Extending to a model structure is not a first-order property

Let $\mathcal{C}$ be a finitely bicomplete category and $\mathcal{W}$ a subcategory. We prove that the existence of a model structure on $\mathcal{C}$ with $\mathcal{W}$ as subcategory of weak equivalence is not first order expressible. Along the way we characterize all model structures where $\mathcal{C}$ is a partial order and show that these are determined by the homotopy categories.