Bertrand Russell · Mathematics
The contradiction Russell found in 1901 - the class of all classes that are not members of themselves - which shattered Frege’s logical system, exposed a crack in the foundations of mathematics, and forced Russell to invent the theory of types to escape it.
At the turn of the twentieth century, mathematicians were rebuilding their subject on the secure-seeming ground of classes, or sets: collections of objects. It felt self-evident that any clearly stated property determines a class - the property is red picks out the class of red things, the property is a number picks out the class of numbers. This principle, later called unrestricted comprehension, was the quiet assumption underlying Frege’s logic and much of the new set theory. In 1901, while thinking about Cantor’s work on infinite sets, Russell noticed that this innocent-looking principle leads straight to contradiction. Some classes are members of themselves and some are not: the class of all abstract ideas is itself an abstract idea, so it belongs to itself, whereas the class of all teacups is not a teacup, so it does not. Then consider the class of all classes that are not members of themselves - and ask the fatal question.
Russell often illustrated the contradiction with a homely story. In a certain village there is a barber who shaves all those, and only those, who do not shave themselves. It sounds like a perfectly ordinary rule until you ask one question: does the barber shave himself? If he does shave himself, then he is one of the people who shave themselves - but the barber shaves only those who do not shave themselves, so he must not shave himself. If he does not shave himself, then he is one of those who do not shave themselves - and the barber shaves all such people, so he must shave himself. Round and round it goes. Russell was careful to note that the barber is only an illustration, not the real paradox: the honest conclusion about the barber is simply that no such barber can exist. But with classes we cannot escape so easily, because the class of all classes that are not members of themselves seems to be defined by a perfectly precise logical property - and yet it too cannot consistently exist.
In June 1902 Russell wrote a now-famous letter to Gottlob Frege, whose Basic Laws of Arithmetic was the most rigorous attempt yet to derive arithmetic from logic. Politely, devastatingly, Russell pointed out that his contradiction could be derived within Frege’s system, from Frege’s fifth basic law, which governed the move from concepts to their extensions. The second volume of Frege’s life’s work was already at the printer. Frege added a hurried appendix that began by admitting that scarcely anything more unwelcome can befall a writer than to have a foundation of his work give way just as it is finished, and confessing that Russell’s letter had put him in exactly that position. He tried to repair the system, but the patch failed, and Frege eventually abandoned the logicist program in something close to despair. The paradox had claimed its first and greatest victim, and Russell, who deeply admired Frege, had been the reluctant executioner.
This is the opening of the lesson. The rest — the dialogue, the primary source, and the recall — is in the app.
You learned that Russell’s paradox arises from the class R of all classes that do not contain themselves: R is a member of itself if and only if it is not. Explain how the contradiction is generated, why it broke Frege’s system, and how the theory of types was meant to block it.
Leads to Georg Cantor.
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