An axiom or postulate is a premise or starting point of reasoning. As classically conceived, an axiom is a premise so evident as to be accepted as true without controversy. The word comes from the Greek axíōma () ‘that which is thought worthy or fit’ or ‘that which commends itself as evident.’ As used in modern logic, an axiom is simply a premise or starting point for reasoning. Axioms define and delimit the realm of analysis; the truth of an axiom is taken for granted within the particular domain of analysis, and serves as a starting point for deducing and inferring other truths. No explicit view regarding the absolute truth of axioms is ever taken in the context of modern mathematics, as such a thing is considered to be irrelevant. In mathematics, the term axiom is used in two related but distinguishable senses: “logical axioms” and “non-logical axioms”. Logical axioms are usually statements that are taken to be true within the system of logic they define (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. As modern mathematics admits multiple, equally “true” systems of logic, precisely the same thing must be said for logical axioms – they both define and are specific to the particular system of logic that is being invoked. 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. In both senses, an axiom is any mathematical statement that serves as a starting point from which other statements are logically derived. Within the system they define, axioms (unless redundant) cannot be derived by principles of deduction, nor are they demonstrable by mathematical proofs, simply because they are starting points; there is nothing else from which they logically follow otherwise they would be classified as theorems. However, an axiom in one system may be a theorem in another, and vice versa.
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