Yes, the "missing axiom" of matroid theory is lost forever

TL;DR

This paper proves that no monadic second-order sentence can characterize matroid representability over infinite fields, contrasting with the finite field case where such a characterization exists due to Rota's Conjecture.

Contribution

It establishes the non-existence of MS0 characterizations for infinite field representability of matroids, highlighting a fundamental difference from finite fields.

Findings

No MS0 sentence characterizes infinite field representability.

A sentence characterizes finite field representability due to Rota's Conjecture.

The result delineates a boundary in logical definability of matroid properties.

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.