# Buchholz psi functions

Buchholz's psi-functions are a hierarchy of single-argument ordinal functions $\psi _{\nu }(\alpha )$ introduced by German mathematician Wilfried Buchholz in 1986. These functions are a simplified version of the $\theta$-functions, but nevertheless have the same strength as those. Later on this approach was extended by Jäger and Schütte.

## Definition

Buchholz defined his functions as follows. Define:
* Ω$_ξ$ = ω$_ξ$ if ξ > 0, Ω$_0$ = 1
The functions ψ$_v$(α) for α an ordinal, v an ordinal at most ω, are defined by induction on α as follows:
* ψ$_v$(α) is the smallest ordinal not in C$_v$(α)
where C$_v$(α) is the smallest set such that
* C$_v$(α) contains all ordinals less than Ω$_v$
* C$_v$(α) is closed under ordinal addition
* C$_v$(α) is closed under the functions ψ$_u$ (for u≤ω) applied to arguments less than α.
The limit of this notation is the Takeuti–Feferman–Buchholz ordinal.

## Properties

Let $P$ be the class of additively principal ordinals. Buchholz showed following properties of this functions:
* $\psi _{\nu }(0)=\Omega _{\nu },$$^{:197}$
* $\psi _{\nu }(\alpha )\in P,$$^{:197}$
* $\psi _{\nu }(\alpha +1)=\min\{\gamma \in P:\psi _{\nu }(\alpha )<\gamma \}{\text{ if }}\alpha \in C_{\nu }(\alpha ),$$^{:199}$
* $\Omega _{\nu }\leq \psi _{\nu }(\alpha )<\Omega _{\nu +1}$$^{:197}$
* $\psi _{0}(\alpha )=\omega ^{\alpha }{\text{ if }}\alpha <\varepsilon _{0},$$^{:199}$
* $\psi _{\nu }(\alpha )=\omega ^{\Omega _{\nu }+\alpha }{\text{ if }}\alpha <\varepsilon _{\Omega _{\nu }+1}{\text{ and }}\nu \neq 0,$$^{:199}$
* $\theta (\varepsilon _{\Omega _{\nu }+1},0)=\psi (\varepsilon _{\Omega _{\nu }+1}){\text{ for }}0<\nu \leq \omega .$$^{:206}$

## Fundamental sequences and normal form for Buchholz's function
### Normal form

The normal form for 0 is 0. If $\alpha$ is a nonzero ordinal number $\alpha <\Omega _{\omega }$ 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 $\nu _{i}\in \mathbb {N} _{0},k\in \mathbb {N} _{>0},\beta _{i}\in C_{\nu _{i}}(\beta _{i})$ and $\psi _{\nu _{1}}(\beta _{1})\geq \psi _{\nu _{2}}(\beta _{2})\geq \cdots \geq \psi _{\nu _{k}}(\beta _{k})$ and each $\beta _{i}$ is also written in normal form.

### Fundamental sequences

The fundamental sequence for an ordinal number $\alpha$ with cofinality $\operatorname {cof} (\alpha )=\beta$ 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 $\operatorname {cof} (\alpha )=1$ and the fundamental sequence has only one element $\alpha [0]=\alpha -1$. If $\alpha$ is a limit ordinal then $\operatorname {cof} (\alpha )\in \{\omega \}\cup \{\Omega _{\mu +1}\mid \mu \geq 0\}$.
For nonzero ordinals $\alpha <\Omega _{\omega }$, written in normal form, fundamental sequences are defined as follows:
* If $\alpha =\psi _{\nu _{1}}(\beta _{1})+\psi _{\nu _{2}}(\beta _{2})+\cdots +\psi _{\nu _{k}}(\beta _{k})$ where $k\geq 2$ then $\operatorname {cof} (\alpha )=\operatorname {cof} (\psi _{\nu _{k}}(\beta _{k}))$ and $\alpha [\eta ]=\psi _{\nu _{1}}(\beta _{1})+\cdots +\psi _{\nu _{k-1}}(\beta _{k-1})+(\psi _{\nu _{k}}(\beta _{k})[\eta ]),$
* If $\alpha =\psi _{0}(0)=1$, then $\operatorname {cof} (\alpha )=1$ and $\alpha [0]=0$,
* If $\alpha =\psi _{\nu +1}(0)$, then $\operatorname {cof} (\alpha )=\Omega _{\nu +1}$ and $\alpha [\eta ]=\Omega _{\nu +1}[\eta ]=\eta$,
* If $\alpha =\psi _{\nu }(\beta +1)$ then $\operatorname {cof} (\alpha )=\omega$ and $\alpha [\eta ]=\psi _{\nu }(\beta )\cdot \eta$ (and note: $\psi _{\nu }(0)=\Omega _{\nu }$),
* 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 $\alpha [\eta ]=\psi _{\nu }(\beta [\eta ])$,
* 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 $\left\{{\begin{array}{lcr}\gamma [0]=\Omega _{\mu }\\\gamma [\eta +1]=\psi _{\mu }(\beta [\gamma [\eta ]])\\\end{array}}\right.$.

