Turing's Cathedral: The Origins of the Digital Universe

by George Dyson

Axiomatization is the reduction of a subject to a minimal set of initial assumptions, sufficient to develop the subject fully without new assumptions having to be introduced along the way. The axiomatization of set theory formed the foundations, mathematically, of everything else. An ambitious previous attempt, by Bertrand Russell and Alfred North Whitehead, despite 1,984 pages extending across three volumes, still left fundamental questions unresolved. Von Neumann started fresh. “The conciseness of the system of axioms is surprising,” comments Stan Ulam. “The axioms take only a little more than one page of print. This is sufficient to build up practically all of the naive set theory and therewith all of modern mathematics … and the formal character of the reasoning employed seems to realize Hilbert’s goal of treating mathematics as a finite game.”32 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. Gödel proved that within any formal system suff...