## FANDOM

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Buchholz's psi functions are ordinal collapsing functions created by German mathematician Wilfried Buchholz. Although there are many Buchholz's psi functions, we explain the most famous one in this community, which is a hierarchy of single-argument functions $$\psi_\nu(\alpha)$$ introduced in 1986,[1] because the other ones are not currently used in this community. These functions are a simplified version of Feferman's $$\theta$$ functions, but nevertheless have the same strength as those.

## Definition

Buchholz defined his functions as follows:

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

where $$\omega$$ is the smallest infinite ordinal,

$$\Omega_\nu=\left\{\begin{array}{lcr}1\text{ if }\nu=0\\\aleph_\nu\text{ if }\nu>0\\\end{array}\right.$$

and $$P(\gamma)=\{\gamma_1,\cdots,\gamma_k\}$$ is the set of additive principal numbers in form $$\omega^\xi$$,

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

the sum of which gives this ordinal $$\gamma$$:

$$\gamma=\alpha_1+\alpha_2+\cdots+\alpha_k$$ where $$\alpha_1\geq\alpha_2\geq\cdots\geq\alpha_k$$ and $$\alpha_1,\alpha_2,\cdots,\alpha_k \in P(\gamma)$$.

Note: Greek letters always denotes ordinals. $$\textrm{On}$$ denotes the class of all ordinals. The convention $$\Omega_0$$ depends on the context even if we are focussing on Buchholz's function. Actually, Buchholz himself uses $$\Omega_0$$ as both of shorthands of $$1$$ and $$\omega$$.[1][2]

The limit of this notation is Takeuti-Feferman-Buchholz ordinal.

## Properties

Buchholz showed following properties of those functions:

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

## Explanation

Buchholz is working in Zermelo–Fraenkel set theory, that means every ordinal $$\alpha$$ is equal to set $$\{\beta|\beta<\alpha\}$$. Then condition $$C_\nu^0(\alpha)=\Omega_\nu$$ means that set $$C_\nu^0(\alpha)$$ includes all ordinals less than $$\Omega_\nu$$ in other words $$C_\nu^0(\alpha)=\{\beta|\beta<\Omega_\nu\}$$.

The condition $$C_\nu^{n+1}(\alpha) = C_\nu^n(\alpha) \cup \{\gamma | P(\gamma) \subseteq C_\nu^n(\alpha)\} \cup \{\psi_\mu(\xi) | \xi \in \alpha \cap C_\nu^n(\alpha) \wedge \mu \leq \omega\}$$ means that set $$C_\nu^{n+1}(\alpha)$$ includes:

1. all ordinals from previous set $$C_\nu^n(\alpha)$$,
2. all ordinals that can be obtained by summation the additively principal ordinals from previous set $$C_\nu^n(\alpha)$$,
3. all ordinals that can be obtained by applying ordinals less than $$\alpha$$ from the previous set $$C_\nu^n(\alpha)$$ as arguments of functions $$\psi_\mu$$, where $$\mu\le\omega$$.

That is why we can rewrite this condition as:

$$C_\nu^{n+1}(\alpha) = \{\beta+\gamma,\psi_\mu(\eta)|\beta, \gamma,\eta\in C_{\nu}^n(\alpha)\wedge\eta<\alpha \wedge \mu \leq \omega\}$$.

Thus union of all sets $$C_\nu^n (\alpha)$$ with $$n<\omega$$ i.e. $$C_\nu(\alpha) = \bigcup_{n < \omega} C_\nu^n (\alpha)$$ denotes the set of all ordinals which can be generated from ordinals $$<\aleph_\nu$$ by the functions + (addition) and $$\psi_{\mu}(\xi)$$, where $$\mu\le\omega$$ and $$\xi<\alpha$$.

Then $$\psi_\nu(\alpha) = \min\{\gamma | \gamma \not\in C_\nu(\alpha)\}$$ is the smallest ordinal that does not belong to this set.

### Examples

Consider the following examples:

$$C_0^0(\alpha)=\{0\} =\{\beta:\beta<1\}$$,

$$C_0(0)=\{0\}$$ (since no functions $$\psi(\eta<0)$$ and 0+0=0).

Then $$\psi_0(0)=1$$.

$$C_0(1)$$ includes $$\psi_0(0)=1$$ and all possible sums of natural numbers:

$$C_0(1)=\{0,1,2,...,\text{googol}, ...,\text{TREE(googol)},...\}$$.

Then $$\psi_0(1)=\omega$$ - first transfinite ordinal, which is greater than all natural numbers by its definition.

$$C_0(2)$$ includes $$\psi_0(0)=1, \psi_0(1)=\omega$$ and all possible sums of them.

Then $$\psi_0(2)=\omega^2$$.

For $$C_0(\omega)$$ we have set $$C_0(\omega)=\{0,\psi(0)=1,...,\psi(1)=\omega,...,\psi(2)=\omega^2,...,\psi(3)=\omega^3,...\}$$.

Then $$\psi_0(\omega)=\omega^\omega$$.

For $$C_0(\Omega)$$ we have set $$C_0(\Omega)=\{0,\psi(0)=1,...,\psi(1)=\omega,...,\psi(\omega)=\omega^\omega,...,\psi(\omega^\omega)=\omega^{\omega^\omega},...\}$$.

