Is the missing axiom of matroid theory lost forever?

Abstract

We conjecture that it is not possible to finitely axiomatize matroid representability in monadic second-order logic for matroids, and we describe some partial progress towards this conjecture. We present a collection of sentences in monadic second-order logic and show that it is possible to finitely axiomatize matroids using only sentences in this collection. Moreover, we can also axiomatize representability over any fixed finite field (assuming Rota's conjecture holds). We prove that it is not possible to finitely axiomatize representability, or representability over any fixed infinite field, using sentences from the collection.