User blog comment:Edwin Shade/Negatively Indexed Epsilon Numbers/@comment-1605058-20171224230129/@comment-28606698-20171227205833

$$C_0(\alpha)=\{0,\Omega\}$$

$$C_{n+1}(\alpha)=\{\psi(\beta)|\beta\in C_n(\alpha), \beta<\alpha\}$$

$$C(\alpha)=\cup_{n=0}^\omega C_n(\alpha)$$

$$\psi(\alpha)=\text{min}\{\beta|\beta\notin C(\alpha)\}$$

then

$$\psi(0)=1$$

$$\psi(1)=2$$

$$\psi(\Omega)=\omega$$

$$\psi(\varepsilon_{\Omega+1})=\varepsilon_0$$