Now that we have structures in front of us, the most pressing need is to start classifying them and their features. Classifying is a kind of defining. Most mathematical classification is by axioms or defining equations – in short, by formulas. This chapter could have been entitled ‘The elementary theory of mathematical classification by formulas’.

Notice three ways in which mathematicians use formulas. First, a mathematician writes the equation ‘y = 4x2’. By writing this equation one names a set of points in the plane, i.e. a set of ordered pairs of real numbers. As a model theorist would put it, the equation defines a 2-ary relation on the reals. We study this kind of definition in section 2.1.

Or second, a mathematician writes down the laws

(*) For all x, y and z, x ≤y and y ≤ z imply x ≤ z;

for all x and y, exactly one of x≤y,y≤x,x = y holds.

By doing this one names a class of relations, namely those relations ≤ for which (*) is true. Section 2.2 lists some more examples of this kind of naming. They cover most branches of algebra.