Gödel's incompleteness theorems
In a formal system, there are axioms. All axioms are supposed to be true, even without a proof. A theorem then serves to come up with other true statements from the axioms, using certain rules. A sequence of such statements is called a 'proof' of a statement, because it shows that the statement is true under all circumstances. Ideally, it should be possible to construct all true statements in the formal system in that manner. A system that has this property is called 'complete', one that does not is called 'incomplete'. Another thing wanted of a theory is that there should be no contradictions.