Paradox Lost

Paul Tissier argues that Russell’s Paradox isn’t really a paradox.

- By Paul Tissier. Photographs by Stephen Leach

Bev and Ali are deep in discussion.

Ali: There can’t be real paradoxes, can there? Those that purport to be paradoxes actually contain ambiguities in meaning, neglected information, hidden assumptions… In short, it’s all smoke and mirrors.

Bev: You might think so. However, Russell’s Paradox in set theory has attracted a lot of serious attention in the literature. In fact, Gottlob Frege, a leading logician and mathematician at the time, felt that this paradox devastated his fundamental work on set theory and the foundations of mathematics.

Ali: So if Russell’s Paradox holds up, it would, rather worryingly, shake the foundations of mathematics?

Bev: Right. So let’s see what is involved. We all know – or think we know – what a ‘set’ is: a collection of all those and only those entities with a defined property. A set’s members can be simple objects, or can be sets themselves. It’s the last possibility we’re interested in here. Sets can also be non-self-membered, or self-membered: they can belong to themselves, or not. Examples of the former would include the set of all finite sets, which is infinite and hence non-self-membered. On the other hand, the set of all infinite sets is itself infinite and so must be included in itself, so it is self-membered. All sets either are non-self-membered or self-membered.

Ali: OK… But I feel like I’m about to be trapped, or perhaps conned.

Bev: Well consider the set of all non-self-membered sets, which for the sake of argument, let’s call ‘N’.

Ali: Hang on! Does such a set even exist?