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A constructive version of the Sylvester-Gallai Theorem

Mark Mandelkern

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.

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.
Paper Structure (2 theorems, 4 equations)

This paper contains 2 theorems, 4 equations.

Key Result

Lemma 1

If a noncollinear set $\mathscr{S}$ of points on the real plane $\mathbb{R}^{2}$ is linearly discrete, then it is discrete.

Theorems & Definitions (8)

  • Example
  • proof
  • proof
  • proof
  • Lemma
  • proof
  • Theorem
  • proof