User:Alemagno12/Pi-w-reflection

Let \(\Pi^m_n\) be the least \(\Pi^m_n\)-indescribable and \(x^{+}\) be the least regular larger than x.

\(C_0(\alpha,\beta) = \beta\cup\{0,\Pi^2_0\}\cup\{\Pi^1_n|n\in\omega\}\)

\(C_{n+1}(\alpha,\beta) = C_n(\alpha,\beta)\cup\{\gamma+\delta,\omega^{\gamma},\gamma^{+},M^{P_m,A}_{\varepsilon}(\gamma),\Psi^{P_m,A}_{\varepsilon}(\gamma)|\gamma,\delta,\varepsilon,A\in C_n(\alpha,\beta)\wedge\zeta < \beta\wedge m\in\omega\}\)

\(C(\alpha,\beta) = \bigcup_{n\in\omega}C_n(\alpha,\beta)\)

\(M^{P_0,A}_{\kappa}(\alpha) = \{\pi|\sup C(\beta,\pi)\cap\kappa = \pi\wedge\pi\in A\}\)

\(M^{P_m,A}_{\kappa}(\alpha) = \{\pi|\sup C(\beta,\pi)\cap\kappa = \pi\wedge \pi~\text{is}~A-\Pi^1_m-\text{indescribable} \}\)

\(\Psi^{P_m,A}_{\kappa}(\alpha) = \min\{x|x\in M^{P_m,A}_{\kappa}(\alpha)\}\)

The definition of \(A-\Pi^1_M\)-indescribables (along with theorems important for understanding how the OCF works) can be found in section 2.1 of Stegert's PhD.

I'll probably write a more detailed explanation of how the OCF works tomorrow. If everything works correctly (which i highly doubt due to the lack of complexity of the OCF), then \(\Psi^{P_1,A}_{\Pi^1_1}(\alpha)\) should correspond to \(\Xi(1+\alpha,0)\) in Deedlit's weakly compact OCF, and \(\Psi^{P_0,M^{P_1,0}_{\Pi^1_1}(\pi)}_{\kappa}(\alpha)\) should correspond to \(\Psi^{\pi}_{\kappa}(\alpha)\).

Furthermore, I conjecture that ΨP0,0ω+(εΠ2 0+1 ) is the PTO of KP + Πω-reflection.

@PsiCubed2 Feel free to scream at me in the comments.