## FANDOM

10,825 Pages

This is a continuation of the work of Cloudy176 here. the w/ operator means that m...m in the array before it is evaluated as (the value to the right) repeats of m. For example, [1...1,2]w/[1] = [1...1,2]w/n will have n 1's in the first array. If they are chained together, they are solved from right to left. Correct it when needed:

### Up to $$\Gamma_0$$

Hyperfactorial array (without the n!) FGH ordinal
[1]w/[1] $\varphi(\omega,0)$
[1]w/[2] $\varphi(\omega+1,0)$
[1]w/[k] $\varphi(\omega+(k-1),0)$
[1]w/[1,2] $\varphi(\omega\times 2,0)$
[1]w/[1,k] $\varphi(\omega\times k,0)$
[1]w/[1,1,2] $\varphi(\omega^2,0)$
[1]w/[1,k,2] $\varphi(\omega^2\times k,0)$
[1]w/[1,1,3] $\varphi(\omega^3,0)$
[1]w/[1,1,k] $\varphi(\omega^k,0)$
[1]w/[1,1,1,2] $\varphi(\omega^{\omega},0)$
[1]w/[1,1,1,1,2] $\varphi(\varepsilon_0,0)$
[1]w/[1,1,1,1,1,2] $\varphi(\zeta_0,0)$
[1]w/[1]w/[1] $\varphi(\varphi(\omega,0),0)$
[1]w/[1]w/[1,2] $\varphi(\varphi(\omega\times 2,0),0)$
[1]w/[1]w/[1,1,2] $\varphi(\varphi(\omega^2,0),0)$
[1]w/[1]w/[1,1,1,2] $\varphi(\varphi(\omega^{\omega},0),0)$
[1]w/[1]w/[1,1,1,2] $\varphi(\varphi(\varepsilon_0,0),0)$
[1]w/[1]w/[1]w/[1] $\varphi(\varphi(\varphi(\omega,0),0),0)$
[1]w/[1]w/[1]w/[1]w/[1] $\varphi(\varphi(\varphi(\varphi(\omega,0),0),0),0)$
[1]w/[1]w/[1]w/[1]w/[1]w/[1] $\varphi(\varphi(\varphi(\varphi(\varphi(\omega,0),0),0),0),0)$
[1(1)2] $\Gamma_0$

The new definitions have rendered most of the stuff beyond this point incorrect.

### Beyond $$\Gamma_0$$

Hyperfactorial array (without the n!) FGH ordinal
[2(1)2] $\Gamma_0+1$
[k(1)2] $\Gamma_0+(k-1)$
[[1,2](1)2] $\Gamma_0+(\omega\times 2)$
[[1,1,2](1)2] $\Gamma_0+(\omega^2)$
[[1,1,2](1)2] $\Gamma_0+(\omega^2)$
[[1,1,1,2](1)2] $\Gamma_0+(\omega^{\omega})$
[[1,1,1,2](1)2] $\Gamma_0+(\omega^{\omega})$
[[1]w/[1](1)2] $\Gamma_0+(\varphi(\omega,0))$
[1,2(1)2] $\Gamma_0\times 2$
[1,k(1)2] $\Gamma_0\times k$
[1,1,2(1)2] $\Gamma_0^2$
[1,1,1,2(1)2] $\Gamma_0^{\Gamma_0}$
[1(1)2]w/[1] $\varphi(\omega,\Gamma_0)$
[1(1)2]w/[1,2] $\varphi(\omega\times 2,\Gamma_0)$
[1(1)2]w/[1,1,2] $\varphi(\omega^2,\Gamma_0)$
[1(1)2]w/[1,1,1,2] $\varphi(\omega^{\omega},\Gamma_0)$
[1(1)2]w/[1]w/[1] $\varphi(\varphi(\omega,0),\Gamma_0)$
[1(1)2]w/[1]w/[1]w/[1] $\varphi(\varphi(\varphi(\omega,0),0),\Gamma_0)$
[1(1)2]w/[1(1)2] $\varphi(\Gamma_0,\Gamma_0)$
[1(1)2]w/[1(1)2]w/[1] $\varphi(\varphi(\omega,\Gamma_0),\Gamma_0)$
[1(1)2]w/[1(1)2]w/[1(1)2]w/[1] $\varphi(\varphi(\varphi(\omega,\Gamma_0),\Gamma_0),\Gamma_0)$
[1(1)2]w/[1(1)2]w/[1(1)2]w/[1(1)2]w/[1] $\varphi(\varphi(\varphi(\varphi(\omega,\Gamma_0),\Gamma_0),\Gamma_0),\Gamma_0)$
[1(1)3] $\Gamma_1$
[1(1)k] $\Gamma_{k-2}$
[1(1)[1]] $\Gamma_{\omega}$
[1(1)[1,2]] $\Gamma_{\omega\times 2}$
[1(1)[1,1,2]] $\Gamma_{\omega^2}$
[1(1)[1(1)2]] $\Gamma_{\Gamma_0}$
[1(1)[1(1)[1(1)2]]] $\Gamma_{\Gamma_{\Gamma_0}}$
[1(1)1,2] $\varphi(1,1,0)$

