User blog comment:P進大好きbot/Rathjen-type Ordinal Notation/@comment-39259101-20190819131438/@comment-35470197-20190819224401

> the standard form might have trouble expressing those ordinals in an addition based system

In the notation associated to this OCF, the algorithm determining standard forms is just a porsion of Buchholz's algorithm using \(G\) function. Therefore it is not so difficult.

For example, I express the ordinals in my analysis above in standard forms. \begin{eqnarray*} \psi_0(0) & = & \omega \\ \psi_0(0+1) & = & \omega \times 2 \\ \psi_0(0+1+1) & = & \omega \times 3 \\ \psi_0(\psi_0(0)) & = & \omega^2 \\ \psi_0(\psi_0(0)+1) & = & \omega^2 + \omega \\ \psi_0(\psi_0(0+1)) & = & \omega^2 \times 2 \\ \psi_0(\psi_0(psi_0(0))) & = & \omega^3 \\ \psi_1(0) & = & \Omega_1 \\ \psi_0(\psi_1(0)) & = & \omega^{\omega} \\ \psi_0(\psi_1(0)+1) & = & \omega^{\omega}+\omega \\ \psi_0(\psi(0)_1+1+1) & = & \omega^{\omega}+\omega \times 2 \\ \psi_1(0+1) & = & \Omega_1 + \omega \\ \psi_0(\psi_1(0+1)) & = & \omega^{\omega}+\omega^2 \\ \psi_1(0+1+1) & = & \Omega_1 + \omega \times 2 \\ \psi_0(\psi_1(0+1+1)) & = & \omega^{\omega}+\omega^2 \times 2 \\ \psi_1(\psi_0(0)) & = & \Omega_1 + \omega^2 \\ \psi_0(\psi_1(\psi_0(0))) & = & \omega^{\omega}+\omega^3 \\ \psi_1(\psi_0(\psi_1(0))) & = & \Omega_1 + \omega^{\omega} \\ \psi_0(\psi_1(\psi_0(\psi_1(0)))) & = & \omega^{\omega} \times 2 \\ \psi_1(\psi_0(\psi_1(\psi_0(\psi_1(0))))) & = & \Omega_1 + \omega^{\omega} \times 2 \\ \psi_0(\psi_1(\psi_0(\psi_1(\psi_0(\psi_1(0)))))) & = & \omega^{\omega} \times 3 \\ \psi_1(\psi_1(0)) & = & \Omega_1 \times 2 \\ \psi_0(\psi_1(\psi_1(0))) & = & \omega^{\omega+1} \\ \psi_2(0) & = & \Omega_2 \\ \psi_1(\psi_2(0)) & = & \Omega_1 \times \omega \\ \psi_0(\psi_2(0)) & = & = \omega^{\omega \times 2} \end{eqnarray*} Since an OCF is an increasing function, it implies \(\psi_0(\Omega_1^{\omega_1}) = \psi_1(\psi_2(0)) = \omega^{\omega \times 2}\). Oh, this differs from your analysis. Could you tell me why your OCF is supposed to catch up with Buchholz's OCF at this point?