FANDOM


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]].

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