Ordinal collapsing function

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. 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. 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 defined in arithmetic. In order to associate an ordinal notation to an OCF, we need 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 the OCF, and construct a primitive recursive relation encoding the \(\in\)-relation restricted to the underlying set of the structure.

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. 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.

There are many OCFs in use, often similar to each other and easily confused (some even use the same symbols).

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

Rathjen suggests a "recast" of the system 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, the proof-theoretic ordinal of Kripke-Platek set theory with axiom of infinity.

Feferman's \(\theta\)
Feferman's \(\theta\)-functions constitute a hierarchy of single-argument functions \(\theta_\alpha: \text{On} \to \text{On}\) for \(\alpha \in \text{On}\). 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 iff 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 Feferman theta function is considered an extension of the two-argument Veblen function &mdash; for \(\alpha < \Gamma_0\), \(\theta_\alpha(\beta) = \varphi_\alpha(\beta)\). For this reason, \(\varphi\) may be used interchangeably with \(\theta\) for \(\alpha < \Gamma_0\). Because of the restriction of \(\xi \in C_n(\alpha, \beta)\) imposed in the definition of \(C_{n+1}(\alpha, \beta)\), which makes \(\theta_{\Gamma_0}\) never used in the calculation of C set when \(\alpha < \Omega\), \(\theta\) function does not grow until \(\alpha < \Omega\). This results in \(\theta_\Omega(0) = \Gamma_0\) while \(\varphi_\Omega(0) = \Omega\). The value of \(\theta_\Omega(0) = \Gamma_0\) can be used above \(\Omega\) because of the definition of \(C_0\) which includes \(\Omega = \omega_1\). The supremum of the range of the function is the Takeuti-Feferman-Buchholz ordinal \(\theta_{\varepsilon_{\Omega_\omega + 1}}(0)\).

Buchholz discusses a set he calls \(\theta(\omega + 1)\), which is the set of all ordinals describable with \(\{0, \omega_1, \omega_2, \ldots, \omega_\omega\}\) and finite applications of \(+\) and \(\theta\).

Buchholz's \(\psi\)
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 \(\psi_v\alpha\). 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. It has a sophisticated extension called extended Buchholz's function (EBF for short) created by wiki user Denis Maksudov.

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. 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.

Bird's \(\theta\)
Chris Bird devised the following shorthand for the extended (zero-indexed) Veblen function \(\phi\):

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

\(\theta(\alpha, 0)\) 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 \(\beta < \Omega_{m+1}\):

\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) \\ \eta \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 \\ \eta \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. 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. 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 &mdash; 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 \(\psi\)
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. It was developed on 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 \(\psi_\pi(\gamma)\in C_\kappa^{n+1}(\alpha)\).

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

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

It has a sophisticated extension created by wiki user Denis Maksudov.

Rathjen's \(\psi\)
Rathjen's \(\psi\) function is based on the least weakly Mahlo cardinal to create large countable ordinals. 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 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 \(\psi\). The ordinal \(\textrm{PTO}(\textrm{KPM})\), which is called "Rathjen's ordinal" in this community, means the proof-theoretic ordinal of \(\textbf{KPM}\), where \(\textrm{KPM}\) is Kripke–Platek set theory \(\textrm{KP}\) augmented by the axiom schema "for any \(\Delta_0\)-formula \(H(x,y)\) satifying \(\forall x, \exists y, H(x,y)\), there exists an addmissible set \(z\) satisfying \(\forall x \in z, \exists y, H(x,y)\)". It coincides with \(\psi_{\omega_1}(\psi_{\chi_{\varepsilon_{\text M+1}}(0)}(0))\) in Rathjen's \(\psi\) function.

