# 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, 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 $$\beta<\alpha$$.

• 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. Feb 16, 2018 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. Feb 17, 2018 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 Apr 20, 2018 at 19:14
• 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. Apr 29, 2018 at 11:45
• @BorisDimitrov I asked another friend about this, they dug up a more specific reference on what KPh is. More specifically "KP+every set is contained in a model of KPi", sourced from here cambridge.org/core/books/abs/logic-colloquium-2004/…. However they did not find $\psi(\Omega_L)$ in there.
– C7X
May 27 at 18:52

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.
• I still don't understand why L wouldn't be an $\Omega$ fixed point. If L is a limit of recursively inaccessibles, then it should be a limit of admissibles and a fixed point of $\alpha\mapsto\omega^{CK}_\alpha$. Did nLab mean $\psi(\varepsilon_{L+1})$? What do you think. Apr 5 at 20:04