Yes, the "missing axiom" of matroid theory is lost forever
arXiv:1412.8399
Abstract
We prove there is no sentence in the monadic second-order language MS0 that characterises when a matroid is representable over at least one field, and no sentence that characterises when a matroid is K-representable, for any infinite field K. By way of contrast, because Rota's Conjecture is true, there is a sentence that characterises F-representable matroids, for any finite field F.
26 pages, 3 figures, this version features revisions of two proofs