Jäger's collapsing functions are a hierarchy of single-argument ordinal functions \(\psi_\pi\) introduced by German mathematician Gerhard Jäger in 1984[1]. It was developed on base of Buchholz's approach [2].

Basic Notions

\(M_0\) is the least Mahlo cardinal, and small Greek letters always denote ordinals less than \(M_0\) in this article. 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 \(M_0\).

Addition

An ordinal \(\alpha\) is said to be 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 \(M_0\).

We define a normal form relation \(=_{NF}\) for the addition as a variadic relation \(\alpha=_{NF}\alpha _{1}+\cdots +\alpha _{n}\) on ordinals \(\alpha,\alpha _{1},\ldots,\alpha _{n}\) (\(n \in \omega \setminus \{0\}\)) in the following way:

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

Veblen function

Enumeration function \(F\) of a set \(X\) of ordinals less than \(M_0\) is the unique strictly increasing function such that \(X=\{F(\alpha)|\alpha\in\text{dom}(F)\}\), where \(\text{dom}(F)\), i.e. the domain of \(F\), is an ordinal number less than \(M_0\). We use \(\text{Enum}(X)\) to denote \(F\). Using this convention, we recall the definition of Veblen function:

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

We define a normal form relation \(=_{NF}\) for Veblen's function as a \(3\)-ary relation \(\alpha=_{NF}\varphi_\beta(\gamma)\) on ordinals \(\alpha,\beta,\gamma\) in the following way:

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

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

We define a finit set \(S(\gamma)\) of ordinals less than \(M_0\) for an ordinal \(\gamma\) less than \(M_0\).

\(S(\gamma)=\{\gamma\}\) 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

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 uncountable regular ordinals less than \(M_0\).

An ordinal \(\alpha\) is said to be weakly inaccessible if \(\alpha\) is a regular limit cardinal larger than \(\omega\). Since we do not use the terminology of strong inaccessibility, we addreviate "weakly inaccessible" to "inaccessible" in this article.

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 greater than \(\omega\), the \(1\)-inaccessible ordinals are the inaccessible ordinals. For each ordinal \(\rho\) less than \(M_0\), we define a function \(I_\rho:M_0 \rightarrow M_0\) enumerating the \(\rho\)-inaccessible ordinals less than \(M_0\) and their limits in the following way:

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

where \(cl(X)=X\cup\{\alpha|0 \neq \alpha=\sup(X\cap\alpha)\}\).

We define a normal form relation \(=_{NF}\) for the enumeration functions \(I_\rho\) as a \(3\)-ary relation \(\alpha=_{NF}I_\beta(\gamma)\) on ordinals \(\alpha,\beta,\gamma\) in the following way:

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

We define an ordinal \(\gamma^{-}\) less than \(M_0\) for a \(\gamma\in R\).

\(\gamma^{-}=0\) if \(\gamma=_{NF}I_\alpha(0)\) for some \(\alpha \in M_0\)

\(\gamma^{-}=I_\alpha(\beta)\) if \(\gamma=_{NF}I_\alpha(\beta+1)\) for some \(\alpha,\beta \in M_0\)

Note that precisely one of the two cases above occurs by the definition of the enumeration functions \(I_\rho\).

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 a regular cardinal \(\kappa\). By the Greek letters \(\kappa\) and \(\pi\), we shall denote uncountable regular cardinals less than \(M_0\).

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

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

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

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

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

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

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

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

We define a normal form relation \(=_{NF}\) for the functions \(\psi_\kappa\) as a \(3\)-ary relation \(\alpha=_{NF}\psi_\kappa(\beta)\) on ordinals \(\alpha,\beta\) and an uncountable regular ordinal \(\kappa\) in the following way:

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

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 \(\alpha=\min\{\xi\in M_0|\forall\eta<\beta(\xi>\alpha[\eta])\}\), where \(\alpha[\eta]\) is the \(\eta\)-th element of this sequence. In the case where \(\alpha\) is a limit ordinal, then the latter condition is equivalent to that \(\alpha\) is the limit of \(\{\alpha[\eta] \mid \eta < \beta\}\).

Inductive Definition of a set \(T\) of ordinals less than \(M_0\).

