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Axiom

In traditional [logic] , an axiom or postulate is a proposition that is not proved or demonstrated but considered to be either [self-evident] , or subject to necessary [decision] . Therefore, its truth is taken for granted, and serves as a starting point for deducing and inferring other (theory dependent) truths.

In [mathematics] , the term axiom is used in two related but distinguishable senses: ["logical axioms"] and ["non-logical axioms"] . In both senses, an axiom is any mathematical statement that serves as a starting point from which other statements are logically derived. Unlike [theorem] s, axioms (unless redundant) cannot be derived by principles of deduction, nor are they demonstrable by [mathematical proof] s, simply because they are starting points; there is nothing else from which they logically follow (otherwise they would be classified as theorems).

Logical axioms are usually statements that are taken to be universally true (e.g., A and B implies A ), while non-logical axioms (e.g., ) are actually defining properties for the domain of a specific mathematical theory (such as [arithmetic] ). When used in the latter sense, "axiom," "postulate", and "assumption" may be used interchangeably. In general, a non-logical axiom is not a self-evident truth, but rather a formal logical expression used in deduction to build a mathematical theory. To axiomatize a system of knowledge is to show that its claims can be derived from a small, well-understood set of sentences (the axioms). There are typically multiple ways to axiomatize a given mathematical domain.

Outside logic and mathematics, the term "axiom" is used loosely for any established principle of some field.

Etymology

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