Preservation of AD via forcings

Abstract

We show that assuming $\mathsf{ZF}+\mathsf{AD}^+ +$ "$V = \mathrm{L} \bigl(\wp (\mathbb{R})\bigr)$", 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 $\mathsf{ZF}+\mathsf{AD}^+ +$ "$V = \mathrm{L} \bigl(\wp (\mathbb{R})\bigr)$" + "$\Theta \text{ is regular}$", 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.