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

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