Let $A\subseteq [N]$ be such that for any pair of distinct subsets $B,C\subset A$, the products $\prod_{b\in B}b$ and $\prod_{c\in C}c$ are distinct. We prove that $|A|\leq π(N)+π(N^{1/2})+o(π(N^{1/2}))$, where $π$ is the prime counting function, answering a question of Erdős.