Cantor's ordinal

Cantor's ordinal \(\zeta_0\) (pronounced "zeta-zero", "zeta-null" or "zeta-nought") is a small countable ordinal, defined as the first fixed point of the function \(\alpha \mapsto \)\(\varepsilon\)\(_\alpha\).

It is equal to \(\varphi(2,0)\) using the Veblen function and to \(\psi(\Omega)\), using Madore's \(\psi\) function.

Larger zeta ordinals
Similarly to the epsilon ordinals, larger zeta ordinals can be defined as larger fixed points of the map \(\alpha\mapsto\varepsilon_\alpha\). For example, \(\zeta_1\) is the next fixed point of this that is greater than \(\zeta_0\), \(\zeta_2\) is the next greater than \(\zeta_1\), etc.

Formally:
 * \(\zeta_0=\textrm{min}(\{\gamma:\gamma=\varepsilon_\gamma\})\)
 * \(\zeta_\alpha=\textrm{min}(\{\gamma:\gamma=\varepsilon_\gamma\land(\forall(\beta<\alpha)(\gamma>\zeta_\beta))\})\)