Buchholz psi functions

Template:More citations needed 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 strengthTemplate:Huh 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, Ω₀ = 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:

• ${\psi }_{\nu }(0)={\mathrm{\Omega }}_{\nu },$ Template:Rp
• ${\psi }_{\nu }(\alpha )\in P,$ Template:Rp
• ${\psi }_{\nu }(\alpha +1)=\min \{\gamma \in P:{\psi }_{\nu }(\alpha )<\gamma \}\text{ if }\alpha \in C_{\nu }(\alpha ),$ Template:Rp
• ${\mathrm{\Omega }}_{\nu }\le {\psi }_{\nu }(\alpha )<{\mathrm{\Omega }}_{\nu +1}$ Template:Rp
• ${\psi }_0(\alpha )={\omega }^{\alpha }\text{ if }\alpha <{\epsilon }_0,$ Template:Rp
• ${\psi }_{\nu }(\alpha )={\omega }^{{\mathrm{\Omega }}_{\nu }+\alpha }\text{ if }\alpha <{\epsilon }_{{\mathrm{\Omega }}_{\nu }+1}\text{ and }\nu \ne 0,$ Template:Rp
• $\theta ({\epsilon }_{{\mathrm{\Omega }}_{\nu }+1},0)=\psi ({\epsilon }_{{\mathrm{\Omega }}_{\nu }+1})\text{ for }0<\nu \le \omega .$ Template:Rp

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 $\alpha <{\mathrm{\Omega }}_{\omega }$ 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 ${\psi }_{{\nu }_1}({\beta }_1)\ge {\psi }_{{\nu }_2}({\beta }_2)\ge \cdots \ge {\psi }_{{\nu }_k}({\beta }_k)$ and each ${\beta }_i$ is also written in normal form.

Fundamental sequences

The fundamental sequence for an ordinal number $\alpha$ with cofinality $\mathrm{cof}(\alpha )=\beta$ 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 $\mathrm{cof}(\alpha )=1$ and the fundamental sequence has only one element $\alpha [0]=\alpha -1$. If $\alpha$ is a limit ordinal then $\mathrm{cof}(\alpha )\in \{\omega \}\cup \{{\mathrm{\Omega }}_{\mu +1}\mid \mu \ge 0\}$.

For nonzero ordinals $\alpha <{\mathrm{\Omega }}_{\omega }$, 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 $k\ge 2$ then $\mathrm{cof}(\alpha )=\mathrm{cof}({\psi }_{{\nu }_k}({\beta }_k))$ 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 $\mathrm{cof}(\alpha )=1$ and $\alpha [0]=0$,
3. If $\alpha ={\psi }_{\nu +1}(0)$, then $\mathrm{cof}(\alpha )={\mathrm{\Omega }}_{\nu +1}$ and $\alpha [\eta ]={\mathrm{\Omega }}_{\nu +1}[\eta ]=\eta$,
4. If $\alpha ={\psi }_{\nu }(\beta +1)$ then $\mathrm{cof}(\alpha )=\omega$ and $\alpha [\eta ]={\psi }_{\nu }(\beta )\cdot \eta$ (and note: ${\psi }_{\nu }(0)={\mathrm{\Omega }}_{\nu }$),
5. If $\alpha ={\psi }_{\nu }(\beta )$ and $\mathrm{cof}(\beta )\in \{\omega \}\cup \{{\mathrm{\Omega }}_{\mu +1}\mid \mu <\nu \}$ then $\mathrm{cof}(\alpha )=\mathrm{cof}(\beta )$ and $\alpha [\eta ]={\psi }_{\nu }(\beta [\eta ])$,
6. If $\alpha ={\psi }_{\nu }(\beta )$ and $\mathrm{cof}(\beta )\in \{{\mathrm{\Omega }}_{\mu +1}\mid \mu \ge \nu \}$ then $\mathrm{cof}(\alpha )=\omega$ and $\alpha [\eta ]={\psi }_{\nu }(\beta [\gamma [\eta ]])$ where $\{\begin{array}{c}\gamma [0]={\mathrm{\Omega }}_{\mu }\\ \gamma [\eta +1]={\psi }_{\mu }(\beta [\gamma [\eta ]])\end{array}$.

Explanation

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

The condition $C_{\nu }^{n+1}(\alpha )=C_{\nu }^n(\alpha )\cup \{\gamma \mid P(\gamma )\subseteq C_{\nu }^n(\alpha )\}\cup \{{\psi }_{\mu }(\xi )\mid \xi \in \alpha \cap C_{\nu }^n(\alpha )\wedge \mu \le \omega \}$ 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 $\mu \le \omega$.

