User blog:Hyp cos/Attempt of OCF up to Stability

These OCFs are more complicated than previous OCFs. They are quite different from the OCFs in some literature, which I do not really understand, so it is probable that I do not follow the right way.

\(\Pi_4\)-Reflection
Not only the definition, but also the expressions of this OCF are highly inductive. A reflection instance is defined as Inductive varibles of a reflection instance (denoted as a set \(IV\mathbb X\) where \(\mathbb X\) is the reflection instance) are defined as Let \(\mathcal K\) be the least \(\Pi_4\)-reflecting ordinal, \begin{eqnarray*} C_0(\alpha,\beta)&=&\beta\cup\{0,\mathcal K\}\\ C_{i+1}(\alpha,\beta)&=&\{\gamma+\delta|\gamma,\delta\in C_i(\alpha,\beta)\}\\ &\cup&\{\omega^\gamma|\gamma\in C_i(\alpha,\beta)\land\gamma>\mathcal K\}\\ &\cup&\{\psi_\pi(\mathbb X,\gamma)|\pi,\gamma\in C_i(\alpha,\beta)\land\gamma<\alpha\land IV\mathbb{X}\subseteq C_i(\alpha,\beta)\cap\alpha\}\\ A_\pi(\alpha)&=&\{\gamma<\pi|C(\alpha,\gamma)\cap\pi\subseteq\gamma\land\\ & &\forall\xi\in C(\alpha,\gamma)\cap\alpha(\gamma\text{ is }\Pi_3\text{-reflecting on }A_\pi(\xi))\}\\ A_\pi(\alpha,0,\mathbb X)&=&A_\pi(\alpha)\cap A_\pi\mathbb X\\ A_\pi(\alpha,\beta,\mathbb X)&=&\{\gamma\in A_\pi(\alpha)|C(\beta,\gamma)\cap\pi\subseteq\gamma\land\\ & &\forall\xi\inC(\beta,\gamma)\cap\beta(\gamma\text{ is }\Pi_2\text{-reflecting on }A_\pi(\alpha,\xi,\mathbb X))\}\\ \psi_\pi(\mathbb X,\alpha)&=&\min(\{\gamma\in A_\pi\mathbb X|C(\alpha,\gamma)\cap\pi\subseteq\gamma\}\cup\{\pi\}) \end{eqnarray*}
 * \((\alpha)\) is a reflection instance.
 * If \(\mathbb X\) is a reflection instance, then \((\alpha,\beta,\mathbb X)\) is also a reflection instance.
 * \(IV(\alpha)=\{\alpha\}\)
 * \(IV(\alpha,\beta,\mathbb X)=IV\mathbb X\cup\{\alpha,\beta\}\)