Bertrand Russell, the Complete Story of His 1901 Paradox

Bertrand Russell found a hole in the foundations of mathematics in 1901, and the repair he built for it is still running inside the tools people use to program quantum computers. That is a strange sentence, and this page explains how it came to be true. Russell was a British philosopher and logician who lived from 1872 to 1970, won the Nobel Prize in Literature, was sacked by two universities and was jailed twice. He is remembered by the public for almost none of the work that mattered most. The through line is a single idea. Russell showed that a system of rules can quietly contain a contradiction, and he invented a way of sorting things into levels so the contradiction cannot be stated. Those levels are what programmers now call types. 1. One letter ended another mathematician’s life work. In 1901 Russell found a contradiction at the heart of the way collections were then defined. He wrote to Gottlob Frege, whose Grundgesetze der Arithmetik rested on exactly that definition. Frege replied with dismay and admiration, and never recovered from it. 2. The paradox is easy to state and hard to escape. Consider the collection of all collections that do…