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.

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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

• Mac Lane and Moerdijk, Sheaves in Geometry and Logic: A First Introduction to Topos Theory, page 4.

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