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 \(\phi\).

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

Here is an example:


 * We begin with \(\vartheta(0)\).
 * First we examine \(C(0, 0)\). We of course have \(0\), then \(\omega^0 = 1\), then \(\omega^1 = \omega\), then \(\omega^\omega\), etc. It can be seen that the supremum of the countable ordinals in \(C(0, 0)\) is \(\varepsilon_0\). More strongly, we have \(C(0, 0) \cap \Omega = \varepsilon_0\).
 * \(C(0, 1) = C(0, 0)\); allowing all the ordinals less than 1 obviously does not help us in the slightest. Indeed, \(C(0, \beta) = C(0, 0)\) for all \(\beta < \varepsilon_0\) because all the ordinals less than these \(\beta\)s are already in \(C(0, 0)\).
 * Something more interesting happens with \(C(0, \varepsilon_0)\). Now we have access to \(\varepsilon_0\), \(\omega^{\varepsilon_0 + 1}\), etc. with a new supremum of \(\varepsilon_1\). Similarly, \(C(0, \varepsilon_1)\) has a supremum of \(\varepsilon_2\), \(C(0, \varepsilon_2)\) has a supremum of \(\varepsilon_3\), etc.
 * \(C(0, \varepsilon_\omega)\), however, reaches a supremum \(\varepsilon_\omega\). Since \(0 \in C(0, \varepsilon_\omega)\) and \(\varepsilon_\omega\) is greater than all the countable ordinals in \(C(0, \varepsilon_\omega)\), we have \(\vartheta(0) = \varepsilon_\omega\).