User blog comment:Edwin Shade/Negatively Indexed Epsilon Numbers/@comment-1605058-20171224230129/@comment-28606698-20171225175047

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

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

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

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

then

$$\psi(0)=\omega$$