Jäger's function

Jäger's collapsing functions are a hierarchy of single-argument ordinal functions \(\psi_\pi\) introduced by German mathematician Gerhard Jäger in 1984. It was developed on base 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 denote \(F\).

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.

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

where $$cl(X)=X\cup\{\alpha|\alpha=\sup(X\cap\alpha)\} $$

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

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 \(\psi_\kappa(\alpha)\).

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

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

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

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

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

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

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

Normal form

\(\alpha=_{NF}\psi_\kappa(\beta):\Leftrightarrow\alpha=\psi_\kappa(\beta)\wedge\beta\in C_\kappa(\beta)\)

Fundamental sequences
The fundamental sequence for an ordinal number \(\alpha\) with cofinality \(\text{cof}(\alpha)=\beta\) is a strictly increasing sequence \((\alpha[\eta])_{\eta<\beta}\) with length \(\beta\) and with limit \(\alpha\), where \(\alpha[\eta]\) is the \(\eta\)-th element of this sequence.

Inductive Definition of \(T\).
 * \(0 \in T\)
 * \(\alpha=_{NF}\alpha _{1}+\cdots +\alpha _{n}\wedge \alpha _{1},... ,\alpha _{n}\in T\Rightarrow\alpha\in T\)
 * \(\alpha=_{NF}\varphi_\beta(\gamma)\wedge\beta,\gamma\in T\Rightarrow\alpha\in T\)
 * \(\alpha=_{NF}I_\beta(\gamma)\wedge\beta,\gamma\in T\Rightarrow\alpha\in T\)
 * \(\alpha=_{NF}\psi_\kappa(\beta)\wedge\kappa, \beta\in T\Rightarrow\alpha\in T\)

Below we write \(I(\alpha,\beta)\) for \(I_\alpha(\beta)\) and \(\varphi(\alpha,\beta)\) for \(\varphi_\alpha(\beta)\)

For non-zero ordinals \(\alpha\in T\) we define the fundamental sequences as follows:


 * If \(\alpha=\varphi(0,\beta+1)\) then \(\text{cof}(\alpha)=\omega\) and \(\alpha[\eta]=\varphi(0,\beta)\times\eta\)
 * If \(\alpha=\varphi(\beta+1,0)\) then \(\text{cof}(\alpha)=\omega\) and \(\alpha[0]=0\) and \(\alpha[\eta+1]=\varphi(\beta,\alpha[\eta])\)
 * If \(\alpha=\varphi(\beta+1,\gamma+1)\) then \(\text{cof}(\alpha)=\omega\) and \(\alpha[0]=\varphi(\beta+1,\gamma)+1\) and \(\alpha[\eta+1]=\varphi(\beta,\alpha[\eta])\)
 * If \(\alpha=\varphi(\beta,0)\) and \(\beta\in L\) then \(\text{cof}(\alpha)=\text{cof}(\beta)\) and \(\alpha[\eta]=\varphi(\beta[\eta],0)\)
 * If \(\alpha=\varphi(\beta,\gamma+1)\) and \(\beta\in L\) then \(\text{cof}(\alpha)=\text{cof}(\beta)\) and \(\alpha[\eta]=\varphi(\beta[\eta],\varphi(\beta,\gamma)+1)\)
 * If \(\alpha=\varphi(\beta,\gamma)\) and \(\gamma\in L\) then \(\text{cof}(\alpha)=\text{cof}(\gamma)\) and \(\alpha[\eta]=\varphi(\beta,\gamma[\eta])\)


 * If \(\alpha=\psi_{I(0,0)}(0)\) then \(\text{cof}(\alpha)=\omega\) and \(\alpha[0]=0\) and \(\alpha[\eta+1]=\varphi(\alpha[\eta],0)\)
 * If \(\alpha=\psi_{I(0,\beta+1)}(0)\) then \(\text{cof}(\alpha)=\omega\) and \(\alpha[0]=I(0,\beta)+1\) and \(\alpha[\eta+1]=\varphi(\alpha[\eta],0)\)
 * If \(\alpha=\psi_{I(0,\beta)}(\gamma+1)\) then \(\text{cof}(\alpha)=\omega\) and \(\alpha[0]=\psi_{I(0,\beta)}(\gamma)+1\) and \(\alpha[\eta+1]=\varphi(\alpha[\eta],0)\)


