FOM: Chow's theorem
Dave Marker
marker at math.uic.edu
Fri Nov 7 00:12:36 EST 1997
There has been some discussion of Chow's theorem.
While I would not call Chow's theorem a result in the foundations of
mathematics, I do believe it has some foundational significance.
First, a precise statement. Let M be a complex manifold. We say that a
subset X of M is "analytic" if for all m in M there is U an open
neighborhood of m and analytic functions f_1,...,f_n on M such that
X intersect U is the set of common zeros of f_1,...f_n. (Analytic sets
are locally the zero sets of systems of analytic equations.)
Let P^n be complex projective n-space. A subset X of P^n is "algebraic" if
there is a finite set of homogeneous polynomials p_1,...,p_m such that
X is the set of common zeros of p_1,...,p_m.
Chow's theorem asserts that every analytic subset of P^n is algebraic.
This implies the same holds for any projective algebraic variety.
Is this a result about the foundations of mathematics? I would say no.
But I think it does have some foundational significance.
Certainly one of the early problems in the foundations of mathematics
is to understand the relationship between algebraic and transcendental
methods. Chow's theorem says that in one important case anything that can
be done by transcendental methods can already be done by algebraic
methods.
I think this is a foundational point that Hilbert would have appreciated.
Dave Marker
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