# Elementary definition

In mathematical logic, an **elementary definition** is a definition that can be made using only finitary first-order logic, and in particular without reference to set theory or using extensions such as plural quantification. Elementary definitions are of particular interest because they admit a complete proof apparatus while still being expressive enough to support most everyday mathematics (via the addition of elementarily-expressible axioms such as Zermelo–Fraenkel set theory (ZFC)).

Saying that a definition is elementary is a weaker condition than saying it is algebraic.

## Related

## References

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

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