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An ordinal collapsing function (OCF for short) is a ordinal function which is studied together with an associated ordinal notation in proof theory. One of the characteristic feature of OCFs is that they output large ordinals using even larger ones, often uncountable ordinals. For example, Buchholz's OCF $$\psi$$ satisfies $$\psi_0($$$$\Omega$$$$) = \varepsilon_0$$ and $$\psi_0(\Omega_2) = \textrm{BHO}$$, where $$\varepsilon_0$$ denotes the least epsilon number and $$\textrm{BHO}$$ denotes the Bachmann-Howard ordinal. In particular, an OCF gives a method of naming large countable ordinals using uncountable ordinals. For the use of an OCF in googology, see also the description of OCFs and why they're needed.

There are many OCFs in use, often similar to each other and easily confused (some even use the same symbols). As such, many analyses can not be trusted due to not pointing out which OCF they use.

## Properties

### Relationship to ordinal notations

The most confusing fact on OCFs is that an OCF itself can never be an ordinal notation, although many googologists who do not know the precise definition of the notion of an ordinal notation state that OCFs are ordinal notations. An OCF is just a function defined in set theory, while an ordinal notation is a notation equipped with a (primitive) recursive well-ordering and hence definable in arithmetic. Since the main use of an OCF in googology is the use in computable googology and an OCF itself is uncomputable, the lack of an ordinal notation is a serius problem.

In order to associate an ordinal notation, one of the easiest ways is to define a recurisve set of formal strings, a syntax on it as a first order language so that the notions of a term and a function symbol makes sense, a "structure" of the language whose underlying set consists of ordinals such that the interpretation of some function symbol is given by actual ordinal functions such as an OCF, and construct a (primitive) recursive relation encoding the $$\in$$-relation for ordinals restricted to the underlying set of the structure. That is why an OCF is useful in the construction of an ordinal notation.

However, to create an associated ordinal notation is much more difficult than to create an OCF. Therefore stating something like "I created an ordinal notation" just by creating an OCF does not literally make sense. Actually, an OCF might not necessarily even yield an ordinal notation, because there is no justification of the existence of a (primitive) recursive interpretation of the $$\in$$-relation. Therefre the situation is similar to the case where a googologist stated "I created a computable function" just by creating a function using an oracle Turing machine, which makes it doubtful that the work is a computable googolism. When we create a computable googolism, we actually need to provide an algorithm to compute it.

For example, googologists tend to "simplify" existing OCFs by dropping important conditions without understanding the reason why they are important. Since many complicated conditions in the original definitions are used in order to define the (primitive) recursive interpretations of the $$\in$$-relatinon, such poor droppings cause the lack of the (primitive) recursive interpretations of the $$\in$$-relations. It might fortunately admit an ordinal notation, but at least the creator need to write down how to define the (primitive) recursive interpretation of the $$\in$$-relation. Otherwise, their is no evidence of the computability of the resulting work and hence it should be just dealt with as an uncomputable googolism.

### Relationship to Mostowski collapse

Many of ordinal collapsing functions have an aspect analogous to Mostowski collapse. Indeed, Rathjen has described a detail of some OCFs, which is referred to as a "pictorial" description[1] of collapse. For example, with respect to Bachmann's $$\psi$$, $$\psi_\Omega(\alpha)$$ can be described as "the $$\alpha$$th collapse of $$\Omega$$". This is because the order type of each $$C^\Omega(\alpha,\psi_\Omega(\alpha))\cap\Omega$$ is $$\psi_\Omega(\alpha)$$, or in Rathjen's words, "the order-type of $$\Omega$$ viewed from within the structure $$C^\Omega(\alpha,\psi_\Omega(\alpha))$$ is actually $$\psi_\Omega(\alpha)$$"[1]. In this way $$\psi_\Omega(\alpha)$$ can be viewed as a (not necessarily Mostowski) collapse of $$\Omega$$. A similar property holds with respect to

and so on.

