In mathematics, an axiomatizable class is a class of mathematical structures which are all models of a fixed set of sentences in formal (typically first-order) logic.

For example, the axiomatic sentences of a multiplicative group are:

$\backslash forall\; xyz\; \backslash ,\; \backslash ,\; (xy)z\; =\; x(yz)$

$\backslash forall\; x\backslash ,\backslash ,\; x\; \backslash cdot\; 1\; =\; x$

$\backslash forall\; x\backslash ,\backslash ,\; x\; \backslash cdot\; x^\{-1\}\; =\; 1.$

The axioms of a left R-module are the axioms of a multiplicative group, together with the additional sentences

$\backslash forall\; xy\; \backslash ,\backslash ,\; r(x+y)=r(x)+r(y)$ for all $r\backslash in\; R$

$\backslash forall\; x\; \backslash ,\backslash ,\; (r+s)(x)=r(x)+s(x)$ for all $r,s\backslash in\; R$

$\backslash forall\; x\; \backslash ,\backslash ,\; (rs)(x)=r(s(x))$ for all $r,s\backslash in\; R$

$\backslash forall\; x\; \backslash ,\backslash ,\; 1(x)=x.$

Many of the common classes of mathematics are easily axiomatizable, including the rings, fields, lattices, boolean algebras and the like.

See also

• category theory
• universal algebra
• elementary class
• basic elementary class
• pseudo-elementary class

References

• Wilfrid Hodges (1997). A shorter model theory. Cambridge University Press. ISBN 0-521-58713-1.

Model theory | Category theory | Set theory