It was not until the nineteenth century, chiefly through the work of Gauss, Bolyai, Lobachevsky, and Riemann, that the impossibility of deducing the parallel axiom from the others was demonstrated. This outcome was of the greatest intellectual importance. In the first place, it called attention in a most impressive way to the fact that a proof can be given of the impossibility of proving certain propositions within a given system.
Need to check out these proofs, hopefully there are beginner level resources to get a taste of it.
