Jäger's collapsing functions and ρ-inaccessible ordinals

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

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

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

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

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

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

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

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

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

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

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