Has this ever happened to you? You're sitting down to prove a lemma, but none of your ideas seem to work out! At the same time, you've tried a number of counterexamples, but none of them seem to disprove your result! In this talk, we'll cover how this may not be your fault: there exist mathematical statements which can neither be proved or disproved from our system of axioms. We'll sketch a technique for proving these sorts of independence results with the classical example, the continuum hypothesis.