A constructive real projective plane
The classical theory of plane projective geometry is examined constructively, using both synthetic and analytic methods. The topics include Desargues’s Theorem, harmonic conjugates, projectivities, involutions, conics, Pascal’s Theorem, poles and polars. The axioms used for the synthetic treatment are constructive versions of the traditional axioms. The analytic construction is used to verify the consistency of the axiom system; it is based on the usual model in three-dimensional Euclidean space, using only constructive properties of the real numbers. The methods of strict constructivism, following principles put forward by Errett Bishop, reveal the hidden constructive content of a portion of classical geometry. A number of open problems remain for future studies.