Bachmann psi function

Bachmann's psi function is an ordinal collapsing function with a limit of the Bachmann-Howard ordinal.

$$\begin{align}&C_0(\alpha,\beta)=\beta\cup\{0,\Omega\} \\ &C_{i+1}(\alpha,\beta)=\{\gamma+\delta,\omega^\gamma,\psi(\eta):\gamma,\delta,\eta\in C_i(\alpha,\beta)\land\eta<\alpha\} \\ &C(\alpha,\beta)=\bigcup_{i\in\mathbb{N}}C_i(\alpha,\beta) \\ &\psi(\alpha)=\text{min}(\{\beta:C(\alpha,\beta)\cap\Omega\subseteq\beta\}) \end{align}$$

Values

 * $$\psi(0)=\varepsilon_0$$
 * $$\psi(1)=\varepsilon_1$$
 * $$\psi(\omega)=\varepsilon_\omega$$
 * $$\psi(\psi(0))=\varepsilon_{\varepsilon_0}$$
 * $$\psi(\psi(\psi(0)))=\varepsilon_{\varepsilon_{\varepsilon_0}}$$
 * $$\psi(\Omega)=\zeta_0$$
 * $$\psi(\Omega+1)=\varepsilon_{\zeta_0+1}$$
 * $$\psi(\Omega+\omega)=\varepsilon_{\zeta_0+\omega}$$
 * $$\psi(\Omega+\psi(\Omega))=\psi(\Omega+\zeta_0)=\varepsilon_{\zeta_02}$$
 * $$\psi(\Omega+\psi(\Omega+\psi(\Omega)))=\varepsilon_{\varepsilon_{\zeta_02}}$$
 * $$\psi(\Omega2)=\zeta_1$$
 * $$\psi(\omega^{\Omega+1})=\zeta_\omega$$
 * $$\psi(\omega^{\Omega+1}+\Omega)=\zeta_{\omega+1}$$
 * $$\psi(\omega^{\Omega+1}2)=\zeta_{\omega2}$$
 * $$\psi(\omega^{\Omega+2})=\zeta_{\omega^2}$$
 * $$\psi(\omega^{\Omega+\omega})=\zeta_{\omega^\omega}$$
 * $$\psi(\omega^{\Omega+\varepsilon_0})=\zeta_{\varepsilon_0}$$
 * $$\psi(\omega^{\Omega+\psi(\Omega)})=\zeta_{\zeta_0}$$
 * $$\psi(\omega^{\Omega+\psi(\omega^{\Omega+\psi(\Omega)})})=\zeta_{\zeta_{\zeta_0}}$$
 * $$\psi(\omega^{\Omega2})=\eta_0$$
 * $$\psi(\omega^{\Omega2}+\Omega)=\zeta_{\eta_0+1}$$
 * $$\psi(\omega^{\Omega2}+\Omega2)=\zeta_{\eta_0+2}$$
 * $$\psi(\omega^{\Omega2}+\omega^{\Omega+1})=\zeta_{\eta_0+\omega}$$
 * $$\psi(\omega^{\Omega2}+\omega^{\Omega+1}+\Omega)=\zeta_{\eta_0+\omega+1}$$
 * $$\psi(\omega^{\Omega2}+\omega^{\Omega+1}2)=\zeta_{\eta_0+\omega2}$$
 * $$\psi(\omega^{\Omega2}+\omega^{\Omega+2})=\zeta_{\eta_0+\omega^2}$$
 * $$\psi(\omega^{\Omega2}+\omega^{\Omega+\psi(\omega^{\Omega2})})=\zeta_{\eta_02}$$
 * $$\psi(\omega^{\Omega2}+\omega^{\Omega+\psi(\omega^{\Omega2}+\omega^{\Omega+\psi(\omega^{\Omega2})})})=\zeta_{\zeta_{\eta_02}}$$
 * $$\psi(\omega^{\Omega2}2)=\eta_1$$
 * $$\psi(\omega^{\Omega2+1})=\eta_\omega$$
 * $$\psi(\omega^{\Omega2+1}+\omega^{\Omega2})=\eta_{\omega+1}$$
 * $$\psi(\omega^{\Omega2+2})=\eta_{\omega^2}$$
 * $$\psi(\omega^{\Omega2+\psi(\omega^{\Omega2})})=\eta_{\eta_0}$$
 * $$\psi(\omega^{\Omega3})=\varphi(4,0)$$
 * $$\psi(\omega^{\Omega n+\gamma_0}\beta_0)=\varphi(n,\omega\gamma_0+\beta_0)$$ with $$n<\omega$$
 * $$\psi(\omega^{\omega^{\Omega+1}})=\varphi(\omega,0)$$
 * $$\psi(\omega^{\omega^{\Omega2}})=\Gamma_0$$
 * $$\psi(\omega^{\omega^{\Omega2}}2)=\Gamma_1$$
 * $$\psi(\omega^{\omega^{\Omega2}+\Omega})=\varphi(1,1,0)$$
 * $$\psi(\omega^{\omega^{\Omega2}+\Omega2})=\varphi(1,2,0)$$
 * $$\psi(\omega^{\omega^{\Omega2}2})=\varphi(2,0,0)$$
 * $$\psi(\omega^{\omega^{\Omega3}})=\varphi(1,0,0,0)$$
 * $$\psi(\omega^{\omega^{\omega^{\Omega+1}}})=\text{SVO}$$