The Kolmogorov complexity of a string is the length of the shortest program which outputs said string. Using this notion, Chaitin formulated a weak version of Gödel's first incompleteness theorem. Subsequently Kikuchi, followed by Kritchman and Raz, used Kolmogorov complexity to prove Gödel's second incompleteness theorem. We introduce the incompleteness theorems and sketch some of these proofs.