User:Binary198/List of ordinal collapsing functions

This is a list of ordinal collapsing functions. Feel free to add any that you invented or others forgot to include to this list!

Bachmann's ψ
The first true OCF, Bachmann's $$\psi$$ was invented by Heinz Bachmann, somewhat cumbersome as it depends on fundamental sequences for all limit ordinals; yet the full definition has been lost over time. Michael Rathjen has suggested a "recast" of the system, which goes like so:


 * Let $$\Omega$$ represent an uncountable ordinal such as $$\omega_1$$;
 * Then define $$C^\Omega_0(\alpha, \beta) = \beta \cup \{0, \Omega\}$$ and $$C^\Omega_{n+1}(\alpha, \beta) = \{\delta + \theta: \delta, \theta \in C^\Omega_n(\alpha, \beta)\} \cup \{\omega^\xi: \xi \in C^\Omega_n(\alpha, \beta)\} \cup \{\psi_\Omega(\xi): \xi \in C^\Omega_n(\alpha, \beta) \land \xi < \alpha\}$$
 * $$C^\Omega(\alpha, \beta) = \bigcup\limits_{n < \omega} C^\Omega_n(\alpha, \beta)$$
 * $$\psi_\Omega(\alpha) = min(\{\rho < \Omega: C^\Omega(\alpha, \rho) \cap \Omega = \rho\})$$

$$\psi_\Omega(\varepsilon_{\Omega+1})$$ is the Bachmann-Howard ordinal, the proof-theoretic ordinal of Kripke-Platek set theory with the axiom of infinity (KP). One can also generalize the definition like so:


 * $$C^\gamma_0(\alpha, \beta) = \beta \cup \{0, \gamma\}$$
 * $$C^\gamma_{n+1}(\alpha, \beta) = \{\delta + \theta: \delta, \theta \in C^\gamma_n(\alpha, \beta)\} \cup \{\omega^\xi: \xi \in C^\gamma_n(\alpha, \beta)\} \cup \{\psi_\gamma(\xi): \xi \in C^\gamma_n(\alpha, \beta) \land \xi < \alpha\}$$
 * $$C^\gamma(\alpha, \beta) = \bigcup\limits_{n < \omega} C^\gamma_n(\alpha, \beta)$$
 * $$\psi_\gamma(\alpha) = min(\{\rho < \Omega: C^\gamma(\alpha, \rho) \cap \Omega = \rho\})$$

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


 * $$C_0(\alpha, \beta) = \beta \cup \{0, \Omega, \Omega_2, ..., \Omega_\omega\}$$
 * $$C_{n+1}(\alpha, \beta) = \{\gamma + \delta, \theta_\xi(\eta) | \gamma, \delta, \xi, \eta \in C_n(\alpha, \beta); \xi < \alpha \}$$
 * $$C(\alpha, \beta) = \bigcup\limits_{n < \omega} C_n(\alpha, \beta)$$
 * $$\theta_\alpha(\beta) = \mathsf{min}(\{\gamma | \gamma \notin C(\alpha, \gamma \land \forall \delta < \beta: \theta_\alpha(\delta) < \gamma \})$$

The supremum of countable ordinals that can be expressed with this function is the Takeuti-Feferman-Buchholz ordinal $$\theta_{\varepsilon_{\Omega_\omega+1}}(0)$$, but an ordinal notation associated to this function is unheard of or nonexistent, unlike for Buchholz's psi function.

Buchholz's ψ
Buchholz's $$\psi$$ is a hierarchy of single-argument functions $$\psi_\nu: \mathsf{On} \rightarrow \mathsf{On}$$, with $$\psi_\nu(\alpha)$$ occasionally abbreviated as $$\psi_\nu\alpha$$. This function is likely the most well known out of all OCFs. The definition is so:


 * Define $$\Omega_0 = 1$$ and $$\Omega_\nu = \aleph_\nu$$ for $$\nu > 0$$.
 * 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 $$\xi \in \mathsf{On}$$)
 * $$C^0_\nu(\alpha) = \Omega_\nu$$
 * $$C^{n+1}_\nu(\alpha) = C^{n}_\nu(\alpha) \cup \{\psi_\nu(\xi) | \xi \in \alpha \cap C^{n}_\nu(\alpha) \land \xi \in C_u(\xi) \land u \leq \omega \}$$
 * $$C_\nu(\alpha) = \bigcup\limits_{n < \omega} C^n_\nu(\alpha)$$
 * $$\psi_\nu(\alpha) = min(\{\gamma | \gamma \notin C_\nu(\alpha)\})$$

The limit of this system is $$\psi_0(\varepsilon_{\Omega_\omega + 1})$$, the Takeuti-Feferman-Buchholz ordinal.

