Godel's Incompleteness Theorem as explained by Whatis.com:

The First Incompleteness Theorem states that any contradiction-free rendition of number theory (a branch of mathematics dealing with the nature and behavior of numbers and number systems) contains propositions that cannot be proven either true or false on the basis of its own postulates.

The Second Incompleteness Theorem states that if a theory of numbers is contradiction-free, then this fact cannot be proven with common reasoning methods.