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Mummy, Mummy, what’s Russell’s Paradox?

Bertrand Russell (1872-1970) was one of the greatest philosophers of the 20th Century. Much of his early work was concerned with the foundations of mathematics and logic. For Russell, one of the basic building blocks of mathematics was set theory. Everything else depended on it. He was therefore disconcerted, to say the least, when in 1901 he discovered the following problem.

Supposing you take the set of all the electric toasters in the world. The set isn’t a member of itself – of course, since it’s a mathematical construct, not a toaster. The same goes for the set of all dogs, the set of all tennis players and so on. On the other hand, the set of all sets is a member of itself. So far so obvious. But what about the set of all those sets which aren’t members of themselves? If it’s a member of itself, then that means it isn’t a member of itself, but if it isn’t a member of itself, well, that makes it a member of itself. Think about it. Then go and sit in a darkened room for a while until your head stops spinning.

Russell himself never found a satisfactory way out of this paradox, and it may have been one of the reasons he finally gave up mathematical philosophy in favour of other pursuits, such as politics, ethical theory and bonking Lady Ottoline Morrell.

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