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

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[3]:

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

The Recursive Notation System (\(T, \triangleleft\))

We shall define a recursive subset \(\mathcal{T}\) of the set \(\mathbb{N}\) of natural numbers and a recursive \(2\)-ary relation \(\triangleleft\) on \(\mathbb{N}\) such that \((\mathcal{T}, \triangleleft)\) is a strict total ordered set isomorphic to \((T, <) \). In particular, \((\mathcal{T}, \triangleleft)\) forms an ordinal notation. Formally the definition is constructive in arithmetic and independent from the classical theory of ordinals.

We shall define simultanously:

  1. Sets of natural numbers \(\mathcal{T}, \mathcal{L}, \mathcal{P}, \mathcal{S}\) and \(\mathcal{R}\)
  2. Two functions \(e,f:\mathcal{T}\to\mathcal{T}\)
  3. A function \(^{-}:\mathcal{R}\to\mathcal{T}\)
  4. Finite coefficient sets \(\mathcal{H}_u(a)\) for \(a \in \mathcal{T}\) and \(u \in \mathcal{R}\)
  5. A 2-place relation \(\triangleleft\) on \(\mathcal{T}\times\mathcal{T}\)

We assume that \(<\cdots>\) is a primitive recursive coding function on the natural numbers such that \(<a_1, \cdots, a_m>=<b_1, \cdots, b_n>\) implies \(m=n\) and \(a_i=b_i\) for \(i=1,...,n\). In order to point out the close connections to the ordinal functions \(+, \varphi, I\) and \(\psi\) we introduce the following abbreviations:

  1. \(a_1 \widetilde{+}\cdots\widetilde{+}a_n:=<2,a_1,...,a_n>\)
  2. \(\widetilde{\varphi}_a(b):=<3,a,b>\) and \(\widetilde{1}:=\widetilde{\varphi}_0(0)\)
  3. \(\widetilde{I}_a(b):=<4,a,b>\)
  4. \(\widetilde{\psi}_a(b):=<5,a,b>\)
  5. \(a \trianglelefteq b :\Leftrightarrow a=b \vee a \triangleleft b\)
  6. \(A\triangleleft a :\Leftrightarrow \forall x\in A(x\triangleleft a)\)

A. Inductive Definition of \(\mathcal{T}, \mathcal{L}, \mathcal{P}, \mathcal{S}, \mathcal{R}\)

  1. \(\mathcal{R}\subset\mathcal{S}\subset\mathcal{P}\subset\mathcal{L}\subset\mathcal{T}\)
  2. \(0 \in\mathcal{T}\)
  3. \(a_1,...,a_n \in \mathcal{H}\wedge n>1 \wedge a_n \trianglelefteq ... \trianglelefteq a_1 \wedge a_n \neq 1 \Rightarrow a_1 \widetilde{+}\cdots\widetilde{+}a_n \in \mathcal{L}\)
  4. \(a_1,...,a_n \in \mathcal{H}\wedge n>1 \wedge a_n \trianglelefteq ... \trianglelefteq a_1 \wedge a_n=1 \Rightarrow a_1 \widetilde{+}\cdots\widetilde{+}a_n \in \mathcal{T}\)
  5. \(a,b \in \mathcal{T}\wedge e(b) \trianglelefteq a \wedge (a \notin \mathcal{S}\vee b \neq 0)\Rightarrow \widetilde{\varphi}_a(b) \in \mathcal{H}\)
  6. \(a,b \in \mathcal{T}\wedge f(b) \trianglelefteq a \wedge b \notin \mathcal{L}\Rightarrow \widetilde{I}_a(b) \in \mathcal{R}\)
  7. \(a,b \in \mathcal{T}\wedge f(b) \trianglelefteq a \wedge b \in \mathcal{L}\Rightarrow \widetilde{I}_a(b) \in \mathcal{S}\)
  8. \(a \in \mathcal{R}\wedge b \in \mathcal{T} \wedge \mathcal{H}_a(b) \triangleleft b \Rightarrow \widetilde{\psi}_a(b) \in \mathcal{S}\)

B. Inductive Definition of \(e(a)\) and \(f(a)\) for \(a \in \mathcal{T}\)

  1. \(e(a):=0\) if \(a \notin \mathcal{P}\)
  2. \(f(a):=0\) if \(a \notin \mathcal{S}\)
  3. \(e(\widetilde{\varphi}_a(b)):=a\)
  4. \(f(\widetilde{I}_a(b)):=a\)
  5. \(e(a):=a\) if \(a \in \mathcal{S}\)
  6. \(f(\widetilde{\psi}_a(b)):=f(a)\)

