Gödel's incompleteness theorems
B. Recall from part A that if we were able to prove the first statement then our system would be inconsistent. Thus to prove the consistency of the system, we must prove that the first statement is not provable (ie: prove the contrapositive of the claim “if we were able to prove the first statement then our system would be inconsistent”). Because of self-reference in the first statement “this statement cannot be proved” (ie: proving the first statement is not provable is proving the first statement), therefore proof of the consistency of the system requires proof of the first statement.