Elementary theory

Short description: Mathematical logic

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 which 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 sentence
* 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.

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