Then \begin{eqnarray*} & & \psi_0(\varepsilon_0)=\psi_0(\varepsilon_0+1) \\ & = & \cdots=\psi_0(\textrm{insert any countable ordinal above } \varepsilon_0 \textrm{which you like very much}) \\ & = & \cdots = \psi_0(\Omega)=\varepsilon_0. \end{eqnarray*} For $$C_0(\Omega+1)$$ we have set $$C_0(\Omega)=\{0,1,...,\psi_0(\Omega)=\varepsilon_0,...,\varepsilon_0+\varepsilon_0,...\psi_1(0)=\Omega,...\}$$.

Then $$\psi_0(\Omega+1)=\varepsilon_0\omega=\omega^{\varepsilon_0+1}$$.

$$\psi_0(\Omega2)=\varepsilon_1$$,

$$\psi_0(\Omega^2)=\zeta_0$$,

$$\psi_0(\Omega^\alpha(1+\beta)) = \varphi(\alpha,\beta)$$,

$$\psi_0(\Omega^\Omega)=\Gamma_0=\theta(\Omega,0)$$, using Feferman theta-function,

$$\psi_0(\Omega^{\Omega^\Omega})$$ is large Veblen ordinal,

$$\psi_0(\varepsilon_{\Omega+1})=\theta(\varepsilon_{\Omega+1},0)$$.

Now let's research how $$\psi_1$$ works:

$$C_1^0(\alpha)=\{\beta:\beta<\Omega_1\}=\{0,\psi(0)=1,2,...\text{googol},...,\psi_0(1)=\omega,...,\psi_0(\Omega)=\varepsilon_0,...$$

$$...,\psi_0(\Omega^\Omega)=\Gamma_0,...,\psi(\Omega^{\Omega^\Omega+\Omega^2}),...\}$$ i.e. includes all countable ordinals.

$$C_1(\alpha)$$ includes all possible sums of all countable ordinals. Then

$$\psi_1(0)=\Omega_1$$ first uncountable ordinal which is greater than all countable ordinal by its definition i.e. smallest number with cardinality $$\aleph_1$$.

$$C_1(1)=\{0,...,\psi_0(0)=\omega,...,\psi_1(0)=\Omega,...,\Omega+\omega,...,\Omega+\Omega,...\}$$

Then $$\psi_1(1)=\Omega\omega=\omega^{\Omega+1}$$.

Then $$\psi_1(2)=\Omega\omega^2=\omega^{\Omega+2}$$,

$$\psi_1(\psi_0(\Omega))=\Omega\varepsilon_0=\omega^{\Omega+\varepsilon_0}$$,

$$\psi_1(\psi_0(\Omega^\Omega))=\Omega\Gamma_0=\omega^{\Omega+\Gamma_0}$$,

$$\psi_1(\psi_1(0))=\psi_1(\Omega)=\Omega^2=\omega^{\Omega+\Omega}$$,

$$\psi_1(\psi_1(\psi_1(0)))=\omega^{\Omega+\omega^{\Omega+\Omega}}=\omega^{\Omega\cdot\Omega}=(\omega^{\Omega})^\Omega=\Omega^\Omega$$,

$$\psi_1^4(0)=\Omega^{\Omega^\Omega}$$,

$$\psi_1(\Omega_2)=\varepsilon_{\Omega+1}$$.

For case $$\psi(\Omega_2)$$ the set $$C_0(\Omega_2)$$ includes functions $$\psi_0$$ with all arguments less than $$\Omega_2$$ i.e. such arguments as $$0, \psi_1(0), \psi_1(\psi_1(0)), \psi_1^3(0),...$$

and then $$\psi_0(\Omega_2)=\psi_0(\psi_1(\Omega_2))=\psi_0(\varepsilon_{\Omega+1})$$.

In general case: $$\psi_0(\Omega_{\nu+1})=\psi_0(\psi_\nu(\Omega_{\nu+1}))=\psi_0(\varepsilon_{\Omega_\nu+1})=\theta(\varepsilon_{\Omega_\nu+1},0)$$.

We also can write:

$$\theta(\Omega_\nu,0)=\psi_0(\Omega_\nu^{\Omega_\nu})$$ ( for $$1\le\nu<\omega$$).

