It is not constructively true, as I conjectured yesterday, that the set of roots of a polynomial over C is Russell-finite. Let X be the subspace of C consisting of 0 and all points whose argument is a rational multiple of \pi, and consider the sheaf of solutions of z^2 - f = 0, where f: X --> C is the inclusion map. It is easy to see that the stalk of this sheaf at 0 is uncountably infinite. Since R-finiteness is preserved by inverse image functors, this yields a counterexample. This doesn't, of course, answer Steve Vickers' original question whether there is a sense in which the *locale* of roots of a polynomial can be said to be finite. But it does indicate that the appropriate notion of finiteness, if it exists, must be a rather delicate one. Peter Johnstone