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