## Functions

### Bachmann's $$\psi$$

Main article: Bachmann's function

Heinz Bachmann's $$\psi$$ function was the first true ordinal collapsing function.[5][6] It is somewhat cumbersome as it depends on fundamental sequences for all limit ordinals[6].

Rathjen suggests a "recast" of the system[7] as follows. Let $$\Omega$$ be an uncountable ordinal such as $$\aleph_1$$. Then define $$C^\Omega(\alpha, \beta)$$ as the closure of $$\beta \cup \{0, \Omega\}$$ under $$+, (\xi \mapsto \omega^\xi), (\xi \mapsto \psi_\Omega(\xi))_{\xi < \alpha}$$, and $$\psi_\Omega(\alpha) = \min \{\rho < \Omega : C^\Omega(\alpha, \rho) \cap \Omega = \rho\}$$.

$$\psi_\Omega(\varepsilon_{\Omega + 1})$$ is the Bachmann-Howard ordinal[citation needed], the proof-theoretic ordinal of Kripke-Platek set theory with axiom of infinity.

### Feferman's $$\theta$$

Main article: Feferman's theta function

Feferman's $$\theta$$-functions constitute a hierarchy of single-argument functions $$\theta_\alpha: \text{On} \to \text{On}$$ for $$\alpha \in \text{On}$$.[2] It is often considered a two-argument function with $$\theta_\alpha(\beta)$$ written as $$\theta\alpha\beta$$. It is defined like so:

\begin{eqnarray*} C_0(\alpha, \beta) &=& \beta \cup \{0, \omega_1, \omega_2, \ldots, \omega_\omega\}\\ C_{n+1}(\alpha, \beta) &=& \{\gamma + \delta, \theta_\xi(\eta) | \gamma, \delta, \xi, \eta \in C_n(\alpha, \beta); \xi < \alpha\} \\ C(\alpha, \beta) &=& \bigcup_{n < \omega} C_n (\alpha, \beta) \\ \theta_\alpha(\beta) &=& \min\{\gamma | \gamma \not\in C(\alpha, \gamma) \wedge \forall \delta < \beta: \theta_\alpha(\delta) < \gamma\} \\ \end{eqnarray*}

Equivalently:

• An ordinal $$\beta$$ is considered $$\alpha$$-critical if it cannot be constructed with the following elements:
• all ordinals less than $$\beta$$,
• all ordinals in the set $$\{0, \omega_1, \omega_2, \ldots, \omega_\omega\}$$,
• the operation $$+$$,
• applications of $$\theta_\xi$$ for $$\xi < \alpha$$.
• $$\theta_\alpha$$ is the enumerating function for all $$\alpha$$-critical ordinals.

The supremum of the countable ordinal expressed by the function is the Takeuti-Feferman-Buchholz ordinal $$\theta_{\varepsilon_{\Omega_\omega + 1}}(0)$$,[2] Theorem 3.7 but an ordinal notation up to the ordinal associated to the function is unknown at least in this community.

### Buchholz's $$\psi$$

Main article: Buchholz's function

Buchholz's $$\psi$$ is a hierarchy of single-argument functions $$\psi_v: \text{On} \to \text{On}$$ for $$v \leq \omega$$, with $$\psi_v(\alpha)$$ abbreviated as $$\psi_v\alpha$$.[2] Define $$\Omega_0 = 1$$ and $$\Omega_v = \aleph_v$$ for $$v > 0$$, and let $$P(\alpha)$$ be the set of distinct terms in the Cantor normal form of $$\alpha$$ (with each term of the form $$\omega^\xi$$ for $$\xi \in \text{On}$$):

\begin{eqnarray*} C_v^0(\alpha) &=& \Omega_v\\ C_v^{n+1}(\alpha) &=& C_v^n(\alpha) \cup \{\gamma | P(\gamma) \subseteq C_v^n(\alpha)\} \cup \{\psi_v\xi | \xi \in \alpha \cap C_v^n(\alpha) \wedge \xi \in C_u(\xi) \wedge u \leq \omega\} \\ C_v(\alpha) &=& \bigcup_{n < \omega} C_v^n (\alpha) \\ \psi_v\alpha &=& \min\{\gamma | \gamma \not\in C_v(\alpha)\} \\ \end{eqnarray*}

