Consider subsets of Euclidean spaces defined by conditions like f(x)=0, g_1(x)>0,...,g_l(x)>0, x in R^n, for well-behaved (in some particular sense) functions f,g_1,...,g_l. Now perform elementary logical, geometric and topological operations on these sets: take (finite) unions, intersections, complements, closures and cartesian products, and project into lower-dimensional Euclidean spaces. If we iterate these operations on the sets that arise, then, roughly speaking, two things can happen: (1) After only a few such operations, a stabilization occurs; performing further operations does not produce any new sets. (2) Ever more complicated sets arise whose properties are further and further removed from those of the sets from which we started. Naturally, it is phenomenon (1) that is of primary interest, but it is often quite difficult to determine which of (1) or (2) holds for a given collection of "primitive" functions. Model theory, a branch of mathematical logic, turns out to be useful in these investigations.