11,578
pages
Not to be confused with Jäger's function, Buchholz's function, or Extended Buchholz's 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.[1] 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

1. 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.