Correct it when needed.
Hyperfactorial array (without the n!) | FGH ordinal |
---|---|
0 | 2 |
1 | 3 |
2 | 4 |
k | k+2 |
[1] = n | \(\omega\) |
[2] | \(\omega + 1\) |
[3] | \(\omega + 2\) |
[k] | \(\omega + k-1\) |
[1,2] = [n] | \(\omega \times 2\) |
[2,2] | \(\omega \times 2 + 1\) |
[k,2] | \(\omega \times 2 + k-1\) |
[1,3] = [n,2] | \(\omega \times 3\) |
[k,3] | \(\omega \times 3 + k-1\) |
[1,4] = [n,3] | \(\omega \times 4\) |
[1,k] = [n,k-1] | \(\omega \times k\) |
Arbitrary section break[]
Hyperfactorial array (without the n!) | FGH ordinal |
---|---|
[1,1,2] = [1,[1,1,1],1] = [1,n] | \(\omega^2\) |
[2,1,2] | \(\omega^2 + 1\) |
[[1],1,2] | \(\omega^2 + \omega\) |
[[2],1,2] | \(\omega^2 + \omega+1\) |
[[1,2],1,2] | \(\omega^2 + \omega \times 2\) |
[[1,3],1,2] | \(\omega^2 + \omega \times 3\) |
[1,2,2] = [[1,1,2],1,2] | \((\omega^2) \times 2\) |
[2,2,2] | \((\omega^2) \times 2 + 1\) |
[[1],2,2] | \((\omega^2) \times 2 + \omega\) |
[[1,2],2,2] | \((\omega^2) \times 2 + \omega \times 2\) |
[1,3,2] = [[1,1,2],2,2] | \((\omega^2) \times 3\) |
[1,k,2] | \((\omega^2) \times k\) |
[1,1,3] = [1,[1,1,1],2] = [1,n,2] | \(\omega^3\) |
[2,1,3] | \(\omega^3 + 1\) |
[[1],1,3] = [n,1,3] | \(\omega^3 + \omega\) |
[[1,2],1,3] = [[n],1,3] | \(\omega^3 + \omega \times 2\) |
[[1,3],1,3] = [[n,2],1,3] | \(\omega^3 + \omega \times 3\) |
[[1,1,2],1,3] | \(\omega^3 + \omega^2\) |
[[1,2,2],1,3] | \(\omega^3 + (\omega^2) \times 2\) |
[1,2,3] = [[1,1,3],1,2] | \((\omega^3) \times 2\) |
[1,3,3] = [[1,1,3],2,2] | \((\omega^3) \times 3\) |
[1,1,4] = [1,[1,1,1],3] = [1,n,3] | \(\omega^4\) |
[1,1,5] = [1,[1,1,1],4] = [1,n,4] | \(\omega^5\) |
[1,1,k] | \(\omega^k\) |
From this part, it becomes a bit erratic[]
Hyperfactorial array (without the n!) | FGH ordinal |
---|---|
[1,1,1,2] = [1,1,[1,1,1,1],1] = [1,1,n] | \(\omega^\omega\) |
[2,1,1,2] | \(\omega^\omega + 1\) |
[[1],1,1,2] = [n,1,1,2] | \(\omega^\omega + \omega\) |
[[1,2],1,1,2] = [[n],1,1,2] | \(\omega^\omega + \omega \times 2\) |
[[1,1,2],1,1,2] = [[1,n],1,1,2] | \(\omega^\omega + \omega^2\) |
[1,2,1,2] = [[1,1,1,2],1,1,2] = [[1,1,n],1,1,2] | \((\omega^\omega) \times 2\) |
[1,3,1,2] = [[1,1,1,2],2,1,2] = [[1,1,n],2,1,2] | \((\omega^\omega) \times 3\) |
[1,[1],1,2] = [1,n,1,2] | \(\omega^{\omega+1}\) |
[[1],[1],1,2] = [n,[1],1,2] | \(\omega^{\omega+1} + \omega\) |
[[1,2],[1],1,2] = [[n],[1],1,2] | \(\omega^{\omega+1} + \omega \times 2\) |
[[1,1,2],[1],1,2] = [[1,n],[1],1,2] | \(\omega^{\omega+1} + \omega^2\) |
[[1,1,1,2],[1],1,2] = [[1,1,n],[1],1,2] | \(\omega^{\omega+1} + \omega^\omega\) |
[1,[2],1,2] = [[1,[1],1,2],[1],1,2] = [[1,n,1,2],[1],1,2] | \((\omega^{\omega+1}) \times 2\) |
[1,[3],1,2] = [[1,[1],1,2],[2],1,2] = [[1,n,1,2],[1],1,2] | \((\omega^{\omega+1}) \times 3\) |
[1,[1,2],1,2] = [1,[n],1,2] | \(\omega^{\omega+2}\) |
[1,[2,2],1,2] = [[1,[1,2],1,2],[1,2],1,2] | \((\omega^{\omega+2}) \times 2\) |
[1,[1,3],1,2] = [1,[n,2],1,2] | \(\omega^{\omega+3}\) |
