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

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

$$C ( \alpha, \beta) = \bigcup_{n\in\omega}C_n ( \alpha, \beta) $$

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

What would the proof-theoretic ordinal of KPh (Kripke-Platek set theory, whose 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.

Note: A hyper-inaccessible cardinal $\kappa$, in this context, is one which is also the $\kappa$th (weakly) inaccessible. In other contexts a "(weakly) hyper-inaccessible cardinal" often means one of the form $\kappa$ that is $\kappa$-(weakly )inaccessible, where all the (weakly) inaccessibles are 0-(weakly )inaccessible and $\alpha$-(weakly )inaccessibles are (weakly) inaccessible and limits of $\beta$-(weakly )inaccessibles for all $\beta<\alpha$.

exactly. $\endgroup$2more comments