FANDOM

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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
$\theta(1,0)$ $\psi(0)$ $\phi(1,0)$ $\epsilon_0$
$\theta(1,1)$ $\psi(1)$ $\phi(1,1)$ $\epsilon_1$
$\theta(1,2)$ $\psi(2)$ $\phi(1,2)$ $\epsilon_2$
$\theta(1,\omega)$ $\psi(\omega)$ $\phi(1,\omega)$ $\epsilon_\omega$
$\theta(1,\theta(1,0))$ $\psi(\psi(0))$ $\phi(1,\phi(1,0))$ $\epsilon_{\epsilon_0}$
$\theta(2,0)$ $\psi(\Omega)$ $\phi(2,0)$ $\zeta_0$
$\theta(1,\theta(2,0)+1)$ $\psi(\Omega+1)$ $\phi(1,\theta(2,0)+1)$ $\epsilon_{\zeta_0+1}$
$\theta(2,1)$ $\psi(\Omega*2)$ $\phi(2,1)$ $\zeta_1$
$\theta(2,\theta(2,0))$ $\psi(\omega^{\Omega+\psi(\Omega)})$ $\phi(2,\phi(2,0))$ $\zeta_{\zeta_0}$
$\theta(3,0)$ $\psi(\omega^{\Omega*2})$ $\phi(3,0)$ $\eta_0$
$\theta(4,0)$ $\psi(\omega^{\Omega*3})$ $\phi(4,0)$
$\theta(\omega,0)$ $\psi(\omega^{\omega^{\Omega+1}})$ $\phi(\omega,0)$
$\theta(\theta(1,0),0)$ $\psi(\omega^{\omega^{\Omega+\psi(0)}})$ $\phi(\theta(1,0),0)$
$\theta(\theta(\omega,0),0)$ $\psi(\omega^{\omega^{\Omega+\psi(\omega^{\omega^{\Omega+1}})}})$ $\phi(\phi(\omega,0),0)$
$\theta(\Omega,0)$ $\psi(\omega^{\omega^{\Omega*2}})$ $\phi(1,0,0)$ $\Gamma_0$
$\theta(1,\theta(\Omega,0)+1)$ $\psi(\omega^{\omega^{\Omega*2}}+1)$ $\phi(1,\phi(1,0,0)+1)$ $\epsilon_{\Gamma_0+1}$
$\theta(2,\theta(\Omega,0)+1)$ $\psi(\omega^{\omega^{\Omega*2}}+\Omega)$ $\phi(2,\phi(1,0,0)+1)$ $\zeta_{\Gamma_0+1}$
$\theta(\omega,\theta(\Omega,0)+1)$ $\psi(\omega^{\omega^{\Omega*2}}+\omega^{\omega^{\Omega+1}})$ $\phi(\omega,\phi(1,0,0)+1)$
$\theta(\theta(1,0),\theta(\Omega,0)+1)$ $\psi(\omega^{\omega^{\Omega*2}}+\omega^{\omega^{\Omega+\psi(0)}})$ $\phi(\phi(1,0),\phi(1,0,0)+1)$
$\theta(\theta(\Omega,0),1)$ $\psi(\omega^{\omega^{\Omega*2}}+\omega^{\omega^{\Omega+\psi(\omega^{\omega^{\Omega*2}})}})$ $\phi(\phi(1,0,0),1)$
$\theta(\theta(\Omega,0),2)$ $\psi(\omega^{\omega^{\Omega*2}}+\omega^{\omega^{\Omega+\psi(\omega^{\omega^{\Omega*2}})}}*2)$ $\phi(\phi(1,0,0),2)$
$\theta(\theta(\Omega,0),\omega)$ $\psi(\omega^{\omega^{\Omega*2}}+\omega^{\omega^{\Omega+\psi(\omega^{\omega^{\Omega*2}})}+1})$ $\phi(\phi(1,0,0),\omega)$
$\theta(\theta(\Omega,0)+1,0)$ $\psi(\omega^{\omega^{\Omega*2}}+\omega^{\omega^{\Omega+\psi(\omega^{\omega^{\Omega*2}})}+\Omega})$ $\phi(\phi(1,0,0)+1,0)$
$\theta(\theta(\Omega,0)+2,0)$ $\psi(\omega^{\omega^{\Omega*2}}+\omega^{\omega^{\Omega+\psi(\omega^{\omega^{\Omega*2}})}+\Omega*2})$ $\phi(\phi(1,0,0)+2,0)$
$\theta(\theta(\theta(\Omega),1),0)$ $\psi(\omega^{\omega^{\Omega*2}}+\omega^{\omega^{\Omega+\psi(\omega^{\omega^{\Omega*2}})}*2})$ $\phi(\phi(\phi(1,0,0),1),0)$
$\theta(\theta(\theta(\Omega),2),0)$ $\psi(\omega^{\omega^{\Omega*2}}+\omega^{\omega^{\Omega+\psi(\omega^{\omega^{\Omega*2}})}*3})$ $\phi(\phi(\phi(1,0,0),2),0)$
$\theta(\theta(\theta(\Omega),\omega),0)$ $\psi(\omega^{\omega^{\Omega*2}}+\omega^{\omega^{\Omega+\psi(\omega^{\omega^{\Omega*2}})+1}})$ $\phi(\phi(\phi(1,0,0),\omega),0)$