User blog comment:Bubby3/FGH on ordinals/@comment-11227630-20171105151344

Then $$f_\omega(\omega)=\varphi(\omega,0)$$, $$f_{\omega+1}(\omega)=\varphi(\omega,\omega)$$, $$f_{\omega+2}(\omega)=\varphi(\omega+1,0)$$, $$f_{\omega2}(\omega)=\varphi(\omega2,0)$$, $$f_{\varepsilon_0}(\omega)=\varphi(\varepsilon_0,0)$$. The limit is $$f_{f_{\cdots f_\omega(\omega)\cdots}(\omega)}(\omega)=\Gamma_0=f_{\Gamma_0}(\omega)$$.