## Comparison between Buchholz’s and Veblen’s functions

### Up to Feferman–Schütte ordinal

Buchholz function Veblen function
a natural number $$n>0$$ is an abbriviation for $$\underbrace{\psi_0(0)+\cdots+\psi_0(0)}_{n\ \psi 's}$$ a natural number $$n>0$$ is an abbriviation for $$\underbrace{\varphi(0,0)+\cdots+\varphi(0,0)}_{n\ \varphi 's}$$
$$\psi_0(0)$$ $$\varphi(0,0)=1$$
$$\psi_0(0)+\psi_0(0)$$ $$\varphi(0,0) +\varphi(0,0)=2$$
$$\psi_0(1)$$ $$\varphi(0,1) =\omega$$
$$\psi_0(2)$$ $$\varphi(0,2) =\omega^2$$
$$\psi_0(\psi_0(1))$$ $$\varphi(0,\varphi(0,1)) =\omega^\omega$$
$$\psi_0(\psi_0(2))$$ $$\varphi(0,\varphi(0,2))=\omega^{\omega^2}$$
$$\psi_0(\psi_0(\psi_0(1)))$$ $$\varphi(0,\varphi(0,\varphi(0,1)))=\omega^{\omega^\omega}$$
$$\psi_0(\psi_1(0))$$ $$\varphi(1,0)=\varepsilon_0$$
$$\psi_0(\psi_1(0)+1)$$ $$\varphi(0,\varphi(1,0)+1)=\omega^{\varepsilon_0+1}=\varepsilon_0\omega$$
$$\psi_0(\psi_1(0)+2)$$ $$\varphi(0,\varphi(1,0)+2)=\omega^{\varepsilon_0+2}=\varepsilon_0\omega^2$$
$$\psi_0(\psi_1(0)+\psi_0(\psi_1(0)))$$ $$\varphi(0,\varphi(1,0)+\varphi(1,0))=\omega^{\varepsilon_0+\varepsilon_0}=\varepsilon_0^2$$
$$\psi_0(\psi_1(0)+\psi_0(\psi_1(0)+1))$$ $$\varphi(0,\varphi(0,\varphi(1,0)+1))=\omega^{\omega^{\varepsilon_0+1}}$$
$$\psi_0(\psi_1(0)+\psi_0(\psi_1(0)+\psi_0(\psi_1(0)+1)))$$ $$\varphi(0,\varphi(0,\varphi(0,\varphi(1,0)+1)))=\omega^{\omega^{\omega^{\varepsilon_0+1}}}$$
$$\psi_0(\psi_1(0)+\psi_1(0))$$ $$\varphi(1,1)=\varepsilon_1$$
$$\psi_0(\psi_1(0)+\psi_1(0)+\psi_1(0))$$ $$\varphi(1,2)=\varepsilon_2$$
$$\psi_0(\psi_1(1))$$ $$\varphi(1,\varphi(0,1))=\varepsilon_\omega$$
$$\psi_0(\psi_1(2))$$ $$\varphi(1,\varphi(0,2))=\varepsilon_{\omega^2}$$
$$\psi_0(\psi_1(\psi_0(\psi_1(0))))$$ $$\varphi(1,\varphi(1,0))=\varepsilon_{\varepsilon_0}$$
$$\psi_0(\psi_1(\psi_0(\psi_1(\psi_0(\psi_1(0))))))$$ $$\varphi(1,\varphi(1,\varphi(1,0)))=\varepsilon_{\varepsilon_{\varepsilon_0}}$$
$$\psi_0(\psi_1(\psi_1(0)))$$ $$\varphi(2,0)=\zeta_0$$
$$\psi_0(\psi_1(\psi_1(0))+1)$$ $$\varphi(0,\varphi(2,0)+1)=\omega^{\zeta_0+1}$$
$$\psi_0(\psi_1(\psi_1(0))+\psi_0(\psi_1(\psi_1(0))+1))$$ $$\varphi(0,\varphi(0,\varphi(2,0)+1))=\omega^{\omega^{\zeta_0+1}}$$
$$\psi_0(\psi_1(\psi_1(0))+\psi_1(0))$$ $$\varphi(1,\varphi(2,0)+1)=\varepsilon_{\zeta_0+1}$$
$$\psi_0(\psi_1(\psi_1(0))+\psi_1(0)+1)$$ $$\varphi(0,\varphi(1,\varphi(2,0)+1)+1)=\omega^{\varepsilon_{\zeta_0+1}+1}$$
$$\psi_0(\psi_1(\psi_1(0))+\psi_1(0)+\psi_1(0))$$ $$\varphi(1,\varphi(2,0)+2)=\varepsilon_{\zeta_0+2}$$
$$\psi_0(\psi_1(\psi_1(0))+\psi_1(1))$$ $$\varphi(1,\varphi(2,0)+\varphi(0,1))=\varepsilon_{\zeta_0+\omega}$$
$$\psi_0(\psi_1(\psi_1(0))+\psi_1(\psi_0(\psi_1(\psi_1(0))+\psi_1(1))))$$ $$\varphi(1,\varphi(1,\varphi(2,0)+\varphi(0,1))=\varepsilon_{\varepsilon_{\zeta_0+\omega}}$$
$$\psi_0(\psi_1(\psi_1(0))+\psi_1(\psi_1(0)))$$ $$\varphi(2,1)=\zeta_1$$
$$\psi_0(\psi_1(\psi_1(0)+1))$$ $$\varphi(2,\varphi(0,1))=\zeta_\omega$$
$$\psi_0(\psi_1(\psi_1(0)+\psi_0(\psi_1(\psi_1(0)+1))))$$ $$\varphi(2,\varphi(2,\varphi(0,1)))=\zeta_{\zeta_\omega}$$
$$\psi_0(\psi_1(\psi_1(0)+\psi_1(0)))$$ $$\varphi(3,0)=\eta_0$$
$$\psi_0(\psi_1(\psi_1(0)+\psi_1(0))+\psi_1(\psi_1(0)+\psi_1(0)))$$ $$\varphi(3,1)=\eta_1$$
$$\psi_0(\psi_1(\psi_1(0)+\psi_1(0)+1))$$ $$\varphi(3,\varphi(0,1))=\eta_{\omega}$$
$$\psi_0(\psi_1(\psi_1(0)+\psi_1(0)+\psi_0(\psi_1(\psi_1(0)+\psi_1(0)+1)))$$ $$\varphi(3,\varphi(3,\varphi(0,1)))=\eta_{\eta_{\omega}}$$
$$\psi_0(\psi_1(\psi_1(0)+\psi_1(0)+\psi_1(0)))$$ $$\varphi(4,0)$$
$$\psi_0(\psi_1(\psi_1(1)))$$ $$\varphi(\varphi(0,1),0) = \varphi(\omega,0)$$
$$\psi_0(\psi_1(\psi_1(\psi_0(\psi_1(\psi_1(1))))))$$ $$\varphi(\varphi(\varphi(0,1),0),0) = \varphi(\varphi(\omega,0),0)$$
$$\psi_0(\psi_1(\psi_1(\psi_1(0))))$$ $$\varphi(1,0,0)=\Gamma_0$$

