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).
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. [1] 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. [2]
Proving independence is often very difficult. Forcing is one commonly used technique. [3]