What is the proof-theoretic ordinal of KPh?

Question

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 $\beta<\alpha$.

Answer 1

Let $\mathsf{KPh}$ denote the theory $\mathsf{KP}+``\textrm{The recursively inaccessibles are unbounded}\! "$. I haven't found an explicit analysis of $\mathsf{KPh}$ in Rathjen's preprints, but there are some implicit and non-detailed claims (e.g. in "Proof theory: From arithmetic to set theory") that the techniques used to analyze $\mathsf{KPi}$ ("admissible proof theory" techniques) carry over to analyzing $\mathsf{KPh}$.

If you accept an answer justified only by conjecture, in Taranovsky's ordinal notation "Degrees of Reflection" (Taranovsky, "Ordinal Notation", section 4.2), it's claimed that when $F$ is some natural property, the proof-theoretic ordinal of $\mathsf{KP}+``\textrm{The universe is }F\! "$ is often $C(C(\Omega,a),0)$, where $a$ is the term assigned to $\textrm{min}\{\alpha\mid L_\alpha\vDash F\}$ - I believe this is either a typo of $\textrm{min}\{\alpha\mid L_\alpha\vDash\mathsf{KP}+F\}$ or a weak claim, since if $F$ is a condition like "the admissibles are unbounded" then $\mathsf{KP}+F$ is already as strong as $\mathsf{KPi}$ even though $\textrm{min}\{\alpha\mid L_\alpha\vDash F\}$ is just the least limit of admissibles.

If this is true for $F$ being "the recursively inaccessibles are unbounded", then the least $\alpha$ such that $L_\alpha\vDash\mathsf{KP}+F$ is the least recursively-hyper-inaccessible. In the notation Degrees of Reflection $C(C(\Omega,C(\Omega,\Omega)),0)$ is assigned to this ordinal, so our proof-theoretic ordinal is conjectured to be $C(C(\Omega,C(C(\Omega,C(\Omega,\Omega)),0)),0)$.

About what this proof-theoretic ordinal would be using the ordinal collapsing function in the question, it depends on what map $\xi\mapsto I_\xi$ is, and also the function may not be adequate for representing the ordinal in the case that it has no way to construct the least hyper-inaccessible.