The limit of this system is $$\psi_0(\varepsilon_{\Omega_\omega + 1})$$, the Takeuti-Feferman-Buchholz ordinal. The ordinal collapsing function admits an ordinal notation system, while those created by googologists do not usually admit ordinal notations (cf. #Relationship to ordinal notations). ​Unlike the unfortunate situations, it has a sophisticated simplification admitting an ordinal notation.

#### Gaoji's simplification

Main article: Weak Buchholz's function

It has a sophisticated simplification called Weak Buchholz's function created by Japanese Googology wiki user Gaoji. It discarded the addition from the closure in the definition of Bucholz's function, and admits an ordinal notation created by Japanese Googology Wiki user p進大好きbot.

#### Denis' extension

Main article: Extended Buchholz's function

It has a sophisticated extension called Extended Buchholz's function (EBF for short) created by wiki user Denis Maksudov.[8] It is based on cardinals below the least omega fixed point, and admits an ordinal notation created by Japanese Googology Wiki user p進大好きbot.

#### Okkuu's simplification of Denis' extension

Main article: Extended Weak Buchholz's function

Denis' extension above also has a sophisticated simplification called Extended Weak Buchholz's function created by Japanese Googology wiki user Okkuu. It discarded the addition from the closure in the definition of Extended Buchholz's function, and admits an ordinal notation created by Japanese Googology Wiki user p進大好きbot.

### Madore's $$\psi$$

David Madore (under "Gro-Tsen" on Wikipedia) defined the following simpler variant of Buchholz's function as a demonstration of how ordinal collapsing functions work.[9] The popularity of the article led to widespread use of the modified function.

\begin{eqnarray*} C_0(\alpha) &=& \{0, 1, \omega, \Omega\}\\ C_{n+1}(\alpha) &=& \{\gamma + \delta, \gamma\delta, \gamma^{\delta}, \psi(\eta) | \gamma, \delta, \eta \in C_n (\alpha); \eta < \alpha\} \\ C(\alpha) &=& \bigcup_{n < \omega} C_n (\alpha) \\ \psi(\alpha) &=& \min\{\beta \in \Omega|\beta \notin C(\alpha)\} \\ \end{eqnarray*}

Informally:

• $$C(\alpha)$$ is the set of all ordinals constructible using only $$0$$, $$1$$, $$\omega$$, $$\Omega$$, and finite applications of the following functions: addition, multiplication, exponentiation, and $$\kappa \mapsto \psi(\kappa)$$ (the latter only if $$\psi(\kappa)$$ has yet been defined).
• $$\psi(\alpha)$$ is the smallest countable ordinal not in $$C(\alpha)$$.

Chris Bird uses this function throughout his googological papers.[10]

### Bird's $$\theta$$

Chris Bird devised the following shorthand for the extended (zero-indexed) Veblen function $$\varphi$$:[11]

$\theta(\Omega^{n - 1}a_{n - 1} + \cdots + \Omega^2a_2 + \Omega a_1 + a_0, b) = \varphi(a_{n - 1}, \ldots, a_2, a_1, a_0, b)$

$$\theta(\alpha, 0)$$ is abbreviated as $$\theta(\alpha)$$. This function is only defined for arguments less than $$\Omega^\omega$$, and its outputs are limited by the small Veblen ordinal. Nevertheless, in his articles, Bird erroneously uses $$\theta(\Omega^\omega)$$ and higher values without properly defining the function. It means that he simply uses expressions like $$\theta(\Omega^{\omega})$$ and $$\theta(\Omega^{\Omega})$$ as shorthands of known ordinals.

