The extender algebra and ${\Sigma }_1^2$-absoluteness

Abstract

We present a self-contained account of Woodin's extender algebra and its use in proving absoluteness results, including a proof of the ${\Sigma }_1^2$-absoluteness theorem. We also include a proof that the existence of an inner model with Woodin limit of Woodin cardinals implies the existence of divergent models of $\AD^+$.

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