Psi function

The psi function is an ordinal collapsing function developed by W. Buchholz. It was developed as a simpler alternative to the theta function, and the two systems are equally powerful.

Let \(\Omega_0 = 1\) and \(\Omega_\alpha = \aleph_alpha\) for \(\alpha > 0\). Let \(P\) be the class of all ordinals of the form \(\omega^\alpha\), and define \(P(0) = \empty\) and \(P(\alpha) = \{\alpha_0, \alpha_1, \ldots, \alpha_n\}\) where all \(\alpha_i \in P\), and \(\alpha_0 + \alpha_1 + \cdots + \alpha_n = \alpha\) and \(\alpha_0 \geq \alpha_1 \geq \cdots \geq \alpha_n\), and all.