Key difference: Axiom and theorem are statements that are most commonly used in mathematics or physics. An axiom is a statement that is accepted as true. It does not need to be proven. A theorem, on the other hand, is a statement that has been proven true.

Axiom and theorem are statements that are most commonly used in mathematics or physics. An axiom is a statement that is accepted as true. It does not need to be proven. A theorem, on the other hand is a statement that has been proven true.

According to Dictionary.com, an axiom is defined as:

• A self-evident truth that requires no proof.
• A universally accepted principle or rule.
• Logic, Mathematics. A proposition that is assumed without proof for the sake of studying the consequences that follow from it.

Essentially, axioms are assumptions that need not be proven. They are generally accepted as true, either because it does not have a contradiction or because we obviously know that it is true. The word axiom is derived from a Greek word that stands for 'that which is thought worthy or fit,' or 'that which commends itself as evident.' Axiom can at times be used interchangeable with postulate or assumption.

A theorem, on the other hand, needs to be proved. Dictionary.com defines theorem as:

• Mathematics. A theoretical proposition, statement, or formula embodying something to be proved from other propositions or formulas.
• A rule or law, especially one expressed by an equation or formula.
• Logic. A proposition that can be deduced from the premises or assumptions of a system.
• An idea, belief, method, or statement generally accepted as true or worthwhile without proof.

A theorem is a statement that has been proven through by testing or calculation. It can be proven based on theorems, which have been previously proven or on the basis of axioms. Theorems are made of two parts: hypotheses and conclusions. 

Image Courtesy: science.nd.edu