If we work in this notation: $$C_0 (\alpha, \beta) = \beta \cup \lbrace 0 \rbrace$$

$$C_{n+1} (\alpha, \beta) = \lbrace \gamma + \delta, \varphi(\gamma, \delta), \Omega_{\gamma}, I_{\gamma}, \psi_\pi(\eta) | \gamma, \delta, \eta, \pi \in C_n (\alpha, \beta); \eta < \alpha; \pi \text{ is a regular cardinal} \rbrace $$

$$C ( \alpha, \beta) = \bigcup_{n = 1}^{\infty} C_n ( \alpha, \beta) $$

$$\psi_\pi (\alpha) = \min\{\beta|\beta\in\pi\land C(\alpha,\beta)\cap\pi\subseteq\beta\land\pi\in C(\alpha,\beta)\}$$

What would the proof-theoretic ordinal of KPh (Kripke-Platek set theory, whos universe is a hyper inaccessible set) in that notation? Me and some of my friends were having a discussion on whether KPh's proof theoretic ordinal would even be a collapse of a hyperinaccessible cardinal.