[1,[1,1,2],1,2] = [1,[1,n],1,2] | \(\omega^{\omega \times 2}\) |
[1,[2,1,2],1,2] = [[1,[1,1,2],1,2],[1,1,2],1,2] | \((\omega^{\omega \times 2}) \times 2\) |
[1,[[1],1,2],1,2] = [1,[n,1,2],1,2] | \(\omega^{\omega \times 2 + 1}\) |
[1,[[2],1,2],1,2] = [[1,[[1],1,2],1,2],[[1],1,2],1,2] | \((\omega^{\omega \times 2 + 1}) \times 2\) |
[1,[[1,2],1,2],1,2] = [1,[[n],1,2],1,2] | \(\omega^{\omega \times 2 + 2}\) |
[1,[[1,3],1,2],1,2] = [1,[[n,2],1,2],1,2] | \(\omega^{\omega \times 2 + 3}\) |
[1,[1,2,2],1,2] = [1,[[1,1,2],1,2],1,2] | \(\omega^{\omega \times 3}\) |
[1,[[1],2,2],1,2] = [1,[n,2,2],1,2] | \(\omega^{\omega \times 3 + 1}\) |
[1,[[1,2],2,2],1,2] = [1,[[n],2,2],1,2] | \(\omega^{\omega \times 3 + 2}\) |
[1,[1,3,2],1,2] = [1,[[1,1,2],2,2],1,2] | \(\omega^{\omega \times 4}\) |
[1,[1,1,3],1,2] = [1,[1,n,2],1,2] | \(\omega^{\omega^2}\) |
[1,[[1],1,3],1,2] = [1,[n,1,3],1,2] | \(\omega^{\omega^2+1}\) |
[1,[[1,2],1,3],1,2] = [1,[[n],1,3],1,2] | \(\omega^{\omega^2+2}\) |
[1,[[1,1,2],1,3],1,2] = [1,[[1,n],1,3],1,2] | \(\omega^{\omega^2+\omega}\) |
[1,[[[1],1,2],1,3],1,2] = [1,[[n,1,2],1,3],1,2] | \(\omega^{\omega^2+\omega+1}\) |
[1,[[[1,2],1,2],1,3],1,2] = [1,[[[n],1,2],1,3],1,2] | \(\omega^{\omega^2+\omega+2}\) |
[1,[[1,2,2],1,3],1,2] = [1,[[[1,1,2],1,2],1,3],1,2] | \(\omega^{\omega^2+\omega \times 2}\) |
[1,[[[1],2,2],1,3],1,2] | \(\omega^{\omega^2+\omega \times 2 + 1}\) |
[1,[[1,3,2],1,3],1,2] = [1,[[[1,1,2],2,2],1,3],1,2] | \(\omega^{\omega^2+\omega \times 3}\) |
[1,[1,2,3],1,2] = [1,[[1,1,3],1,3],1,2] | \(\omega^{(\omega^2) \times 2}\) |
[1,[[1],2,3],1,2] = [1,[n,2,3],1,2] | \(\omega^{(\omega^2) \times 2 + 1}\) |
[1,[[1,1,2],2,3],1,2] | \(\omega^{(\omega^2) \times 2 + \omega}\) |
[1,[[1,2,2],2,3],1,2] | \(\omega^{(\omega^2) \times 2 + \omega \times 2}\) |
[1,[1,3,3],1,2] = [1,[[1,1,3],2,3],1,2] | \(\omega^{(\omega^2) \times 3}\) |
[1,[1,1,4],1,2] = [1,[1,n,3],1,2] | \(\omega^{\omega^3}\) |
[1,[[1],1,4],1,2] | \(\omega^{\omega^3 + 1}\) |
[1,[[1,1,2],1,4],1,2] | \(\omega^{\omega^3 + \omega}\) |
[1,[[1,1,3],1,4],1,2] | \(\omega^{\omega^3 + \omega^2}\) |
[1,[1,2,4],1,2] = [1,[[1,1,4],1,4],1,2] | \(\omega^{(\omega^3) \times 2}\) |
[1,[1,1,5],1,2] | \(\omega^{\omega^4}\) |
[1,[1,1,k],1,2] | \(\omega^{\omega^{k-1}}\) |
[1,1,2,2] = [1,[1,1,1,2],1,2] = [1,[1,1,n],1,2] | \(\omega^{\omega^\omega}\) |
[1,2,2,2] = [[1,1,2,2],1,2,2] | \((\omega^{\omega^\omega}) \times 2\) |
[1,[1],2,2] = [1,n,2,2] | \(\omega^{\omega^\omega + 1}\) |
[1,[1,2],2,2] = [1,[n],2,2] | \(\omega^{\omega^\omega + 2}\) |
[1,[1,1,2],2,2] = [1,[1,n],2,2] | \(\omega^{\omega^\omega + \omega}\) |
[1,[1,1,3],2,2] = [1,[1,n,2],2,2] | \(\omega^{\omega^\omega + \omega^2}\) |
[1,1,3,2] = [1,[1,1,1,2],2,2] = [1,[1,1,n],2,2] | \(\omega^{(\omega^\omega) \times 2}\) |
[1,[1],3,2] = [1,n,3,2] | \(\omega^{(\omega^\omega) \times 2 + 1}\) |
[1,[1,2],3,2] = [1,[n],3,2] | \(\omega^{(\omega^\omega) \times 2 + 2}\) |
