Jäger-Buchholz function

The Jäger-Buchholz ψ function is an ordinal collapsing function that uses the inaccessible cardinal defined by Buchholz, as a restriction of Jäger's function.W. Buchholz, A simplified version of local predicativity, Proof Theory : A Selection of Papers from the Leeds Proof Theory Programme 1990 (1992), pp. 115--147. This article is a direct translation of a very small portion of the Japanese article, and you can read it here for the full detailed explanation.
== Definition ==
In the following, we work in \(\textsf{ZFC} + \textsf{IC}\), that is, assuming the existence of a weakly inaccessible cardinal. Let \(\mathbb{I}\) denote the least weakly inaccessible cardinal. Let \(\mathrm{Reg}_\mathbb{I}\) denote the set of all regular cardinals less than or equal to \(\mathbb{I}\). A class function \(\psi: \mathrm{Reg}_\mathbb{I} \times \mathrm{On} \rightarrow \mathrm{On}\) and the set \(\mathrm{Cl}(\alpha, \beta) \subseteq \mathrm{On}\) for ordinals \(\alpha\) and \(\beta\) are defined by the following mutual transfinite recursion:
\(\begin{align*}\mathrm{Cl}^0(\alpha,\beta):=&\beta\cup\{0,\mathbb{I}\}\\ \mathrm{Cl}^{n+1}(\alpha,\beta):=&\{\xi+\zeta\mid\xi,\zeta\in\mathrm{Cl}^n(\alpha,\beta)\}\cup\\ &\{\varphi_\xi(\zeta)\mid\xi,\zeta\in\mathrm{Cl}^n(\alpha,\beta)\}\cup\\ &\{\Omega_\xi\mid\xi\in\mathrm{Cl}^n(\alpha,\beta)\}\cup\\ &\{\psi_\pi(\xi)\mid\xi\in\mathrm{Cl}^n(\alpha,\beta)\cap\alpha\land\pi\in\mathrm{Cl}^n(\alpha,\beta)\cap\mathrm{Reg}_\mathbb{I}\}\\ \mathrm{Cl}(\alpha,\beta):=&\bigcup_{n<\omega}\mathrm{Cl}^n(\alpha,\beta)\\ \psi_\kappa(\alpha):=&\min\{\xi\in\mathrm{On}\mid\kappa\in\mathrm{Cl}(\alpha,\xi)\land\mathrm{Cl}(\alpha,\xi)\cap\kappa\subseteq\xi\}\end{align*}\)
Here, \(\varphi\) is the Veblen function, \(\Omega_0 := 0, \Omega_\alpha := \aleph_\alpha (\alpha > 0)\), and we abbreviate \(\psi_\kappa(\alpha) := \psi(\kappa, \alpha)\). 
== Sources ==
== See also ==
ja:イェーガー・ブーフホルツのψ関数
Category:Set theory
Category:Ordinal collapsing functions