Be careful that there are several common mistakes on Rathjen's \(\psi\) in googology:
 * The limit of countable ordinals expressed by Rathjen's \(\psi\) function is often confounded with \(\textrm{PTO}(\textrm{KPM})\), but they are distinct ordinals. The former one is strictly greater than the limit of \(T(\text{M})\), while the latter one is strictly smaller than it.
 * Rathjen's \(\psi\) function is often confounded with another simplified Rathjen's \(\psi\), but they are distinct OCFs. The former one admits a published associated ordinal notation, while the latter one seems not. The former one uses indices in subscripts, while the latter one uses indices in superscripts.
 * Rathjen's \(\psi\) function is often confounded with another Rathjen's function symbol \(\psi\), but they are distinct notions. The former one is a published OCF, while the latter one is just a function symbol in an ordinal notation associated to an unpublished OCF.
 * Rathjen's \(\psi\) function is often confounded with the unspecified OCF in the title of the article of ψ(ψᵢ(0)), but they are distinct OCFs. The former one satisfies \(\psi(\psi_I(0)) > \psi(\Omega_{\Omega_{\Omega_{\cdot_{\cdot_{\cdot}}}}})\), while the latter one is intended to satisfy \(\psi(\psi_I(0)) = \psi(\Omega_{\Omega_{\Omega_{\cdot_{\cdot_{\cdot}}}}})\), where \(\psi\) is the abbreviation of \(\psi_{\Omega}\) and \(\Omega_{\Omega_{\Omega_{\cdot_{\cdot_{\cdot}}}}}\) denotes the least omega fixed point.
 * Rathjen's \(\psi\) function is often confounded with unspecified OCFs in 2000 steps, but they are distinct OCFs. The former one does not fit the analyses of the latter ones.
 * Several complicated conditions in the definition of Rathjen's \(\psi\) function are often considered as droppable ones if we are interested only in googology, but they are actually used to define the ordinal notation associated to it, which is needed to create a computable large number.

Due to the complexity of the original notation, the one provided below has been simplified slightly, but still emulates the main properties of the original notation. In order to avoid the confusion similar to above ones, we emphasise that the simplified OCF below is different from the original one, and hence we do not have an ordinal notation associated to it.

We restrict \(\pi\) to uncountable regular ordinals.

\(\text{enum}(X)\) is the unique increasing function \(f\) such that the range of \(f\) is exactly \(X\).

\(\text{cl}(X)\) is the closure of \(X\) under the order topology; that is \(\text{cl}(X):=X\cup\{\beta:\sup(X\cap\beta)=\beta\}\)

\(\text B_0(\alpha,\beta):=\beta\cup\{0,\text M\}\)

\(\text B_{n+1}(\alpha,\beta):=\{\gamma+\delta,\varphi_\gamma(\delta),\chi_\mu(\delta):\gamma,\delta,\mu\in\text B_n(\alpha,\beta)\wedge\mu<\alpha\}\)

\(\text B(\alpha,\beta):=\cup_{n<\omega}\text B_n(\alpha,\beta)\)

\(\chi_\alpha:=\text{enum}(\text{cl}(\{\pi:\text B(\alpha,\pi)\cap\text M\subseteq \pi\wedge\alpha\in\text B(\alpha,\pi)\}))\)

\(\text C_0(\alpha,\beta):=\beta\cup\{0,\text M\}\)

\(\text C_{n+1}(\alpha,\beta):=\{\gamma+\delta,\varphi_\gamma(\delta),\chi_\gamma(\delta),\psi_\pi(\mu):\gamma,\delta,\mu,\pi\in\text C_n(\alpha,\beta)\wedge\mu<\alpha\}\)

\(\text C(\alpha,\beta):=\cup_{n<\omega}\text C_n(\alpha,\beta)\)

\(\psi_\pi(\alpha):=\min\{\beta:\text C(\alpha,\beta)\cap\pi\subseteq\beta\wedge\alpha\in\text C(\alpha,\beta)\}\)

Rathjen originally defined the \(\psi\) function in more complicated a way in order to create an ordinal notation associated to it. Therefore it is not certain whether the simplified OCF above yields an ordinal notation or not.

