Jäger-Buchholz function

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Not to be confused with Jäger's function or 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. 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 \(\psi_\kappa(\alpha) := \psi(\kappa, \alpha)\).

See also

Ordinals, ordinal analysis, and set theory

Basics: cardinal numbers · ordinal numbers · limit ordinals · fundamental sequence · normal form · transfinite induction · ordinal notation · Absolute infinity
Theories: Robinson arithmetic · Presburger arithmetic · Peano arithmetic · KP · second-order arithmetic · ZFC
Model Theoretic Concepts: structure · elementary embedding
Countable ordinals: \(\omega\) · \(\varepsilon_0\) · \(\zeta_0\) · \(\eta_0\) · \(\Gamma_0\) (Feferman–Schütte ordinal) · \(\varphi(1,0,0,0)\) (Ackermann ordinal) · \(\psi_0(\Omega^{\Omega^\omega})\) (small Veblen ordinal) · \(\psi_0(\Omega^{\Omega^\Omega})\) (large Veblen ordinal) · \(\psi_0(\varepsilon_{\Omega + 1}) = \psi_0(\Omega_2)\) (Bachmann-Howard ordinal) · \(\psi_0(\Omega_\omega)\) with respect to Buchholz's ψ · \(\psi_0(\varepsilon_{\Omega_\omega + 1})\) (Takeuti-Feferman-Buchholz ordinal) · \(\psi_0(\Omega_{\Omega_{\cdot_{\cdot_{\cdot}}}})\) (countable limit of Extended Buchholz's function)‎ · \(\omega_1^\mathfrak{Ch}\) (Omega one of chess) · \(\omega_1^{\text{CK}}\) (Church-Kleene ordinal) · \(\omega_\alpha^\text{CK}\) (admissible ordinal) · recursively inaccessible ordinal · recursively Mahlo ordinal · reflecting ordinal · stable ordinal · \(\lambda,\gamma,\zeta,\Sigma\) (Infinite time Turing machine ordinals) · gap ordinal · List of countable ordinals
Ordinal hierarchies: Fast-growing hierarchy · Hardy hierarchy · Slow-growing hierarchy · Middle-growing hierarchy · N-growing hierarchy
Ordinal functions: enumeration · normal function · derivative · Veblen function · ordinal collapsing function · Weak Buchholz's function · Bachmann's function · Madore's function · Feferman's theta function · Buchholz's function · Extended Weak Buchholz's function · Extended Buchholz's function · Jäger-Buchholz function · Jäger's function · Rathjen's psi function · Rathjen's Psi function · Stegert's Psi function · Arai's psi function
Uncountable cardinals: \(\omega_1\) · omega fixed point · inaccessible cardinal \(I\) · Mahlo cardinal \(M\) · weakly compact cardinal \(K\) · indescribable cardinal · rank-into-rank cardinal · Berkeley cardinal · (Club Berkeley cardinal · limit club Berkeley cardinal)
Classes: \(\textrm{Card}\) · \(\textrm{On}\) · \(V\) · \(L\) · \(\textrm{Lim}\) · \(\textrm{AP}\) · Class (set theory)