0\backslash 0\backslash A\}\) | \(\varphi(4,\lambda)\) |
| \(\{0\backslash 0\backslash 0\backslash 0\backslash 0\{0\backslash 0\backslash 0\backslash 0\backslash 1\}1\}\) | \(\varphi(4,\varphi(4,0))\) |
| \(\{0\backslash 0\backslash 0\backslash 0\backslash 0\backslash 1\}\) | \(\varphi(5,0)\) |
| \(\{a\backslash b\backslash c.....i\backslash j\}\) W/ \(n\) numbers | \(\varphi(1,\varphi(2,\varphi(...... \\ \varphi(n-2,\varphi(n-1,j-1)+i).....)+c)+b)^a\) |
| \(\{0\backslash 0\backslash 0.......0\backslash 1\}\) W/ \(n\) zeros | \(\varphi(n,0)\) |
Limits[]
The ordinal limits for ABHAN so far go as follows:
- Linear arrays (l-ABHAN) - \(\omega^\omega\)
- Multidimentional arrays (md-ABHAN) - \(\omega^{\omega^\omega}\)
- Hyperdimentional arrays (hd-ABHAN) - \(\varepsilon_0\)
Nested arrays are not yet completely analysed,though it should reach as far as \(\psi(\varepsilon_{\Omega +1})\),the Bachmann-Howard Ordinal!