The original \(\chi\) functions used in Rathjen's original OCF are not so easy to understand, and differ from the \(\chi\) functions defined above. Instead, we explain the \(\text I\) functions, which is deeply related to the original \(\chi\) functions. The \(\text I_0\) function enumerates the uncountable cardinals less than \(\text M\). The \(\text I_1\) function enumerates the weakly inaccessible cardinals less than \(\text M\), and their limits. Similarly, for each \(\alpha<\text M\), \(\text I_{1+\alpha}\) enumerates the weakly \(\alpha\)-inaccessible cardinals less than \(\text M\) (and their limits, because the function is normal), and the function \(\text I_{\text M}\) diagonalises over these, and enumerates the weakly hyper-inaccessible cardinals and their limits less than \(\text M\). \(\text I_{\text M2}\) enumerates the weakly hyper-hyper-inaccessible cardinals and their limits less than \(\text M\), and so on and so fourth. Rathjen verified that \(\chi_{\alpha}(\beta)\) (with respect to the original \(\chi\) functions) coincides with \(\textrm I_{\alpha}(\beta)\) for any \(\alpha < \Lambda_0 := \chi_{\chi_{\chi_{\cdot_{\cdot_{\cdot}}}(0)}(0)}(0)\) and \(\beta < \text M\).

Although it may not be obvious why, the limits of these large cardinals are added to the \(\text I\) function to make it much easier to notate, say, the suprenum of the first weakly inaccessible cardinal, the second weakly inaccessible cardinal, the third weakly inaccessible cardinal, \(\ldots\), as values like these don't otherwise have any way of being notated easily. Be careful that some author denote by \(\text I_\alpha\) the \(\alpha\)th weakly inaccessible cardinal, while Rathjen denoted by \(\text I_\alpha\) a function.

Note that it is possible to define the same ordinal using only a single \(\text C\) function instead of both \(\text B\) and \(\text C\), however this is not done in the original text, as it is more transparent and easier to manage values when the functions are defined separately. Moreover, such a "simplification" might not admit an ordinal notation associated to it.

Rathjen's \(\Psi\)
Rathjen's \(\Psi\) function is based on the least weakly compact cardinal to create large countable ordinals. 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.

\(M^0=K\cap\text{Class of limit ordinals}\)

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

Stegert's \(\Psi\)
As part of his 2010 Ph.D. dissertation, Jan-Carl Stegert introduced collapsing hierarchies \(\mathcal{M}\) and two collapsing functions \(\Psi\), and developed a powerful ordinal notation based on the work of Rathjen. <!--Define the following:


 * \(\Xi\) is a \(\Pi_0^2\)-indescribable cardinal.
 * \(\text{Card}\) is the class of cardinals.
 * \(=_\text{NF}\) denotes that the right-hand side is the Cantor normal form of the left-hand side.
 * For a cardinal \(\kappa\), \(\kappa^+\) denotes its cardinal successor.

A reflection instance is a type of string defined as follows. The alphabet consists of the symbols \((\,)\,;\,\text{-}\,\epsilon\,\mathsf{M}\,\mathsf{P}\) and ordinals, and the syntax is one of \((\pi;\mathsf{P}_m;\vec{R};\mathbb{Z};\alpha)\) or \((\pi;\mathsf{M}^\xi_{\mathbb{M}(\vec{\nu})};\vec{R};\mathbb{Z};\alpha)\) where \(\pi,\alpha\) are ordinals and \(\vec{R}\) is a string in the alphabet. A reflection configuration is a function from a finite sequence of ordinals to reflection instances -- in this way a reflection instance can be considered a reflection configuration with zero arguments.

