Ordinal collapsing function

An ordinal collapsing function (OCF for short) is a ordinal function which is studied 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 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.

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*}

Informally:


 * 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. In his papers, Bird erroneously uses \(\theta(\Omega^\omega)\) and higher values without properly defining the function.

However, this function is equivalent to Feferman's \(\theta\) function (defined above) where both are defined, so it may be that he was merely using an extension of this \(\theta\) function.

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.

\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 \(C(\alpha, \beta)\).

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.

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.

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 \(X\); 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.

Rathjen's ordinal is \(\psi_{\omega_1}(\psi_{\chi_{\varepsilon_{\text M+1}}(0)}(0))\) in Rathjen's original \(\psi\) function, and coincides with the proof-theoretic ordinal of \(\textrm{KPM}\), where \(\textrm{KPM}\) is Kripke–Platek set theory 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 \(\forall x \in z, \exists y, H(x,y)\)".

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.

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.

Stegert's \(\Psi\)
As part of his 2010 Ph.D. dissertation, Jan-Carl Stegert introduced collapsing hierarchies \(\mathcal{M}\) and 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.