# 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 $$\Omega_{\alpha+1}=\aleph_{\alpha+1}$$.

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.

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

## 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 $$\mu \le \omega$$.

## 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

Now we assign a fundamental sequence for each limit ordinal below the Bachmann-Howard ordinal. 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. If $$\alpha$$ is a countable limit ordinal (i.e. $$\alpha$$ is a limit ordinal less than $$\Omega$$) then $$\text{cof}(\alpha)=\omega$$. The first uncountable ordinal $$\Omega$$ is the least ordinal whose cofinality greater than $$\omega$$ since $$\text{cof}(\Omega)=\Omega$$.

At first we define the normal form for ordinals

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

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

After this 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 the define 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 $$\gamma[\eta+1]=\psi_\mu(\beta[\gamma[\eta]])$$