User blog:Dchew89/New Custom Ordinal Notation

NOT DONE

\(\omega=sup\{0,1,2,...\}\)

\(\varepsilon=sup\{\omega,\omega^\omega,\omega^{\omega^\omega},...\}\)

\(\zeta=sup\{\varepsilon,\varepsilon^\varepsilon,\varepsilon^{\varepsilon^\varepsilon},...\}\)

\(\varphi(\omega)=sup\{\omega,\varepsilon,\zeta,...\}\)

\(\varphi(1,0)=sup\{\varphi(\omega),\varphi(\varphi(\omega)),...\}\)

\(\varphi(2,0)=sup\{\varphi(1,0),\varphi(\varphi(1,0)),...\}\)

\(\varphi(\omega,0)=sup\{\varphi(\omega),\varphi(1,0),\varphi(2,0),...\}\)

\(\varphi(1,0,0)=sup\{\varphi(\omega,0),\varphi(\varphi(\omega,0),0),...\}\)

\(\varphi(1,0,0,0)=sup\{\varphi(\omega,0,0),\varphi(\varphi(\omega,0,0),0,0),...\}\)

\(\varphi_{0}(\omega)=sup\{\varphi(\omega),\varphi(\omega,0),\varphi(\omega,0,0),...\}\)

\(\varphi_{0}(1,0)=sup\{\varphi_{0}(\omega),\varphi_{0}(\varphi_{0}(\omega)),...\}\)

\(\varphi_{0}(1,0,0)=sup\{\varphi_{0}(\omega,0),\varphi_{0}(\varphi_{0}(\omega,0),0),...\}\)

\(\varphi_{1}(\omega)=sup\{\varphi_{0}(\omega),\varphi_{0}(\omega,0),\varphi_{0}(\omega,0,0),...\}\)

\(\varphi_{1}(1,0)=sup\{\varphi_{1}(\omega),\varphi_{1}(\varphi_{1}(\omega)),...\}\)

\(\varphi_{\omega}(\omega)=sup\{\varphi(\omega),\varphi_{0}(\omega),\varphi_{1}(\omega),...\}\)

\(\varphi_{(1,0)}(\omega)=sup\{\varphi_{\omega}(\omega),\varphi_{\varphi_{\omega}(\omega)}(\omega),...\}\)

\(\varphi_{(1,0)}(1,0)=sup\{\varphi_{(1,0)}(\omega),\varphi_{(1,0)}(\varphi_{(1,0)}(\omega)),...\}\)

\(\varphi_{(2,0)}(\omega)=sup\{\varphi_{(1,0)}(\omega),\varphi_{\varphi_{(1,0)}(\omega)}(\omega),...\}\)

\(\varphi_{(1,0,0)}(\omega)=sup\{\varphi_{(\omega,0)}(\omega),\varphi_{(\varphi_{(\omega,0)}(\omega),0)}(\omega),...\}\)

\(\{0\}\varphi_{0}(\omega)=sup\{\varphi_{\omega}(\omega),\varphi_{(\omega,0)}(\omega),\varphi_{(\omega,0,0)}(\omega),...\}\)

\(\{0\}\varphi_{0}(1,0)=sup\{\{0\}\varphi_{0}(\omega),\{0\}\varphi_{0}(\{0\}\varphi_{0}(\omega)),...\}\)

\(\{1\}\varphi_{0}(\omega)=sup\{\{0\}\varphi_{\omega}(\omega),\{0\}\varphi_{(\omega,0)}(\omega),\{0\}\varphi_{(\omega,0,0)}(\omega),...\}\)

\(\{(1,0)\}\varphi_{0}(\omega)=sup\{\{\omega\}\varphi_{0}(\omega),{\{\omega\}\varphi_{0}(\{\omega)\}\varphi_{0}(\omega),...\}\)

\(\{0\}\varphi_{0}^{0}(\omega)=sup\{\{\omega\}\varphi_{0}(0),\{(\omega,0)\}\varphi_{0}(0),\{(\omega,0,0)\}\varphi_{0}(0),...\}\)

\(\psi(\omega)=sup\{\varphi_{0}(\omega),\{0\}\varphi_{0}(\omega),\{0\}\varphi_{0}^{0}(\omega),...\}\)

\(\psi(1,0)=sup\{psi(\omega),psi(psi(\omega)),...\}\)

NOT DONE