User blog comment:Ikosarakt1/Fast-growing hierarchy/@comment-157.193.53.9-20130620180435

Maybe the following is of some interest: There are representations of $\epsilon_0$ in terms of unary $\psi$ and unary $\theta$-functions without involving the addition function. For the $\psi$-functions Schütte Simpson is the reference. For the additionfree $\theta$-functions I recently calculated the order type together with my PhD student Jeroen Van der Meeren. The result is $psi_0(\Omega_{n+1}=\theta_0\Omega_n$ for $n>1$. Moreover, Arai has shown that $\psi_0\Omega_\Omega$ is equal to $\Gamma_0$ in this context. The precise relation between $\psi_0$ with and without involving the addition function is unknown. The first match between these will be at the first fixed point of $\xi\mapsto \Omega_\xi$. My conjecture is that $\psi_0\Omega_{\Omega_\omega}$ (without addition function) is equal to $\psi_0\Omega_\omega$ when the addition function is contained in the system. What is lacking is a proof for $\leq$.