In formal mathematics an axiom is a formula or schema of formulas that is stipulated as true (and therefore not requiring proof). Axioms are the counterpart in mathematics of suppositions, assumptions, or premises in ordinary syllogistic logic.

A mathematical theory may be understood as consisting of all of the statements that may be derived from its axioms. For example, Euclidean-geometry is based on Euclid’s well-known five axioms (typically called postulates in geometry), and common set theory is usually taken as the theory of the Zermelo-Fränkel axioms.