$L(\mathbb{R})$ with Determinacy Satisfies the Suslin Hypothesis

Abstract

The Suslin hypothesis states that there are no nonseparable complete dense linear orderings without endpoints which have the countable chain condition. $\mathsf{ZF}+\mathsf{A}{\mathsf{D}}^{+}+\mathsf{V}=\mathsf{L}(\mathcal{P}(\mathbb{R}))$ proves the Suslin hypothesis. In particular, if $L(\mathbb{R})\models \mathsf{AD}$, then $L(\mathbb{R})$ satisfies the Suslin hypothesis, which answers a question of Foreman.