Math Detours · Grade 7 · Around the corner · 15

The barber who shaves everyone

He shaves all those who do not shave themselves. Who shaves the barber? A puzzle that shook the foundations of mathematics.

Name ____________________Date ____________

Tonight's idea

A village has one barber, a man who lives in the village. The sign in his window says:

I shave every man in this village who does not shave himself, and nobody else.

It sounds sensible. Ali shaves himself, so the barber leaves him alone. Ben does not, so the barber shaves him. Now ask: who shaves the barber?

Suppose he shaves himself. Then he is a man who shaves himself, and the sign says the barber shaves no such man. So he does not shave himself.

Suppose he does not. Then he is a man who does not shave himself, and the sign says the barber shaves every such man. So he does shave himself.

He shaves himself He does not so the sign says he must not so the sign says he must
Each answer forces the other. There is nowhere to stop.

"Yes" forces "no", and "no" forces "yes". A statement that fights itself like this is called a contradiction. The way out is not a clever answer. It is that the sign cannot be true: no such barber can exist. Plain words can describe something impossible.

Watch one: Russell's set

A set is a collection of things, called its members. Just after 1900, many mathematicians assumed that any description gives a set: "all teaspoons", even "all sets". Most sets are not members of themselves: the set of all teaspoons is not a teaspoon. Bertrand Russell asked about this one:

\(R\) = the set of all sets that are not members of themselves.

Is \(R\) a member of \(R\)? If yes, it breaks its own entry rule, so no. If no, it fits the rule, so yes. The barber again. So \(R\) cannot exist: not every description defines a set. Mathematicians had to write more careful rules for building sets.

Your turn

  1. The barber's cousin lives in the next village. He hangs the same sign, and it is still about the men of this village. (a) Does he shave Ali? Does he shave Ben? (b) Is there a contradiction now? Why, or why not?
  2. Back to a barber who lives in the village. Can he exist if his sign says (a) "I shave every man here who does not shave himself", and he may shave others too? (b) "I shave only men here who do not shave themselves", but maybe not all of them? In each case say whether he shaves himself.
  3. A card says on the front: "The sentence on the back is true." On the back: "The sentence on the front is false." Suppose the front is true and follow it round. Then suppose the front is false. What do you find?
  4. In a notebook, each page holds a list of page numbers. Page 1 lists "1, 2". Page 2 lists "1". Page 3 is headed: "Every page whose list does not include its own number." (a) Should 1 be on page 3's list? Should 2? (b) Should 3? (c) What does that say about page 3?

Stretch: a shy word

"Short" is a short word, so it describes itself. "Long" is not a long word, so it does not. Call a word shy if it does not describe itself. So "long" is shy and "short" is not. Is the word "shy" shy?

Answers

  1. (a) Not Ali, who shaves himself. Yes Ben, who does not. (b) No contradiction. The cousin is not a man of this village, so the sign says nothing about who shaves him. The trouble only starts when the rule is turned on the barber himself.
  2. (a) Yes, he can exist, and he shaves himself. If he did not, the sign would say he must. If he does, no rule is broken, because this sign lets him shave self-shavers too. (b) Yes, he can exist, and he does not shave himself. If he did, he would be shaving a man who shaves himself. Someone else shaves him, or he grows a beard. Each half of the sign is harmless. Only both together are impossible.
  3. Front true: then the back is true, and the back says the front is false. Front false: then the back is false, so the front is not false, so it is true. Either way the front comes out both true and false. The pair of sentences cannot be given truth values at all.
  4. (a) Not 1: page 1's list includes 1. Yes 2: page 2's list is just "1". (b) If 3 is on the list, page 3 includes its own number, so it should not be there. If 3 is not on the list, page 3 does not include its own number, so it should be. (c) Nobody can write page 3 correctly. The heading describes a list that cannot exist. It is Russell's set in a notebook.

Stretch. If "shy" is shy, it does not describe itself, so it is not shy. If "shy" is not shy, it does describe itself, so it is shy. Both fail. The definition works for ordinary words but breaks when turned on itself, so "shy" is not a properly defined property of every word.

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