[1,[1,1,2],3,2] = [1,[n,2],3,2] | \(\omega^{(\omega^\omega) \times 2 + \omega}\) |
[1,1,4,2] = [1,[1,1,1,2],2,2] = [1,[1,1,n],3,2] | \(\omega^{(\omega^\omega) \times 3}\) |
[1,1,k,2] | \(\omega^{(\omega^\omega) \times (k-1)}\) |
More![]
Hyperfactorial array (without the n!) | FGH ordinal |
---|---|
[1,1,1,3] = [1,1,[1,1,1,1],2] = [1,1,n,2] | \(\omega^{\omega^{\omega+1}}\) |
[1,2,1,3] | \((\omega^{\omega^{\omega+1}}) \times 2\) |
[1,[1],1,3] = [1,n,1,3] | \(\omega^{\omega^{\omega+1}+1}\) |
[1,[2],1,3] | \((\omega^{\omega^{\omega+1}+1}) \times 2\) |
[1,[1,2],1,3] | \(\omega^{\omega^{\omega+1}+2}\) |
[1,[1,1,2],1,3] | \(\omega^{\omega^{\omega+1}+\omega}\) |
[1,[1,1,3],1,3] | \(\omega^{\omega^{\omega+1}+\omega^2}\) |
[1,[1,1,1,2],1,3] | \(\omega^{\omega^{\omega+1}+\omega^\omega}\) |
[1,[[1],1,1,2],1,3] = [1,[n,1,1,2],1,3] | \(\omega^{\omega^{\omega+1}+\omega^\omega+1}\) |
[1,[1,2,1,2],1,3] = [1,[[1,1,1,2],1,1,2],1,3] | \(\omega^{\omega^{\omega+1}+(\omega^\omega) \times 2}\) |
[1,[1,[1],1,2],1,3] = [1,[1,n,1,2],1,3] | \(\omega^{(\omega^{\omega+1}) \times 2}\) |
[1,[1,[2],1,2],1,3] = [1,[[1,[1],1,2],[1],1,2],1,3] | \(\omega^{(\omega^{\omega+1}) \times 3}\) |
[1,[1,[1,2],1,2],1,3] = [1,[1,[n],1,2],1,3] | \(\omega^{\omega^{\omega+2}}\) |
[1,[1,[2,2],1,2],1,3] | \(\omega^{(\omega^{\omega+2}) \times 2}\) |
[1,[1,[1,3],1,2],1,3] | \(\omega^{\omega^{\omega+3}}\) |
[1,[1,[1,1,2],1,2],1,3] | \(\omega^{\omega^{\omega \times 2}}\) |
[1,[1,[1,2,2],1,2],1,3] = [1,[1,[[1,1,2],1,2],1,2],1,3] | \(\omega^{\omega^{\omega \times 3}}\) |
[1,[1,[1,1,3],1,2],1,3] | \(\omega^{\omega^{\omega^2}}\) |
[1,[1,[1,1,4],1,2],1,3] | \(\omega^{\omega^{\omega^3}}\) |
[1,[1,1,2,2],1,3] | \(\omega^{\omega^{\omega^\omega}}\) |
[1,[1,1,3,2],1,3] | \(\omega^{\omega^{(\omega^\omega) \times 2}}\) |
[1,1,2,3] = [1,[1,1,1,3],1,3] | \(\omega^{\omega^{\omega^{\omega+1}}}\) |
[1,[1],2,3] = [1,n,2,3] | \(\omega^{\omega^{\omega^{\omega+1}}+1}\) |
[1,[1,1,1,2],2,3] | \(\omega^{\omega^{\omega^{\omega+1}}+\omega^\omega}\) |
[1,[[1],1,1,2],2,3] | \(\omega^{\omega^{\omega^{\omega+1}}+\omega^\omega+1}\) |
[1,[1,2,1,2],2,3] | \(\omega^{\omega^{\omega^{\omega+1}}+(\omega^\omega) \times 2}\) |
[1,[1,[1],1,2],2,3] | \(\omega^{\omega^{\omega^{\omega+1}}+\omega^{\omega+1}}\) |
[1,[1,[1,2],1,2],2,3] | \(\omega^{\omega^{\omega^{\omega+1}}+\omega^{\omega+2}}\) |
[1,[1,[1,1,2],1,2],2,3] | \(\omega^{\omega^{\omega^{\omega+1}}+\omega^{\omega \times 2}}\) |
[1,[1,[1,1,3],1,2],2,3] | \(\omega^{\omega^{\omega^{\omega+1}}+\omega^{\omega^2}}\) |
[1,[1,1,2,2],2,3] | \(\omega^{\omega^{\omega^{\omega+1}}+\omega^{\omega^\omega}}\) |
[1,1,3,3] = [1,[1,1,1,3],2,3] | \(\omega^{(\omega^{\omega^{\omega+1}}) \times 2}\) |
[1,1,4,3] = [1,[1,1,1,3],3,3] | \(\omega^{(\omega^{\omega^{\omega+1}}) \times 3}\) |
[1,1,1,4] = [1,1,n,3] | \(\omega^{\omega^{\omega^{\omega+1}+1}}\) |
[1,[1],1,4] = [1,n,1,4] | \(\omega^{\omega^{\omega^{\omega+1}+1}+1}\) |
