Elliptic Geometry - Self-consistency

Self-consistency

Because spherical elliptic geometry can be modeled as, for example, a spherical subspace of a Euclidean space, it follows that if Euclidean geometry is self-consistent, so is spherical elliptic geometry. Therefore it is not possible to prove the parallel postulate based on the other four postulates of Euclidean geometry.

Tarski proved that elementary Euclidean geometry is complete in a certain sense: there is an algorithm which, for every proposition, can show it to be either true or false. (This does not violate Gödel's theorem, because Euclidean geometry cannot describe a sufficient amount of arithmetic for the theorem to apply.) It therefore follows that elementary elliptic geometry is also self-consistent and complete.