Axiomatization and undecidability results for metrizable betweenness relations

Abstract:

Let d be a metric on a nonempty set A. The ternary betweenness relation ${T_d}$ induced by d on A is defined by \[ {T_d}(x,y,z) \Leftrightarrow d(x,y) + d(y,z) = d(x,z)\] for $x,y,z \in A$. Allowing the range of d to vary over some "reasonable" ordered additive algebraic structures (not just the real numbers), we will prove that the class $\mathcal {M}$ of all metrizable ternary structures, i.e., the class of all structures $(A,{T_d})$, where d is some metric on A, is an elementary class which can be axiomatized by a set of universal Horn sentences. Further, using an algorithm of linear programming, we will show that the first-order theory of $\mathcal {M}$ is recursively axiomatizable and its universal part is decidable. On the other hand, the theory of $\mathcal {M}$ is not finitely axiomatizable and the theory of finite members of $\mathcal {M}$ is hereditarily undecidable.