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An axiom P is independent if there is no other axiom Q such that Q implies P.

In many cases independency 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 is usually a simple logical task. If we are trying to prove an axiom Q independent, then the set of all the other axioms P can't imply Q. One way of doing this is by proving that the negation of the set of axioms P implies Q, it then follow by the law of contradiction that P can't imply Q, because if that were the case then P and not P would both imply Q, and that would be a logical contradiction.

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Last updated on Wednesday November 14, 2007 at 22:50:20 PST (GMT -0800)

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This article is licensed under the GNU Free Documentation License.

Last updated on Wednesday November 14, 2007 at 22:50:20 PST (GMT -0800)

View this article at Wikipedia.org - Edit this article at Wikipedia.org - Donate to the Wikimedia Foundation

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