Editing Synthetic geometry (section)

===Properties of axiom sets===
There is no fixed axiom set for geometry, as more than one [[consistency|consistent set]] can be chosen. Each such set may lead to a different geometry, while there are also examples of different sets giving the same geometry. With this plethora of possibilities, it is no longer appropriate to speak of "geometry" in the singular.

Historically, Euclid's [[parallel postulate]] has turned out to be [[Independence (mathematical logic)|independent]] of the other axioms. Simply discarding it gives [[absolute geometry]], while negating it yields [[hyperbolic geometry]].<!-- note in absolute geometry non intersecting lines and infinite long lines exists--> Other [[consistency|consistent axiom sets]] can yield other geometries, such as [[projective geometry|projective]], [[elliptic geometry|elliptic]], [[spherical geometry|spherical]] or [[affine geometry|affine]] geometry.

Axioms of continuity and "betweenness" are also optional, for example, [[discrete geometry|discrete geometries]] may be created by discarding or modifying them.

Following the [[Erlangen program]] of [[Felix Klein|Klein]], the nature of any given geometry can be seen as the connection between [[symmetry]] and the content of the propositions, rather than the style of development.