Ordinal notation

An ordinal collapsing function (OCF) is a method of naming large ordinals using even larger ones. More specifically, ordinal collapsing functions take the structures found in large (often uncountable) ordinals and mirror those structures onto smaller ordinals. OCFs are employed as notations for large recursive ordinals, for which they have the most relevance to googology.

There are many OCFs in use, often similar to each other and easily confused (some even use the same symbols), but most are nearly or exactly equal. Popular systems include Pohler's \(\psi\), Feferman's \(\theta\), Weiermann's \(\vartheta\), Buchholz's extension of Pohler's \(\psi\), and an uncountable extension of the Veblen function \(\phi\).

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.

Feferman's \(\theta\)
Feferman's \(\theta\)-functions constitute a hierarchy of single-argument functions \(\theta_\alpha: \text{On} \mapsto \text{On}\) for \(\alpha \in \text{On}\) It is often considered a two-argument function with \(\theta_\alpha(\beta)\) written as \(\theta\alpha\beta\).

If \(\theta\xi\eta\) has been defined for all \(\xi < \alpha\) and \(\eta \in \text{On}\), then for all \(\beta \in \text{On}\), \(C(\alpha, \beta)\) is the set of all ordinals that can be generated from ordinals less than \(\beta\) and the constants \(0, \aleph_1, \aleph_2, \ldots, \aleph_\omega\) by finite applications of \(+\) and \(\theta \upharpoonright \alpha \times \text{On}\). An ordinal \(\beta\) is considered \(\alpha\)-critical iff \(\beta \not \in C(\alpha, \beta)\), and finally \(\theta_\alpha\) is defined as the enumerating function for all \(\alpha\)-critical ordinals.

Buchholz's \(\psi\)
Buchholz's \(\psi\) is a hierarchy of single-argument functions \(\psi_\alpha: \text{On} \mapsto \text{On}\) for \(\alpha \in \text{On}\).

Gro-Tsen's \(\psi\)
A Wikipedia user under the name "Gro-Tsen" defined the following simpler variant of one of Buchholz's functions 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, \beta); \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 \(C(\alpha)\).

Bird's \(\theta\)
Chris Bird devised the following shorthand for the extended 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.

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*}

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