[1,[1,2],1,4] | \(\omega^{\omega^{\omega^{\omega+1}+1}+2}\) |
[1,[1,1,2],1,4] | \(\omega^{\omega^{\omega^{\omega+1}+1}+\omega}\) |
[1,[1,1,1,2],1,4] | \(\omega^{\omega^{\omega^{\omega+1}+1}+\omega^\omega}\) |
[1,[1,1,2,2],1,4] | \(\omega^{\omega^{\omega^{\omega+1}+1}+\omega^{\omega^\omega}}\) |
[1,[1,1,3,2],1,4] | \(\omega^{\omega^{\omega^{\omega+1}+1}+\omega^{(\omega^\omega) \times 2}}\) |
[1,[1,1,1,3],1,4] | \(\omega^{\omega^{\omega^{\omega+1}+1}+\omega^{\omega^{\omega+1}}}\) |
[1,[1,[1],1,3],1,4] | \(\omega^{(\omega^{\omega^{\omega+1}+1}) \times 2}\) |
[1,[1,[1,2],1,3],1,4] | \(\omega^{\omega^{\omega^{\omega+1}+2}}\) |
[1,[1,[1,1,1,2],1,3],1,4] | \(\omega^{\omega^{\omega^{\omega+1}+\omega^\omega}}\) |
[1,[1,[1,[1],1,2],1,3],1,4] | \(\omega^{\omega^{(\omega^{\omega+1}+1) \times 2}}\) |
[1,[1,[1,[2],1,2],1,3],1,4] | \(\omega^{\omega^{(\omega^{\omega+1}+1) \times 3}}\) |
[1,[1,[1,[1,2],1,2],1,3],1,4] | \(\omega^{\omega^{\omega^{\omega+2}}}\) |
[1,[1,[1,[1,3],1,2],1,3],1,4] | \(\omega^{\omega^{\omega^{\omega+3}}}\) |
[1,[1,[1,[1,1,2],1,2],1,3],1,4] | \(\omega^{\omega^{\omega^{\omega \times 2}}}\) |
[1,[1,[1,[1,2,2],1,2],1,3],1,4] | \(\omega^{\omega^{\omega^{\omega \times 3}}}\) |
[1,[1,[1,[1,1,3],1,2],1,3],1,4] | \(\omega^{\omega^{\omega^{\omega^2}}}\) |
[1,[1,[1,[1,1,4],1,2],1,3],1,4] | \(\omega^{\omega^{\omega^{\omega^3}}}\) |
[1,[1,[1,1,2,2],1,3],1,4] | \(\omega^{\omega^{\omega^{\omega^\omega}}}\) |
[1,[1,[1,1,3,2],1,3],1,4] | \(\omega^{\omega^{\omega^{(\omega^\omega) \times 2}}}\) |
[1,[1,1,2,3],1,4] = [1,[1,[1,1,1,3],1,3],1,4] | \(\omega^{\omega^{\omega^{\omega^{\omega+1}}}}\) |
[1,[1,1,3,3],1,4] = [1,[1,[1,1,1,3],2,3],1,4] | \(\omega^{\omega^{(\omega^{\omega^{\omega+1}}) \times 2}}\) |
[1,[1,1,4,3],1,4] = [1,[1,[1,1,1,3],3,3],1,4] | \(\omega^{\omega^{(\omega^{\omega^{\omega+1}}) \times 3}}\) |
[1,1,2,4] = [1,[1,1,1,4],1,4] | \(\omega^{\omega^{\omega^{\omega^{\omega+1}+1}}}\) |
[1,[1],2,4] = [1,n,2,4] | \(\omega^{\omega^{\omega^{\omega^{\omega+1}+1}}+1}\) |
[1,[1,1,1,2],2,4] | \(\omega^{\omega^{\omega^{\omega^{\omega+1}+1}}+\omega^\omega}\) |
[1,[1,1,2,2],2,4] | \(\omega^{\omega^{\omega^{\omega^{\omega+1}+1}}+\omega^{\omega^\omega}}\) |
[1,[1,1,1,3],2,4] | \(\omega^{\omega^{\omega^{\omega^{\omega+1}+1}}+\omega^{\omega^{\omega+1}}}\) |
[1,[1,1,2,3],2,4] | \(\omega^{\omega^{\omega^{\omega^{\omega+1}+1}}+\omega^{\omega^{\omega^{\omega+1}}}}\) |
[1,1,3,4] = [1,[1,1,1,4],2,4] | \(\omega^{(\omega^{\omega^{\omega^{\omega+1}+1}}) \times 2}\) |
[1,1,4,4] = [1,[1,1,1,4],3,4] | \(\omega^{(\omega^{\omega^{\omega^{\omega+1}+1}}) \times 3}\) |
[1,1,1,5] = [1,1,n,4] | \(\omega^{\omega^{\omega^{\omega^{\omega+1}+1}+1}}\) |
[1,1,1,6] = [1,1,n,5] | \(\omega^{\omega^{\omega^{\omega^{\omega^{\omega+1}+1}+1}+1}}\) |
Beyond epsilon 0...[]
Hyperfactorial array (without the n!) | FGH ordinal |
---|---|
[1,1,1,1,2] = [1,1,1,[1,1,1,1,1],1] = [1,1,1,n] | \(\varepsilon_0\) |
