User blog comment:P進大好きbot/Rathjen-type Ordinal Notation/@comment-39259101-20190819131438/@comment-35470197-20190819143359

I have never considered that stratesy, i.e. using the successor and the enumeration of limit ordinals. Personally, I always use the addition and the enumeration of additive principal ordinals, because my work is based on papers by Buchholz and Rathjen.

So if we only use the successor, then the resulting OCF will satisfy the following: \begin{eqnarray*} \psi_0(0) & = & \omega \\ \psi_0(1) & = & \omega \times 2 \\ \psi_0(2) & = & \omega \times 3 \\ \psi_0(\omega) & = & \omega^2 \\ \psi_0(\omega+1) & = & \omega^2 + \omega \\ \psi_0(\omega \times 2) & = & \omega^2 \times 2 \\ \psi_0(\omega^2) & = & \omega^3 \\ \psi_1(0) & = & \Omega_1 \\ \psi_0(\Omega_1) & = & \omega^{\omega} \\ \psi_0(\Omega_1+1) & = & \omega^{\omega}+\omega \\ \psi_0(\Omega_1+2) & = & \omega^{\omega}+\omega \times 2 \\ \psi_1(1) & = & \Omega_1 + \omega \\ \psi_0(\Omega_1 + \omega) & = & \omega^{\omega}+\omega^2 \\ \psi_1(2) & = & \Omega_1 + \omega \times 2 \\ \psi_0(\Omega_1 + \omega \times 2) & = & \omega^{\omega}+\omega^2 \times 2 \\ \psi_1(\omega) & = & \Omega_1 + \omega^2 \\ \psi_0(\Omega_1 + \omega^2) & = & \omega^{\omega}+\omega^3 \\ \psi_1(\omega^{\omega}) & = & \Omega_1 + \omega^{\omega} \\ \psi_0(\Omega_1 + \omega^{\omega}) & = & \omega^{\omega} \times 2 \\ \psi_1(\omega^{\omega} \times 2) & = & \Omega_1 + \omega^{\omega} \times 2 \\ \psi_0(\Omega_1 + \omega^{\omega} \times 2) & = & \omega^{\omega} \times 3 \\ \psi_1(\Omega_1) & = & \Omega_1 \times 2 \\ \psi_0(\Omega_1 \times 2) & = & \omega^{\omega+1} \\ \psi_2(0) & = & \Omega_2 \\ \psi_1(\Omega_2) & = & \Omega_1 \times \omega \\ \psi_0(\Omega \times \omega) & = & \omega^{\omega \times 2} \end{eqnarray*} If I am correct, the system actually works, but it grows so slowly at first. Maybe this is the main reason why people use other base fucntions. We traditionally require that \(\epsilon_0\) should be expressed easily.