### Up to Large Veblen ordinal

Buchholz function Veblen function
$$\psi_0(\psi_1(\psi_1(\psi_1(0)))+1)$$ $$\varphi(0,\varphi(1,0,0)+1)=\omega^{\Gamma_0+1}$$
$$\psi_0(\psi_1(\psi_1(\psi_1(0)))+\psi_1(0))$$ $$\varphi(1,\varphi(1,0,0)+1)=\varepsilon_{\Gamma_0+1}$$
$$\psi_0(\psi_1(\psi_1(\psi_1(0)))+\psi_1(1))$$ $$\varphi(1,\varphi(1,0,0)+\varphi(0,1))=\varepsilon_{\Gamma_0+\omega}$$
$$\psi_0(\psi_1(\psi_1(\psi_1(0)))+\psi_1(\psi_1(0)))$$ $$\varphi(2,\varphi(1,0,0)+1)=\zeta_{\Gamma_0+1}$$
$$\psi_0(\psi_1(\psi_1(\psi_1(0)))+\psi_1(\psi_1(0)+1))$$ $$\varphi(2,\varphi(1,0,0)+\varphi(0,1))=\zeta_{\Gamma_0+\omega}$$
$$\psi_0(\psi_1(\psi_1(\psi_1(0)))+\psi_1(\psi_1(0)+\psi_1(0)))$$ $$\varphi(3,\varphi(1,0,0)+1)=\eta_{\Gamma_0+1}$$
$$\psi_0(\psi_1(\psi_1(\psi_1(0)))+\psi_1(\psi_1(1)))$$ $$\varphi(\varphi(0,1),\varphi(1,0,0)+1)=\varphi(\omega,\Gamma_0+1)$$
$$\psi_0(\psi_1(\psi_1(\psi_1(0)))+\psi_1(\psi_1(\psi_0(\psi_1(\psi_1(1))))))$$ $$\varphi(\varphi(\varphi(0,1),0),\varphi(1,0,0)+1)=\varphi(\varphi(\omega,0),\Gamma_0+1)$$
$$\psi_0(\psi_1(\psi_1(\psi_1(0)))+\psi_1(\psi_1(\psi_0(\psi_1(\psi_1(\psi_1(0)))))))$$ $$\varphi(\varphi(1,0,0),1)=\varphi(\Gamma_0,1)$$
$$\psi_0(\psi_1(\psi_1(\psi_1(0)))+\psi_1(\psi_1(\psi_0(\psi_1(\psi_1(\psi_1(0)))))+1))$$ $$\varphi(\varphi(1,0,0),\varphi(0,1))=\varphi(\Gamma_0,\omega)$$
$$\psi_0(\psi_1(\psi_1(\psi_1(0)))+\psi_1(\psi_1(\psi_0(\psi_1(\psi_1(\psi_1(0)))))$$

