An axiom P is independent if there are no other axioms Q such that Q implies P.
In many cases independence is desired, either to reach the conclusion of a reduced set of axioms, or to be able to replace an independent axiom to create a more concise system (for example, the parallel postulate is independent of Euclid's Axioms, and can provide interesting results when a negated or manipulated form of the postulate is put into its place).
Proving Independence
If the original axioms Q are not consistent, then no new axiom is independent. If they are consistent, then P can be shown independent of them if adding P to them, or adding the negation of P, both yield consistent sets of axioms. For example, Euclid's Axioms, with the parallel postulate included, yields Euclidean geometry, and with the parallel postulate negated, yields non-Euclidean (spherical or hyperbolic) geometry. Both of these are consistent systems, showing that the parallel postulate is independent of the other axioms of geometry.
Proving independence is often very difficult. Forcing is one commonly used technique.