Is this system of hyperbolic geometry a consistent mathematical system? Does it lead to a contradiction? This is a question of metamathematics; i.e., a question outside of a mathematical system about the system itself. The question is not about lines and points, but about the whole system itself. If it is inconsistent, then an ordinary mathematical argument would derive a contradiction. This is what Saccheri and Lambert attempted to do, and failed.
Theorem 16.1: If Euclidean geometry is consistent, so is hyperbolic geometry.
Corollary: If Euclidean geometry is consistent, then no proof or disproof of the parallel postulate from the rest of Hilbert's postulates will ever be found, i.e., the parallel postulate is independent of the other postulates.
Proof: Assume not, then there is a proof of the parallel postulate. This implies that hyperbolic geometry is inconsistent since the Hyperbolic Axiom contradicts a proven result. Our above theorem asserts that hyperbolic geometry is as consistent as Euclidean geometry. Thus, Euclidean geometry is inconsistent. This contradiction implies that no proof o the parallel postulate exists. Note also that the hypothesis that euclidean geometry is consistent implies that do disproof exists either.
You should recognize that had mathematicians succeeded in proving Euclid's Fifth Postulate from the other axioms with the intention of making Euclidean geometry more secure and elegant, they would have completely destroyed Euclidean geometry as a consistent body of thought.
What we will do now, in order to see that hyperbolic geometry is as
consistent as Euclidean geometry, is to construct a model of hyperbolic
geometry. We will look at three different models for 
.