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I will create an FGH Ordinals.

  1. \(f_0(\beta) = \beta + 1\)
  2. \(f_{\alpha + 1}(\beta) = \underbrace{f_{\alpha}(f_{\alpha}(... f_{\alpha}(f_{\alpha}(\beta)) ...))}_{\beta}\)
  3. \(f_{\alpha}(\beta) = f_{\alpha[\beta]}(\beta)\)

\(f_0(\omega) = \omega\)

\(f_1(\omega) = \omega 2\)

\(f_2(\omega) = \omega 2^\omega = \omega^\omega\)

\(f_n(\omega) = \varphi(n-2,0)\)

\(f_\omega(\omega) = \varphi(\omega,0)\)

\(f_{f_0(\omega)}(\omega) = \varphi(1,0,0)\)

\(f_{f_1(\omega)}(\omega) = \varphi(1,\omega,0)\)

\(f_{\omega^2}(\omega) = \varphi(\omega,0,0)\)

\(f_{f_2(\omega)}(\omega) = \vartheta(\Omega^\omega)\)

\(f_{f_\omega(\omega)}(\omega) = \vartheta(\varphi(\omega,\Omega + 1))\)

\(f_{f_{f_1(\omega)}(\omega)}(\omega) = \vartheta(\Omega_2)\)

\(f_{f_{f_\omega(\omega)}(\omega)}(\omega) = \vartheta(\varphi(\omega,\Omega_2 + 1))\)

\(\alpha = f_{\alpha}(\omega) = \vartheta(\Omega_\omega)\)

1st Extension

4. \(f_{0,0...0,0,\alpha + 1 \#}(\beta) = \underbrace{f_{0,0...0,f_{0,0...0,... f_{0,0...0,f_{0,0...0,0,\alpha \#}(\beta),\alpha \#}(\beta) ...,\alpha \#}(\beta),\alpha \#}(\beta)}_{\beta}\)

So \(\alpha = f_{0,1}(\omega)\).

\(f_{1,1}(\omega) = \vartheta(\Omega_\Omega)\)

\(f_{2,1}(\omega) = \vartheta(\Omega_\Omega + 1)\)

\(f_{\omega,1}(\omega) = \vartheta(\Omega_\Omega + \omega)\)

\(f_{f_{0,1}(\omega),1}(\omega) = \vartheta(\Omega_\Omega + \Omega_\omega)\)

\(f_{f_{0,1}(\omega)\omega,1}(\omega) = \vartheta(\Omega_\Omega\omega)\)

\(f_{f_{0,1}(\omega)^2,1}(\omega) = \vartheta(\Omega_\Omega\Omega_\omega)\)

\(f_{f_3(f_{0,1}(\omega)),1}(\omega) = \vartheta(\varepsilon_{\Omega_\Omega + 1})\)

\(f_{f_{f_1(\omega)}(f_{0,1}(\omega)),1}(\omega) = \vartheta(\Gamma_{\Omega_\Omega + 1})\)

\(f_{f_{0,1}(\omega + 1),1}(\omega) = \vartheta(\Omega_{\Omega + 1})\)

\(f_{f_{0,1}(\omega 2),1}(\omega) = \vartheta(\Omega_{\Omega + \omega})\)

\(f_{f_{1,1}(\omega),1}(\omega) = \vartheta(\Omega_{\Omega_2})\)

\(f_{f_{f_{1,1}(\omega),1}(\omega),1}(\omega) = \vartheta(\Omega_{\Omega_3})\)

\(f_{0,2}(\omega) = \vartheta(\Omega_{\Omega_\omega})\)

\(f_{0,\omega}(\alpha) = C(\alpha)\)

\(f_{1,\omega}(\omega) = C(\Omega)\)

\(f_{0,\omega + 1}(\omega) = C(\Omega_\omega)\)

\(f_{0,\omega 2}(\omega) = C(\psi_I(0))\)

Let \(C^{*}(0) = \Omega_\omega\), \(C^{*}(1) = \Omega_{\Omega_\omega}\) etc.

\(f_{0,\omega 3}(\omega) = C(C^{*}(\Omega))\)

\(f_{0,\omega^2}(\omega) = C(C^{*}(C^{*}(...)))\)

Let D(n) = W_W...W_w (n W's) and D(w+a) = C*(D(a)).

