## Ideal OCF attempt

Ideal OCF \(\kappa\) is hereditarily \(\Pi_n\)-reflecting if and only if it is \(\Pi_n\)-reflecting or \(\Pi_m\)-reflecting on the hereditarily \(\Pi_n\)-reflecting ordinals. \(C_\kappa(\alpha)\) is the closure of \(\kappa\cup{\text{least }\Pi_n\text{-reflecting}\mid n>1}\) under \(\beta,\gamma\mapsto\beta+\gamma\) and \(\beta,\gamma,\delta\mapsto\chi_\beta(\gamma,\delta)\) provided \(\gamma<\alpha\). Suppose \(\kappa\) is \(\Pi_{n+1}\)-reflecting (\(n>0\)) but not \(\Pi_{n+2}\)-reflecting. Then set \(\mathcal A_\kappa^\alpha\) to be the set of \(\pi<\kappa\) such that: If […]