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.
Gödel said that every non-trivial (interesting) formal system is either incomplete or inconsistent:
- There will always be questions that cannot be answered, using a certain set of axioms;
- You cannot prove that a system of axioms is consistent, unless you use a different set of axioms.
Those theorems are important to mathematicians because they prove that it is impossible to create a set of axioms that explains everything in maths.