Russell’s Paradox
Bertrand Russell · 1901
The puzzle
Let R be the set of all sets that don’t contain themselves. Does R contain itself?
Note
If yes, it shouldn’t; if no, it should. Russell found the paradox in 1901 while working on his Principles of Mathematics; the next year he wrote to Frege pointing out that it undermined the reduction of arithmetic to logic Frege had attempted in Grundgesetze. Frege wrote, in the appendix to volume two of his life’s work, that arithmetic now had no foundation he could see. Russell and Whitehead spent ten years on the Principia Mathematica trying to repair the damage with a theory of types. The eventual professional solution was Zermelo–Fraenkel set theory’s axiom schema of separation, which forbids forming a set with arbitrary properties. A clean formal repair, but no one has explained why naive set theory was wrong; only that it was. Mathematics rebuilt itself around the absence.