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

The authors would like to thank Jonathan Campbell and Wesley Calvert for their thoughts on the paper, as well as the anonymous referee whose comments on the exposition (including the definitions of ``semi-(co)fibrant'' and W^χ_f) greatly improved the paper. Zakharevich was supported in part by NSF grant DMS-1654522.