To fill in some of the details: the so-called 'Second Incompleteness Theorem' states that recursively inumerable systems containing arithmetic are insufficient

To fill in some of the details: the so-called 'Second Incompleteness Theorem' states that recursively inumerable systems containing arithmetic are insufficient for proving their own consistency. It is possible to prove the consistency of arithmetic, but to do so you must use an even more complex system of axioms whose consistency can itself only be proved with an even more complex system, and so on. So strictly speaking, the theorem doesn't rule out consistency proofs, but it does rule out proving the consistency of complex systems by simpler systems, meaning that the consistency of the consistency proofs can't be trusted any more than the consistency of the original system!