A constructive version of the Sylvester-Gallai Theorem

Abstract

The Sylvester-Gallai Theorem, stated as a problem by J. J. Sylvester in 1893, asserts that for any finite, noncollinear set of points on a plane, there exists a line passing through exactly two points of the set. First, it is shown that for the real plane, the theorem is constructively invalid. Then, a well-known classical proof is examined from a constructive standpoint, locating the nonconstructivities. Finally, a constructive version of the theorem is established for the plane; this reveals the hidden constructive content of the classical theorem. The constructive methods used are those proposed by Errett Bishop.