User blog:Googleaarex/My Ordinals

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.

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

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

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

The fastest recursive ordinal is \(\beta\).