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.
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 is impossible to create a set of axioms that explains everything in math.