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==Logical synthesis==
The process of logical synthesis begins with some arbitrary but definite starting point. This starting point is the introduction of primitive notions or primitives and axioms about these primitives:
* '''[[Primitive notion|Primitives]]''' are the most basic ideas. Typically they include both objects and relationships. In geometry, the objects are things such as ''points'', ''lines'' and ''planes'', while a fundamental relationship is that of ''incidence'' – of one object meeting or joining with another. The terms themselves are undefined. [[David Hilbert|Hilbert]] once remarked that instead of points, lines and planes one might just as well talk of tables, chairs and beer mugs,<ref>{{harvnb|Greenberg|1974|loc=p. 59}}</ref> the point being that the primitive terms are just empty [[Free variables and bound variables|placeholders]] and have no intrinsic properties.
* '''[[Axiom]]s''' are statements about these primitives; for example, ''any two points are together incident with just one line'' (i.e. that for any two points, there is just one line which passes through both of them). Axioms are assumed true, and not proven. They are  the ''building blocks'' of geometric concepts, since they specify the properties that the primitives have.

From a given set of axioms, synthesis proceeds as a carefully constructed logical argument. When a significant result is proved rigorously, it becomes a '''[[theorem]]'''.

===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.