In Chapter Forty (Bringing it Back), Brian the university librarian took exception to his fellow staff attempting to deal with books that couldn’t be classified, by introducing a new category called “unclassified”.
This chapter was a small homage to the most famous paradox in mathematical logic. The issue related to the definition of normal and abnormal sets within the mathematical discipline of naïve set theory:
- a normal set exists where the classification isn’t a member of the set, for example, “the class of all motorbikes” isn’t a motorbike.
- “the class of all things that aren’t motorbikes” is an abnormal set, because whatever this term means, it’s not a motorbike, so the definition contains itself.
(Other abnormal set examples include the class of all classes or the property of being a property.)
These two concepts formed the bedrock of class and set theory at the start of the 20th century until a young Viscount Russell asked us to consider: “the set of all sets that are not members of themselves.” Paradoxically, such a set appears to be a member of itself if and only if it is not a member of itself.
A popular illustration of this quandary is known as the Barber’s Paradox:
Suppose you are in a town with a clean-shaven chap who tells you: “I’m the town barber, I shave every man in the town that does not shave himself, and no one else.”
The question that this raises is: does the barber shave himself?
- If he does, then he mustn’t, because he doesn’t shave men who shave themselves.
- If he doesn’t, then he must, because he shaves every man who doesn’t shave himself.