[1,2,1,1,2] = [[1,1,1,1,2],1,1,1,2] | \(\varepsilon_0 \times 2\) |
[1,[1],1,1,2] = [1,n,1,1,2] | \(\omega^{\varepsilon_0+1}\) |
[1,[2],1,1,2] = [[1,[1],1,1,2],[1],1,1,2] | \((\omega^{\varepsilon_0+1}) \times 2\) |
[1,[1,2],1,1,2] = [1,[n],1,1,2] | \(\omega^{\varepsilon_0+2}\) |
[1,[1,3],1,1,2] = [1,[n,2],1,1,2] | \(\omega^{\varepsilon_0+3}\) |
[1,[1,1,2],1,1,2] = [1,[1,n],1,1,2] | \(\omega^{\varepsilon_0+\omega}\) |
[1,[2,1,2],1,1,2] | \((\omega^{\varepsilon_0+\omega}) \times 2\) |
[1,[[1],1,2],1,1,2] = [1,[n,1,2],1,1,2] | \(\omega^{\varepsilon_0+\omega+1}\) |
[1,[[1,2],1,2],1,1,2] = [1,[[n],1,2],1,1,2] | \(\omega^{\varepsilon_0+\omega+2}\) |
[1,[1,2,2],1,1,2] = [1,[[1,1,2],1,2],1,1,2] | \(\omega^{\varepsilon_0+\omega \times 2}\) |
[1,[1,3,2],1,1,2] = [1,[[1,1,2],2,2],1,1,2] | \(\omega^{\varepsilon_0+\omega \times 3}\) |
[1,[1,1,3],1,1,2] = [1,[1,n,2],1,1,2] | \(\omega^{\varepsilon_0+\omega^2}\) |
[1,[1,2,3],1,1,2] = [1,[[1,1,3],1,3],1,1,2] | \(\omega^{\varepsilon_0+(\omega^2) \times 2}\) |
[1,[1,1,4],1,1,2] = [1,[1,n,3],1,1,2] | \(\omega^{\varepsilon_0+\omega^3}\) |
[1,[1,1,1,2],1,1,2] | \(\omega^{\varepsilon_0+\omega^\omega}\) |
[1,[1,2,1,2],1,1,2] = [1,[[1,1,1,2],1,1,2],1,1,2] | \(\omega^{\varepsilon_0+(\omega^\omega) \times 2}\) |
[1,[1,[1],1,2],1,1,2] = [1,[1,n,1,2],1,1,2] | \(\omega^{\varepsilon_0+\omega^{\omega+1}}\) |
[1,[1,[1,2],1,2],1,1,2] | \(\omega^{\varepsilon_0+\omega^{\omega+2}}\) |
[1,[1,[1,1,2],1,2],1,1,2] | \(\omega^{\varepsilon_0+\omega^{\omega \times 2}}\) |
[1,[1,[1,2,2],1,2],1,1,2] | \(\omega^{\varepsilon_0+\omega^{\omega \times 3}}\) |
[1,[1,[1,1,3],1,2],1,1,2] | \(\omega^{\varepsilon_0+\omega^{\omega^2}}\) |
[1,[1,[1,1,4],1,2],1,1,2] | \(\omega^{\varepsilon_0+\omega^{\omega^3}}\) |
[1,[1,1,2,2],1,1,2] | \(\omega^{\varepsilon_0+\omega^{\omega^\omega}}\) |
[1,[1,1,3,2],1,1,2] | \(\omega^{\varepsilon_0+\omega^{(\omega^\omega) \times 2}}\) |
[1,[1,1,1,3],1,1,2] | \(\omega^{\varepsilon_0+\omega^{\omega^{\omega+1}}}\) |
[1,[1,1,1,4],1,1,2] | \(\omega^{\varepsilon_0+\omega^{\omega^{\omega^{\omega+1}+1}}}\) |
[1,1,2,1,2] = [1,[1,1,1,1,2],1,1,2] | \(\omega^{\varepsilon_0 \times 2}\) |
[1,[1],2,1,2] = [1,n,2,1,2] | \(\omega^{\varepsilon_0 \times 2+1}\) |
[1,[1,1,1,2],2,1,2] | \(\omega^{\varepsilon_0 \times 2+\omega^\omega}\) |
[1,1,3,1,2] = [1,[1,1,1,1,2],2,1,2] | \(\omega^{\varepsilon_0 \times 3}\) |
[1,1,4,1,2] = [1,[1,1,1,1,2],3,1,2] | \(\omega^{\varepsilon_0 \times 4}\) |
[1,1,[1],1,2] = [1,1,n,1,2] | \(\omega^{\omega^{\varepsilon_0+1}}\) |
[1,[1],[1],1,2] = [1,n,[1],1,2] | \(\omega^{\omega^{\varepsilon_0+1}+1}\) |
[1,[1,1,1,1,2],[1],1,2] | \(\omega^{\omega^{\varepsilon_0+1}+\varepsilon_0}\) |
[1,[[1],1,1,1,2],[1],1,2] | \(\omega^{\omega^{\varepsilon_0+1}+\varepsilon_0+1}\) |
[1,[1,2,1,1,2],[1],1,2] = [1,[[1,1,1,1,2],1,1,1,2],[1],1,2] | \(\omega^{\omega^{\varepsilon_0+1}+\varepsilon_0 \times 2}\) |
[1,[1,3,1,1,2],[1],1,2] = [1,[[1,1,1,1,2],2,1,1,2],[1],1,2] | \(\omega^{\omega^{\varepsilon_0+1}+\varepsilon_0 \times 3}\) |
