$$\eta_0$$, written as "eta-zero", is a small countable ordinal defined as the first fixed point of the function $$\alpha\mapsto$$$$\,\zeta$$$$_\alpha$$[1]. It's equal to $$\varphi(3,0)$$ in Veblen's function, $$\psi_0(\Omega^3)=\psi_0(\psi_1(\psi_1(0)+\psi_1(0)))$$ in Buchholz's function, and $$\psi(\Omega^2)$$ in Madore's function.