The barber that shaves all and only those who do not shave themselves

This is not a posting about shaving/barber.. but this may change your perspective about shaving.. forever. Okay I exaggerated, that’s only me I guess 😀 This is actually a posting about a logical puzzle and paradox, and no shaving experience is required to understand this paradox 😆 The antimony itself is the following:

Suppose there is this hypothetical village where all male citizens keep their face clean from hair by shaving. There’s only one barber there, and this barber shaves all and only those men who do not shave themselves. The question is, given that the barber is a male.. who shave the barber?

The only possible answer is.. the barber himself, of course. Since from the sentence “… this barber shaves all and only those men who do not shave themselves” we know that any man in the village is either shaved by himself or by the barber. So the barber can only be shaved by himself (the barber) or by the barber. However, is it true that the barber shave himself. Owh.. come on Hafiz, how often have I said to you that when you have eliminated the impossible, whatever remains, however improbable, must be the truth?

Wait a minute Sherlock, we still have not eliminated all the impossible. If the barber shave himself, then by the rule he must not be shaved by the barber. A contradiction, such a village can never exist then. This is what is known as Barber paradox, a modification of Russell’s paradox by famous philosopher and mathematician Bertrand Russell.

Russell’s paradox itself is a paradox about set theory, one very important underlying part of mathematics. Foundations of mathematics is actually not my playground, but I stumbled upon this paradox a long time ago when I read the book Analysis by Terence Tao. The paradox itself is pretty much the same with the barber version. From wikipedia:

“ Let us call a set “abnormal” if it is a member of itself, and “normal” otherwise. For example, take the set of all squares. That set is not itself a square, and therefore is not a member of the set of all squares. So it is “normal”. On the other hand, if we take the complementary set that contains all non-squares, that set is itself not a square and so should be one of its own members. It is “abnormal”.Now we consider the set of all normal sets, R. Attempting to determine whether R is normal or abnormal is impossible: If R were a normal set, it would be contained in the set of normal sets (itself), and therefore be abnormal; and if it were abnormal, it would not be contained in the set of normal sets (itself), and therefore be normal. This leads to the conclusion that R is both normal and abnormal: Russell’s paradox. ”

Can you see how the barber paradox is derived from Russell’s paradox? 😉 This paradox is used by Russell to show that naive set theory which lies in the building block of mathematics is not flawless. But you may relax, most likely none of any set defining you have and will ever do in your life is wrong (esp if you’re not an Algebraist). So yes, mathematics is still the same for us. The only thing I’m afraid of is that.. shaving will never feel the same again 😆


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