The Destruction of the Axiom of Determinacy by Forcings on $\mathbb{R}$ when $\Theta$ is Regular

Abstract

$\mathsf{ZF}+\mathsf{AD}$ proves that for all nontrivial forcings $\mathbb{P}$ on a wellorderable set of cardinality less than $\Theta$, $1_{\mathbb{P}}{⊩}_{\mathbb{P}}\neg \mathsf{AD}$. $\mathsf{ZF}+\mathsf{AD}+\Theta$ is regular proves that for all nontrivial forcing $\mathbb{P}$ which is a surjective image of $\mathbb{R}$, $1_{\mathbb{P}}{⊩}_{\mathbb{P}}\neg \mathsf{AD}$. In particular, $\mathsf{ZF}+\mathsf{AD}+\mathsf{V}=\mathsf{L}(\mathbb{R})$ proves that for every nontrivial forcing $\mathbb{P}\in L_{\Theta }(\mathbb{R})$, $1_{\mathbb{P}}{⊩}_{\mathbb{P}}\neg \mathsf{AD}$.