$$+\psi_0(\psi_1(\psi_1(\psi_1(0))))))$$

$$\varphi(\varphi(1,0,0),\varphi(1,0,0))=\varphi(\Gamma_0,\Gamma_0)$$
$$\psi_0(\psi_1(\psi_1(\psi_1(0)))+\psi_1(\psi_1(\psi_0(\psi_1(\psi_1(\psi_1(0)))))+\psi_1(0)))$$ $$\varphi(\varphi(1,0,0)+1,0)=\varphi(\Gamma_0+1,0)$$
$$\psi_0(\psi_1(\psi_1(\psi_1(0)))+\psi_1(\psi_1(\psi_0(\psi_1(\psi_1(\psi_1(0))))+1)))$$ $$\varphi(\varphi(0,\varphi(1,0,0)+1),0)=\varphi(\omega^{\Gamma_0+1},0)$$
$$\psi_0(\psi_1(\psi_1(\psi_1(0)))+\psi_1(\psi_1(\psi_0(\psi_1(\psi_1(\psi_1(0)))+1))))$$ $$\varphi(\varphi(0,\varphi(0,\varphi(1,0,0)+1)),0)=\varphi(\omega^{\omega^{\Gamma_0+1}},0)$$
$$\psi_0(\psi_1(\psi_1(\psi_1(0)))+\psi_1(\psi_1(\psi_0(\psi_1(\psi_1(\psi_1(0)))+\psi_1(0)))))$$ $$\varphi(\varphi(1,\varphi(1,0,0)+1),0)=\varphi(\varepsilon_{\Gamma_0+1},0)$$
$$\psi_0(\psi_1(\psi_1(\psi_1(0)))+\psi_1(\psi_1(\psi_0(\psi_1(\psi_1(\psi_1(0)))+\psi_1(1)))))$$ $$\varphi(\varphi(1,\varphi(1,0,0)+\varphi(0,1)),0)=\varphi(\varepsilon_{\Gamma_0+\omega},0)$$
$$\psi_0(\psi_1(\psi_1(\psi_1(0)))+\psi_1(\psi_1(\psi_0(\psi_1(\psi_1(\psi_1(0)))+\psi_1(\psi_1(0))))))$$ $$\varphi(\varphi(2,\varphi(1,0,0)+1),0)=\varphi(\zeta_{\Gamma_0+1},0)$$
$$\psi_0(\psi_1(\psi_1(\psi_1(0)))+\psi_1(\psi_1(\psi_0(\psi_1(\psi_1(\psi_1(0)))+\psi_1(\psi_1(0)+1)))))$$ $$\varphi(\varphi(2,\varphi(1,0,0)+\varphi(0,1)),0)=\varphi(\zeta_{\Gamma_0+\omega},0)$$
$$\psi_0(\psi_1(\psi_1(\psi_1(0)))+\psi_1(\psi_1(\psi_0(\psi_1(\psi_1(\psi_1(0)))+\psi_1(\psi_1(0)+\psi_1(0))))))$$ $$\varphi(\varphi(3,\varphi(1,0,0)+1),0)=\varphi(\eta_{\Gamma_0+1},0)$$
$$\psi_0(\psi_1(\psi_1(\psi_1(0)))+\psi_1(\psi_1(\psi_0(\psi_1(\psi_1(\psi_1(0)))+\psi_1(\psi_1(1))))))$$ $$\varphi(\varphi(\varphi(0,1),\varphi(1,0,0)+1),0)=\varphi(\varphi(\omega,\Gamma_0+1),0)$$
$$\psi_0(\psi_1(\psi_1(\psi_1(0)))+\psi_1(\psi_1(\psi_1(0))))$$ $$\varphi(1,0,1)=\Gamma_1$$
$$\psi_0(\psi_1(\psi_1(\psi_1(0))+1))$$ $$\varphi(1,0,\varphi(0,1))=\Gamma_{\omega}$$
$$\psi_0(\psi_1(\psi_1(\psi_1(0))+\psi_0(\psi_1(\psi_1(\psi_1(0))+1))))$$ $$\varphi(1,0,\varphi(1,0,\varphi(0,1)))=\Gamma_{\Gamma_{\omega}}$$
$$\psi_0(\psi_1(\psi_1(\psi_1(0))+\psi_1(0)))$$ $$\varphi(1,1,0)$$
$$\psi_0(\psi_1(\psi_1(\psi_1(0))+\psi_1(0))+\psi_1(\psi_1(\psi_1(0))+\psi_1(0)))$$ $$\varphi(1,1,1)$$
$$\psi_0(\psi_1(\psi_1(\psi_1(0))+\psi_1(0)+1))$$ $$\varphi(1,1,\varphi(0,1)) = \varphi(1,1,\omega)$$
$$\psi_0(\psi_1(\psi_1(\psi_1(0))+\psi_1(0)+\psi_1(0)))$$ $$\varphi(1,2,0)$$
$$\psi_0(\psi_1(\psi_1(\psi_1(0))+\psi_1(1)))$$ $$\varphi(1,\varphi(0,1),0) = \varphi(1,\omega,0)$$
$$\psi_0(\psi_1(\psi_1(\psi_1(0))+\psi_1(\psi_1(0))))$$ $$\varphi(2,0,0)$$
$$\psi_0(\psi_1(\psi_1(\psi_1(0)+1)))$$ $$\varphi(\varphi(0,1),0,0) = \varphi(\omega,0,0)$$
$$\psi_0(\psi_1(\psi_1(\psi_1(0)+\psi_0(\psi_1(\psi_1(\psi_1(0)+1))))))$$ $$\varphi(\varphi(\varphi(0,1),0,0),0,0) = \varphi(\varphi(\omega,0,0),0,0)$$
$$\psi_0(\psi_1(\psi_1(\psi_1(0)+\psi_1(0))))$$ $$\varphi(1,0,0,0)$$
$$\psi_0(\psi_1(\psi_1(\psi_1(0)+\psi_1(0)+\psi_1(0))))$$ $$\varphi(1,0,0,0,0)$$
$$\psi_0(\psi_1(\psi_1(\psi_1(1))))$$ Small Veblen ordinal
$$\psi(\psi_1(\psi_1(\psi_1(\psi_1(0)))))$$ Large Veblen ordinal

## Ordinal notation

Buchholz defined an ordinal notation $$(OT,<)$$ associated to $$\psi$$ as an array notation.[1] We explain the original definition of $$(OT,<)$$.

We simultaneously define the sets $$T$$ and $$PT$$ of formal strings consisting of $$0$$, $$D_v$$ indexed by an $$v \in \omega+1$$, braces, and commas in the following recursive way:

1. $$0 \in T$$ and $$0 \in PT$$.
2. For any $$(v,a) \in (\omega+1) \times T$$, $$D_va \in T$$ and $$D_va \in PT$$.
3. For any $$(a_i)_{i=0}^{k} \in PT^{k+1}$$ with $$k \in \mathbb{N} \setminus \{0\}$$, $$(a_0,\ldots,a_k) \in T$$. More precisely, this condition can be formulated without using ellipses in the following way:
1. For any $$(a_0,a_1) \in PT^2$$, $$(a_0,a_1) \in T$$.
2. For any $$(a_0,a_1) \in T \times PT$$, if there exists formal strings $$s$$ such that $$a_0 = (s)$$, then $$(s,a_1) \in T$$.

Namely, $$T$$ and $$PT$$ are the smallest sets satisfying the conditions above. An element of $$T$$ is called a term, and an element of $$PT$$ is called a principal term. By the definition, $$T$$ is a recursive set, $$PT$$ is a recursive subset of $$T$$, and every term is $$0$$, a principal term, or an array of principal terms of length $$\geq 2$$ in braces.

We also denote an $$a \in PT$$ by $$(a)$$. Since the clause 3 in the definition of $$T$$ and $$PT$$ is applicable only to arrays of length $$\geq 2$$, the additional convention does not cause serious ambiguity. By the convention, every term can be uniquely expressed as either $$0$$ or a non-empty array of principal terms in braces.

