# What is the Proof-Theoretic Ordinal of KPh?

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, 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.

Note: A hyper-inaccessible cardinal $\kappa$, in this context, is one which is also the $\kappa$th (weakly) inaccessible.

• Context: According to nLab, KPh has PTO which is the collapse of the limit of the first $\omega$ (recursively) inaccessibles. However, since KPh is a theory stating that the universe is a limit of recursively inaccessible sets, i.e. it's hyperinaccessible, it seems more natural to expect its PTO to correspondingly be a collapse of a (recursively) hyperinaccesible. – Wojowu Feb 16 '18 at 20:49
• +Wojowu That's exactly what I'm trying to figure out - whether it is a collapse of the limit of the first $\omega$ recursively inaccessibles, or if it is a collapse of a hyperinaccessible ordinal. – Boris Dimitrov Feb 17 '18 at 13:20
• I'm suspicious too about the weird nLab claim ($\psi(\Omega_L)$ with $L$ limit of the first $\omega$ recursively inaccessibles). I would really like a reference – Julian Barathieu Apr 20 '18 at 19:14
• Yeah that's the weird thing and there isn't a reference. – Boris Dimitrov Apr 29 '18 at 9:40
• As Wikipedia points out, the term "hyper-inaccessible" is ambiguous, at least when referring to cardinals, but I suppose in the "recursively large ordinal" context also. So maybe you should clarify what KPh means exactly. – Gro-Tsen Apr 29 '18 at 11:45