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

TL;DR

This paper explains Woodin's extender algebra and demonstrates its application in proving $oldsymbol{oldsymbol{ ext{Sigma}}^2_1}$-absoluteness, linking inner models with Woodin cardinals to models of $oldsymbol{ ext{AD}}^+$.

Contribution

It provides a self-contained explanation of the extender algebra and new proofs connecting Woodin cardinals to $ ext{AD}^+$ models.

Findings

Proof of $oldsymbol{ ext{Sigma}}^2_1$-absoluteness

Inner model existence implies divergent $ ext{AD}^+$ models

Clarification of Woodin's extender algebra framework

Abstract

We present a self-contained account of Woodin's extender algebra and its use in proving absoluteness results, including a proof of the $\Sigma^2_1$-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 $\AD^+$.