C. Inductive Definition of \(u^{-}\) for \(u \in \mathcal{R}\)

  1. \(u^{-}:=0\) if \(u=\widetilde{I}_a(0)\)
  2. For \(u=\widetilde{I}_a(b_1\widetilde{+}\cdots\widetilde{+}b_{n-1}\widetilde{+}\widetilde{1})\) we define:

\(u^{-}:=\left\{\begin{array}{lcr} \widetilde{I}_a(b_1\widetilde{+}\cdots\widetilde{+}b_{n-1}) \text{ if }f(b_1\widetilde{+}\cdots\widetilde{+}b_{n-1})\trianglelefteq a\\ b_1\widetilde{+}\cdots\widetilde{+}b_{n-1}\text{ if } a \triangleleft f(b_1\widetilde{+}\cdots\widetilde{+}b_{n-1})\\ \end{array}\right.\)

D. Inductive Definition of \(\mathcal{H}_u(a)\) for \(a \in \mathcal{T}\) and \(u \in \mathcal{R}\)

  1. \(\mathcal{H}_u(0):=\emptyset\)
  2. \(\mathcal{H}_u(a_1\widetilde{+}\cdots\widetilde{+}a_n):=\mathcal{H}_u(a_1)\cup ... \cup \mathcal{H}_u(a_n)\)
  3. \(\mathcal{H}_u(\widetilde{\varphi}_a(b)):=\mathcal{H}_u(\widetilde{I}_a(b)):=\mathcal{H}_u(a)\cup\mathcal{H}_u(b)\)
  4. \(\mathcal{H}_u(\widetilde{\psi}_\nu(b)):=\left\{\begin{array}{lcr} \emptyset \text{ if }\widetilde{\psi}_\nu(b)\trianglelefteq u^{-}\\ \mathcal{H}_u(\nu)\text{ if }u^{-}\triangleleft \widetilde{\psi}_\nu(b)\wedge\nu\triangleleft u\\ \{b\}\cup \mathcal{H}_u(b)\cup \mathcal{H}_u(\nu) \text{ if }u^{-}\triangleleft \widetilde{\psi}_\nu(b)\wedge u\trianglelefteq \nu\\ \end{array}\right.\)

E. Inductive Definition of \(a \triangleleft b\) for \(a,b \in \mathcal{T}\)

  1. \(0 \triangleleft a\) if \(a \neq 0\)
  2. \(a_1\widetilde{+}\cdots\widetilde{+}a_m\triangleleft b_1\widetilde{+}\cdots\widetilde{+}b_n (1 \le m, 1\le n, 2<m+n) \) iff we have one of the following cases:
    1. \(m<n\wedge a_i=b_i\) for \(1\le i\le m\)
    2. there exists a \(k\) such that \(1 \le k \le \min(m,n)\wedge a_k \triangleleft b_k \wedge a_i=b_i\) for \(1\le i<k\)
  3. \(c \in \mathcal{S}\wedge a,b \triangleleft c \Rightarrow \widetilde{\varphi}_a(b)\triangleleft c\)
  4. \(c \in \mathcal{S}\wedge (c \trianglelefteq a \vee c \trianglelefteq b)\Rightarrow c\triangleleft\widetilde{\varphi}_a(b) \)
  5. \(\widetilde{\varphi}_{a_1}(b_1)\triangleleft\widetilde{\varphi}_{a_2}(b_2)\) iff we have one of the following cases:
    1. \(a_1\triangleleft a_2\wedge b_1 \triangleleft\widetilde{\varphi}_{a_2}(b_2)\)
    2. \(a_1=a_2\wedge b_1 \triangleleft b_2\)
    3. \(a_2\triangleleft a_1\wedge \widetilde{\varphi}_{a_1}(b_1) \triangleleft b_2\)
  6. \(\widetilde{I}_{a_1}(b_1)\triangleleft\widetilde{I}_{a_2}(b_2)\) iff we have one of the following cases:
    1. \(a_1\triangleleft a_2\wedge b_1 \triangleleft\widetilde{I}_{a_2}(b_2)\)
    2. \(a_1=a_2\wedge b_1 \triangleleft b_2\)
    3. \(a_2\triangleleft a_1\wedge \widetilde{I}_{a_1}(b_1) \triangleleft b_2\)
  7. \(\widetilde{\psi}_{u_1}(b_1)\triangleleft\widetilde{\psi}_{u_2}(b_2)\) iff we have one of the following cases:
    1. \(u_1\triangleleft u_2\wedge u_1 \triangleleft\widetilde{\psi}_{u_2}(b_2)\)
    2. \(u_1=u_2\wedge b_1 \triangleleft b_2\)
    3. \(u_2\triangleleft u_1\wedge \widetilde{\psi}_{u_1}(b_1) \triangleleft u_2\)
  8. \((b \triangleleft f(u) \wedge c \triangleleft \widetilde{\psi}_u(a)) \vee (f(u)\trianglelefteq b \wedge \widetilde{I}_b(c)\triangleleft u) \Rightarrow \widetilde{I}_b(c) \triangleleft \widetilde{\psi}_u(a)\)
  9. \((b\triangleleft f(u)\wedge \widetilde{\psi}_u(a)\trianglelefteq c)\vee(f(u)\trianglelefteq b\wedge u\trianglelefteq \widetilde{I}_b(c))\Rightarrow\widetilde{\psi}_u(a)\triangleleft\widetilde{I}_b(c)\)

