Jäger's collapsing functions and ρ-inaccessible ordinals

Jäger's collapsing functions are a hierarchy of single-argument ordinal functions $\psi_\pi$ 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 $\psi_\kappa(\alpha)$.

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

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

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

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

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

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

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

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 $\gamma[\eta+1]=\psi_{\rho}(\beta[\gamma[\eta]])$

Limit of this notation $\lambda$. If $\alpha=\lambda$ then $\text{cof}(\alpha)=\omega$ and $\alpha[0]=0$ and $\alpha[\eta+1]=I(\alpha[\eta],0)$

See also

Other ordinal collapsing functions:

Madore's ψ function

Buchholz's ψ functions

collapsing functions based on a weakly Mahlo cardinal

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