Gödel's First Incompleteness Theorem
Any adequate axiomatisable theory is incomplete. In particular the sentence "This sentence is not provable" is true but not provable in the theory.
Gödel's Second Incompleteness Theorem
In any consistent axiomatisable theory (axiomatisable means the axioms can be computably generated) which can encode sequences of numbers (and thus the syntactic notions of "formula", "sentence", "proof") the consistency of the system is not provable in the system.