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.

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axiom

In mathematics or logic, an unprovable rule or first principle accepted as true because it is self-evident or particularly useful (e.g., “Nothing can both be and not be at the same time and in the same respect”). The term is often used interchangeably with postulate, though the latter term is sometimes reserved for mathematical applications (such as the postulates of Euclidean geometry). It should be contrasted with a theorem, which requires a rigorous proof.

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1.	(language)	AXIOM - A commercially available subset of Scratchpad, from IBM. ["Axiom - The Scientific Computing System", R. Jenks et al, Springer 1992].
2.	(logic)	axiom - A well-formed formula which is taken to be true without proof in the construction of a theory. Compare: lemma.