The mathematical landscape of the early twentieth century was dominated by Göttingen’s David Hilbert, who believed that from a strictly limited set of axioms, all mathematical truths could be reached by a sequence of well-defined logical steps. Hilbert’s challenge, taken up by von Neumann, led directly both to Kurt Gödel’s results on the incompleteness of formal systems of 1931 and Alan Turing’s results on the existence of noncomputable functions (and universal computation) of 1936. Von Neumann set the stage for these two revolutions, but missed taking the decisive steps himself.
