axiom An axiom, postulate, or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word (), meaning 'that which is thought worthy or f ...

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 In geometry, the parallel postulate, also called Euclid's fifth postulate because it is the fifth postulate in Euclid's ''Elements'', is a distinctive axiom in Euclidean geometry. It states that, in two-dimensional geometry: ''If a line segment ...

is independent of other axioms of

Euclidean geometry Euclidean geometry is a mathematical system attributed to ancient Greek mathematics, Greek mathematician Euclid, which he described in his textbook on geometry: the ''Euclid's Elements, Elements''. Euclid's approach consists in assuming a small ...

, and provides interesting results when negated or replaced).

Proving Independence

If the original axioms Q are not

consistent In classical deductive logic, a consistent theory is one that does not lead to a logical contradiction. The lack of contradiction can be defined in either semantic or syntactic terms. The semantic definition states that a theory is consistent i ...

, 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 including the parallel postulate yield Euclidean geometry, and with the parallel postulate negated, yields

non-Euclidean geometry In mathematics, non-Euclidean geometry consists of two geometries based on axioms closely related to those that specify Euclidean geometry. As Euclidean geometry lies at the intersection of metric geometry and affine geometry, non-Euclidean geo ...

. For examples,

elliptic geometry Elliptic geometry is an example of a geometry in which Euclid's parallel postulate does not hold. Instead, as in spherical geometry, there are no parallel lines since any two lines must intersect. However, unlike in spherical geometry, two lines a ...

(no parallels) and

hyperbolic geometry In mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai– Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with: :For any given line ''R'' and point ''P'' ...

(many parallels). Both elliptic and hyperbolic geometry are consistent systems, showing that the parallel postulate is independent of the other axioms. Proving independence is often very difficult. Forcing is one commonly used technique. Kenneth Kunen, ''Set Theory: An Introduction to Independence Proofs'', pages 184-237

References

External links

* {{DEFAULTSORT:Axiom Independence Metalogic