Axiomatizable classes of finite models and definability of linear order

Abstract

It may happen that a first order formula with two free variables over a signature defines a linear order of some finite structure of the signature. Then, naturally, this finite structure is rigid, i.e. admits the single (trivial) automorphism. Also, the class of all the finite structures such that the formula defines a linear order on any of them, is finitely axiomatizable in the class of all finite structures (of the signature). It is shown that the inverse is not true, i.e. that there exists a finitely axiomatizable class of rigid finite structures, such that no first-order formula defines a linear order on all the structures of the class. To illustrate possible applications of the result in finite model theory, it is shown that Y. Gurevich's (1984) result that E.W. Beth's (1953) definability theorem fails for finite models is an immediate corollary.<>