\(L(\mathbb{R})\) 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. \(\mathsf{ZF + AD^+ + V = L(\mathscr{P}(\mathbb{R}))}\) 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.