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^[1] and Schütte.^[2]

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:

$\psi _{\nu }(0)=\Omega _{\nu },$ ^[3]^: 197
$\psi _{\nu }(\alpha )\in P,$ ^[3]^: 197
$\psi _{\nu }(\alpha +1)=\min\{\gamma \in P:\psi _{\nu }(\alpha )<\gamma \}{\text{ if }}\alpha \in C_{\nu }(\alpha ),$ ^[3]^: 199
$\Omega _{\nu }\leq \psi _{\nu }(\alpha )<\Omega _{\nu +1}$ ^[3]^: 197
$\psi _{0}(\alpha )=\omega ^{\alpha }{\text{ if }}\alpha <\varepsilon _{0},$ ^[3]^: 199
$\psi _{\nu }(\alpha )=\omega ^{\Omega _{\nu }+\alpha }{\text{ if }}\alpha <\varepsilon _{\Omega _{\nu }+1}{\text{ and }}\nu \neq 0,$ ^[3]^: 199
$\theta (\varepsilon _{\Omega _{\nu }+1},0)=\psi (\varepsilon _{\Omega _{\nu }+1}){\text{ for }}0<\nu \leq \omega .$ ^[3]^: 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's function,

$\psi _{0}(\Omega ^{\Omega })=\Gamma _{0}=\varphi (1,0,0)=\theta (\Omega ,0)$ , where $\theta$ denotes the Feferman's function,

\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^[3] 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 $a\neq 0\land b\neq 0$ $a\neq 0\land b\neq 0$ , then:
- If $a=D_{u}a'\land b=D_{v}b'$ $a=D_{u}a'\land b=D_{v}b'$ for some $((u,a'),(v,b'))\in ((\omega +1)\times T)^{2}$ $((u,a'),(v,b'))\in ((\omega +1)\times T)^{2}$ , then $a<b$ $a<b$ is true if either of the following are true:
  - $u<v$
  - $u=v\land a'<b'$
- If $a=(a_{0},...,a_{n})\land b=(b_{0},...,b_{m})$ $a=(a_{0},...,a_{n})\land b=(b_{0},...,b_{m})$ for some $(n,m)\in \mathbb {N} ^{2}\land 1\leq n+m$ $(n,m)\in \mathbb {N} ^{2}\land 1\leq n+m$ , then $a<b$ $a<b$ 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 $a=D_{v}a'$ $a=D_{v}a'$ for some $(v,a')\in (\omega +1)\times T$ $(v,a')\in (\omega +1)\times T$ , 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.

References

↑ Jäger, G (1984). "ρ-inaccessible ordinals, collapsing functions and a recursive notation system". Archiv für Mathematische Logik und Grundlagenforschung. 24 (1): 49–62. doi:10.1007/BF02007140. S2CID 38619369.
↑ Buchholz, W.; Schütte, K. (1983). "Ein Ordinalzahlensystem ftir die beweistheoretische Abgrenzung der H~-Separation und Bar-Induktion". Sitzungsberichte der Bayerischen Akademie der Wissenschaften, Math.-Naturw. Klasse.
1 2 3 4 5 6 7 8 Buchholz, W. (1986). "A new system of proof-theoretic ordinal functions". Annals of Pure and Applied Logic. 32: 195–207. doi:10.1016/0168-0072(86)90052-7.

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[1] Jäger, G (1984). "ρ-inaccessible ordinals, collapsing functions and a recursive notation system". Archiv für Mathematische Logik und Grundlagenforschung. 24 (1): 49–62. doi:10.1007/BF02007140. S2CID 38619369.

[2] Buchholz, W.; Schütte, K. (1983). "Ein Ordinalzahlensystem ftir die beweistheoretische Abgrenzung der H~-Separation und Bar-Induktion". Sitzungsberichte der Bayerischen Akademie der Wissenschaften, Math.-Naturw. Klasse.

[buchholz84-3] 1 2 3 4 5 6 7 8 Buchholz, W. (1986). "A new system of proof-theoretic ordinal functions". Annals of Pure and Applied Logic. 32: 195–207. doi:10.1016/0168-0072(86)90052-7.

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