Definition of Axioms and Theorems
Definition
A proposition that is regarded as not requiring proof is called an axiom, and a proposition that has been proved is called a theorem.
Explanation
If you trace back the theorems needed to prove a theorem, you eventually reach propositions that cannot be proved on the basis of other statements; these are axioms.
For reference, a postulate is a term similar to “an axiom used in a particular field”, and indeed it appears in Geometry in systems such as Euclid’s axiom system or Birkhoff’s axiom system, but in essence it is not distinguished from an axiom.