In mathematical logic, an elementary theory is a theory that involves axioms using only finitary first-order logic, without reference to set theory or using any axioms that have consistency strength equal to set theory. Saying that a theory is elementary is a weaker condition than saying it is algebraic.
Examples
Examples of elementary theories include:
The theory of groups The theory of finite groups The theory of abelian groups The theory of fields The theory of finite fields The theory of real closed fields Axiomization of Euclidean geometry
Related Elementary definition Elementary theory of the reals
References Mac Lane and Moerdijk, Sheaves in Geometry and Logic: A First Introduction to Topos Theory, page 4.