Axiom
High school
An axiom is a statement that is accepted without proof as a starting point, from which other statements are proved.
Undergraduate (logic)
Consider a mathematical theory, such as set theory or the theory of groups. An axiom of the theory is a statement that is taken as given, rather than proved, and from which the theory’s theorems are derived.
Examples
Examples
- Euclid’s parallel postulate High school
Non-examples
Things that may be mistaken for examples.
-
The statement “$f$ is continuous on $[a, b]$,” in the intermediate value theorem
Undergraduate
The assumption “$f$ is continuous on $[a, b]$” is not an axiom, because it is a condition on $f$, not a foundation of a theory.
Properties
- Axioms are not claimed to be true about the world; they define what the theory is about. For example, the group axioms define what a group is.