Axiom independence

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).