Note that w/ will take precedence over entries in higher spaces (anything not in the 1st row in this case).

Thanks to Googleaarex for helping with the following comparisons. I have used the $$\psi$$ function as it was developed after the theta one to be a simpler version of it. The fact that it lags behing one $$\Omega$$ in the power tower really makes no difference in the long run.

### Up to $$\psi(\Omega^{\Omega^{\Omega}})$$, the large Veblen ordinal

Hyperfactorial array (without the n!) FGH ordinal using $$\phi$$ FGH ordinal using $$\psi$$
[1,2(1)1,2] $\varphi(1,1,0)\times 2$ $\psi(\Omega^{\Omega+1})\times 2$
[1,1,2(1)1,2] $\varphi(1,1,0)^2$ $\psi(\Omega^{\Omega+1})^2$
[1,1,1,2(1)1,2] $\varphi(1,1,0)^{\varphi(1,1,0)}$ $\psi(\Omega^{\Omega+1})^{\psi(\Omega^{\Omega+1})}$
[1,1,1,1,2(1)1,2] $\varepsilon_{\varphi(1,1,0)+1}$ $\psi(\Omega^{\Omega+1}+1)$
[1(1)1,2]w/[1] $\varphi(\omega,\varphi(1,1,0)+1)$ $\psi(\Omega^{\omega}\times\psi(\Omega^{\Omega+1}))$
[1(1)1,2]w/[1(1)2] $\varphi(1,0,\varphi(1,1,0)+1)$ $\psi(\Omega^{\Omega}\times\psi(\Omega^{\Omega+1}))$
[1(1)2,2] $\varphi(1,1,1)$ $\psi(\Omega^{\Omega+1}\times 2)$
[1(1)3,2] $\varphi(1,1,2)$ $\psi(\Omega^{\Omega+1}\times 3)$
[1(1)[1],2] $\varphi(1,1,\omega)$ $\psi(\Omega^{\Omega+1}\times\omega)$
[1(1)1,3] $\varphi(1,2,0)$ $\psi(\Omega^{\Omega+2})$
[1(1)1,[1]] $\varphi(1,\omega,0)$ $\psi(\Omega^{\Omega+\omega})$
[1(1)1,1,2] $\varphi(2,0,0)$ $\psi(\Omega^{\Omega\times 2})$
[1(1)1,1,3] $\varphi(3,0,0)$ $\psi(\Omega^{\Omega\times 3})$
[1(1)1,1,[1]] $\varphi(\omega,0,0)$ $\psi(\Omega^{\Omega\times\omega})$
[1(1)1,1,1,2] $\varphi(1,0,0,0)$ $\psi(\Omega^{\Omega^2})$
[1(1)1,1,2,2] $\varphi(1,1,0,0)$ $\psi(\Omega^{\Omega^2+\Omega})$
[1(1)1,1,1,3] $\varphi(2,0,0,0)$ $\psi(\Omega^{\Omega^2\times 2})$
[1(1)1,1,1,1,2] $\varphi(1,0,0,0,0)$ $\psi(\Omega^{\Omega^3})$

[1(1)1]w/k This w/ and all lower ones on this table will affect the second row.

