Constructive finiteness

["Dr. P.T. Johnstone" <P.T.Johnstone@dpmms.cam.ac.uk>]

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