 * If \(\alpha=\psi_{I(\beta+1,0)}(0)\) then \(\text{cof}(\alpha)=\omega\) and \(\alpha[0]=0\) and \(\alpha[\eta+1]=I(\beta,\alpha[\eta])\)
 * If \(\alpha=\psi_{I(\beta+1,\gamma+1)}(0)\) then \(\text{cof}(\alpha)=\omega\) and \(\alpha[0]=I(\beta+1,\gamma)+1\) and \(\alpha[\eta+1]=I(\beta,\alpha[\eta])\)
 * If \(\alpha=\psi_{I(\beta+1,\gamma)}(\delta+1)\) then \(\text{cof}(\alpha)=\omega\) and \(\alpha[0]=\psi_{I(\beta+1,\gamma)}(\delta)+1\) and \(\alpha[\eta+1]=I(\beta,\alpha[\eta])\)


 * If \(\alpha=\psi_{I(\beta,0)}(0)\) and \(\beta\in L\) then \(\text{cof}(\alpha)=\text{cof}(\beta)\) and \(\alpha[\eta]=I(\beta[\eta],0)\)
 * If \(\alpha=\psi_{I(\beta,\gamma+1)}(0)\) and \(\beta\in L\) then \(\text{cof}(\alpha)=\text{cof}(\beta)\) and \(\alpha[\eta]=I(\beta[\eta],I(\beta,\gamma)+1)\)
 * If \(\alpha=\psi_{I(\beta,\gamma)}(\delta+1)\) and \(\beta\in L\) then \(\text{cof}(\alpha)=\text{cof}(\beta)\) and \(\alpha[\eta]=I(\beta[\eta],\psi_{I(\beta,\gamma)}(\delta)+1)\)


 * If \(\alpha=\alpha_1+\alpha_2+\cdots+\alpha_n\) with \(n\geq 2\) then \(\text{cof}(\alpha)=\text{cof}(\alpha_n)\) and \(\alpha[\eta]=\alpha_1+\alpha_2+\cdots+(\alpha_n[\eta])\)
 * If \(\alpha=\varphi(0,0)\) then \(\text{cof}(\alpha)=\alpha=1\) and \(\alpha[0]=0\)
 * If \(\alpha=I(\beta,0)\) or \(\alpha=I(\beta,\gamma+1)\) then \(\text{cof}(\alpha)=\alpha\) and \(\alpha[\eta]=\eta\)
 * If \(\alpha=I(\beta,\gamma)\) and \(\gamma\in L\) then \(\text{cof}(\alpha)=\text{cof}(\gamma)\) and \(\alpha[\eta]=I(\beta,\gamma[\eta])\)
 * If \(\alpha=\psi_\pi(\beta)\) and \(\omega\le\text{cof}(\beta)<\pi\) then \(\text{cof}(\alpha)=\text{cof}(\beta)\) and \(\alpha[\eta]=\psi_\pi(\beta[\eta])\)
 * If \(\alpha=\psi_\pi(\beta)\) and \(\text{cof}(\beta)=\rho\geq\pi\) then \(\text{cof}(\alpha)=\omega\) and \(\alpha[\eta]=\psi_\pi(\beta[\gamma[\eta]])\) with \(\gamma[0]=1\) and \(\gamma[\eta+1]=\psi_{\rho}(\beta[\gamma[\eta]])\)

Let math>\lambda=\min\{xi|I(\xi, 0)=\xi\}\). If \(\alpha=\lambda\) then \(\text{cof}(\alpha)=\omega\) and \(\alpha[0]=0\) and \(\alpha[\eta+1]=I(\alpha[\eta],0)\)