John von Neumann met Kurt Gödel. It was at a 1930 conference in Königsberg in a discussion of von Neumann's talk that Gödel for the first time in public mentioned his incompleteness theorem. Von Neumann in his 1932 book on quantum mechanics introduced the quantum measurement problem to the physics literature. The main question of this paper is: Is Gödel's incompleteness theorem related in any way to the quantum measurement problem? Is perhaps self-reference at the heart of both?