Axiomatization and undecidability results for linear betweenness relations
Robert Mendris, Pavol Jan Zlatos · Czech digital mathematics library · 1996
Let V be a vector space over an ordered field F. The ternary betweenness relation T v on V, induced by the linear structure of V and the ordering of F, is defined byfor x,y,z G V. We will prove that the class C of all linear ternary structures, i.e., the class of all structures (A,T) with a single ternary relation T which can be embedded into (V, T v ) for some vector space V over an arbitrary ordered field F (not just the real numbers), is an elementary class which can be axiomatized by a set of universal sentences.Further, we will show that the first-order theory of C is recursively axiomatizable, and its universal part is decidable.On the other hand, the theory of C is not finitely axiomatizable, and the theory of finite members of C is hereditarily undecidable.