User blog comment:Alemagno12/I made a Pi w-reflection OCF, how horribly wrong am I?/@comment-35470197-20180812220655/@comment-25601061-20180813142301

"I asked the question in order to know whether there is a way to define an ordina notation associated to it or not." Oh. Well, we can define a notation system \(T(\Pi^2_0)\) for the OCF as follows:

\(0,\Pi^1_n,\Pi^2_0\in T(\Pi^2_0)\)

\(\alpha\in T(\Pi^2_0)\implies\alpha^{+},\omega^{\alpha}\in T(\Pi^2_0)\)

\(\alpha,\beta\in T(\Pi^2_0)\implies\alpha+\beta\in T(\Pi^2_0)\)

\(\alpha,\beta,A\in T(\Pi^2_0)\implies\M^{P_m,A}_{\beta}(\alpha),\Psi^{P_m,A}_{\beta}(\alpha)\in (\Pi^2_0)\)

"Such a expression-based technique is not useful in this case, because we do not have \(\alpha=\sup_{n<\omega}\alpha[n]\) in general with respect to the FS you defined."

Why not?