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Jäger's collapsing psi-functions are a hierarchy of single-argument ordinal functions \(\psi_\pi(\alpha)\) introduced by German mathematician Gerhard Jäger in 1984. This is an extension of Buchholz's notation.

Basic Notions

\(M_0\) is the least Mahlo cardinal, small Greek letters denote ordinals less than \(M_0\). Each ordinal \(\alpha\) is identified with the set of its predecessors \(\alpha=\{\beta|\beta<\alpha\}\).

\(L\) denotes the set of all limit ordinals less than \(M_0\).

An ordinal \(\alpha\) is an additive principal number if \(\alpha>0\) and \(\xi+\eta<\alpha\) for all \(\xi,\eta<\alpha\). Let \(P\) denote the set of all additive principal numbers less than \(M_0\).

\(\alpha=_{NF}\alpha _{1}+\cdots +\alpha _{n}:\Leftrightarrow \alpha =\alpha _{1}+\cdots +\alpha _{n}\wedge \alpha _{1}\geq \cdots \geq \alpha _{n}\wedge \alpha _{1},... ,\alpha _{n}\in P\)

Cofinality \(\text{cof}(\alpha)\) of an ordinal \(\alpha\) is the least \(\beta\) such that there exists a function \(f:\beta\rightarrow\alpha\) with \(\text{sup}\{f(\xi )|\xi <\beta \}=\alpha\). An ordinal \(\alpha\) is regular, if \(\alpha\) is a limit ordinal and \(\text{cof}(\alpha)=\alpha\). Let \(R\) denote the set of all regular ordinals \(\in(\omega, M_0)\).

An ordinal \(\alpha\) is (weakly) inaccessible if \(\alpha\) is a regular limit cardinal larger than \(\omega\).

Enumeration function \(F\) of class of ordinals \(X\) is the unique increasing function such that \(X=\{F(\alpha)|\alpha\in\text{dom}(F)\}\) where domain of \(F\), \(\text{dom}(F)\) is an ordinal number. We use \(\text{Enum}(X)\) to donate \(F\).

Veblen function

\(\varphi_\alpha=\text{Enum}(\{\beta\in P|\forall\gamma<\alpha(\varphi_\gamma(\beta)=\beta)\})\)

Normal form

\(\alpha=_{NF}\varphi_\beta(\gamma):\Leftrightarrow\alpha=\varphi_\beta(\gamma)\wedge\beta,\gamma<\alpha\)

An ordinal \(\alpha\) is a strongly critical if \(\varphi(\alpha,0)=\alpha\). Let \(S\) denote the set of all strongly critical ordinals less than \(M_0\).

Definition of \(S(\gamma)\) for arbitrary \(\gamma\).

\(S(\gamma)=\{\gamma\}\) if \(\gamma\in S\cup\{0\}\)

\(S(\gamma)=\{\alpha_1,...,\alpha_n\}\) if \(\gamma=_{NF}\alpha_1+\cdots+\alpha_n\notin P\)

\(S(\gamma)=\{\alpha,\beta\}\) if \(\gamma=_{NF}\varphi_\alpha(\beta)\notin S\)

\(\rho\)-Inaccessible Ordinals

An ordinal is \(\rho\)-inaccessible if it is a regular cardinal and limit of \(\alpha\)-inaccessible ordinals for all \(\alpha<\rho\). So the 0-inaccessible ordinals are exactly the regular cardinals \(>\omega\), the 1-inaccessible ordinals are the inaccessible ordinals. Functions \(I_\rho:M_0 \rightarrow M_0\) enumerate the \(\rho\)-inaccessible ordinals less than \(M_0\) and their limits.

\(I_\alpha=\text{Enum}(\{\beta\in R|\forall\gamma<\alpha(I_\gamma(\beta)=\beta)\}) \)

Normal form

\(\alpha=_{NF}I_\beta(\gamma):\Leftrightarrow\alpha=I_\beta(\gamma)\wedge\gamma\notin L\)

Definition of \(\gamma^{-}\) for \(\gamma\in R\).

\(\gamma^{-}=0\) if \(\gamma=_{NF}I_\alpha(0)\)

\(\gamma^{-}=I_\alpha(\beta)\) if \(\gamma=_{NF}I_\alpha(\beta+1)\)

Properties

Veblen function \(\rho\)-Inaccessible Ordinals
\(\varphi_\alpha(\beta)\in P\) \(I_\alpha(0), I_\alpha(\beta+1)\in R\)
\(\gamma<\alpha\Rightarrow\varphi_\gamma(\varphi_\alpha(\beta))=\varphi_\alpha(\beta)\) \(\gamma<\alpha\Rightarrow I_\gamma(I_\alpha(\beta))=I_\alpha(\beta)\)
\(\beta<\gamma\Rightarrow\varphi_\alpha(\beta)<\varphi_\alpha(\gamma)\) \(\beta<\gamma\Rightarrow I_\alpha(\beta)<I_\alpha(\gamma)\)
\(\alpha<\beta\Rightarrow\varphi_\alpha(0)<\varphi_\beta(0)\) \(\alpha<\beta\Rightarrow I_\alpha(0)<I_\beta(0)\)

The Ordinal Functions \(\psi_\kappa\)

Every \(\psi_\kappa\) is a function from \(M_0\) to \(\kappa\) which "collapses" the elements of \(M_0\) below \(\kappa\). By the Greek letters \(\kappa\) and \(\pi\) we shall denote uncountable regular cardinals less than \(M_0\).

Inductive Definition of \(C_\kappa(\alpha)\) and \(\psi_\kappa(\alpha)\).

\(\{\kappa^{-}\}\cup\kappa^{-}\subset C_\kappa^n(\alpha)\)

\(S(\gamma)\subset C_\kappa^n(\alpha)\Rightarrow\gamma\in C_\kappa^{n+1}(\alpha)\)

\(\beta,\gamma\in C_\kappa^n(\alpha)\Rightarrow I_\beta(\gamma)\in C_\kappa^{n+1}(\alpha)\)

\(\gamma<\pi<\kappa\wedge\pi\in C_\kappa^n(\alpha)\Rightarrow \gamma\in C_\kappa^{n+1}(\alpha)\)

\(\gamma<\alpha\wedge\gamma,\pi\in C_\kappa^n(\alpha)\wedge\gamma\in C_\pi(\gamma)\Rightarrow \psi_\pi(\gamma)\in C_\kappa^{n+1}(\alpha)\)

\(C_\kappa(\alpha)=\cup\{C_\kappa^n(\alpha)|n<\omega\}\)

\(\psi_\kappa(\alpha)=\text{min}\{\xi|\xi\notin C_\kappa(\alpha)\}\)


References

1. W.Buchholz. A New System of Proof-Theoretic Ordinal Functions. Annals of Pure and Applied Logic (1986),32

2. M.Jäger. \(\rho\)-inaccessible ordinals, collapsing functions and a recursive notation system. Arch. Math. Logik Grundlagenforsch (1984),24