Questions about axiomatic systems for geometry. Use this tag if you're looking for a proof starting directly from some set of axioms (e.g., Hilbert's axioms for Euclidean geometry), or if you have a question about the axioms themselves.
Historically, Euclid's Elements were the first attempt to put geometry (or any other branch of mathematics) on a rigorous footing, proving theorems starting from a fixed group of postulates and common notions.
Since then, many mathematicians have asked questions such as:
- Are all the axioms used by Euclid really necessary, or can some of them (notably the parallel postulate) be proven from the other axioms?
- What sort of objects satisfy some or all of the axioms of geometry?
- What collections of axioms can completely specify Euclidean geometry?
- What sort of geometries do we obtain, and which theorems can we still prove, by dropping or negating some of these axioms?
Such questions are the domain of axiomatic geometry.
More modern axiomatizations of geometry include Hilbert's axioms (which are possibly the most widely used today) as well as other systems proposed by Birkhoff and Tarski.
In addition to Euclidean geometry, objects studied in axiomatic geometry include hyperbolic geometry, elliptic geometry, and even finite incidence structures.
