  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.