Since this function is equivalent to Feferman's $$\theta$$ function where both are defined, it may be that he was merely using an extension of this $$\theta$$ function. It does not mean that this function is well-defined. Beginners should be careful if people use Bird's $$\theta$$ function for arguments greater than or equal to $$\Omega^{\omega}$$, because this function is ill-defined for ordinals greater than or equal to $$\Omega^{\omega}$$.

### Wilken's $$\vartheta$$

Wilken's $$\vartheta$$ is more generic than other OCFs. Let $$\Omega_0$$ be either $$1$$ or a number of the form $$\varepsilon_\alpha$$, let $$\Omega_1 > \Omega_0$$ be an uncountable regular cardinal and for $$0 < i < \omega$$ let $$\Omega_{i+1}$$ be the successor cardinal to $$\Omega_i$$. Then, for $$0 < n < \omega$$ and $$0 \leq m < n$$, define the following for $$\beta < \Omega_{m+1}$$:[12]

\begin{eqnarray*} \Omega_m \cup \beta &\subseteq& C_m^n(\alpha, \beta) \\ \xi, \eta \in C_m^n(\alpha, \beta) &\Rightarrow& \xi + \eta \in C_m^n(\alpha, \beta) \\ \xi \in C_m^n(\alpha, \beta) \cap \Omega_{k + 2} &\Rightarrow& \vartheta_k^n(\xi) \in C_m^n(\alpha, \beta) \text{ for } m < k < n \\ \xi \in C_m^n(\alpha, \beta) \cap \alpha &\Rightarrow& \vartheta_m^n(\xi) \in C_m^n(\alpha, \beta) \\ \vartheta_m^n(\alpha) &=& \min(\{\xi < \Omega_{m+1}|C_m^n(\alpha, \xi) \cap \Omega_{m+1} \subseteq \xi \wedge \alpha \in C_m^n(\alpha, \xi)\} \cup \{\Omega_{m+1}\}) \\ \end{eqnarray*}

$$n$$ is needed to define the function, but the actual value of $$n$$ does not affect the function's behavior. So, for example, $$\vartheta_0^1(\alpha) = \vartheta_0^2(\alpha) = \vartheta_0^3(\alpha) = \cdots$$ So we can safely eliminate $$n$$ and express the function using only two arguments $$\vartheta_m(\alpha)$$.

### Wilken and Weiermann's $$\bar\vartheta$$

Wilken and Weiermann's $$\bar\vartheta$$ is closely related to Wilken's $$\vartheta$$, and their paper closely analyzes the relationship between the two.[13] As before, let $$\Omega_0$$ be either $$1$$ or a number of the form $$\varepsilon_\alpha$$, let $$\Omega_1 > \Omega_0$$ be an uncountable regular cardinal and for $$0 < i < \omega$$ let $$\Omega_{i+1}$$ be the successor cardinal to $$\Omega_i$$, and finally let $$\Omega_\omega = \sup_{i < \omega} \Omega_i$$. For all $$\beta \leq \Omega_{i+1}$$ define the following:

\begin{eqnarray*} \Omega_i \cup \beta &\subseteq& \bar{C}_i(\alpha, \beta) \\ \xi, \eta \in \bar{C}_i (\alpha, \beta) &\Rightarrow& \xi + \eta \in \bar{C}_i(\alpha, \beta) \\ j \leq i < \omega \wedge \xi \in \bar{C}_j(\xi, \Omega_{j + 1}) \cap \bar{C}_i(\alpha, \beta) \cap \alpha &\Rightarrow& \bar{\vartheta}_j(\xi) \in \bar{C}_i(\alpha, \beta) \\ \bar{\vartheta}_i(\alpha) &=& \min(\{\xi < \Omega_{i + 1} | \alpha \in \bar{C}_i(\alpha, \xi) \wedge \bar{C}_i(\alpha, \xi) \cap \Omega_{i + 1} \subseteq \xi\} \cup \{\Omega_{i + 1}\}) \\ \end{eqnarray*}

