FANDOM


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 \(\phi(\omega,0)\).

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