The author deals with three proofs of the Sylvester-Gallai theorem which lead to three different and incompatible axiom systems. The theorem can be stated as follows: If the points of a finite set $S$ are not all on a line, then there is a line through exactly two of the points. The present, very interesting, article shows many historical facts.
After an introduction the author studies the Steinberg-Coxeter proof, Kelly’s proof, and Moszyńska geometries. The author writes in the paper’s abstract: “In particular, we show that proofs respecting the purity of the method, using only notions considered to be part of the statement of the theorem to be proved, are not always the simplest, as they may require axioms which proofs using extraneous predicates do not rely upon.”