That is why we can rewrite this condition as:

$C_{\nu }^{n+1}(\alpha )=\{\beta +\gamma ,{\psi }_{\mu }(\eta )\mid \beta ,\gamma ,\eta \in C_{\nu }^n(\alpha )\wedge \eta <\alpha \wedge \mu \le \omega \}.$

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 $<{\mathrm{\aleph }}_{\nu }$ by the functions + (addition) and ${\psi }_{\mu }(\eta )$, where $\mu \le \omega$ and $\eta <\alpha$.

Then ${\psi }_{\nu }(\alpha )=\min \{\gamma \mid \gamma \in ̸C_{\nu }(\alpha )\}$ 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 $C_0(\alpha )=\{0,\psi (0)=1,\dots ,\psi (1)=\omega ,\dots ,\psi (2)={\omega }^2,\dots ,\psi (3)={\omega }^3,\dots \}$ and ${\psi }_0(\omega )={\omega }^{\omega }$.

If $\alpha =\mathrm{\Omega }$ then $C_0(\alpha )=\{0,\psi (0)=1,\dots ,\psi (1)=\omega ,\dots ,\psi (\omega )={\omega }^{\omega },\dots ,\psi ({\omega }^{\omega })={\omega }^{{\omega }^{\omega }},\dots \}$ and ${\psi }_0(\mathrm{\Omega })={\epsilon }_0$ – the smallest epsilon number i.e. first fixed point of $\alpha ={\omega }^{\alpha }$.

If $\alpha =\mathrm{\Omega }+1$ then $C_0(\alpha )=\{0,1,\dots ,{\psi }_0(\mathrm{\Omega })={\epsilon }_0,\dots ,{\epsilon }_0+{\epsilon }_0,\dots ,{\psi }_1(0)=\mathrm{\Omega },\dots \}$ and ${\psi }_0(\mathrm{\Omega }+1)={\epsilon }_0\omega ={\omega }^{{\epsilon }_0+1}$.

${\psi }_0(\mathrm{\Omega }2)={\epsilon }_1$ the second epsilon number,

${\psi }_0({\mathrm{\Omega }}^2)={\epsilon }_{{\epsilon }_{\cdots }}={\zeta }_0$ i.e. first fixed point of $\alpha ={\epsilon }_{\alpha }$,

$\phi (\alpha ,\beta )={\psi }_0({\mathrm{\Omega }}^{\alpha }(1+\beta ))$, where $\phi$ denotes the Veblen function,

${\psi }_0({\mathrm{\Omega }}^{\mathrm{\Omega }})={\mathrm{\Gamma }}_0=\phi (1,0,0)=\theta (\mathrm{\Omega },0)$, where $\theta$ denotes the Feferman's function and ${\mathrm{\Gamma }}_0$ is the Feferman–Schütte ordinal,

${\psi }_0({\mathrm{\Omega }}^{{\mathrm{\Omega }}^2})=\phi (1,0,0,0)$ is the Ackermann ordinal,

${\psi }_0({\mathrm{\Omega }}^{{\mathrm{\Omega }}^{\omega }})$ is the small Veblen ordinal,

${\psi }_0({\mathrm{\Omega }}^{{\mathrm{\Omega }}^{\mathrm{\Omega }}})$ is the large Veblen ordinal,

${\psi }_0(\mathrm{\Omega }\uparrow \uparrow \omega )={\psi }_0({\epsilon }_{\mathrm{\Omega }+1})=\theta ({\epsilon }_{\mathrm{\Omega }+1},0).$

Now let's research how ${\psi }_1$ works:

$C_1^0(0)=\{\beta \mid \beta <{\mathrm{\Omega }}_1\}=\{0,\psi (0)=1,2,\dots \text{googol},\dots ,{\psi }_0(1)=\omega ,\dots ,{\psi }_0(\mathrm{\Omega })={\epsilon }_0,\dots$

$\dots ,{\psi }_0({\mathrm{\Omega }}^{\mathrm{\Omega }})={\mathrm{\Gamma }}_0,\dots ,\psi ({\mathrm{\Omega }}^{{\mathrm{\Omega }}^{\mathrm{\Omega }}+{\mathrm{\Omega }}^2}),\dots \}$ i.e. includes all countable ordinals. And therefore $C_1(0)$ includes all possible sums of all countable ordinals and ${\psi }_1(0)={\mathrm{\Omega }}_1$ first uncountable ordinal which is greater than all countable ordinal by its definition i.e. smallest number with cardinality ${\mathrm{\aleph }}_1$.

