User blog comment:P進大好きbot/Rathjen-type Ordinal Notation/@comment-39259101-20190819131438/@comment-35470197-20190820004926

In order to discuss your OCF, we should clarify the definition. I guess that you are talking about the following: \begin{eqnarray*} \psi \colon \omega \times \textrm{On} & \to & \textrm{On} \\ (n,\alpha) & \mapsto & \psi_n(\alpha) := \min \textrm{On} \setminus C_n(\alpha) \\ C \colon \omega \times \textrm{On} & \to &2^{\textrm{On}} \\ (n,\alpha) & \mapsto & C_n(\alpha) := \bigcup_{m \in \mathbb{N}} C_n(\alpha)_m \\ C_n(\alpha)_0 & := & \Omega_n \\ C_n(\alpha)_{m+1} & := & C_n(\alpha)_m \cup \{\beta+1 \mid \beta \in C_n(\alpha)_m \\ & & \cup \{\psi_k(\beta) \mid k \in \omega \land \beta \in C_n(\alpha) \cap \alpha\} \end{eqnarray*} Is it correct? If you are using a non-equivalent definition, please tell me the precise definition.