Preservation of AD via forcings

Abstract

We show that assuming $\mathsf{ZF}+{\mathsf{AD}}^{+}+$ "$V=\mathrm{L}(\wp (\mathbb{R}))$", any poset which increases $\Theta$ does not preserve the truth of $\mathsf{AD}$. We also show that in $\mathsf{ZF}+\mathsf{AD}$, any non-trivial poset on $\mathbb{R}$ does not preserve the truth of $\mathsf{AD}$. This answers the question of Chan and Jackson. Furthermore, we show that under the assumptions $\mathsf{ZF}+{\mathsf{AD}}^{+}+$ "$V=\mathrm{L}(\wp (\mathbb{R}))$" + "$\Theta \text{ is regular}$", there is a poset on $\Theta$ which adds a new subset of $\Theta$ while preserving the truth of $\mathsf{AD}$. This answers the question of Cunningham.