FANDOM


I'm presenting the comparisons between 4 known transfinite ordinal notations below. I'm not certain about comparisons past \(\Gamma_0\) in psi function.

Theta function Psi function Phi function Indexed-letter notation
\theta(1,0) \psi(0) \phi(1,0) \epsilon_0
\theta(1,1) \psi(1) \phi(1,1) \epsilon_1
\theta(1,2) \psi(2) \phi(1,2) \epsilon_2
\theta(1,\omega) \psi(\omega) \phi(1,\omega) \epsilon_\omega
\theta(1,\theta(1,0)) \psi(\psi(0)) \phi(1,\phi(1,0)) \epsilon_{\epsilon_0}
\theta(2,0) \psi(\Omega) \phi(2,0) \zeta_0
\theta(1,\theta(2,0)+1) \psi(\Omega+1) \phi(1,\theta(2,0)+1) \epsilon_{\zeta_0+1}
\theta(2,1) \psi(\Omega*2) \phi(2,1) \zeta_1
\theta(2,\theta(2,0)) \psi(\omega^{\Omega+\psi(\Omega)}) \phi(2,\phi(2,0)) \zeta_{\zeta_0}
\theta(3,0) \psi(\omega^{\Omega*2}) \phi(3,0) \eta_0
\theta(4,0) \psi(\omega^{\Omega*3}) \phi(4,0)
\theta(\omega,0) \psi(\omega^{\omega^{\Omega+1}}) \phi(\omega,0)
\theta(\theta(1,0),0) \psi(\omega^{\omega^{\Omega+\psi(0)}}) \phi(\theta(1,0),0)
\theta(\theta(\omega,0),0) \psi(\omega^{\omega^{\Omega+\psi(\omega^{\omega^{\Omega+1}})}}) \phi(\phi(\omega,0),0)
\theta(\Omega,0) \psi(\omega^{\omega^{\Omega*2}}) \phi(1,0,0) \Gamma_0
\theta(1,\theta(\Omega,0)+1) \psi(\omega^{\omega^{\Omega*2}}+1) \phi(1,\phi(1,0,0)+1) \epsilon_{\Gamma_0+1}
\theta(2,\theta(\Omega,0)+1) \psi(\omega^{\omega^{\Omega*2}}+\Omega) \phi(2,\phi(1,0,0)+1) \zeta_{\Gamma_0+1}
\theta(\omega,\theta(\Omega,0)+1) \psi(\omega^{\omega^{\Omega*2}}+\omega^{\omega^{\Omega+1}}) \phi(\omega,\phi(1,0,0)+1)
\theta(\theta(1,0),\theta(\Omega,0)+1) \psi(\omega^{\omega^{\Omega*2}}+\omega^{\omega^{\Omega+\psi(0)}}) \phi(\phi(1,0),\phi(1,0,0)+1)
\theta(\theta(\Omega,0),1) \psi(\omega^{\omega^{\Omega*2}}+\omega^{\omega^{\Omega+\psi(\omega^{\omega^{\Omega*2}})}}) \phi(\phi(1,0,0),1)
\theta(\theta(\Omega,0),2) \psi(\omega^{\omega^{\Omega*2}}+\omega^{\omega^{\Omega+\psi(\omega^{\omega^{\Omega*2}})}}*2) \phi(\phi(1,0,0),2)
\theta(\theta(\Omega,0),\omega) \psi(\omega^{\omega^{\Omega*2}}+\omega^{\omega^{\Omega+\psi(\omega^{\omega^{\Omega*2}})}+1}) \phi(\phi(1,0,0),\omega)
\theta(\theta(\Omega,0)+1,0) \psi(\omega^{\omega^{\Omega*2}}+\omega^{\omega^{\Omega+\psi(\omega^{\omega^{\Omega*2}})}+\Omega}) \phi(\phi(1,0,0)+1,0)
\theta(\theta(\Omega,0)+2,0) \psi(\omega^{\omega^{\Omega*2}}+\omega^{\omega^{\Omega+\psi(\omega^{\omega^{\Omega*2}})}+\Omega*2}) \phi(\phi(1,0,0)+2,0)
\theta(\theta(\theta(\Omega),1),0) \psi(\omega^{\omega^{\Omega*2}}+\omega^{\omega^{\Omega+\psi(\omega^{\omega^{\Omega*2}})}*2}) \phi(\phi(\phi(1,0,0),1),0)
\theta(\theta(\theta(\Omega),2),0) \psi(\omega^{\omega^{\Omega*2}}+\omega^{\omega^{\Omega+\psi(\omega^{\omega^{\Omega*2}})}*3}) \phi(\phi(\phi(1,0,0),2),0)
\theta(\theta(\theta(\Omega),\omega),0) \psi(\omega^{\omega^{\Omega*2}}+\omega^{\omega^{\Omega+\psi(\omega^{\omega^{\Omega*2}})+1}}) \phi(\phi(\phi(1,0,0),\omega),0)

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