## Explanation

Buchholz is working in Zermelo–Fraenkel set theory, that means every ordinal $\alpha$ is equal to set $\{\beta \mid \beta <\alpha \}$. 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 $C_{\nu }^{0}(\alpha )=\{\beta \mid \beta <\Omega _{\nu }\}$.
The condition $C_{\nu }^{n+1}(\alpha )=C_{\nu }^{n}(\alpha )\cup \{\gamma \mid P(\gamma )\subseteq C_{\nu }^{n}(\alpha )\}\cup \{\psi _{\mu }(\xi )\mid \xi \in \alpha \cap C_{\nu }^{n}(\alpha )\wedge \mu \leq \omega \}$ means that set $C_{\nu }^{n+1}(\alpha )$ includes:
* all ordinals from previous set $C_{\nu }^{n}(\alpha )$,
* all ordinals that can be obtained by summation the additively principal ordinals from previous set $C_{\nu }^{n}(\alpha )$,
* 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 $\mu \leq \omega$.
That is why we can rewrite this condition as:
    $C_{\nu }^{n+1}(\alpha )=\{\beta +\gamma ,\psi _{\mu }(\eta )\mid \beta ,\gamma ,\eta \in C_{\nu }^{n}(\alpha )\wedge \eta <\alpha \wedge \mu \leq \omega \}.$
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 $\psi _{\mu }(\eta )$, where $\mu \leq \omega$ and $\eta <\alpha$.
Then $\psi _{\nu }(\alpha )=\min\{\gamma \mid \gamma \not \in C_{\nu }(\alpha )\}$ is the smallest ordinal that does not belong to this set.
Examples
Consider the following examples:
    $C_{0}^{0}(\alpha )=\{0\}=\{\beta \mid \beta <1\},$
    $C_{0}(0)=\{0\}$ (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 and therefore $\psi _{0}(1)=\omega$ – 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 and therefore $\psi _{0}(2)=\omega ^{2}$.
If $\alpha =\omega$ then $C_{0}(\alpha )=\{0,\psi (0)=1,\ldots ,\psi (1)=\omega ,\ldots ,\psi (2)=\omega ^{2},\ldots ,\psi (3)=\omega ^{3},\ldots \}$ and $\psi _{0}(\omega )=\omega ^{\omega }$.
If $\alpha =\Omega$ then $C_{0}(\alpha )=\{0,\psi (0)=1,\ldots ,\psi (1)=\omega ,\ldots ,\psi (\omega )=\omega ^{\omega },\ldots ,\psi (\omega ^{\omega })=\omega ^{\omega ^{\omega }},\ldots \}$ and $\psi _{0}(\Omega )=\varepsilon _{0}$ – the smallest epsilon number i.e. first fixed point of $\alpha =\omega ^{\alpha }$.
If $\alpha =\Omega +1$ then $C_{0}(\alpha )=\{0,1,\ldots ,\psi _{0}(\Omega )=\varepsilon _{0},\ldots ,\varepsilon _{0}+\varepsilon _{0},\ldots ,\psi _{1}(0)=\Omega ,\ldots \}$ and $\psi _{0}(\Omega +1)=\varepsilon _{0}\omega =\omega ^{\varepsilon _{0}+1}$.
$\psi _{0}(\Omega 2)=\varepsilon _{1}$ the second epsilon number,
    $\psi _{0}(\Omega ^{2})=\varepsilon _{\varepsilon _{\cdots }}=\zeta _{0}$ i.e. first fixed point of $\alpha =\varepsilon _{\alpha }$,
