Axiom

An axiom is a statement accepted as true without proof, serving as a starting premise from which the rest of a mathematical system is built by deduction; since nothing can be deduced from nothing, some starting assumptions are unavoidable in any deductive system. The classic example is Euclid's own five postulates for geometry, often described as four plus one because the fifth, the parallel postulate, was long suspected of being provable from the other four; two thousand years of failed attempts to prove it, and its eventual demonstrated independence in the nineteenth century, showed that a different, equally consistent choice of axiom yields a genuinely different, non-Euclidean geometry. David Hilbert's 1899 Grundlagen der Geometrie later set out a complete system of axioms for Euclidean geometry on a fully rigorous logical footing.