### Weiermann's $$\vartheta$$

The $$\vartheta$$ function has the advantage of having only a single argument, at the cost of some added complexity.[14] This OCF is similar in some ways to Bachmann's $$\psi$$ and its recase by Rathjen.

\begin{eqnarray*} C_0(\alpha, \beta) &=& \beta \cup \{0, \Omega\}\\ C_{n+1}(\alpha, \beta) &=& \{\gamma + \delta, \omega^{\gamma}, \vartheta(\eta) | \gamma, \delta, \eta \in C_n (\alpha, \beta); \eta < \alpha\} \\ C(\alpha, \beta) &=& \bigcup_{n < \omega} C_n (\alpha, \beta) \\ \vartheta(\alpha) &=& \min \{\beta < \Omega | C(\alpha, \beta) \cap \Omega \subseteq \beta \wedge \alpha \in C(\alpha, \beta)\} \\ \end{eqnarray*}

Rathjen and Weiermann showed that $$\vartheta(\alpha)$$ is defined for all $$\alpha < \varepsilon_{\Omega+1}$$, but do not discuss higher values.

$$\vartheta$$ follows the archetype of many ordinal collapsing functions — it is defined inductively with a "marriage" to the $$C$$ function. Interpreting the equations:

• $$C(\alpha, \beta)$$ is the set of all ordinals constructible using only the following:
• Zero, all ordinals less than $$\beta$$, and $$\Omega$$.
• Finite applications of addition, $$\kappa \mapsto \omega^\kappa$$, $$\kappa \mapsto \vartheta(\kappa)$$ (the latter only if $$\vartheta(\kappa)$$ has yet been defined).
• $$\vartheta(\alpha)$$ is the smallest ordinal $$\beta$$ so that $$\alpha \in C(\alpha, \beta)$$, and $$\beta$$ is greater than all the countable ordinals in $$C(\alpha, \beta)$$.

### Jäger's $$\psi$$

Main article: Jäger's function

Jäger's $$\psi$$ is a hierarchy of single-argument ordinal functions $$\psi_\kappa$$ indexed by uncountable regular cardinals $$\kappa$$ smaller than the least weakly Mahlo cardinal $$M_0$$ introduced by German mathematician Gerhard Jäger in 1984[15]. It was developed on the base of Buchholz's approach.

Every $$\psi_\kappa$$ is a function from $$M_0$$ to $$\kappa$$ which "collapses" the elements of $$M_0$$ below $$\kappa$$.

$$\kappa^{-}=0$$ if $$\kappa=I_\alpha(0)$$ for some $$\alpha \in \kappa$$

$$\kappa^{-}=I_\alpha(\beta)$$ if $$\kappa=I_\alpha(\beta+1)$$ for some $$\alpha,\beta \in \kappa$$

$$C_\kappa^0(\alpha) = \{\kappa^{-}\}\cup\kappa^{-}$$

For any $$n < \omega$$, $$C_\kappa^{n+1}(\alpha) \subset M_0$$ is the smallest set satisfying the following:

1. The sum of any finitely many ordinals in $$C_\kappa^n(\alpha)$$ belongs to $$C_\kappa^{n+1}(\alpha)$$.
2. For any $$\beta,\gamma\in C_\kappa^n(\alpha)$$, $$\varphi_\beta(\gamma)\in C_\kappa^{n+1}(\alpha)$$.
3. For any $$\beta,\gamma\in C_\kappa^n(\alpha)$$, $$I_\beta(\gamma)\in C_\kappa^{n+1}(\alpha)$$.
4. For any ordinal $$\gamma$$ and uncountable regular cardinal $$\pi \in C_\kappa^n(\alpha)$$, $$\gamma<\pi<\kappa$$ implies $$\gamma\in C_\kappa^{n+1}(\alpha)$$.
5. For any $$\gamma \in \alpha \cap C_\kappa^n(\alpha)$$ and uncountable regular cardinal $$\pi\in C_\kappa^n(\alpha)$$, $$\gamma\in C_\pi(\gamma)$$ implies $$\psi_\pi(\gamma)\in C_\kappa^{n+1}(\alpha)$$.