Extended Buchholz's ψ
This OCF is a sophisticated extension of Buchholz's $$\psi$$ by mathematician Denis Maksudov. The limit of this system is much greater, equal to $$\psi_0(\Omega_{\Omega_{\Omega_{...}}})$$ where $$\Omega_{\Omega_{\Omega_{...}}}$$ denotes the first omega fixed point, sometimes referred to as Extended Buchholz's ordinal. The function is defined as follows:


 * Define $$\Omega_0 = 1$$ and $$\Omega_\nu = \aleph_\nu$$ for $$\nu > 0$$.
 * $$C^0_\nu(\alpha) = \{\beta | \beta < \Omega_\nu\}$$
 * $$C^{n+1}_\nu(\alpha) = \{\beta + \gamma, \psi_\mu(\eta) | \mu, \beta, \gamma, \eta \in C^{n}_\nu(\alpha) \land \eta < \alpha\}$$
 * $$C_\nu(\alpha) = \bigcup\limits_{n < \omega} C^n_\nu(\alpha)$$
 * $$\psi_\nu(\alpha) = min(\{\gamma | \gamma \notin C_\nu(\alpha)\})$$

Madore's ψ
This OCF was the same as the ψ function previously used throughout this article; it is a simpler, more efficient version of Buchholz's ψ function defined by David Madore. Its use in this article lead to widespread use of the function.


 * $$C_0(\alpha) = \{0, 1, \omega, \Omega\}$$
 * $$C_{n+1}(\alpha) = \{\gamma + \delta, \gamma\delta, \gamma^\delta, \psi(\eta) | \gamma, \delta, \eta \in C_n(\alpha); \eta < \alpha\}$$
 * $$C(\alpha) = \bigcup\limits_{n < \omega} C_n(\alpha)$$
 * $$\psi(\alpha) = min(\{\beta \in \Omega | \beta \notin C(\alpha)\})$$

This function was used by Chris Bird, who also invented the next OCF.

Bird's θ
Chris Bird devised the following shorthand for the extended Veblen function $$\varphi$$:


 * $$\theta(\Omega^{n-1}a_{n-1} + ... + \Omega^2a_2 + \Omega a_1 + a_0, b) = \varphi(a_{n-1}, ..., a_2, a_1, a_0, b)$$
 * $$\theta(\alpha, 0)$$ is abbreviated $$\theta(\alpha)$$

This function is only defined for arguments less than $$\Omega^\omega$$, and its outputs are limited by the small Veblen ordinal.

Wilken's ϑ
Wilken's ϑ is more generic than other OCFs:


 * Let $$\Omega_0$$ be either 1 or an epsilon number.
 * Let $$\Omega_1 > \Omega_0$$ be an uncountable regular cardinal.
 * For $$0 < i < \omega$$, let $$\Omega_{i + 1}$$ be the successor cardinal to $$\Omega_i$$.
 * For finite $$n$$ and $$0 \leq m < n$$, define the following for $$\beta < \Omega_{m + 1}$$:
 * $$\Omega_m \cup \beta \subseteq C^n_m(\alpha, \beta)$$
 * $$\xi, \eta \in C^n_m(\alpha, \beta) \Rightarrow \xi + \eta \in C^n_m(\alpha, \beta)$$
 * $$\eta \in C^n_m(\alpha, \beta) \cap \Omega_{k + 2} \Rightarrow \vartheta^n_k \in C^n_m(\alpha, \beta)$$ for m < k < n
 * $$\eta \in C^n_m(\alpha, \beta) \cap \alpha \Rightarrow \vartheta^n_m \in C^n_m(\alpha, \beta)$$
 * $$\vartheta^n_m(\alpha) = min(\{\xi < \Omega_{m + 1} | C^n_m(\alpha, \xi) \cap \Omega_{m + 1} \subseteq \xi \land \alpha \in C^n_m(\alpha, \xi)\} \cup \{\Omega_{m + 1}\})$$

n is needed to define the function, but n does not actually affect the function's behaviour. Therefore, one may safely eliminate n and simply write $$\vartheta_m(\alpha)$$.

Wilken and Weiermann's ϑ–
Wilken and Weiermann's ϑ– is closely related to Wilken's ϑ, and their paper closely analyzes the relationship between the two.