We define a binary relation $$a < b$$ on $$T$$ in the following recursive way:

1. If $$b = 0$$, then $$a < b$$ is false.
2. If $$a = 0$$, then $$a < b$$ is equivalent to $$b \neq 0$$.
3. Suppose $$a \neq 0$$ and $$b \neq 0$$.
1. If $$a = D_ua'$$ and $$b = D_vb'$$ for some $$((u,a'),(v,b')) \in ((\omega+1) \times T)^2$$, $$a < b$$ is equivalent to that either one of the following holds:
1. $$u < v$$.
2. $$u = v$$ and $$a' < b'$$.
2. If $$a = (a_0,\ldots,a_n)$$ and $$b = (b_0,\ldots,b_m)$$ for some $$(n,m) \in \mathbb{N}^2$$ with $$1 \leq n+m$$, $$a < b$$ is equivalent to that either one of the following holds:
1. $$n < m$$ and $$a_i = b_i$$ for any $$i \in \mathbb{N}$$ with $$i \leq n$$
2. There exists some $$k \in \mathbb{N}$$ such that $$k \leq \min\{n,m\}$$, $$a_k < b_k$$, and $$a_i = b_i$$ for any $$i \in \mathbb{N}$$ with $$i < k$$.

By the definition, $$<$$ is a recursive strict total ordering on $$T$$. We abbreviate $$(a < b) \lor (a = b)$$ to $$a \leq b$$. Although $$\leq$$ itself is not a well-ordering, its restriction to a recursive subset $$OT \subset T$$, which will be introduced later, forms a well-ordering.

In order to define $$OT$$, we define a subset $$G_ua \subset T$$ for each $$(u,a) \in (\omega+1) \times T$$ in the following recursive way:

1. If $$a = 0$$, then $$G_ua = \emptyset$$.
2. Suppose that $$a = D_va'$$ for some $$(v,a') \in (\omega+1) \times T$$.
1. If $$u \leq v$$, then $$G_ua = \{a'\} \cup G_ua'$$.
2. If $$u > v$$, then $$G_ua = \emptyset$$.
3. If $$a = (a_0,\ldots,a_k)$$ for some $$(a_i)_{i=0}^{k} \in PT^{k+1}$$ with $$k \in \mathbb{N} \setminus \{0\}$$,$$G_ua = \bigcup_{i=0}^{k} G_ua_i$$.

By the definition, $$b \in G_ua$$ is a recursive relation on $$(u,a,b) \in (\omega+1) \times T^2$$.

Finally, we define a subset $$OT \subset T$$ in the following recursive way:

1. $$0 \in OT$$.
2. For any $$(v,a) \in (\omega+1) \times T$$, $$D_va \in OT$$ is equivalent to $$a \in OT$$ and $$a' < a$$ for any $$a' \in G_va$$.
3. For any $$(a_i)_{i=0}^{k} \in PT^{k+1}$$, $$(a_0,\ldots,a_k) \in OT$$ is equivalent to $$(a_i)_{i=0}^{k} \in OT^{k+1}$$ and $$a_k \leq \cdots \leq a_0$$.

By the definition, $$OT$$ is a recursive subset of $$T$$. An element of $$OT$$ is called an ordinal term.

We define a map \begin{eqnarray*} o \colon OT & \to & C_0(\epsilon_{\Omega_{\omega}+1}) \\ a & \mapsto & o(a) \end{eqnarray*} in the following inductive way:

1. If $$a = 0$$, then $$o(a) = 0$$.
2. If $$a = D_va'$$ for some $$(v,a') \in (\omega+1) \times OT$$, then $$o(a) = \psi_v(o(a'))$$.
3. If $$a = (a_0,\ldots,a_k)$$ for some $$(a_i)_{i=0}^{k} \in (OT \cap PT)^{k+1}$$ with $$k \in \mathbb{N} \setminus \{0\}$$, then $$o(a) = o(a_0) \# \cdots \# o(a_k)$$, where $$\#$$ denotes the descending sum of ordinals, which coincides with the usual addition by the definition of $$OT$$.

Buchholz verified that the map $$o$$ satisfies the following:[1]

• The map $$o$$ is an order-preserving bijective map with respect to $$<$$ and $$\in$$. In particular, $$<$$ is a recursive strict well-ordering on $$OT$$.
• For any $$a \in OT$$ satisfying $$a < D_10$$, $$o(a)$$ coincides with the ordinal type of $$<$$ restricted to the countable subset $$\{x \in OT \mid x < a\}$$.
• The ordinal $$\psi_0(\varepsilon_{\Omega_{\omega}+1})$$ coincides with the ordinal type of $$<$$ restricted to the recursive subset $$\{x \in OT \mid x < D_10\}$$. In particular, $$(\{x \in OT \mid x < D_10\},<)$$ is an ordinal notation equivalent to $$\psi_0(\varepsilon_{\Omega_{\omega}+1})$$, and hence $$(OT,<)$$ is an ordinal notation whose ordinal type is strictly greater than the countable limit of $$\psi$$.
• For any $$v \in \mathbb{N} \setminus \{0\}$$, the well-foundedness of $$<$$ restricted to the recursive subset $$\{x \in OT \mid x < D_0D_{v+1}0\}$$ in the sense of the non-existence of a primitive recursive infinite descending sequence is not provable under $$\textrm{ID}_v$$.
• The well-foundedness of $$<$$ restricted to the recursive subset $$\{x \in OT \mid x < D_0D_{\omega}0\}$$ in the same sense above is not provable under $$\Pi_1^1-\textrm{CA}_0$$.

