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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 $\beta<\alpha$.

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    $\begingroup$ 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. $\endgroup$
    – Wojowu
    Feb 16, 2018 at 20:49
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    $\begingroup$ +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. $\endgroup$ Feb 17, 2018 at 13:20
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    $\begingroup$ 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 $\endgroup$ Apr 20, 2018 at 19:14
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    $\begingroup$ 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. $\endgroup$
    – Gro-Tsen
    Apr 29, 2018 at 11:45
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    $\begingroup$ @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. $\endgroup$
    – C7X
    May 27 at 18:52

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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.

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  • $\begingroup$ 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. $\endgroup$ Apr 5 at 20:04
  • $\begingroup$ I have asked a few more people and unfortunately we might have to wait for another answer. $\endgroup$
    – C7X
    Apr 8 at 4:57

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