 * As before, let $$\Omega_0$$ be either 1 or an epsilon number.
 * Let $$\Omega_1 > \Omega_0$$ be an uncountable regular cardinal.
 * For $$0 < i < \omega$$, let $$\Omega_{i + 1}$$ be the successor cardinal to $$\Omega_i$$.
 * Let $$\Omega_\omega = sup_{i < \omega}\Omega_i$$
 * For all $$\beta < \Omega_{i + 1}$$, define the following:
 * $$\Omega_i \cup \beta \subseteq C^{-}_i(\alpha, \beta)$$
 * $$\xi, \eta \in C^{-}_i(\alpha, \beta) \Rightarrow \xi + \eta \in C^{-}_i(\alpha, \beta)$$
 * $$j \leq i < \omega \land C^{-}_j(\xi, \omega_{j + 1}) \cap C^{-}_i(\alpha, \beta) \cap \alpha \Rightarrow \vartheta^{-}_j(\xi) \in C^{-}_i(\alpha, \beta)$$
 * $$\vartheta^{-}_i(\alpha) = min(\{\xi < \Omega_{m + 1} | C^n_m(\alpha, \xi) \cap \Omega_{m + 1} \subseteq \xi \land \alpha \in C^n_m(\alpha, \xi)\} \cup \{\Omega_{m + 1}\})$$

Weiermann's ϑ
This ϑ 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 ψ and its recast by Rathjen.


 * $$C_0(\alpha + \beta) = \beta \cup \{0, \Omega\}$$
 * $$C_{n + 1}(\alpha + \beta) = \{\gamma + \delta, \omega^{\gamma}, \vartheta(\eta) | \gamma, \delta, \eta \in C_n(\alpha, \beta); \eta < \alpha\}$$
 * $$C(\alpha, \beta) = \bigcup\limits_{n < \omega} C_n(\alpha, \beta)$$
 * $$\vartheta(\alpha) = min(\{\beta < \Omega | C(\alpha, \beta) \cap \Omega \subseteq \beta \land \alpha \in C(\alpha, \beta)\})$$

Rathjen and Weiermann showed that $$\vartheta(\alpha)$$ is defined for all $$\alpha < \varepsilon_{\Omega + 1}$$, but do not discuss higher values. ϑ follows the archetype of many ordinal collapsing functions — it is defined inductively with a "marriage" to the C function.

Jäger's ψ
Jäger's ψ is a hierarchy of single-argument ordinal functions ψκ indexed by uncountable regular cardinals κ smaller than the least weakly Mahlo cardinal M0 introduced by German mathematician Gerhard Jäger in 1984. It was developed on the base of Buchholz's approach.


 * If $$\kappa = I_\alpha(0)$$ for some α < κ, $$\kappa^{-} = 0$$.
 * If $$\kappa = I_\alpha(\beta + 1)$$ for some α, β < κ, $$\kappa^{-} = I_\alpha(\beta)$$.
 * $$C^0_\kappa(\alpha) = \{\kappa^{-}\} \cup \kappa^{-}$$
 * For any finite n, $$C^{n + 1}_\kappa(\alpha) \subset M_0$$ is the smallest set satisfying the following:
 * The sum of any finitely many ordinals in $$C^{n}_\kappa(\alpha) \subset M_0$$ belongs to $$C^{n + 1}_\kappa(\alpha) \subset M_0$$.
 * For any $$\beta, \gamma \in C^{n}_\kappa(\alpha)$$, $$\varphi_\beta(\gamma) \in C^{n + 1}_\kappa(\alpha)$$.
 * For any $$\beta, \gamma \in C^{n}_\kappa(\alpha)$$, $$I_\beta(\gamma) \in C^{n + 1}_\kappa(\alpha)$$.
 * For any ordinal γ and uncountable regular cardinal $$\pi \in C^n_\kappa(\alpha)$$, $$\gamma < \pi < \kappa \Rightarrow \gamma \in C^{n + 1}_\kappa(\alpha)$$.
 * For any $$\gamma \in \alpha \cap C^n_\kappa(\alpha)$$ and uncountable regular cardinal $$\pi \in C^n_\kappa(\alpha)$$, $$\gamma \in C_\pi(\gamma) \Rightarrow \psi_\pi(\gamma) \in C^{n + 1}_\kappa(\alpha)$$.
 * $$C_\kappa(\alpha) = \bigcup\limits_{n < \omega} C^n_\kappa(\alpha)$$
 * $$\psi_\kappa(\alpha) = min(\{\xi \in \kappa | \xi \notin C_\kappa(\alpha)\})$$

