axiom

axiom in mathematics and logic, general statement accepted without proof as the basis for logically deducing other statements (theorems). Examples of axioms used widely in mathematics are those related to equality (e.g., "Two things equal to the same thing are equal to each other" ; "If equals are added to equals, the sums are equal" ) and those related to operations (e.g., the associative law and the commutative law ). A postulate, like an axiom, is a statement that is accepted without proof; however, it deals with specific subject matter (e.g., properties of geometrical figures) and thus is not so general as an axiom. It is sometimes said that an axiom or postulate is a "self-evident" statement, but the truth of the statement need not be evident and may in some cases even seem to contradict common sense. Moreover, a statement may be an axiom or postulate in one deductive system and may instead be derived from other statements in another system. A set of axioms on which a system is based is often wished to be independent; i.e., no one of its members can be deduced from any combination of the others. (Historically, the development of non-Euclidean geometry grew out of attempts to prove or disprove the independence of the parallel postulate of Euclid.) The axioms should also be consistent; i.e., it should not be possible to deduce contradictory statements from them. Completeness is another property sometimes mentioned in connection with a set of axioms; if the set is complete, then any true statement within the system described by the axioms may be deduced from them.

axiom Assumption used as a basis for deductive reasoning. The axiomatic method is fundamental to the philosophy of modern mathematics: it was used by the Greeks and formalized early in the 20th century by David Hilbert. In an axiomatic system, certain undefined entities (terms) are taken and described by a set of axioms. Other, often unsuspected, relationships (theorems) are then deduced by logical reasoning. For example, the points, lines and angles of Euclidean geometry are connected by postulates; theorems, such as Pythagoras' theorem, can be deduced. This geometry describes measurements of position, distance and angle in space.

axiom In logic, a statement that is stipulated to be true for a particular chain of reasoning. See deduction.