A constructive version of the Sylvester–Gallai theorem

Abstract

The Sylvester–Gallai Theorem, stated as a problem by James Joseph 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 \({{\mathbb{R}^{2}}}\) 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 \({{\mathbb{R}^{2}}}\); this reveals the hidden constructive content of the classical theorem. The constructive methods used are those proposed by Errett Bishop.