\begin{align*} C^0(\alpha, \pi) &= \pi \cup \{0, \Xi\} \\ C^{n+1}(\alpha, \pi) &= \bigcup \begin{cases} C^n(\alpha, \pi) \\ \{\gamma + \omega^\delta \mid \gamma, \delta \in C^n(\alpha, \pi) \wedge \gamma =_\text{NF} \omega^{\gamma_1} + \cdots + \omega^{\gamma_n} \wedge \gamma_n \geq \delta\} \\ \{\varphi(\xi, \eta) \mid \xi, \eta \in C^n(\alpha, \pi)\} \\ \{\kappa^+ \mid \kappa \in C^n(\alpha, \pi) \cap \text{Card} \cap \Xi\} \\ \{\Psi_\mathbb{X}^\gamma \mid \mathbb{X}, \gamma \in C^n(\alpha, \pi) \wedge \gamma < \alpha \wedge \Psi_\mathbb{X}^\gamma \text{ is well-defined.}\} \end{cases} \\ C(\alpha, \pi) &= \bigcup_{n < \omega} C^n(\alpha, \pi) \\ \end{align*}

The reflection configuration \(\mathbb{A}\) is defined as \(\mathbb{A}(m) = (\Xi;\mathsf{P}_m;\epsilon;\epsilon;\omega)\) and with a domain of \(\{m \mid 1 \leq m < \omega\}\) (type 1). For every \(\omega \leq \kappa < \Xi,\,\kappa \in \text{Card}\), we define the reflection instance \((\kappa^+;\mathsf{P}_0;\epsilon;\epsilon;0)\) (type 2).


 * Type 1. Let \(\mathbb{X} = \mathbb{A}(m + 1)\). For \(\alpha \geq \omega\), define \(\mathfrak{M}_\mathbb{X}^\alpha\) as the set of all ordinals \(\kappa < \Xi\) such that \(C(\alpha, \kappa) \cap \Xi = \kappa\), \(\mathbb{X}, \alpha \in C(\kappa)\), \(\kappa\) is \(\Pi_1^m\)-indescribable, and \(\kappa \vDash \mathsf{M}_{\mathbb{A}}^{<\alpha} \text{-} \mathsf{P}_m\). We have \(\Psi_\mathbb{X}^\alpha = \min(\mathfrak{M}_\mathbb{X}^\alpha)\) (and if \(\mathfrak{M}_\mathbb{X}^\alpha\) is empty then it is not well-defined).
 * Let \(\alpha = \omega\). Then we define the reflection instance \((\Psi_\mathbb{X}^\alpha;\mathsf{P}_m;(\mathsf{M}_\mathbb{A}^{<\alpha}\text{-}\mathsf{P}_{m-1},\ldots,\mathsf{M}_\mathbb{A}^{<\alpha}\text{-}\mathsf{P}_0);\mathbb{X};\omega + 1)\) (type 2).
 * Let \(\alpha > \omega\). Then we define the reflection configuration \(\mathbb{G}\) as \(\text{dom}(\mathbb{G}) = [\omega,\alpha)_{C(\Psi_\mathbb{X}^\alpha)} \times (m, \omega)\) and \(\mathbb{G}(\zeta, n) = (\Psi_\mathbb{X}^\alpha;\mathsf{M}_{\mathbb{A}(n)}^\zeta\text{-}\mathsf{P}_m;(\mathsf{M}_\mathbb{A}^{<\alpha}\text{-}\mathsf{P}_{m-1},\ldots,\mathsf{M}_\mathbb{A}^{<\alpha}\text{-}\mathsf{P}_0);\mathbb{X};\alpha + 1)\) (type 3).
 * Type 2. Let \(\mathbb{X} = (\pi;\mathsf{P}_m;\vec{R};\mathbb{Z};\delta)\). For \(\alpha \geq \delta\), define \(\mathfrak{M}_\mathbb{X}^\alpha\) as the set of all ordinals \(\kappa < \pi\) such that \(C(\alpha, \kappa) \cap \pi = \kappa\), \(\mathbb{X}, \alpha \in C(\kappa)\), and if \(m > 0\) then \(\kappa \vDash \vec{R}\) and \(\kappa \vDash \mathsf{M}_\mathbb{X}^{<\alpha}\text{-}\mathsf{P}_{m-1}\). Again \(\Psi_\mathbb{X}^\alpha = \min(\mathfrak{M}_\mathbb{X}^\alpha)\). We proceed to these next subcases iff \(m > 0\).

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Arai's \(\psi\)
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. 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.