[Math Lair] Axioms and Postulates

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An axiom is a statement, usually considered to be self-evident, that is assumed to be true without proof. It is used as a starting point in mathematical proof for deducing other truths.

What is the Difference Between Axioms and Postulates?

Classically, axioms were considered different from postulates. An axiom would refer to a self-evident assumption common to many areas of inquiry, while a postulate referred to a hypothesis specific to a certain line of inquiry, that was accepted without proof. As an example, in Euclid's Elements, you can compare "common notions" (axioms) with postulates.

In much of modern mathematics, however, there is generally no difference between what were classically referred to as "axioms" and "postulates". The word "assumption" is sometimes used as well; in this context, it means the same as both "axiom" and "postulate." Modern mathematics does distinguish between logical axioms and non-logical axioms, with the latter sometimes being referred to as postulates.

See also: mathematical systems.