Axiomatizable class

In mathematics, an axiomatizable class is a class whose standard definition can be expressed as a sentence of formal symbols. The resulting sentences that can be built out of the axioms are the topic of study of model theory.

Thus, for example, the axiomatic sentences of a multiplicative group are:

<math>\forall xyz \, \, (xy)z = x(yz)<math>

<math>\forall x\,\, x \cdot 1 = x<math>

<math>\forall x\,\, x \cdot x^{-1} = 1.<math>

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

<math>\forall xy \,\, r(x+y)=r(x)+r(y)<math> for all <math>r\in R<math>

<math>\forall x \,\, (r+s)(x)=r(x)+s(x)<math> for all <math>r,s\in R<math>

<math>\forall x \,\, (rs)(x)=r(s(x))<math> for all <math>r,s\in R<math>

<math>\forall x \,\, 1(x)=x.<math>

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

References

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