The Destruction of the Axiom of Determinacy by Forcings on $\mathbb{R}$ when $Θ$ is Regular
William Chan, Stephen C. Jackson · arXiv (Cornell University) · 2019
$\mathsf{ZF + AD}$ proves that for all nontrivial forcings $\mathbb{P}$ on a wellorderable set of cardinality less than $Θ$, $1_{\mathbb{P}} \Vdash_{\mathbb{P}} eg\mathsf{AD}$. $\mathsf{ZF + AD} + Θ$ is regular proves that for all nontrivial forcing $\mathbb{P}$ which is a surjective image of $\mathbb{R}$, $1_{\mathbb{P}} \Vdash_{\mathbb{P}} eg\mathsf{AD}$. In particular, $\mathsf{ZF + AD + V = L(\mathbb{R})}$ proves that for every nontrivial forcing $\mathbb{P} \in L_Θ(\mathbb{R})$, $1_{\mathbb{P}} \Vdash_{\mathbb{P}} eg\mathsf{AD}$.