[1,[1,[1],1,1,2],[1],1,2] | \(\omega^{(\omega^{\varepsilon_0+1}) \times 2}\) |
[1,[1,[2],1,1,2],[1],1,2] | \(\omega^{(\omega^{\varepsilon_0+1}) \times 3}\) |
[1,[1,[1,2],1,1,2],[1],1,2] | \(\omega^{\omega^{\varepsilon_0+2}}\) |
[1,[1,[2,2],1,1,2],[1],1,2] | \(\omega^{(\omega^{\varepsilon_0+2}) \times 2}\) |
[1,[1,[1,3],1,1,2],[1],1,2] | \(\omega^{\omega^{\varepsilon_0+3}}\) |
[1,[1,[1,1,2],1,1,2],[1],1,2] | \(\omega^{\omega^{\varepsilon_0+\omega}}\) |
[1,[1,[1,1,1,2],1,1,2],[1],1,2] | \(\omega^{\omega^{\varepsilon_0+\omega^\omega}}\) |
[1,[1,[1,1,1,3],1,1,2],[1],1,2] | \(\omega^{\omega^{\varepsilon_0+\omega^{\omega^{\omega+1}}}}\) |
[1,[1,1,2,1,2],[1],1,2] = [1,[1,[1,1,1,1,2],1,1,2],[1],1,2] | \(\omega^{\omega^{\varepsilon_0 \times 2}}\) |
[1,[1,1,3,1,2],[1],1,2] = [1,[1,[1,1,1,1,2],2,1,2],[1],1,2] | \(\omega^{\omega^{\varepsilon_0 \times 3}}\) |
[1,1,[2],1,2] = [1,[1,1,[1],1,2],[1],1,2] | \(\omega^{\omega^{\omega^{\varepsilon_0+1}}}\) |
[1,1,[3],1,2] = [1,[1,1,[1],1,2],[2],1,2] | \(\omega^{(\omega^{\omega^{\varepsilon_0+1}}) \times 2}\) |
[1,1,[4],1,2] = [1,[1,1,[1],1,2],[3],1,2] | \(\omega^{(\omega^{\omega^{\varepsilon_0+1}}) \times 3}\) |
[1,1,[1,2],1,2] = [1,1,[n],1,2] | \(\omega^{\omega^{\omega^{\varepsilon_0+1}+1}}\) |
[1,[1,1,1,1,2],[1,2],1,2] | \(\omega^{\omega^{\omega^{\varepsilon_0+1}+1}+\varepsilon_0}\) |
[1,[1,1,[1],1,2],[1,2],1,2] | \(\omega^{\omega^{\omega^{\varepsilon_0+1}+1}+\omega^{\omega^{\varepsilon_0+1}}}\) |
[1,[1,[1],[1],1,2],[1,2],1,2] | \(\omega^{(\omega^{\omega^{\varepsilon_0+1}+1}) \times 2}\) |
[1,[1,[2],[1],1,2],[1,2],1,2] | \(\omega^{(\omega^{\omega^{\varepsilon_0+1}+1}) \times 3}\) |
[1,[1,[1,2],[1],1,2],[1,2],1,2] | \(\omega^{\omega^{\omega^{\varepsilon_0+1}+2}}\) |
[1,[1,[1,1,1,1,2],[1],1,2],[1,2],1,2] | \(\omega^{\omega^{\omega^{\varepsilon_0+1}+\varepsilon_0}}\) |
[1,[1,[1,[1],1,1,2],[1],1,2],[1,2],1,2] | \(\omega^{\omega^{(\omega^{\varepsilon_0+1}) \times 2}}\) |
[1,[1,[1,[1,2],1,1,2],[1],1,2],[1,2],1,2] | \(\omega^{\omega^{\omega^{\varepsilon_0+2}}}\) |
[1,[1,[1,1,2,1,2],[1],1,2],[1,2],1,2] = [1,[1,[1,[1,1,1,1,2],1,1,2],[1],1,2],[1,2],1,2] | \(\omega^{\omega^{\omega^{\varepsilon_0 \times 2}}}\) |
[1,[1,[1,1,3,1,2],[1],1,2],[1,2],1,2] | \(\omega^{\omega^{\omega^{\varepsilon_0 \times 3}}}\) |
[1,[1,1,[2],1,2],[1,2],1,2] = [1,[1,[1,1,[1],1,2],[1],1,2],[1,2],1,2] | \(\omega^{\omega^{\omega^{\omega^{\varepsilon_0+1}}}}\) |
[1,[1,1,[3],1,2],[1,2],1,2] | \(\omega^{\omega^{(\omega^{\omega^{\varepsilon_0+1}}) \times 2}}\) |
[1,1,[2,2],1,2] = [1,[1,1,[1,2],1,2],[1,2],1,2] | \(\omega^{\omega^{\omega^{\omega^{\varepsilon_0+1}+1}}}\) |
[1,1,[3,2],1,2] = [1,[1,1,[1,2],1,2],[2,2],1,2] | \(\omega^{(\omega^{\omega^{\omega^{\varepsilon_0+1}+1}}) \times 2}\) |
[1,1,[1,3],1,2] = [1,1,[n,2],1,2] | \(\omega^{\omega^{\omega^{\omega^{\varepsilon_0+1}+1}+1}}\) |
[1,1,[1,1,2],1,2] = [1,1,[1,n],1,2] | \(\varepsilon_1\) |
