Jäger's collapsing functions and ρ-inaccessible ordinals

• Basic Notions
• Veblen function
• $\rho$-Inaccessible Ordinals
• The Ordinal Functions ${\psi }_{\kappa }$
• Fundamental sequences
• See also
• References

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 $M_0$.

$\alpha {=}_{NF}{\alpha }_1+\cdots +{\alpha }_n:\iff \alpha ={\alpha }_1+\cdots +{\alpha }_n\wedge {\alpha }_1\ge \cdots \ge {\alpha }_n\wedge {\alpha }_1,\dots ,{\alpha }_n\in P$

Cofinality $\text{cof}(\alpha )$ of an ordinal $\alpha$ is the least $\beta$ such that there exists a function $f:\beta \to \alpha$ 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

${\phi }_{\alpha }=\text{Enum}(\{\beta \in P|\mathrm{\forall }\gamma <\alpha ({\phi }_{\gamma }(\beta )=\beta )\})$

Normal form

$\alpha {=}_{NF}{\phi }_{\beta }(\gamma ):\iff \alpha ={\phi }_{\beta }(\gamma )\wedge \beta ,\gamma <\alpha$

An ordinal $\alpha$ is a strongly critical if $\phi (\alpha ,0)=\alpha$. 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 $\gamma \in S\cup \{0\}$

$S(\gamma )=\{{\alpha }_1,\dots ,{\alpha }_n\}$ if $\gamma {=}_{NF}{\alpha }_1+\cdots +{\alpha }_n\notin P$

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

$\rho$-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 $I_{\rho }:M_0\to M_0$ enumerate the $\rho$-inaccessible ordinals less than $M_0$ and their limits.

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

Normal form

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

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
${\phi }_{\alpha }(\beta )\in P$	$I_{\alpha }(0),I_{\alpha }(\beta +1)\in R$
$\gamma <\alpha \Rightarrow {\phi }_{\gamma }({\phi }_{\alpha }(\beta ))={\phi }_{\alpha }(\beta )$	$\gamma <\alpha \Rightarrow I_{\gamma }(I_{\alpha }(\beta ))=I_{\alpha }(\beta )$
$\beta <\gamma \Rightarrow {\phi }_{\alpha }(\beta )<{\phi }_{\alpha }(\gamma )$	$\beta <\gamma \Rightarrow I_{\alpha }(\beta )<I_{\alpha }(\gamma )$
$\alpha <\beta \Rightarrow {\phi }_{\alpha }(0)<{\phi }_{\beta }(0)$	$\alpha <\beta \Rightarrow I_{\alpha }(0)<I_{\beta }(0)$

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

$\alpha {=}_{NF}{\psi }_{\kappa }(\beta ):\iff \alpha ={\psi }_{\kappa }(\beta )\wedge \beta \in C_{\kappa }(\beta )$

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 $T$.

• $0\in T$
• $\alpha {=}_{NF}{\alpha }_1+\cdots +{\alpha }_n\wedge {\alpha }_1,\dots ,{\alpha }_n\in T\Rightarrow \alpha \in T$
• $\alpha {=}_{NF}{\phi }_{\beta }(\gamma )\wedge \beta ,\gamma \in T\Rightarrow \alpha \in T$
• $\alpha {=}_{NF}{I}_{\beta }(\gamma )\wedge \beta ,\gamma \in T\Rightarrow \alpha \in T$
• $\alpha {=}_{NF}{\psi }_{\kappa }(\beta )\wedge \kappa ,\beta \in T\Rightarrow \alpha \in T$

Below we write $I(\alpha ,\beta )$ for $I_{\alpha }(\beta )$ and $\phi (\alpha ,\beta )$ for ${\phi }_{\alpha }(\beta )$

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

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

• If $\alpha =\phi (\beta ,\gamma )$ and $\gamma \in L$ then $\text{cof}(\alpha )=\text{cof}(\gamma )$ and $\alpha [\eta ]=\phi (\beta ,\gamma [\eta ])$

• If $\alpha ={\psi }_{I(0,0)}(0)$ then $\text{cof}(\alpha )=\omega$ and $\alpha [0]=0$ and $\alpha [\eta +1]=\phi (\alpha [\eta ],0)$
• If $\alpha ={\psi }_{I(0,\beta +1)}(0)$ then $\text{cof}(\alpha )=\omega$ and $\alpha [0]=I(0,\beta )+1$ and $\alpha [\eta +1]=\phi (\alpha [\eta ],0)$

• If $\alpha ={\psi }_{I(0,\beta )}(\gamma +1)$ then $\text{cof}(\alpha )=\omega$ and $\alpha [0]={\psi }_{I(0,\beta )}(\gamma )+1$ and $\alpha [\eta +1]=\phi (\alpha [\eta ],0)$

• 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 $n\ge 2$ then $\text{cof}(\alpha )=\text{cof}({\alpha }_n)$ and $\alpha [\eta ]={\alpha }_1+{\alpha }_2+\cdots +({\alpha }_n[\eta ])$
• If $\alpha =\phi (0,0)$ 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 $\text{cof}(\beta )=\rho \ge \pi$ 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