Simplified Jäger's ψ
This is a sophisticated simplification of Jäger's ψ created by Denis Maksudov. An ordinal is α-weakly inaccessible if it is uncountable, regular and it is a limit of γ-weakly inaccessible cardinals for γ < α. Let I(α, 0) be the first α-weakly inaccessible cardinal, I(α, β + 1) be the first α-weakly inaccessible cardinal after I(α, β) and I(α, β) = $$sup(\{I(\alpha, \gamma) | \gamma < \beta\})$$ for limit β. Restrict ρ and π to uncountable regular ordinals of the form I(α, 0) or I(α, β + 1). Then,


 * $$C_0(\alpha, \beta) = \beta \cup \{0\}$$
 * $$C_{n + 1}(\alpha, \beta) = \{\gamma + \delta | \gamma, \delta \in C_n(\alpha, \beta)\} \cup \{I(\gamma, \delta) | \gamma, \delta \in C_n(\alpha, \beta)\} \cup \{\psi_\pi(\gamma) | \pi, \gamma, \in C_n(\alpha, \beta) \land \gamma < \alpha\}$$
 * $$C(\alpha, \beta) = \bigcup\limits_{n < \omega} C_n(\alpha, \beta)$$
 * $$\psi_\pi(\alpha) = min(\{\beta < \pi | C(\alpha, \beta) \cap \pi \subseteq \beta \})$$



Rathjen's Ψ
Rathjen's Ψ function is based on the least weakly compact cardinal to create large countable ordinals. For a weakly compact cardinal K, the functions $$M^\alpha$$, $$C(\alpha, \pi)$$, $$\Xi(\alpha)$$, and $$\Psi^\xi_\pi(\alpha)$$ are defined in mutual recursion in the following way:


 * M0 = $$K \cap \mathsf{Lim}$$, where Lim denotes the class of limit ordinals.
 * For α > 0, Mα is the set $$\{\pi < K | C(\alpha, \pi) \cap K = \pi \land \forall \xi \in C(\alpha, \pi) \cap \alpha, M^\xi \mathsf{} $$ is stationary in $$\pi \land \alpha \in C(\alpha, \pi)\} $$
 * $$C(\alpha, \beta) $$ is the closure of $$\beta \cup \{0, K\} $$ under addition, $$(\xi, \eta) \rightarrow \varphi(\xi, \eta) $$, $$\xi \rightarrow \Omega_\xi $$ given ξ < K, $$\xi \rightarrow \Xi(\xi) $$ given ξ < α, and $$(\xi, \pi, \delta) \rightarrow \Psi^\xi_\pi(\delta) $$ given $$\xi \leq \delta < \alpha $$.
 * $$\Xi(\alpha) = min(M^\alpha \cup \{K\}) $$.
 * For $$\xi \leq \alpha $$, $$\Psi^\xi_\pi(\alpha) = min(\{\rho \in M^\xi \cap \pi: C(\alpha, \rho) \cap \pi = \rho \land \pi \land \alpha \in C(\alpha, \rho)\} \cup \{\pi\}) $$.

Bachmann and Howard's ϑ
Note: This OCF does not seem to have an official name and the above is simply a nickname.

This OCF was introduced by "Emlightened" in an article on the Googology Wikia about ordinal collapsing functions. The definition is like so:


 * Let $$\Omega $$ represent ω1, the first uncountable ordinal.
 * $$C_0(\alpha, \beta) = \beta \cup \{\Omega\} $$
 * $$C_{n+1}(\alpha, \beta) = \{\gamma + \delta: \gamma, \delta \in C_n(\alpha, \beta)\} \cup \{\omega^\gamma: \gamma \in C_n(\alpha, \beta) \land \gamma \geq \Omega \} \cup \{\vartheta(\gamma): \gamma \in C_n(\alpha, \beta) \land \gamma < \alpha \} $$
 * $$C(\alpha, \beta) = \bigcup\limits_{n < \omega} C_n(\alpha, \beta)$$
 * $$\vartheta(\alpha, \beta) = min(\{\beta > 0: \Omega \cap C(\alpha, \beta) \subseteq \beta\})$$

AndrasKovacs' ψ
This is a variant of Madore's ψ introduced by AndrasKovacs in a piece of code describing large countable ordinals in Agda. $$\psi_\alpha(\beta) = \psi(\Omega_\alpha \cdot \beta)$$