Is the missing axiom of matroid theory lost forever?

TL;DR

This paper explores the limitations of finitely axiomatizing matroid representability in monadic second-order logic, providing partial progress and showing certain axiomatizations are impossible.

Contribution

It introduces a collection of logical sentences for axiomatizing matroids and proves the impossibility of finite axiomatization for representability over fields.

Findings

Finitely axiomatizing matroid representability in monadic second-order logic is conjectured to be impossible.

A collection of sentences can finitely axiomatize matroids and representability over fixed finite fields (assuming Rota's conjecture).

It is proven that finite axiomatization of representability over any field is impossible.

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.