Wednesday, 28 May 2008

101 Philosophy Problems Part 4: The Hairdresser of Hindu Kush

Or, more traditionally, Russel's Paradox, or the Barber's Paradox.

The paradox as presented by Cohen is horribly worded, and doesn't even manage to reach the status of contradiction, but that's not really anything new. In it's standard "barber" form, it goes:

Imagine a town with only one male barber, the Elders of the town are seriously committed to everyone having neat hair, so they require *by law* that the barber cuts the hair of everyone, and only those people, who does not cut their own hair. No-one else in the town is allowed to do any hair cutting. Everyone's hair must be cut.

So, does the barber cut his own hair? Obviously he can't, because then he wouldn't be allowed to cut it by law, but he must, because no-one else can cut it either. Blah, blah. The conclusion, one would think, is that this is a really silly law. Bertrand Russel certainly thought so (and never endorsed the "paradox" in this form).

However, it is supposed to be a version of Russel's real paradox - "The set of all sets which do not contain themselves". This set, it seems can neither contain itself or not contain itself, as either would lead to a contradiction. However, to simply declare that, like the barber, this set doesn't exist is rather rash - it undermines the axiomatic set theory that Russel spent a lot of his life developing. Luckily, it doesn't undermine Zermelo-Fraenkel Set Theory, so mathematics didn't entirely collapse in on itself.