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From: "Dr. P.T. Johnstone"
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Subject: Constructive finiteness
Date: Sun, 9 Dec 2001 15:20:47 +0000 (GMT)
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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