If $\alpha =1$ then $C_1(\alpha )=\{0,\dots ,{\psi }_0(0)=\omega ,\dots ,{\psi }_1(0)=\mathrm{\Omega },\dots ,\mathrm{\Omega }+\omega ,\dots ,\mathrm{\Omega }+\mathrm{\Omega },\dots \}$ and ${\psi }_1(1)=\mathrm{\Omega }\omega ={\omega }^{\mathrm{\Omega }+1}$.

${\psi }_1(2)=\mathrm{\Omega }{\omega }^2={\omega }^{\mathrm{\Omega }+2},$

${\psi }_1({\psi }_1(0))={\psi }_1(\mathrm{\Omega })={\mathrm{\Omega }}^2={\omega }^{\mathrm{\Omega }+\mathrm{\Omega }},$

${\psi }_1({\psi }_1({\psi }_1(0)))={\omega }^{\mathrm{\Omega }+{\omega }^{\mathrm{\Omega }+\mathrm{\Omega }}}={\omega }^{\mathrm{\Omega }\cdot \mathrm{\Omega }}=({\omega }^{\mathrm{\Omega }}{)}^{\mathrm{\Omega }}={\mathrm{\Omega }}^{\mathrm{\Omega }},$

${\psi }_1^4(0)={\mathrm{\Omega }}^{{\mathrm{\Omega }}^{\mathrm{\Omega }}},$

${\psi }_1^{k+1}(0)=\mathrm{\Omega }\uparrow \uparrow k$ where $k$ is a natural number, $k\ge 2$,

${\psi }_1({\mathrm{\Omega }}_2)={\psi }_1^{\omega }(0)=\mathrm{\Omega }\uparrow \uparrow \omega ={\epsilon }_{\mathrm{\Omega }+1}.$

For case $\psi ({\psi }_2(0))=\psi ({\mathrm{\Omega }}_2)$ the set $C_0({\mathrm{\Omega }}_2)$ includes functions ${\psi }_0$ with all arguments less than ${\mathrm{\Omega }}_2$ i.e. such arguments as $0,{\psi }_1(0),{\psi }_1({\psi }_1(0)),{\psi }_1^3(0),\dots ,{\psi }_1^{\omega }(0)$

and then

${\psi }_0({\mathrm{\Omega }}_2)={\psi }_0({\psi }_1({\mathrm{\Omega }}_2))={\psi }_0({\epsilon }_{\mathrm{\Omega }+1}).$

In the general case:

${\psi }_0({\mathrm{\Omega }}_{\nu +1})={\psi }_0({\psi }_{\nu }({\mathrm{\Omega }}_{\nu +1}))={\psi }_0({\epsilon }_{{\mathrm{\Omega }}_{\nu }+1})=\theta ({\epsilon }_{{\mathrm{\Omega }}_{\nu }+1},0).$

We also can write:

$\theta ({\mathrm{\Omega }}_{\nu },0)={\psi }_0({\mathrm{\Omega }}_{\nu }^{\mathrm{\Omega }})\text{ (for }1\le \nu )<\omega$

Ordinal notation

Buchholz defined an ordinal notation $\mathsf{(}𝖮𝖳,<\mathsf{)}$ 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:

• $0\in T\wedge 0\in PT$,
• $\mathrm{\forall }(v,a)\in (\omega +1)\times T(D_{v}a\in T\wedge D_{v}a\in PT)$.
• $\mathrm{\forall }(a_0,a_1)\in P{T}^2((a_0,a_1)\in T)$.
• $\mathrm{\exists }s(a_0=s)\wedge (a_0,a_1)\in T\times PT\to (s,a_1)\in T$.

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 $\ge 2$. 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 $\ge 2$, this convention does not cause serious ambiguity.

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

• $b=0\to a<b=\mathrm{\perp }$
• $a=0\to (a<b\leftrightarrow b\ne 0)$
• If $a\ne 0\wedge b\ne 0$, then:
  • If $a=D_u{a}^{\prime }\wedge b=D_v{b}^{\prime }$ for some $((u,a^{\prime }),(v,b^{\prime }))\in ((\omega +1)\times T{)}^2$, then $a<b$ is true if either of the following are true:
    • $u<v$
    • $u=v\wedge a^{\prime }<b^{\prime }$
  • If $a=(a_0,...,a_n)\wedge b=(b_0,...,b_m)$ for some $(n,m)\in {\mathbb{N}}^2\wedge 1\le n+m$, then $a<b$ is true if either of the following are true:
    • $\mathrm{\forall }i\in \mathbb{N}\wedge i\le n(n<m\wedge a_i=b_i)$
    • $\mathrm{\exists }k\in \mathbb{N}\mathrm{\forall }i\in \mathbb{N}\wedge i<k(k\le min\{n,m\}\wedge a_k<b_k\wedge a_i=b_1)$

