According to Gödel, Escher, Bach: An Eternal Golden Braid, an axiom is a 'free' theorem used at the beginning of a formal system.

For example, in an exam situation, one might be given an axiom (cheese is good) and asked to derive the proof that ice-cream is good.

Ax"i*om (?), n. [L. axioma, Gr. that which is thought worthy, that which is assumed, a basis of demonstration, a principle, fr. to think worthy, fr. worthy, weighing as much as; cf. to lead, drive, also to weigh so much: cf F. axiome. See Agent, a.]

1. Logic & Math.

A self-evident and necessary truth, or a proposition whose truth is so evident as first sight that no reasoning or demonstration can make it plainer; a proposition which it is necessary to take for granted; as, "The whole is greater than a part;" "A thing can not, at the same time, be and not be."

2.

An established principle in some art or science, which, though not a necessary truth, is universally received; as, the axioms of political economy.

Syn. -- Axiom, Maxim, Aphorism, Adage. An axiom is a self-evident truth which is taken for granted as the basis of reasoning. A maxim is a guiding principle sanctioned by experience, and relating especially to the practical concerns of life. An aphorism is a short sentence pithily expressing some valuable and general truth or sentiment. An adage is a saying of long-established authority and of universal application.

 

© Webster 1913.

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