hilbert's vision of mathematics was of a vast game governed by simple rules, where all facts and proofs could be deduced systematically by finitely many applications of these rules. throughout the twentieth century, we have seen dramatic counterexamples to this vision. however, it is still meaningful to study that part of mathematics that can be described in this way. in this talk, we will discuss the notion of a decision procedure and the related idea of computability. we will see examples of interesting mathematical problems that are decidable and, on the flip side, think about what it would mean for a problem to be undecidable. coming full circle, we return to hilbert and to his famous list of problems. in particular, his 10th problem proved to be a milestone in undecidability theory. we will trace through the history of its solution and notice the various consequences of undecidability that crop up.