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". These systems are based on sets of "axioms". Axioms are statements that are accepted as true, and need no proof.