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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 $$\alpha \mapsto$$$$\varepsilon$$$$_\alpha$$.[1]. 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[2], 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[3].

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.