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arXiv · 2402.03662

A constructive version of the Sylvester-Gallai Theorem

Mark Mandelkern
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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.

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Mark Mandelkern. 2024-02-06. A constructive version of the Sylvester-Gallai Theorem. https://arxiv.org/abs/2402.03662

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