$\varphi(1,0...0)w/k$ $\psi(\Omega^{\Omega^{k-1}})$
[1(1)1]w/[1] $\varphi(1,0...0)w/\omega$, the SVO. $\psi(\Omega^{\Omega^{\omega}})$
[1(1)1]w/[1(1)2] $\varphi(1,0...0)w/\Gamma_0$ $\psi(\Omega^{\Omega^{\Gamma_0}})$
[1(1)1]w/[1(1)1,2] $\varphi(1,0...0)w/\varphi(1,1,0)$ $\psi(\Omega^{\Omega^{\psi(\Omega^{\Omega+1})}})$
[1(1)1]w/[1(1)1,1,2] $\varphi(1,0...0)w/\varphi(2,0,0)$ $\psi(\Omega^{\Omega^{\psi(\Omega^{\Omega\times 2})}})$
[1(1)1]w/[1(1)1]w/[1] $\varphi(1,0...0)w/\varphi(1,0...0)w/\omega$ $\psi(\Omega^{\Omega^{\psi(\Omega^{\Omega^{\omega}})}})$
[1(1)1]w/[1(1)1]w/[1(1)1]w/[1] $\varphi(1,0...0)w/\varphi(1,0...0)w/\varphi(1,0...0)w/\omega$ $\psi(\Omega^{\Omega^{\psi(\Omega^{\Omega^{\psi(\Omega^{\Omega^{\omega}})}})}})$
[1(1)1(1)2] the LVO $\psi(\Omega^{\Omega^{\Omega}})$

From now on, I am not really certain about the comparisons, but they should be somewhere near the truth.

### Up to 2D arrays

Hyperfactorial array (without the n!) FGH ordinal using $$\psi$$
[1(1)2(1)2] $\psi(\Omega^{\Omega}\times\psi(\Omega^{\Omega^{\Omega}}))$
[1(1)1,2(1)2] $\psi(\Omega^{\Omega+1}\times\psi(\Omega^{\Omega^{\Omega}}))$
[1(1)1,k(1)2] $\psi(\Omega^{\Omega+(k-1)}\times\psi(\Omega^{\Omega^{\Omega}}))$
[1(1)1,1,2(1)2] $\psi(\Omega^{\Omega\times 2}\times\psi(\Omega^{\Omega^{\Omega}}))$
[1(1)1(1)3] $\psi(\Omega^{\Omega^{\Omega}}\times\psi(\Omega^{\Omega^{\Omega}}))$
[1(1)1(1)[1]] $\psi(\Omega^{(\Omega^{\Omega}+1)})$
[1(1)1(1)[1,2]] $\psi(\Omega^{\Omega^{\Omega}+2})$
[1(1)1(1)[1,k]] $\psi(\Omega^{\Omega^{\Omega}+k})$
[1(1)1(1)[1,1,2]] $\psi(\Omega^{\Omega^{\Omega}+\omega})$
[1(1)1(1)[1(1)1(1)2]] $\psi(\Omega^{\Omega^{\Omega}+\psi(\Omega^{\Omega^{\Omega}})})$
[1(1)1(1)1,2] $\psi(\Omega^{(\Omega^{\Omega}+\Omega)})$
[1(1)1(1)1,3] $\psi(\Omega^{(\Omega^{\Omega}+\Omega\times 2)})$
[1(1)1(1)1,1,2] $\psi(\Omega^{(\Omega^{\Omega}+\Omega^2)})$
[1(1)1(1)1,1,3] $\psi(\Omega^{(\Omega^{\Omega}+\Omega^2\times 2)})$
[1(1)1(1)1,1,[1]] $\psi(\Omega^{(\Omega^{\Omega}+\Omega^2\times\omega)})$
[1(1)1(1)1,1,1,2] $\psi(\Omega^{(\Omega^{\Omega}+\Omega^3)})$
[1(1)1(1)1]w/[1] affecting 3rd row. $\psi(\Omega^{(\Omega^{\Omega}+\Omega^{\omega})})$
[1(1)1(1)1(1)2] $\psi(\Omega^{(\Omega^{\Omega}\times 2)})$
[1(1)1(1)1(1)3] $\psi(\Omega^{(\Omega^{\Omega}\times 2)}\times 2)$
[1(1)1(1)1(1)1,2] $\psi(\Omega^{(\Omega^{\Omega}\times 2+1)})$
[1(1)1(1)1(1)1(1)2] $\psi(\Omega^{(\Omega^{\Omega}\times 3)})$
[1(1)1(1)1(1)1(1)1(1)2] $\psi(\Omega^{(\Omega^{\Omega}\times 4)})$
[1]w(1)/[1] $\psi(\Omega^{(\Omega^{\Omega}\times\omega)})$
[1(2)2] $\psi(\Omega^{(\Omega^{\Omega+1})})$