$\varphi (\alpha ,\beta )=\psi _{0}(\Omega ^{\alpha }(1+\beta ))$, where $\varphi$ denotes the Veblen function,
$\psi _{0}(\Omega ^{\Omega })=\Gamma _{0}=\varphi (1,0,0)=\theta (\Omega ,0)$, where $\theta$ denotes the Feferman's function and $\Gamma _{0}$ is the Feferman–Schütte ordinal,
    $\psi _{0}(\Omega ^{\Omega ^{2}})=\varphi (1,0,0,0)$ is the Ackermann ordinal,
    $\psi _{0}(\Omega ^{\Omega ^{\omega }})$ is the small Veblen ordinal,
    $\psi _{0}(\Omega ^{\Omega ^{\Omega }})$ is the large Veblen ordinal,
    $\psi _{0}(\Omega \uparrow \uparrow \omega )=\psi _{0}(\varepsilon _{\Omega +1})=\theta (\varepsilon _{\Omega +1},0).$
Now let's research how $\psi _{1}$ works:
    $C_{1}^{0}(0)=\{\beta \mid \beta <\Omega _{1}\}=\{0,\psi (0)=1,2,\ldots {\text{googol}},\ldots ,\psi _{0}(1)=\omega ,\ldots ,\psi _{0}(\Omega )=\varepsilon _{0},\ldots$
$\ldots ,\psi _{0}(\Omega ^{\Omega })=\Gamma _{0},\ldots ,\psi (\Omega ^{\Omega ^{\Omega }+\Omega ^{2}}),\ldots \$ i.e. includes all countable ordinals. And therefore $C_{1}(0)$ includes all possible sums of all countable ordinals and $\psi _{1}(0)=\Omega _{1}$ first uncountable ordinal which is greater than all countable ordinal by its definition i.e. smallest number with cardinality $\aleph _{1}$.
If $\alpha =1$ then $C_{1}(\alpha )=\{0,\ldots ,\psi _{0}(0)=\omega ,\ldots ,\psi _{1}(0)=\Omega ,\ldots ,\Omega +\omega ,\ldots ,\Omega +\Omega ,\ldots \}$ and $\psi _{1}(1)=\Omega \omega =\omega ^{\Omega +1}$.
    $\psi _{1}(2)=\Omega \omega ^{2}=\omega ^{\Omega +2},$
    $\psi _{1}(\psi _{1}(0))=\psi _{1}(\Omega )=\Omega ^{2}=\omega ^{\Omega +\Omega },$
    $\psi _{1}(\psi _{1}(\psi _{1}(0)))=\omega ^{\Omega +\omega ^{\Omega +\Omega }}=\omega ^{\Omega \cdot \Omega }=(\omega ^{\Omega })^{\Omega }=\Omega ^{\Omega },$
    $\psi _{1}^{4}(0)=\Omega ^{\Omega ^{\Omega }},$
    $\psi _{1}^{k+1}(0)=\Omega \uparrow \uparrow k$ where $k$ is a natural number, $k\geq 2$,
    $\psi _{1}(\Omega _{2})=\psi _{1}^{\omega }(0)=\Omega \uparrow \uparrow \omega =\varepsilon _{\Omega +1}.$
For case $\psi (\psi _{2}(0))=\psi (\Omega _{2})$ the set $C_{0}(\Omega _{2})$ includes functions $\psi _{0}$ with all arguments less than $\Omega _{2}$ i.e. such arguments as $0,\psi _{1}(0),\psi _{1}(\psi _{1}(0)),\psi _{1}^{3}(0),\ldots ,\psi _{1}^{\omega }(0)$
and then
    $\psi _{0}(\Omega _{2})=\psi _{0}(\psi _{1}(\Omega _{2}))=\psi _{0}(\varepsilon _{\Omega +1}).$
In the 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:
    $\theta (\Omega _{\nu },0)=\psi _{0}(\Omega _{\nu }^{\Omega }){\text{ (for }}1\leq \nu )<\omega$

