
On Saturday, a jailor made an interesting promise to his condemned prisoner: You will be hanged one morning next week (the week runs from Sunday through Saturday) and you will not expect the hangman on the particular morning when he will come for you.
The prisoner had some time to ponder his fate. The first thing he realized is that if he survives to Saturday morning � the last possible day on which he can be hanged — then he will expect the hangman, and so the jailor's promise would not be kept. So Saturday is not a possible execution day. This makes Friday the last possible day on which he can be hanged. By the same reasoning, then, the prisoner realized Friday is not a possible execution day. Continuing this line of thought, Thursday, Wednesday, Tuesday, and Monday are not possible execution days, which leaves only Sunday. So he will expect the hangman to arrive on Sunday, but this, too, violates the jailor's promise.
Elated, the prisoner reasoned that the jailor can't possibly keep his promise! Brimming with joy at his reprieve, the prisoner whiled away the time in his cell, having determined there is no day the hangman could possibly arrive without being expected on that day. His smile faded on Wednesday morning, however, when to his utter astonishment, the hangman arrived for him.
What was wrong with the prisoner's logic? I think it's that the jailor's promise isn't meaningful. The spreading out of the promise to seven days obscures the lack of meaning. What if there were just two days, and the promise was "you will be hanged either tomorrow or the next day, and you won't expect it that day." The prisoner's reasoning would be the same, and he wouldn't expect it either day. So when the hangman shows up, the prisoner will be surprised, fulfilling the promise. In fact, if the promise were restricted to a single day, the paradox would be the same: the jailor would say "you will be hanged tomorrow, and it will be a surprise to you." The prisoner would reason that the promise can't be kept, and for that very reason, he will be surprised when the hangman shows up in his cell.
So then here's what was wrong with the prisoner's logic: he at first relied on the jailor's statement, and this resulted in a chain of reasoning resulting in the refutation of that very statement. Then, relying on his belief that the statement was false, he found to his surprise that the statement came true! The root of the problem is not so much in the prisoner's logic, but rather in the jailor's promise, which we now see has no truth value at all!
The proof that there are no uninteresting numbers is similar to the unexpected hanging. Think of it as a promise that there will be an uninteresting number. Then as each number is examined in turn, it strikes you as interesting for one reason or another. Finally, though, you may run out of reasons for finding numbers uninteresting, and so you will eventually hit an uninteresting number. Of course, if that ever happens, it will be very interesting! So the promise that a number will be uninteresting is, like the jailor's promise, both true and false, and therefore neither true nor false.
The webmaster and author of this Math Help site is Graeme McRae.