To continue from here, notice that the effect of adding a new row is to multiply the power of the previous row inside the psi function by $$\Omega^{\Omega^{\Omega}}$$. Therefore:

### Up to $$\psi(\Omega^{\Omega^{\Omega\times 2}})$$

Hyperfactorial array (without the n!) FGH ordinal using $$\psi$$
[1(2)1(1)2] $\psi(\Omega^{(\Omega^{\Omega+1}+\Omega^{\Omega})})$
[1(2)1(1)1(1)2] $\psi(\Omega^{(\Omega^{\Omega+1}+\Omega^{\Omega}\times 2)})$
[1(2)1(2)2] $\psi(\Omega^{\Omega^{\Omega+1}\times 2})$
[1(2)1(2)1(2)2] $\psi(\Omega^{\Omega^{\Omega+1}\times 3})$
[1]w(2)/[1] $\psi(\Omega^{\Omega^{\Omega+1}\times\omega})$
[1(3)2] $\psi(\Omega^{\Omega^{\Omega+2}})$
[1(3)1(2)2] $\psi(\Omega^{\Omega^{\Omega+2}+\Omega^{\Omega+1}})$
[1(3)1(2)1(2)2] $\psi(\Omega^{\Omega^{\Omega+2}+\Omega^{\Omega+1}\times 2})$
[1(3)1(3)2] $\psi(\Omega^{\Omega^{\Omega+2}\times 3})$
[1]w(3)/[1] $\psi(\Omega^{\Omega^{\Omega+2}\times\omega})$
[1(4)2] $\psi(\Omega^{\Omega^{\Omega+3}})$
[1(4)1(4)2] $\psi(\Omega^{\Omega^{\Omega+3}\times 2})$
[1(5)2] $\psi(\Omega^{\Omega^{\Omega+4}})$
[1(5)1(5)2] $\psi(\Omega^{\Omega^{\Omega+4}\times 2})$
[1(6)2] $\psi(\Omega^{\Omega^{\Omega+5}})$
[1(6)1(6)2] $\psi(\Omega^{\Omega^{\Omega+5}\times 2})$
[1(7)2] $\psi(\Omega^{\Omega^{\Omega+6}})$
[1(8)2] $\psi(\Omega^{\Omega^{\Omega+7}})$
[1(9)2] $\psi(\Omega^{\Omega^{\Omega+8}})$
[1([1])2] $\psi(\Omega^{\Omega^{\Omega+\omega}})$
[1([1([1([...([1([1([1])2])2])...])2])2])2] $\psi(\Omega^{\Omega^{\Omega\times 2}})$

This is not as big as I used to think it was (I used to think that it went up to the Bachmann-Howard), but even so, [1]w(1)/[1] is still at the level of BEAF, and with the extended array notation with bracket types, the Bachmann-Howard has now been delayed until [1(1)[21,1,1,1,2]], and the TFB to [1(1)[[2]1,1,1,1,2]].