\(f_{0,\alpha}(\omega) = C(D(\alpha))\)

\(f_{0,0,1}(\omega) = C(D(\Omega))\)

\(f_{0,0,2}(\omega) = C(D(D(\Omega)))\)

\(f_{0,0,\omega}(\omega) = C(D^\omega(\Omega))\)

\(f_{0,0,\omega + 1}(\omega) = C(D^{D(\Omega)}(\Omega))\)

\(f_{0,0,\omega^2}(\omega) = C(D^{D^{...}(\Omega)}(\Omega))\)

Let D_2(n) = D(D_2(n-1)) and D_2(w+a) = D^{D_2(a)}(W)

\(f_{0,0,0,1}(\omega) = C(D_2(\Omega))\)

\(f_{0,0,0,0,1}(\omega) = C(D_3(\Omega))\)

The limit of extension is \(\beta = f_{0,0...0,1}(\omega) = C(D_\omega(\Omega))\).

2nd Extension

4. \(f_{\# f^{\gamma + 1}\alpha + 1}(\beta) = \underbrace{f_{\# f^{\gamma + 1}\alpha + f^{\gamma}f_{\# f^{\gamma + 1}\alpha + f^{\gamma}... f_{\# f^{\gamma + 1}\alpha + f^{\gamma}f_{\# f^{\gamma + 1}\alpha + f^{\gamma}0}(\beta)}(\beta) ...}(\beta)}(\beta)}_{\beta}\)

The limit of this extension is \(f_{\gamma}(\omega)\), where \(\gamma\) is \(f_\gamma(f)\).

3rd Extension

Support \(f_{f_2}(\omega) = f_{\underbrace{f_{f_{...}(f)}(f)}_{\omega}}(\omega)\), \(f_{f_3}(\omega) = f_{\underbrace{f_{f_{...}(f_2)}(f_2)}_{\omega}}(\omega)\), etc.

The limit of this extension is \(f_{\delta}(\omega)\), where \(\delta\) is \(f_\delta\).

4th Extension

The next extension is \(f\)-array.

So \(f_0,f_1(n) = f_{f_f...f_f}(n)\) with (n \(f\)'s), \(f_0,f_2(n) = f_{f_f...f_f},f_1(n)\) with (n \(f\)'s), etc.

Then \(f_0,f_0,f_1(n) = f_0,f_{f_f...f_f}(n)\), \(f_0,f_0,f_0,f_1(n) = f_0,f_0,f_{f_f...f_f}(n)\), etc.

We have dimensions. So [1] = comma and \(f_0[n+1]f_{m+1}(n) = \underbrace{f_0,f_0...f_0}_{n},f_1[n+1]f_m(n)\)

The limit of this extension is \(\epsilon(\omega)\), where \(\epsilon\) is \(f_0[\epsilon]f_1\).

5th Extension

Support \(f_0[0],[1]f_1(n) = \underbrace{f_0[f_0[... f_0[f_0[]f_1]f_1 ...]f_1]f_1}_{n}(n) = \underbrace{f_0[f_0[... f_0[f_1]f_1 ...]f_1]f_1}_{n}(n)\) and \(f_0[0],[1]f_{a+1}(n) = \underbrace{f_0[f_0[... f_0[f_0[]f_1[0],[1]f_a]f_1[0],[1]f_a ...]f_1[0],[1]f_a]f_1[0],[1]f_a}_{n}(n) = \underbrace{f_0[f_0[... f_0[f_1[0],[1]f_a]f_1[0],[1]f_a ...]f_1[0],[1]f_a]f_1[0],[1]f_a}_{n}(n)\).

Then \(f_0[0],[2]f_1(n) = \underbrace{f_0[... f_0[0],[1]f_1 ...],[1]f_1}_{n}(n)\) and \(f_0[0],[0],[1]f_1(n) = \underbrace{f_0[0],[... f_1 ...]f_1}_{n}(n)\).

Finally we have \(f_0[0]([2])[1]f_1(n) = f_0\underbrace{[0],[0]...[0]}_{n},[1]f_1(n)\).

The limit of this extension is \(f_0 \zeta f_1(\omega)\), where \(\zeta\) is \([0](\zeta)[1]\). That marks fastest limit ordinal ever.

6th Extension?

Coming Soon! Can you extend fifth extension of FGH?

This is WIP project forever!

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