$$C_\kappa(\alpha)=\bigcup\{C_\kappa^n(\alpha)|n<\omega\}$$

$$\psi_\kappa(\alpha)=\text{min}\{\xi \in \kappa|\xi\notin C_\kappa(\alpha)\}$$

The ordinal collapsing function admits an ordinal notation system, while those created by googologists do not usually admit ordinal notations (cf. #Relationship to ordinal notations). Unlike the unfortunate situations, it has sophisticated simplifications admitting ordinal notations.

#### Buchholz's simplification

Main article: Jäger-Buchholz function

Buchholz created a sophisticated simplification.[16] It is based on the least weakly inaccessible cardinal, and is called Jäger-Buchholz function in Japanese googology community.

#### Denis' simplification

Googology wiki user Denis Maksudov also created a sophisticated simplification.[17] It is based on $$2$$-ary enumeration function of weakly higher inaccessible cardinals.

### Rathjen's $$\psi$$

Main article: Rathjen's psi function

Rathjen's $$\psi$$ function is based on the least weakly Mahlo cardinal to create large countable ordinals.[3] A weakly Mahlo cardinal can be defined as a cardinal $$\text M$$ such that all normal functions closed in $$\text M$$ have a regular fixed point. He uses this to diagonalise over the weakly inaccessible hierarchy. It admits an associated ordinal notation $$T(\text{M})$$ whose limit (i.e. ordinal type) is $$\psi_{\Omega}(\chi_{\varepsilon_{\text{M}+1}}(0))$$, which is strictly greater than $$\textrm{PTO}(\textrm{KPM})$$ and is strictly smaller than the limit of countable ordinals expressed by Rathjen's $$\psi$$. For the full definition, see the main article.

### Rathjen's $$\Psi$$

Main article: Rathjen's Psi function

Rathjen's $$\Psi$$ function is based on the least weakly compact cardinal to create large countable ordinals.[4] A weakly compact cardinal can be defined as a cardinal $$\mathcal{K}$$ such that it is $$\Pi_1^1$$-indescribable. He uses this to diagonalise over the weakly Mahlo hierarchy. The functions $$M^{\alpha}$$, $$C(\alpha,\pi)$$, $$\Xi(\alpha)$$, and $$\Psi^{\xi}_{\pi}(\alpha)$$ are defined in mutual recursion in the following way:

$$M^0=K\cap\text{Lim}$$, where $$\text{Lim}$$ denotes the class of limit ordinals.

For $$\alpha>0$$, $$M^\alpha$$ is the set of $$\pi<K$$ such that $$\pi$$ satisfies these 3 conditions:

1. $$C(\alpha,\pi)\cap K = \pi$$
2. $$\forall\xi\in C(\alpha,\pi) \cap \alpha, M^{\xi} \text{ stationary in }\pi$$
3. $$\alpha\in C(\alpha,\pi)$$

$$C(\alpha,\beta)$$ is the closure of $$\beta\cup\{0,K\}$$ under addition, $$(\xi,\eta)\mapsto\varphi(\xi,\eta)$$, $$\xi\mapsto\Omega_\xi$$ given $$\xi<K$$, $$\xi\mapsto\Xi(\xi)$$ given $$\xi<\alpha$$, and $$(\xi,\pi,\delta)\mapsto\Psi^\xi_\pi(\delta)$$ given $$\xi\le\delta<\alpha$$.

$$\Xi(\alpha)=\min(M^\alpha \cup\{K\})$$.

For $$\xi\le\alpha$$, $$\Psi^\xi_\pi(\alpha)=\min(\{\rho\in M^\xi\cap\pi \colon C(\alpha,\rho) \cap \pi = \rho \land \pi \land \alpha\in C(\alpha,\rho)\} \cup \{\pi\})$$.