[1,1,[2,1,2],1,2] = [1,[1,1,[1,1,2],1,2],[1,1,2],1,2] | \(\omega^{\varepsilon_1 \times 2}\) |
[1,1,[3,1,2],1,2] = [1,[1,1,[1,1,2],1,2],[2,1,2],1,2] | \(\omega^{\varepsilon_1 \times 3}\) |
[1,1,[[1],1,2],1,2] = [1,1,[n,1,2],1,2] | \(\omega^{\omega^{\varepsilon_1+1}}\) |
[1,1,[[2],1,2],1,2] = [1,[1,1,[[1],1,2],1,2],[[1],1,2],1,2] | \(\omega^{\omega^{\omega^{\varepsilon_1+1}}}\) |
[1,1,[[3],1,2],1,2] | \(\omega^{(\omega^{\omega^{\varepsilon_1+1}}) \times 2}\) |
[1,1,[[1,2],1,2],1,2] | \(\omega^{\omega^{\omega^{\varepsilon_1+1}+1}}\) |
[1,1,[[1,3],1,2],1,2] | \(\omega^{\omega^{\omega^{\omega^{\varepsilon_1+1}+1}+1}}\) |
[1,1,[1,2,2],1,2] = [1,1,[[1,1,2],1,2],1,2] | \(\varepsilon_2\) |
[1,1,[[1],2,2],1,2] = [1,1,[n,2,2],1,2] | \(\omega^{\omega^{\varepsilon_2+1}}\) |
[1,1,[1,3,2],1,2] = [1,1,[[1,1,2],2,2],1,2] | \(\varepsilon_3\) |
[1,1,[1,1,3],1,2] = [1,1,[1,n,2],1,2] | \(\varepsilon_\omega\) |
[1,1,[[1,1,2],1,3],1,2] | \(\varepsilon_{\omega+1}\) |
[1,1,[[1,2,2],1,3],1,2] | \(\varepsilon_{\omega+2}\) |
[1,1,[1,2,3],1,2] | \(\varepsilon_{\omega \times 2}\) |
[1,1,[1,1,4],1,2] | \(\varepsilon(\omega^2)\) |
[1,1,[1,1,1,2],1,2] | \(\varepsilon(\omega^\omega)\) |
[1,1,[1,2,1,2],1,2] | \(\varepsilon((\omega^\omega) \times 2)\) |
[1,1,[1,[1],1,2],1,2] | \(\varepsilon(\omega^{\omega+1})\) |
[1,1,[1,[1,2],1,2],1,2] | \(\varepsilon(\omega^{\omega+2})\) |
[1,1,[1,1,2,2],1,2] = [1,1,[1,[1,1,1,2],1,2],1,2] | \(\varepsilon(\omega^{\omega^\omega})\) |
[1,1,[1,1,1,3],1,2] | \(\varepsilon(\omega^{\omega^{\omega+1}})\) |
[1,1,[1,1,1,4],1,2] | \(\varepsilon(\omega^{\omega^{\omega^{\omega+1}+1}})\) |
[1,1,1,2,2] = [1,1,[1,1,1,1,2],1,2] | \(\varepsilon(\varepsilon_0)\) |
[1,1,2,2,2] = [1,[1,1,1,2,2],1,2,2] | \(\omega^{(\varepsilon(\varepsilon_0)) \times 2}\) |
[1,1,[1],2,2] | \(\omega^{\omega^{\varepsilon(\varepsilon_0)+1}}\) |
[1,1,[1,1,2],2,2] | \(\varepsilon(\varepsilon_0+1)\) |
[1,1,1,3,2] = [1,1,[1,1,1,1,2],2,2] | \(\varepsilon(\varepsilon_0 \times 2)\) |
[1,1,[1,1,2],3,2] | \(\varepsilon(\varepsilon_0 \times 2+1)\) |
[1,1,1,4,2] = [1,1,[1,1,1,1,2],3,2] | \(\varepsilon(\varepsilon_0 \times 3)\) |
[1,1,1,1,3] = [1,1,1,n,2] | \(\varepsilon(\omega^{\varepsilon_0+1})\) |
[1,1,[1,1,2],1,3] | \(\varepsilon(\omega^{\varepsilon_0+1}+1)\) |
[1,1,[1,1,1,1,2],1,3] | \(\varepsilon(\omega^{\varepsilon_0+1}+\varepsilon_0)\) |
[1,1,[1,[1,2],1,1,2],1,3] | \(\varepsilon(\omega^{\varepsilon_0+2})\) |
[1,1,[1,1,2,1,2],1,3] | \(\varepsilon(\omega^{\varepsilon_0 \times 2})\) |
[1,1,[1,1,[1],1,2],1,3] | \(\varepsilon(\omega^{\omega^{\varepsilon_0+1}})\) |
[1,1,[1,1,[2],1,2],1,3] | \(\varepsilon(\omega^{\omega^{\omega^{\varepsilon_0+1}}})\) |
[1,1,[1,1,[1,2],1,2],1,3] | \(\varepsilon(\omega^{\omega^{\omega^{\varepsilon_0+1}+1}})\) |
[1,1,[1,1,[1,1,2],1,2],1,3] | \(\varepsilon(\varepsilon_1)\) |
[1,1,[1,1,[1,2,2],1,2],1,3] | \(\varepsilon(\varepsilon_2)\) |
[1,1,[1,1,[1,1,3],1,2],1,3] | \(\varepsilon(\varepsilon_\omega)\) |
