Sylvester's conjecture states that, given n distinct noncollinear points in a plane, there exists a connecting line of two of the points such that no other point is incident with the line. First a proof is given of the six-point Sylvester conjecture from a constructive axiomatization of plane incidence geometry. Next ordering principles are studied that are needed for the seven-point case. This results in a symmetrically ordered plane affine geometry. A corollary is the axiom of complete quadrangles. Finally, it is shown that the problem admits of an arithmetic translation by which Sylvester's conjcture is decidable for any n.

Sylvester's conjecture, constructive geometry, ordered geometry