Buchholz's ψ functions

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Buchholz's \(\psi\) function is an ordinal collapsing function introduced by German mathematician Wilfried Buchholz's in 1981.

Basic Notions

Small Greek letters always denote ordinals. Each ordinal \(\alpha\) is identified with the set of its predecessors \(\alpha=\{\beta|\beta<\alpha\}\).

\(On\) denotes the class of all ordinals.

We define \(\Omega_0=1\) and \(\Omega_{\nu}=\aleph_{\nu}\) for \(\nu>0\).

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 i.e.

\(P=\{\alpha\in On|0<\alpha\wedge\forall\xi,\eta<\alpha(\xi+\eta\in\alpha)\}=\{\omega^\beta|\beta\in On\}\)

For every \(\alpha\notin P\) there exist unique set \(P(\alpha)=\{\alpha_1, \alpha_2, ... ,\alpha_n\}\) such that \(\alpha=\alpha_1+\alpha_2+ \cdots+\alpha_n\) and \(\alpha>\alpha_1\geq\alpha_2\geq \cdots\geq\alpha_n\) and \(\alpha_1, \alpha_2, ... ,\alpha_n\in P\)

\(\alpha=_{NF}\alpha_1+\alpha_2+\cdots+\alpha_n\) iff \(\alpha=\alpha_1+\alpha_2+\cdots+\alpha_n\) and \(\alpha_1\geq\alpha_2\geq\cdots\geq\alpha_n\) and \(\alpha_1,\alpha_2,...,\alpha_n\in P\)

Definition

Buchholz's \(\psi\) function is defined as follows:

  • \(C_\nu^0(\alpha) = \Omega_\nu\),
  • \(C_\nu^{n+1}(\alpha) = C_\nu^n(\alpha) \cup \{\gamma | P(\gamma) \subseteq C_\nu^n(\alpha)\} \cup \{\psi_\mu(\xi) | \xi \in \alpha \cap C_\nu^n(\alpha) \wedge \xi \in C_\mu(\xi) \wedge \mu \leq \omega\}\),
  • \(C_\nu(\alpha) = \bigcup_{n < \omega} C_\nu^n (\alpha)\),
  • \(\psi_\nu(\alpha) = \min\{\gamma | \gamma \not\in C_\nu(\alpha)\}\),

In other words \(\psi_\nu(\alpha)\) is the least ordinal number which cannot be generated from ordinals less than \(\Omega_\nu\) by applying of addition and the functions \(\psi_{\mu}(\eta)\) with \(\eta < \alpha\) and \(\mu \le \omega\).

We define \(\alpha=_{NF}\psi_\nu(\beta)\) iff \(\alpha=\psi_\nu(\beta)\) and \(\beta\in C_\nu(\beta)\)

Properties

Buchholz showed following properties of those functions:

  • \(\psi_\nu(0)=\Omega_\nu\),
  • \(\psi_\nu(\alpha)\in P\),
  • \(\psi_\nu(\alpha+1)=\text{min}\{\gamma\in P| \psi_\nu(\alpha)<\gamma\}\text{ if }\alpha\in C_\nu(\alpha)\),
  • \(\Omega_\nu\le\psi_\nu(\alpha)<\Omega_{\nu+1}\),
  • \(\alpha\le\beta\Rightarrow\psi_\nu(\alpha)\le\psi_\nu(\beta)\),
  • \(\psi_0(\alpha)=\omega^\alpha \text{ if }\alpha<\varepsilon_0\),
  • \(\psi_\nu(\alpha)=\omega^{\Omega_\nu+\alpha} \text{ if }\alpha<\varepsilon_{\Omega_\nu+1} \text{ and } \nu\neq 0\).

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.

We define the set \(T\) consisting of zero and all ordinals expressible using Buchholz's function

  1. \(0 \in T\)
  2. if \(\alpha=_{NF}\alpha_1+\alpha_2+\cdots+\alpha_n\) and \(\alpha_1,\alpha_2,...,\alpha_n\in T\) then \(\alpha \in T\)
  3. if \(\alpha=_{NF}\psi_\nu(\beta)\) and \(\nu,\beta\in T\) and \(\nu\le\omega\) then \(\alpha \in T\)

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

  1. if \(\alpha=\alpha_1+\alpha_2+\cdots+\alpha_n\) then \(\text{cof} (\alpha)= \text{cof} (\alpha_n)\) and \(\alpha[\eta]=\alpha_1+\alpha_2+\cdots+(\alpha_n[\eta])\)
  2. if \(\alpha=\psi_0(0)\) or \(\alpha=\psi_{\nu+1}(0)\) then \(\operatorname{cof}(\alpha)=\alpha\) and \(\alpha[\eta]=\eta\)
  3. if \(\alpha=\psi_\omega(0)\) then \(\operatorname{cof}(\alpha)=\omega\) and \(\alpha[\eta]=\psi_\eta(0)\)
  4. if \(\alpha=\psi_{\nu}(\beta+1)\) then \(\operatorname{cof}(\alpha)=\omega\) and \(\alpha[\eta]=\psi_{\nu}(\beta)\cdot \eta\)
  5. if \(\alpha=\psi_{\nu}(\beta)\) and \(\operatorname{cof}(\beta)\in\{\omega\}\cup\{\Omega_{\mu+1}\mid\mu<\nu\}\) then \(\operatorname{cof}(\alpha)=\operatorname{cof}(\beta)\) and \(\alpha[\eta]=\psi_{\nu}(\beta[\eta])\)
  6. if \(\alpha=\psi_{\nu}(\beta)\) and \(\operatorname{cof}(\beta)\in\{\Omega_{\mu+1}\mid\mu\geq\nu\}\) then \(\operatorname{cof}(\alpha)=\omega\) and \(\alpha[\eta]=\psi_\nu(\beta[\gamma[\eta]])\) where \(\gamma[0]=\Omega_\mu\) and \(\gamma[\eta+1]=\psi_\mu(\beta[\gamma[\eta]])\)

See also

Other ordinal collapsing functions:

Madore's ψ function

Jäger'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