User blog comment:MilkyWay90/Help understand the zeta ordinals/@comment-5529393-20180620055127/@comment-5529393-20180621054617

Yes to both \(\eta_\alpha = \zeta_{\eta_\alpha}\) and \(\eta_\alpha = \varepsilon_{\eta_\alpha}\).

More generally, if \(\gamma < \alpha\), then \(\varphi(\alpha,\beta) = \varphi(\gamma, \varphi(\alpha, \beta))\), and in fact for \(\alpha > 0\), we can define \(\varphi(\alpha, \beta\) as the \(\beta\) th ordinal such that \(\varphi(\alpha,\beta) = \varphi(\gamma, \varphi(\alpha, \beta))\).