## Extension

We introduce the extension of Buchholz's definition by Googology WIki user Denis Maksudov as follows[3]:

• $$C_\nu^0(\alpha) = \{\beta|\beta<\Omega_\nu\}$$,
• $$C_\nu^{n+1}(\alpha) = \{\beta+\gamma,\psi_\mu(\eta)|\mu,\beta, \gamma,\eta\in C_{\nu}^n(\alpha)\wedge\eta<\alpha\}$$,
• $$C_\nu(\alpha) = \bigcup_{n < \omega} C_\nu^n (\alpha)$$,
• $$\psi_\nu(\alpha) = \min\{\gamma | \gamma \not\in C_\nu(\alpha)\}$$,

where

$$\Omega_\nu=\left\{\begin{array}{lcr} 1\text{ if }\nu=0 \\ \text{smallest ordinal with cardinality }\aleph_\nu \text{ if }\nu>0 \\ \end{array}\right.$$

There is only one little detail difference with original Buchholz definition: ordinal $$\mu$$ is not limited by $$\omega$$, now ordinal $$\mu$$ belongs to previous set $$C_n$$.

For example if $$C_0^0(1)=\{0\}$$ then $$C_0^1(1)=\{0,\psi_0(0)=1\}$$ and $$C_0^2(1)=\{0,...,\psi_1(0)=\Omega\}$$ and $$C_0^3(1)=\{0,...,\psi_\Omega(0)=\Omega_\Omega\}$$ and so on.

Limit of this notation must be omega fixed point $$\psi_0(\Omega_{\Omega_{\Omega_{\cdots}}})=\psi_0(\psi_{\psi_{\cdots}(0)}(0)) = \psi_0(\Lambda)$$ (see the next section).

## Normal form and fundamental sequences

### Normal form

The normal form for 0 is 0. If $$\alpha$$ is a nonzero ordinal number $$\alpha<\Lambda=\text{min}\{\beta|\psi_\beta(0)=\beta\}$$ then the normal form for $$\alpha$$ is $$\alpha=\psi_{\nu_1}(\beta_1)+\psi_{\nu_2}(\beta_2)+\cdots+\psi_{\nu_k}(\beta_k)$$ where $$k$$ is a positive integer and $$\psi_{\nu_1}(\beta_1)\geq\psi_{\nu_2}(\beta_2)\geq\cdots\geq\psi_{\nu_k}(\beta_k)$$ and each $$\nu_i$$, $$\beta_i$$ are ordinals satisfying $$\beta_i \in C_{\nu_i}(\beta_i)$$ also written in normal form. More precisely, the normality of an expression of an ordinal can be described in a recursive way with respect to the corresponding ordinal notation system extending the original ordinal notation system $$(OT,<)$$ explained above.

### 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. If $$\alpha$$ is a successor ordinal then $$\text{cof}(\alpha)=1$$ and the fundamental sequence has only one element $$\alpha[0]=\alpha-1$$. If $$\alpha$$ is a limit ordinal then $$\text{cof}(\alpha)\in\{\omega\}\cup\{\Omega_{\mu+1}|\mu\geq 0\}$$.

Although a system of fundamental sequences is not unique, there is a canonical choice of fundamental sequences in this community given by Denis. For nonzero ordinals $$\alpha<\Lambda$$, written in normal form, fundamental sequences are defined as follows:

1. If $$\alpha=\psi_{\nu_1}(\beta_1)+\psi_{\nu_2}(\beta_2)+\cdots+\psi_{\nu_k}(\beta_k)$$ where $$k\geq2$$ then $$\text{cof}(\alpha)=\text{cof}(\psi_{\nu_k}(\beta_k))$$ and $$\alpha[\eta]=\psi_{\nu_1}(\beta_1)+\cdots+\psi_{\nu_{k-1}}(\beta_{k-1})+(\psi_{\nu_k}(\beta_k)[\eta])$$,
2. If $$\alpha=\psi_{0}(0)=1$$, then $$\text{cof}(\alpha)=1$$ and $$\alpha[0]=0$$,
3. If $$\alpha=\psi_{\nu+1}(0)$$, then $$\text{cof}(\alpha)=\Omega_{\nu+1}$$ and $$\alpha[\eta]=\Omega_{\nu+1}[\eta]=\eta$$,
4. If $$\alpha=\psi_{\nu}(0)$$ and $$\text{cof}(\nu)\in\{\omega\}\cup\{\Omega_{\mu+1}|\mu\geq 0\}$$, then $$\text{cof}(\alpha)=\text{cof}(\nu)$$ and $$\alpha[\eta]=\psi_{\nu[\eta]}(0)=\Omega_{\nu[\eta]}$$,
5. If $$\alpha=\psi_{\nu}(\beta+1)$$ then $$\text{cof}(\alpha)=\omega$$ and $$\alpha[\eta]=\psi_{\nu}(\beta)\cdot \eta$$ (and note: $$\psi_\nu(0)=\Omega_\nu$$),
6. If $$\alpha=\psi_{\nu}(\beta)$$ and $$\text{cof}(\beta)\in\{\omega\}\cup\{\Omega_{\mu+1}|\mu<\nu\}$$ then $$\text{cof}(\alpha)=\text{cof}(\beta)$$ and $$\alpha[\eta]=\psi_{\nu}(\beta[\eta])$$,
7. If $$\alpha=\psi_{\nu}(\beta)$$ and $$\text{cof}(\beta) = \Omega_{\mu+1}$$ for a $$\mu\geq\nu$$ then $$\text{cof}(\alpha)=\omega$$ and $$\alpha[\eta]=\psi_{\nu}(\beta[\gamma[\eta]])$$ where $$\left\{\begin{array}{lcr} \gamma[0]=\Omega_\mu \\ \gamma[\eta+1]=\psi_\mu(\beta[\gamma[\eta]])\\ \end{array}\right.$$.

