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Definition of Axiom. Axiom is a rule or a statement that is accepted as true without proof. An axiom is also called a postulate. Examples of Axiom. Here is an axiom of addition and multiplication. Let x and y be real numbers. Then x + y is also a real number and xy is also a real number.
However, an axiom in one system may be a theorem in another, and vice versa. Whether it is meaningful (and, if so, what it means) for an axiom, or any mathematical statement, to be true is an open question in the philosophy of mathematics.