Euclidean geometry is a first-order theory. That is, it allows statements such as those that begin as "for all triangles ...," but it is incapable of forming statements such as "for all sets of triangles ..." Statements of the latter type are deemed to be outside the scope of the theory.

We owe much of our present understanding of the properties of the logical and metamathematical properties of Euclidean geometry to the work of Alfred Tarski and his students, beginning in the 1920s. Tarski proved his axiomatic formulation of Euclidean geometry to be complete in a certain sense: there is an algorithm which, for every proposition, can be show it to be either true or false. Gödel's theorem showed the futility of Hilbert's program of proving the consistency of all of mathematics using finitistic reasoning. Tarski's findings do not violate Gödel's theorem, because Euclidean geometry cannot describe a sufficient amount of arithmetic for the theorem to apply.[3]

Although complete in the formal sense used in modern logic, there are things that Euclidean geometry cannot accomplish. For example, the problem of trisecting an angle with a compass and straightedge is one that naturally occurs within the theory, since the axioms refer to constructive operations that can be carried out with those tools. However, centuries of efforts failed to find a solution to this problem, until Pierre Wantzel published a proof in 1837 that such a construction was impossible.

Absolute geometry, formed by removing the parallel postulate, is also a consistent theory (provided that Euclidean geometry is), as is non-Euclidean geometry (provided Euclidean geometry is), formed by alterations of the parallel postulate. Non-Euclidean geometries are consistent (provided Euclidean geometry is) because there are Euclidean models of non-Euclidean geometry. For example, geometry on the surface of a sphere is a model of an elliptical geometry, carried out within a self-contained subset of a three-dimensional Euclidean space.

http://en.wikipedia.org/wiki/Euclidean_geometry