$<$ is a recursive strict total ordering on $T$. We abbreviate $(a<b)\vee (a=b)$ as $a\le b$. Although $\le$ 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:

• $a=0\to G_{u}a=\varnothing$
• Suppose that $a=D_v{a}^{\prime }$ for some $(v,a^{\prime })\in (\omega +1)\times T$, then:
  • $u\le v\to G_{u}a=\{a^{\prime }\}\cup G_u{a}^{\prime }$
  • $u>v\to G_{u}a=\varnothing$

• If $a=(a_0,...,a_k)$ for some $(a_i{)}_{i=0}^k\in P{T}^{k+1}$ for some $k\in \mathbb{N}\mathrm{\setminus }\{0\},G_{u}a=\bigcup _{i=0}^k{G}_u{a}_i$.

$b\in G_{u}a$ 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 $a^{\prime }\in G_{v}a$, $a^{\prime }<a$.
• For any $(a_i{)}_{i=0}^k\in P{T}^{k+1}$, $(a_0,...,a_k)\in OT$ is equivalent to $(a_i{)}_{i=0}^k\in OT$ and $a_k\le ...\le a_0$.

$OT$ is a recursive subset of $T$, and an element of $OT$ is called an ordinal term. We can then define a map $o:OT\to C_0({\epsilon }_{{\mathrm{\Omega }}_{\omega }+1})$ in the following way:

• $a=0\to o(a)=0$
• If $a=D_v{a}^{\prime }$ for some $(v,a^{\prime })\in (\omega +1)\times OT$, then $o(a)={\psi }_v(o(a^{\prime }))$.
• If $a=(a_0,...,a_k)$ for some $(a_i{)}_{i=0}^k\in (OT\cap PT{)}^{k+1}$ with $k\in \mathbb{N}\mathrm{\setminus }\{0\}$, then $o(a)=o(a_0)\mathrm{\#}...\mathrm{\#}o(a_k)$, where $\mathrm{\#}$ 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 $v\in \mathbb{N}\mathrm{\setminus }\{0\}$, 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 ${𝖨𝖣}_v$.
• 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 ${\mathrm{\Pi }}_1^1-C{A}_0$.

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 $C_0({\epsilon }_{{\mathrm{\Omega }}_{\omega }+1})$ is defined in the following way:

• The predicate $\alpha {=}_{NF}0$ on an ordinal $\alpha \in C_0({\epsilon }_{{\mathrm{\Omega }}_{\omega }+1})$ defined as $\alpha =0$ belongs to $NF$.

• The predicate ${\alpha }_0{=}_{NF}{\psi }_{k_1}({\alpha }_1)$ on ordinals ${\alpha }_0,{\alpha }_1\in C_0({\epsilon }_{{\mathrm{\Omega }}_{\omega }+1})$ 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 ${\alpha }_0,{\alpha }_1,...,{\alpha }_n\in C_0({\epsilon }_{{\mathrm{\Omega }}_{\omega }+1})$ with an integer $n>1$ defined as ${\alpha }_0={\alpha }_1+...+{\alpha }_n$, ${\alpha }_1\ge ...\ge {\alpha }_n$, 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 ${\alpha }_0,{\alpha }_1,...,{\alpha }_n\in C_0({\epsilon }_{{\mathrm{\Omega }}_{\omega }+1})$ with an integer $n>1$ and $k_1,...,k_n\in \omega +1$ defined as ${\alpha }_0={\psi }_{k_1}({\alpha }_1)+...+{\psi }_{k_n}({\alpha }_n)$, ${\psi }_{k_1}({\alpha }_1)\ge ...\ge {\psi }_{k_n}({\alpha }_n)$, 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, ${\psi }_0(\mathrm{\Omega })+1{=}_{NF}{\psi }_0({\zeta }_1)+{\psi }_0(0)$ is a valid formula which is false with respect to the second formulation because of ${\zeta }_1\ne C_0({\zeta }_1)$, while it is a valid formula which is true with respect to the first formulation because of ${\psi }_0(\mathrm{\Omega })+1={\psi }_0({\zeta }_1)+{\psi }_0(0)$, ${\psi }_0({\zeta }_1),{\psi }_0(0)\in AP$, and ${\psi }_0({\zeta }_1)\ge {\psi }_0(0)$. In this way, the notion of normal form heavily depends on the context. In both formulations, every ordinal in $C_0({\epsilon }_{{\mathrm{\Omega }}_{\omega }+1})$ can be uniquely expressed in normal form for Buchholz's function.