A constructive version of the Sylvester-Gallai Theorem
Metric Geometry
2024-02-07 v1
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.
Cite
@article{arxiv.2402.03662,
title = {A constructive version of the Sylvester-Gallai Theorem},
author = {Mark Mandelkern},
journal= {arXiv preprint arXiv:2402.03662},
year = {2024}
}