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

TL;DR

This paper proves that under certain set-theoretic assumptions involving determinacy and the constructible universe over the reals, the Suslin hypothesis holds, ruling out the existence of certain complex linear orders.

Contribution

It demonstrates that $L(R)$ models the Suslin hypothesis assuming $ ext{ZF} + ext{AD}^+ + V=L(P(R))$, answering a longstanding question.

Findings

$L(R)$ with determinacy satisfies the Suslin hypothesis.

The result holds under $ ext{ZF} + ext{AD}^+ + V=L(R)$.

Provides a set-theoretic consistency proof for the Suslin hypothesis in this context.

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.