[1,1,[1,1,[1,1,1,2],1,2],1,3] | \(\varepsilon(\varepsilon(\omega^\omega))\) |
[1,1,[1,1,1,2,2],1,3] | \(\varepsilon(\varepsilon(\varepsilon_0))\) |
[1,1,[1,1,1,3,2],1,3] | \(\varepsilon(\varepsilon(\varepsilon_0 \times 2))\) |
[1,1,1,2,3] = [1,1,[1,1,1,1,3],1,3] | \(\varepsilon(\varepsilon(\omega^{\varepsilon_0+1}))\) |
[1,1,1,3,3] = [1,1,[1,1,1,1,3],2,3] | \(\varepsilon(\varepsilon(\omega^{\varepsilon_0+1}) \times 2)\) |
[1,1,1,1,4] = [1,1,1,n,3] | \(\varepsilon(\omega^{\varepsilon(\omega^{\varepsilon_0+1})+1})\) |
Last one on this blog post[]
Hyperfactorial array (without the n!) | FGH ordinal |
---|---|
[1,1,1,1,1,2] = [1,1,1,1,n] | \(\zeta_0\) |
[1,1,2,1,1,2] | \(\omega^{\zeta_0 \times 2}\) |
[1,1,[1],1,1,2] | \(\omega^{\omega^{\zeta_0+1}}\) |
[1,1,[1,2],1,1,2] | \(\omega^{\omega^{\omega^{\zeta_0+1}+1}}\) |
[1,1,[1,1,2],1,1,2] | \(\varepsilon(\zeta_0+1)\) |
[1,1,[1,2,2],1,1,2] | \(\varepsilon(\zeta_0+2)\) |
[1,1,[1,1,3],1,1,2] | \(\varepsilon(\zeta_0+\omega)\) |
[1,1,[[1,1,2],1,3],1,1,2] | \(\varepsilon(\zeta_0+\omega+1)\) |
[1,1,[1,2,3],1,1,2] | \(\varepsilon(\zeta_0+\omega \times 2)\) |
[1,1,[1,1,4],1,1,2] | \(\varepsilon(\zeta_0+\omega^2)\) |
[1,1,[1,1,1,2],1,1,2] | \(\varepsilon(\zeta_0+\omega^\omega)\) |
[1,1,[1,1,1,1,2],1,1,2] | \(\varepsilon(\zeta_0+\varepsilon_0)\) |
[1,1,1,2,1,2] = [1,1,[1,1,1,1,1,2],1,1,2] | \(\varepsilon(\zeta_0 \times 2)\) |
[1,1,1,3,1,2] | \(\varepsilon(\zeta_0 \times 3)\) |
[1,1,1,[1],1,2] | \(\varepsilon(\omega^{\zeta_0+1})\) |
[1,1,1,[2],1,2] | \(\varepsilon(\varepsilon(\omega^{\zeta_0+1}))\) |
[1,1,1,[3],1,2] | \(\varepsilon(\varepsilon(\omega^{\zeta_0+1}) \times 2)\) |
[1,1,1,[1,2],1,2] | \(\varepsilon(\omega^{\varepsilon(\omega^{\zeta_0+1})+1})\) |
[1,1,1,[1,3],1,2] | \(\varepsilon(\omega^{\varepsilon(\omega^{\varepsilon(\omega^{\zeta_0+1})+1})+1})\) |
[1,1,1,[1,1,2],1,2] | \(\zeta_1\) |
[1,1,1,[1,2,2],1,2] | \(\zeta_2\) |
[1,1,1,[1,1,3],1,2] | \(\zeta_\omega\) |
[1,1,1,[1,1,1,1,2],1,2] | \(\zeta(\varepsilon_0)\) |
[1,1,1,[1,1,1,1,3],1,2] | \(\zeta(\varepsilon(\omega^{\varepsilon_0+1}))\) |
[1,1,1,1,2,2] = [1,1,1,[1,1,1,1,1,2],1,2] | \(\zeta(\zeta_0)\) |
[1,1,1,2,2,2] | \(\varepsilon(\zeta(\zeta_0) \times 2)\) |
[1,1,1,[1],2,2] | \(\varepsilon(\omega^{\zeta(\zeta_0)+1})\) |
[1,1,1,[1,1,2],2,2] | \(\zeta(\zeta_0+1)\) |
[1,1,1,1,3,2] = [1,1,1,[1,1,1,1,1,2],2,2] | \(\zeta(\zeta_0 \times 2)\) |
[1,1,1,1,4,2] = [1,1,1,[1,1,1,1,1,2],3,2] | \(\zeta(\zeta_0 \times 3)\) |
[1,1,1,1,1,3] = [1,1,1,1,n,2] | \(\zeta(\omega^{\zeta_0+1})\) |
[1,1,1,1,2,3] | \(\zeta(\zeta(\omega^{\zeta_0+1}))\) |
[1,1,1,1,3,3] | \(\zeta(\zeta(\omega^{\zeta_0+1}) \times 2)\) |
[1,1,1,1,1,4] | \(\zeta(\omega^{\zeta(\omega^{\zeta_0+1})+1})\) |
[1,1,1,1,1,1,2] = [1,1,1,1,1,n] | \(\phi(3,0)\) |
[1,1,1,1,1,1,1,2] | \(\phi(4,0)\) |
This marks the end of linear hyperfactorial arrays. Somewhat erratic behavior, but still, the limit ordinal is \(\phi(\omega,0)\).