Gödel's incompleteness theorems

Gödel's incompleteness theorems is the name given to two theorems (true mathematical statements), proved by Kurt Gödel in 1931. They are theorems in mathematical logic.
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":
Those theorems are important to mathematicians because they prove that it is impossible to create a set of axioms that explains everything in maths.