Russell Wrote Frege a Letter That Broke His Life's Work

Bertrand Russell discovered the paradox now bearing his name in May 1901, while working on his Principles of Mathematics. The argument is short. Consider the set R of all sets that are not members of themselves. Is R itself a member of R? If yes, then by definition it shouldn't be; if no, then by definition it should be. Either answer contradicts itself. The paradox lives at the heart of any set theory powerful enough to define such a collection at all.

The German logician Gottlob Frege had spent the previous twenty years building exactly such a system. His Grundgesetze der Arithmetik — Foundational Laws of Arithmetic — aimed to derive the whole of arithmetic from purely logical principles, and the second volume of the work was already at the printer when, on June 16, 1902, Russell sent him a letter. Russell apologized as gently as he could and laid out the contradiction. Frege replied within six days. "Hardly anything more unfortunate," he wrote, "can befall a scientific writer than to have one of the foundations of his edifice shaken after the work is finished." He added a hastily drafted appendix to the second volume acknowledging that his Basic Law V was untenable.

The paradox forced a generation of logicians to redesign set theory from scratch. Ernest Zermelo, who had stumbled into a version of the paradox independently around 1899 and never published it, formalized the modern axiomatic version. Russell himself responded by spending the next decade with Whitehead writing Principia Mathematica, whose elaborate "theory of types" was meant to outlaw the offending self-referential sets. The fix is widely considered an ugly one. Set theory survived; Frege's program did not.