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Categories: Model theory | Category theory | Set theory

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:

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

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

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

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

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

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

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

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

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

References

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

Categories: Model theory | Category theory | Set theory

Last updated: 05-26-2005 16:49:18

Science Fair Project Encyclopedia Contents Page

10-26-2009 08:16:03
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