Simplification

Googology wiki user Denis Maksudov simplified Jäger's function by dropping Veblen function in the definition of the functions \(\psi_{\kappa}\).[4] Unlike many other naive "simplifications" of standard ordinal collapsing functions, it is quite meaningful because Denis Maksudov has also defined an associated ordinal notation, which plays one of the most important role in computable googology.

References

  1. M. Jäger, "ρ-inaccessible ordinals, collapsing functions and a recursive notation system", Archiv für mathematische Logik und Grundlagenforschung, volume 24 (1984), pp. 49--62.
  2. Buchholz, W. "A New System of Proof-Theoretic Ordinal Functions"Annals of Pure and Applied Logic, vol. 32. Retrieved 2020-02-15.
  3. Denis Maksudov. Jäger's collapsing functions and ρ-inaccessible ordinals (archived version of the article from "Cantor’s Attic" ).
  4. Traveling To The Infinity

See also

Basics: cardinal numbers · ordinal numbers · limit ordinals · fundamental sequence · normal form · ordinal notation · transfinite induction
Theories: Robinson arithmetic · Presburger arithmetic · Peano arithmetic · KP · second-order arithmetic · ZFC
Model Theoretic Concepts: structure · elementary embedding
Countable ordinals: \(\omega\) · \(\varepsilon_0\) · \(\zeta_0\) · \(\eta_0\) · \(\Gamma_0\) (Feferman–Schütte ordinal) · \(\varphi(1,0,0,0)\) (Ackermann ordinal) · \(\psi_0(\Omega^{\Omega^\omega})\) (small Veblen ordinal) · \(\psi_0(\Omega^{\Omega^\Omega})\) (large Veblen ordinal) · \(\psi_0(\varepsilon_{\Omega + 1}) = \psi_0(\Omega_2)\) (Bachmann-Howard ordinal) · \(\psi_0(\Omega_\omega)\) with respect to Buchholz's ψ · \(\psi_0(\varepsilon_{\Omega_\omega + 1})\) (Takeuti-Feferman-Buchholz ordinal) · \(\psi_0(\Omega_{\Omega_{\cdot_{\cdot_{\cdot}}}})\) (countable limit of Extended Buchholz's function)‎ · \(\omega_1^\mathfrak{Ch}\) · \(\omega_1^{\text{CK}}\) (Church-Kleene ordinal) · \(\omega_\alpha^\text{CK}\) (admissible ordinal) · recursively inaccessible ordinal · recursively Mahlo ordinal · reflecting ordinal · stable ordinal · \(\lambda,\zeta,\Sigma,\gamma\) (ordinals on infinite time Turing machine) · gap ordinal · List of countable ordinals
Ordinal hierarchies: Fast-growing hierarchy · Slow-growing hierarchy · Hardy hierarchy · Middle-growing hierarchy · N-growing hierarchy
Ordinal functions: enumeration · normal function · derivative · Veblen function · ordinal collapsing function · Bachmann's function · Madore's function · Feferman's theta function · Buchholz's function · Extended Buchholz's function · Jäger's function · Rathjen's psi function · Rathjen's Psi function · Stegert's Psi function
Uncountable cardinals: \(\omega_1\) · omega fixed point · inaccessible cardinal \(I\) · Mahlo cardinal \(M\) · weakly compact cardinal \(K\) · indescribable cardinal · rank-into-rank cardinal
Classes: \(V\) · \(L\) · \(\textrm{On}\) · \(\textrm{Lim}\) · \(\textrm{AP}\) · Class (set theory)

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