### Duchhardt's $$\Psi$$

Christoph Duchhardt has created a 6-ary OCF $$\Psi$$ using a $$\Pi_4$$-reflecting ordinal[18], and proved bounds on the proof-theoretic ordinal of KP set theory augmented by a certain $$\Pi_4$$-reflection principle.

### Stegert's $$\Psi$$

Main article: Stegert's Psi function

As part of his 2010 Ph.D. dissertation, Jan-Carl Stegert introduced collapsing hierarchies $$\mathfrak{M}$$ and two collapsing functions $$\Psi$$, and developed a powerful ordinal notation based on the work of Rathjen.[19]

### Arai's $$\psi$$

Main article: Arai's psi function

A Japanese professional mathematician Arai introduced an OCF $$\psi$$ based on first order reflection under an extension $$\textrm{ZFLK}_N$$ of $$\textrm{ZFC}$$ and a new simplified ordinal notation associated to it.[20] Here, $$N$$ is a natural number greater than $$2$$. Arai verified $$\psi_{\Omega_1}(\varepsilon_{\mathbb{K}+1})$$ is the proof theoretic ordinal of $$\textrm{KP}\Pi_N$$, where $$\mathbb{K}$$ is the least $$\Pi_N$$-reflecting ordinal and $$\textrm{KP}\Pi_N$$ is Kripke–Platek set theory for the $$\Pi_N$$-reflecting universe. This seems to be the current state of the art in the field of ordinal analysis.

## Sources

1. M. Rathjen, Proof Theory (Stanford encyclopedia of Philosophy), special case of definition 5.11. Accessed 2021-11-08
2. W. Buchholz, A New System of Proof-Theoretic Ordinal Functions, Annals of Pure and Applied Logic, vol. 32, pp.195-207, (1986).
3. Rathjen, Michael. "Ordinal Notations Based on a Weakly Mahlo Cardinal", Archive for Mathematical Logic 29 (1990) 249--263.
4. Rathjen, Michael. "Proof Theory of Reflection", Annals of Pure and Applied Logic 68, 181--224 (1994).
5. Bachmann, Heinz. "Die Normalfunktionen und das Problem der ausgezeichneten Folgen von Ordnungszahlen"Vierteljahrsschrift der Naturforschenden Gesellschaft in Zürich, vol. 95. Zürich: Orell Füssli Graphische Betriebe AG.. Retrieved 2014-08-29. Translation at https://arxiv.org/pdf/1903.04609.pdf
6. M. Rathjen, "A history of ordinal representations" (notes). Archived 2007-06-12.
7. Rathjen, Michael. "The Art of Ordinal Analysis"
8. Maksudov, Denis. The extension of Buchholz's functionTraveling To The Infinity. Retrieved 2017-05-18.
9. "Ordinal Collapsing Function". Wikipedia. Retrieved 2014-08-29.
10. Chris Bird, pers. comm
11. Bird, Chris. "The Fast-Growing Hierarchy in Terms of Bird's Array Notations". Retrieved 2014-08-29.
12. Wilken, Gunnar. "Ordinal arithmetic based on Skolem hulling". Annals of Pure and Applied Logic. Retrieved 2014-09-21.
13. Weiermann, Andreas and Wilken, Gunnar. "Ordinal arithmetic with simultaneously defined theta-functions". Wiley Online. Retrieved 2014-09-21.
14. Rathjen, Michael and Weiermann, Andreas. "Proof Theoretic Investigations of Kruskal's Theorem"
15. Jäger, M.. Arch. Math. Logik Grundlagenforsch (1984),24
16. W. Buchholz, A simplified version of local predicativity, Proof Theory : A Selection of Papers from the Leeds Proof Theory Programme 1990 (1992), pp. 115--147.
17. Traveling To The Infinity
18. C. Duchhardt, Thinning operators and $$\Pi_4$$-reflection (2008)

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