It is an extension of the system of fundamental sequences up to $$\psi_0(\varepsilon_{\Omega_{\omega}+1})$$ in Buchholz hierarchy given by modifying the rule ([].5) (ii) in recursive definition of the $$\textrm{dom}$$ function and $$[]$$ in Buchholz's original paper[1] by the rule 6 in the definition of $$[]$$ in p.6 in Buchholz's another paper[2] applied to the convention $$\Omega_0 = 1$$ except for the minor differences related to the difference $$\omega[n] = n+1$$ in the original definition and $$\omega[n] = n$$ in the definition here. (Please remember that $$\Omega_0$$ is defined as $$1$$ in the original paper, while it is defined as $$\omega$$ in the other paper.) More precisely, the fundamental sequence of $$\psi_0(2) = \omega \times \omega$$ is given as $$\omega \times \omega [n] = \omega \times (n+1)$$ in the original definition while we have $$\omega \times \omega[n] = \omega \times n$$ in the definition here, and the fundamental sequence of $$\psi_{\omega}(0) = \Omega_{\omega}$$ is given as $$\Omega_{\omega}[n] = \Omega_{n+1}$$ while we have $$\Omega_{\omega}[n] = \Omega_n$$ in the definition here.

If $$\alpha=\Lambda$$ then $$\text{cof}(\alpha)=\omega$$ and $$\alpha[0]=0$$ and $$\alpha[\eta+1]=\psi_{\alpha[\eta]}(0)=\Omega_{\alpha[\eta]}$$.

For comparison of ordinals written in normal form use the following property:

if $$\alpha<\beta$$ and $$1\le\eta<\omega$$ then $$\left\{\begin{array}{lcr} 0<\psi_\alpha(\gamma)\cdot\eta <\psi_\beta(\delta)\\ 0<\psi_\gamma(\alpha)\cdot\eta<\psi_\gamma(\beta)\\ \end{array}\right.$$

The comparison of expressions of ordinals can be described in a recursive way with respect to the corresponding ordinal notation system extending the original ordinal notation system $$OT$$.[1] In particular, the system of fundamental sequences above induce a recursive system of fundamental sequences on the corresponding ordinal notation, and hence the fast-growing hierarchy associated to it is recursive.

## Common Misconceptions

Here is a list of some common misconceptions regarding Buchholz's function, which appear in many introductory articles on Buchholz's function:

Common misconceptions Fact Reason
Buchholz's function is computable. Buchholz's function is not a computable function, despite the fact that the associated ordinal notation is computable. Turing machine defines a function whose domain and codomain are sets of tuple of natural numbers, neither of which includes transfinite ordinals.
$$\psi_0(\Omega_{\omega}+1)$$ is ill-defined. The ordinal $$\psi_{\nu}(\alpha)$$ is defined for all $$(\nu,\alpha) \in (\omega+1) \times \textrm{On}$$. Buchholz's original definition uses transfinite recursion on $$\alpha$$ and restricts $$v\le\omega$$.
$$\psi_0(\alpha)$$ with respect to Buchholz's function coincides with $$\psi_0(\alpha)$$ with respect to extended Buchholz's function. Buchholz's function is neither a restriction of extended Buchholz's function nor extended Buchholz's function itself. However, some values correspond, such as $$\psi_0(0)$$. Some ordinals don't correspond from extended Buchholz function to Buchholz function, such as $$\psi_0(\psi_{\psi_1(0)}(0))$$.
$$\psi_0(\varepsilon_0+1) = \varepsilon_0 \times \omega$$ $$\psi_0(\varepsilon_0+1) = \varepsilon_0$$. This is due to the fact that the $$\psi_0$$ function equals $$\varepsilon_0$$ for all inputs between $$\varepsilon_0$$ and $$\Omega$$ inclusive. This is because $$\varepsilon_0\notin C_0(\varepsilon_0+1)$$.
$$\psi_0(\psi_1(\psi_2(\psi_3(0)))) = \psi_0(\psi_3(0))$$ $$\psi_0(\psi_1(\psi_2(\psi_3(0)))) = \psi_0(\psi_2(0))$$. In fact, for any ordinal $$\alpha\geq\psi_2(0)$$, we have $$\psi_0(\psi_1(\alpha))=\psi_0(\psi_2(0))$$. This is because $$\psi_1(\psi_2(\psi_3(0)))\notin C_0(\psi_1(\psi_2(\psi_3(0))))$$.
The sequence $$\psi_0(0), \psi_0(\psi_1(0)), \psi_0(\psi_1(\psi_2(0))), \psi_0(\psi_1(\psi_2(\psi_3(0)))), \ldots$$ has a limit of $$\psi_0(\psi_{\omega}(0))$$. The sequence is an eventually constant sequence with limit $$\psi_0(\psi_2(0))$$. This is because for $$n\ge 3$$, $$\psi_1(\psi_2(\cdots\psi_n(0)\cdots))\notin C_0(\psi_1(\psi_2(\cdots\psi_n(0)\cdots)))$$.
It equals the least omega fixed point. See also $$\psi_0(\psi_I(0))$$. The ordinal $$\psi_I(0)$$ equals $$I$$. This is because $$C^0_I(0)$$ is defined as $$\Omega_I=I$$.
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