Elementary theory

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