## Ordinal notation

Buchholz defined an ordinal notation ${\mathsf {(OT,<)}}$ associated to the $\psi$ function. We simultaneously define the sets $T$ and $PT$ as formal strings consisting of $0,D_{v}$ indexed by $v\in \omega +1$, braces and commas in the following way:
* $0\in T\land 0\in PT$,
* $\forall (v,a)\in (\omega +1)\times T(D_{v}a\in T\land D_{v}a\in PT)$.
* $\forall (a_{0},a_{1})\in PT^{2}((a_{0},a_{1})\in T)$.
* $\exists s(a_{0}=s)\land (a_{0},a_{1})\in T\times PT\rightarrow (s,a_{1})\in T$.
An element of $T$ is called a term, and an element of $PT$ is called a principal term. By definition, $T$ is a recursive set and $PT$ is a recursive subset of $T$. Every term is either $0$, a principal term or a braced array of principal terms of length $\geq 2$. We denote $a\in PT$ by $(a)$. By convention, every term can be uniquely expressed as either $0$ or a braced, non-empty array of principal terms. Since clauses 3 and 4 in the definition of $T$ and $PT$ are applicable only to arrays of length $\geq 2$, this convention does not cause serious ambiguity.
We then define a binary relation $a<b$ on $T$ in the following way:
* $b=0\rightarrow a<b=\bot$
* $a=0\rightarrow (a<b\leftrightarrow b\neq 0)$
* If $a\neq 0\land b\neq 0$, then:
** If $a=D_{u}a'\land b=D_{v}b'$ for some $((u,a'),(v,b'))\in ((\omega +1)\times T)^{2}$, then $a<b$ is true if either of the following are true:
*** $u<v$
*** $u=v\land a'<b'$
** If $a=(a_{0},...,a_{n})\land b=(b_{0},...,b_{m})$ for some $(n,m)\in \mathbb {N} ^{2}\land 1\leq n+m$, then $a<b$ is true if either of the following are true:
*** $\forall i\in \mathbb {N} \land i\leq n(n<m\land a_{i}=b_{i})$
*** $\exists k\in \mathbb {N} \forall i\in \mathbb {N} \land i<k(k\leq min\{n,m\}\land a_{k}<b_{k}\land a_{i}=b_{1})$
$<$ is a recursive strict total ordering on $T$. We abbreviate $(a<b)\lor (a=b)$ as $a\leq b$. Although $\leq$ itself is not a well-ordering, its restriction to a recursive subset $OT\subset T$, which will be described later, forms a well-ordering. In order to define $OT$, we define a subset $G_{u}a\subset T$ for each $(u,a)\in (\omega +1)\times T$ in the following way:
* $a=0\rightarrow G_{u}a=\varnothing$
* Suppose that $a=D_{v}a'$ for some $(v,a')\in (\omega +1)\times T$, then:
** $u\leq v\rightarrow G_{u}a=\{a'\}\cup G_{u}a'$
** $u>v\rightarrow G_{u}a=\varnothing$
* If $a=(a_{0},...,a_{k})$ for some $(a_{i})_{i=0}^{k}\in PT^{k+1}$ for some $k\in \mathbb {N} \backslash \{0\},G_{u}a=\bigcup _{i=0}^{k}G_{u}a_{i}$.
$b\in G_{u}a$ 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 way:
* $0\in OT$
* For any $(v,a)\in (\omega +1)\times T$, $D_{v}a\in OT$ is equivalent to $a\in OT$ and, for any $a'\in G_{v}a$, $a'<a$.
* For any $(a_{i})_{i=0}^{k}\in PT^{k+1}$, $(a_{0},...,a_{k})\in OT$ is equivalent to $(a_{i})_{i=0}^{k}\in OT$ and $a_{k}\leq ...\leq a_{0}$.
$OT$ is a recursive subset of $T$, and an element of $OT$ is called an ordinal term. We can then define a map $o:OT\rightarrow C_{0}(\varepsilon _{\Omega _{\omega }+1})$ in the following way:
* $a=0\rightarrow o(a)=0$
* If $a=D_{v}a'$ for some $(v,a')\in (\omega +1)\times OT$, then $o(a)=\psi _{v}(o(a'))$.
* If $a=(a_{0},...,a_{k})$ for some $(a_{i})_{i=0}^{k}\in (OT\cap PT)^{k+1}$ with $k\in \mathbb {N} \backslash \{0\}$, then $o(a)=o(a_{0})\#...\#o(a_{k})$, where $\#$ denotes the descending sum of ordinals, which coincides with the usual addition by the definition of $OT$.
Buchholz verified the following properties of $o$:
* The map $o$ is an order-preserving bijective map with respect to $<$ and $\in$. In particular, $o$ is a recursive strict well-ordering on $OT$.
* For any $a\in OT$ satisfying $a<D_{1}0$, $o(a)$ coincides with the ordinal type of $<$ restricted to the countable subset $\{x\in OT\;|\;x<a\}$.
* The Takeuti-Feferman-Buchholz ordinal coincides with the ordinal type of $<$ restricted to the recursive subset $\{x\in OT\;|\;x<D_{1}0\}$.
* The ordinal type of $(OT,<)$ is greater than the Takeuti-Feferman-Buchholz ordinal.
* For any $v\in \mathbb {N} \;\backslash \;\{0\}$, the well-foundedness of $<$ restricted to the recursive subset $\{x\in OT\;|\;x<D_{0}D_{v+1}0\}$ in the sense of the non-existence of a primitive recursive infinite descending sequence is not provable under ${\mathsf {ID}}_{v}$.
* The well-foundedness of $<$ restricted to the recursive subset$\{x\in OT\;|\;x<D_{0}D_{\omega }0\}$ in the same sense as above is not provable under $\Pi _{1}^{1}-CA_{0}$.

