  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.
