4 problems

Let $\mathcal{D}(V,\kappa)$ denote the derived model at $\kappa$, let $\Theta^{D(V,\kappa)}$ be its supremum of ordinals surjected by its reals, and let $\kappa^+$ be the successor…

Let $V$ satisfy PFA, let $\kappa$ be a limit of Woodin cardinals, and let $D(V,\kappa)$ denote the derived model at $\kappa$. Let $\Theta^{D(V,\kappa)}$ be its Solovay ordinal. Wil…

Let $V$ satisfy PFA, let $\nobreak\bkappa$ be a limit of Woodin cardinals, and let $D(V,\kappa)$ denote the derived model at $\kappa$. Write $\Theta_0^{D(V,\kappa)}$ for the first…

Let $\delta$ be a Woodin cardinal that is a limit of Woodin cardinals. Let $g\text{?}$ be $V$-generic for $\mathrm{Col}(\omega,{<}\delta)$, and let $L(\Gamma^*_g,\mathbb{R}^*_g)$ b…