People are interested in replacing more axioms by fewer axioms, especially shorter axioms. An example is Robbins algebra. Robbins algebra consists of three axioms.
Boolean algebra includes five axioms.
The open problem, Robbins algebra is Boolean algebra, has been solved. In the twenties, Heyting attempted at axiomatizing constructive geometry. Recently, von Plato used different concepts to axiomatize constructive geometry. He used point apartness instead of point equality, and line apartness instead of line equality, apartness of a point from a line instead of incidence of a point with a line, convergence of two lines instead of parallelism of two lines. It is why he would call the geometry apartness geometry rather than incidence geometry, according to the classical terminology.