Editing First-order logic (section)

===Example: ordered abelian groups===
In mathematics, the language of ordered [[abelian group]]s has one constant symbol 0, one unary function symbol −, one binary function symbol +, and one binary relation symbol ≤. Then:
*The expressions +(''x'', ''y'') and +(''x'', +(''y'', −(''z''))) are ''terms''. These are usually written as ''x'' + ''y'' and ''x'' + ''y'' − ''z''.
*The expressions +(''x'', ''y'') = 0 and ≤(+(''x'', +(''y'', −(''z''))), +(''x'', ''y'')) are ''atomic formulas''. These are usually written as ''x'' + ''y'' = 0 and ''x'' + ''y'' − ''z'' &nbsp;≤&nbsp; ''x'' + ''y''.
*The expression <math>(\forall x \forall y \, [\mathop{\leq}(\mathop{+}(x, y), z) \to \forall x\, \forall y\, \mathop{+}(x, y) = 0)]</math> is a ''formula'', which is usually written as <math>\forall x \forall y ( x + y \leq z) \to \forall x \forall y (x+y = 0).</math> This formula has one free variable, ''z''.

The axioms for ordered abelian groups can be expressed as a set of sentences in the language. For example, the axiom stating that the group is commutative is usually written <math>(\forall x)(\forall y)[x+ y = y + x].</math>