Axiom

An axiom is any sentence, proposition, statement or rule that forms the basis of a formal system. Unlike theorems, axioms are neither derived by principles of deduction, nor are they demonstrable by formal proofs. Instead, an axiom is taken for granted as valid, and serves as a necessary starting point for deducing and inferencing logically consistent propositions. In many usages, "axiom," "postulate," and "assumption" are used interchangeably.

In certain epistemological theories, an axiom is a self-evident truth upon which other knowledge must rest, and from which other knowledge is built up. An axiom in this sense can be known before one knows any of these other propositions. Not all epistemologists agree that any axioms, understood in that sense, exist.

In logic and mathematics, an axiom is not necessarily a self-evident truth, but rather a formal logical expression used in a deduction to yield further results. To axiomatize a system of knowledge is to show that all of its claims can be derived from a small set of sentences that are independent of one another. This does not imply that they could have been known independently; and there are typically multiple ways to axiomatize a given system of knowledge (such as arithmetic). Mathematics distinguishes two types of axioms: logical axioms and non-logical axioms.