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, 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. In particular, note that an OCF itself can never be an ordinal notation. An OCF is just a function defined in set theory, while an ordinal notation is a notation equipped with a (usually 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 recursive relation encoding the \(\in\)-relation for ordinals restricted to the underlying set of the structure. Creating an associated ordinal notation is usually much more difficult than creating an OCF.

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 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 \(\theta_{\varepsilon_{\Omega_\omega + 1}}(0)\), Theorem 3.7 but an ordinal notation up to the ordinal associated to the function is unknown at least in this community.

Buchholz's \(\psi\)
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 \(\varphi\):

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

\(\theta(\alpha, 0)\) 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.

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 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 \(\psi_\pi(\gamma)\in C_\kappa^{n+1}(\alpha)\).

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

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

It has a sophisticated simplification 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 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 \(\Psi\)
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. 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, and proved bounds on the proof-theoretic ordinal of KP set theory augmented by a certain \(\Pi_4\)-reflection principle.

Stegert's \(\Psi\)
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. <!--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.