### Normal form

The normal form for Buchholz's function can be defined by the pull-back of standard form for the ordinal notation associated to it by $o$. Namely, the set $NF$ of predicates on ordinals in $C_{0}(\varepsilon _{\Omega _{\omega }+1})$ is defined in the following way:
* The predicate $\alpha =_{NF}0$ on an ordinal $\alpha \in C_{0}(\varepsilon _{\Omega _{\omega }+1})$ defined as $\alpha =0$ belongs to $NF$.
* The predicate $\alpha _{0}=_{NF}\psi _{k_{1}}(\alpha _{1})$ on ordinals $\alpha _{0},\alpha _{1}\in C_{0}(\varepsilon _{\Omega _{\omega }+1})$ with $k_{1}=\omega +1$ defined as $\alpha _{0}=\psi _{k_{1}}(\alpha _{1})$ and $\alpha _{1}=C_{k_{1}}(\alpha _{1})$ belongs to $NF$.
* The predicate $\alpha _{0}=_{NF}\alpha _{1}+...+\alpha _{n}$ on ordinals $\alpha _{0},\alpha _{1},...,\alpha _{n}\in C_{0}(\varepsilon _{\Omega _{\omega }+1})$ with an integer $n>1$ defined as $\alpha _{0}=\alpha _{1}+...+\alpha _{n}$, $\alpha _{1}\geq ...\geq \alpha _{n}$, and $\alpha _{1},...,\alpha _{n}\in AP$ belongs to $NF$. Here $+$ is a function symbol rather than an actual addition, and $AP$ denotes the class of additive principal numbers.
It is also useful to replace the third case by the following one obtained by combining the second condition:
* The predicate $\alpha _{0}=_{NF}\psi _{k_{1}}(\alpha _{1})+...+\psi _{k_{n}}(\alpha _{n})$ on ordinals $\alpha _{0},\alpha _{1},...,\alpha _{n}\in C_{0}(\varepsilon _{\Omega _{\omega }+1})$ with an integer $n>1$ and $k_{1},...,k_{n}\in \omega +1$ defined as $\alpha _{0}=\psi _{k_{1}}(\alpha _{1})+...+\psi _{k_{n}}(\alpha _{n})$, $\psi _{k_{1}}(\alpha _{1})\geq ...\geq \psi _{k_{n}}(\alpha _{n})$, and $\alpha _{1}\in C_{k_{1}}(\alpha _{1}),...,\alpha _{n}\in C_{k_{n}}(\alpha _{n})\in AP$ belongs to $NF$.
We note that those two formulations are not equivalent. For example, $\psi _{0}(\Omega )+1=_{NF}\psi _{0}(\zeta _{1})+\psi _{0}(0)$ is a valid formula which is false with respect to the second formulation because of $\zeta _{1}\neq C_{0}(\zeta _{1})$, while it is a valid formula which is true with respect to the first formulation because of $\psi _{0}(\Omega )+1=\psi _{0}(\zeta _{1})+\psi _{0}(0)$, $\psi _{0}(\zeta _{1}),\psi _{0}(0)\in AP$, and $\psi _{0}(\zeta _{1})\geq \psi _{0}(0)$. In this way, the notion of normal form heavily depends on the context. In both formulations, every ordinal in $C_{0}(\varepsilon _{\Omega _{\omega }+1})$ can be uniquely expressed in normal form for Buchholz's function.