Gödel's incompleteness theorems
Mathematicians once thought that everything that is true has a mathematical proof. A system that has this property is called complete; one that does not is called incomplete. Also, mathematical ideas should not have contradictions. This means that they should not be true and false at the same time. A system that does not include contradictions is called consistent. A system is a collection of theorems (logical consequences) based on axioms (basic assumptions). Axioms are statements that are accepted as true, and need no proof.