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 \(\alpha \mapsto \)\(\varepsilon\)\(_\alpha\). . Note that some sources give the name "\(\zeta_0\)" to other ordinals, for example Rathjen has used \(\zeta_0\) to denote the Feferman-Schutte ordinal, and Sbiis Saibian has reserved the name for an ordinal \(``\omega\uparrow\uparrow\uparrow\omega\!"\) for a sufficient - although not currently complete - extension of hyper operators to ordinals.

It is equal to \(\varphi(2,0)\) using the Veblen function, \(\psi(\Omega)\) using Madore's \(\psi\) function, and \(\psi_0(\Omega^2) = \psi_0(\psi_1(\psi_1(0)))\) using Buchholz's 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:
 * \(\zeta_0=\textrm{min}(\{\gamma:\gamma=\varepsilon_\gamma\})\)
 * \(\zeta_\alpha=\textrm{min}(\{\gamma:\gamma=\varepsilon_\gamma\land(\forall(\beta<\alpha)(\gamma>\zeta_\beta))\})\)
 * One fundamental sequence for \(\zeta_1\) is \(\zeta_1[0]\)=\(\zeta_0+1\) and \(\zeta_1[n+1]=\varepsilon_{\zeta_1[n]}\), and, in general, \(\zeta_{\alpha+1}[0]\)=\(\zeta_{\alpha}+1\) and \(\zeta_{\alpha+1}[n+1]=\varepsilon_{\zeta_{\alpha+1}[n]}\).

Fixed points
Fixed points of the zeta function \(\lambda\alpha.\zeta_\alpha\) are sometimes called \(\eta\)-ordinals, and also are equivalent to ordinals of the form \(\varphi(3,\beta)\) where \(\varphi\) denotes the Veblen function.