The British Library sends out instructions that every library in the country has to make a catalogue of all its books. Each librarian makes their catalogue and are then faced with a choice: the catalogue is, after all, a book in their library; should the title of the catalogue be included in the catalogue itself or not? Some librarians decide to include it, others not to.
In the course of time the catalogues are sent to the British Library and the chief librarian there has the job of making a catalogue of these catalogues. But they find that they have two different sorts of catalogues to deal with: those that mention themselves inside the catalogue and those that don’t. That is, they have catalogues that contain their own titles and catalogues that don’t. So the chief librarian decides to make two different catalogues corresponding to these two different kinds. With the catalogue of all those catalogues that include themselves the librarian has the choice to include the title of the catalogue in itself or not. No problem there. But with the catalogue of catalogues that don’t include themselves the librarian is faced with a dilemma: should they include the title of the catalogue in the catalogue or not? if they do then it is not a catalogue that does not contain its own title and so it shouldn’t be included; if they don’t put it in then it is a catalogue that doesn’t contains its own title and so should be included. Either way, it should contain itself if it doesn’t and shouldn’t contain itself if it does!
This paradox is a version of Russell’s Paradox. It came about from Bertrand Russell thinking about the notion of a set, or class, or collection of things, and whether a set can be a member of itself or not. For example, think about the totality of cats in the world: this is the set of cats. Is this set a member of itself or not. Clearly not – it’s a set, an abstract object, not a cat. But now think about all of the things in the world that are not cats – dogs, chairs, books, violin sonatas, . . . and sets. This set is a member of itself. Now it is far more usual for a set not to be a member of itself than for it to be a member of itself. Let’s call sets that don’t belong to themselves normal. Now Russell’s problem is: is the set of all normal sets a member of itself or not? If it is then it isn’t. But if it isn’t then it is…
Russell’s own solution to the problem is his Theory of Types. This can be explained as follows. Imagine that five people get together to form a five-a-side football team. This team then joins a local league, and the league in turn is part of a regional association. Clearly, an individual can only be member of a team and cannot be a member of a league or an association: it is the wrong type of thing, as only teams can be members of leagues and leagues members of associations. Similarly Russell thought that individuals, type C, were members of sets or classes of type 1. These sets could only be members of sets of type 2, and sets of type 2 could only be members of the higher type 3, and so on. Specifically, a set could not belong to another set of the same type, as it has to belong to a set of the next highest type, and so a set can never be a member of itself.
While the Theory of Types does work as a solution to the Paradox many logicians feel that it has an ad hoc air about it, and isn’t as intuitively obvious as is required for the foundations of mathematics: after all, Russell’s solution may seem plausible for football teams but mathematics does consider sets that are formed from individuals, sets of individuals, sets of sets of individuals and so on, and counting out such sets seems to be a high price to pay for avoiding the Paradox.
Readers may wonder whether Russell’s Paradox is a problem only in the foundations of mathematics or if it actually occurs in real life. I recently saw a policy concerning disability which contained a number of clauses naming specific disabilities which were included under the policy. The last clause, however, contained the statement “Other disabilities not on this is list”. Now consider a specific disability not named in the list: if it was not on this list then this clause includes it in the list so it is on the list; if it is on the list then the clause says it is not on the list. Without care, paradoxes can occur where we least expect them.
