Paradoxes

What is a paradox? It is defined as a statement that is seemingly contradictory or opposed to common sense and yet is perhaps true . Here's an example :

  • Happiness or a ham sandwich?

    Which is better, eternal happiness or a ham sandwich? It would appear that eternal happiness is better, but this is really not so! After all, nothing is better than eternal happiness, and a ham sandwich is certainly better than nothing. Therefore a ham sandwich is better than eternal happiness.

    Does that make any sense? Or is it total bullshit? Still as blur as a sotong? Let me hit you with another one.

  • Proof that I am Dracula
    Assume :
    1. Everyone is afraid of Dracula.
    2. Dracula is afraid of only me.
        Therefore I am Dracula.

    How the hell did I arrive at that?Doesn't that argument sound like just a silly joke? Well it isn't; it is valid. Since everyone is afraid of Dracula, then Dracula is afraid of Dracula. So Dracula is afraid of Dracula, but also is afraid of no one but me. Therefore I must be Dracula!

    Come visit a hotel with me. I'll show you yet another one. Don't get lost now!

  • Hilbert's hotel paradox

    Imagine a hotel with a finite number of rooms, and assume that all the rooms are occupied. A new guest arrives and asks for a room. "Sorry" - says the proprietor - "but all the rooms are occupied." Now let us imagine a hotel with an infinite number of rooms, and all the rooms are occupied. To this hotel, too, comes a new guest and asks for a room. "But of course!" - exclaims the proprietor, and he moves the person previously occupying room N1 into room N2, the person from room N2 into room N3, the person from room N3 into room N4, and so on... And the new customer receives room N1, which becomes free as a result of these transpositions.

    Let us imagine now a hotel with an infinite number of rooms, all taken up, and an infinite number of new guests who come in, and ask for rooms.

    "Certainly, gentlemen," says the proprietor, "just wait a minute." He moves the occupant of N1 into N2, the occupant of N2 into N4, the occupant of N3 into N6, and so on, and so on...

    Now all odd numbered rooms become free and the infinity of new guests can easily be accommodated in them.

    Not the best of examples..do I hear? This one should be quite straightforward. If you don't get it..then...read it over and over till you do?

  • The barber paradox

    In a certain village there is a man, so the paradox runs, who is a barber; this barber shaves all and only those men in the village who do not shave themselves. Query: Does the barber shave himself?

    Any man in this village is shaved by the barber if and only if he is not shaved by himself. Therefore in particular the barber shaves himself if and only if he does not. We are in trouble if we say the barber shaves himself and we are in trouble if we say he does not.

    Comments : We are asked to swallow a story about a village and a man in it who shaves all and only those men in the village who do not shave themselves. This is the source of our trouble; grant this and we end up saying, absurdly, that the barber shaves himself only if he does not. The proper conclusion to draw is just that there is no such barber. We disprove the barber by assuming him and deducing the absurdity that he shaves himself if and only if he does not. The paradox is simply a proof that no village can contain a man who shaves all and only those men in it who do not shave themselves.

  • The unexpected hanging

    A man condemned to be hanged was sentenced on Saturday. "The hanging will take place at noon," said the judge to the prisoner, "on one of the seven days of next week. But you will not know which day it is until you are so informed on the morning of the day of the hanging."

    The judge was known to be a man who always kept his word. The prisoner, accompanied by his lawyer, went back to his cell. As soon as the two men were alone, the lawyer broke into a grin. "Don't you see?" he exclaimed. "The judge's sentence cannot possibly be carried out."

    "I don't see," said the prisoner.

    "Let me explain They obviously can't hang you next Saturday. Saturday is the last day of the week. On Friday afternoon you would still be alive and you would know with absolute certainty that the hanging would be on Saturday. You would know this before you were told so on Saturday morning. That would violate the judge's decree."

    "True," said the prisoner.

    "Saturday, then is positively ruled out," continued the lawyer. "This leaves Friday as the last day they can hang you. But they can't hang you on Friday because by Thursday only two days would remain: Friday and Saturday. Since Saturday is not a possible day, the hanging would have to be on Friday. Your knowledge of that fact would violate the judge's decree again. So Friday is out. This leaves Thursday as the last possible day. But Thursday is out because if you're alive Wednesday afternoon, you'll know that Thursday is to be the day."

    "I get it," said the prisoner, who was beginning to feel much better. "In exactly the same way I can rule out Wednesday, Tuesday and Monday. That leaves only tomorrow. But they can't hang me tomorrow because I know it today!"

    ... He is convinced, by what appears to be unimpeachable logic, that he cannot be hanged without contradicting the conditions specified in his sentence. Then on Thursday morning, to his great surprise, the hangman arrives. Clearly he did not expect him. What is more surprising, the judge's decree is now seen to be perfectly correctly. The sentence can be carried out exactly as stated.

    Here is a version with an explanation :

    A judge tells a condemned prisoner two statements:

  • You will be hanged at noon one day next week, Monday through Friday.
  • The choice of day will be a surprise to you.

    The prisoner thinks about that for a while, then smiles. If the hanging were on Friday, then it wouldn't be a surprise. He would wake up Friday morning expecting it, since it's the last day. Therefore the hanging absolutely can't be on Friday.

    Now that he knows it can't be Friday, what will he think if he wakes up Thursday morning? Since it can't be Friday, it must be today. So a Thursday hanging wouldn't be unexpected either. Therefore a Thursday hanging is impossible too. Continuing the reasoning, the prisoner concludes that logically, he can't be hanged any day next week. He returns to his cell confident in his safety.

    The next week, he is hanged on Wednesday, which is a complete surprise. Everything the judge said turned out to be true, so where is the flaw in the prisoner's reasoning?

    A Simpler Form of the Paradox :

    There is a simpler form of this paradox. The judge tells the condemned these two statements:

  • You will be hanged at noon next week on Friday.
  • The choice of day will be a surprise to you.

    The prisoner exclaims "how can it be a surprise, if you've already told me it will be on Friday? That's contradictory! I can't logically conclude anything rational from such an inconsistent system. This is just nonsense. Since I obviously can't trust you, I don't know when I'll be hanged, or even whether I will be hanged at all. I've learned nothing from your words!"

    The next Friday, the prisoner is hanged. He wasn't at all sure that would happen, so it was a surprise. Everything the judge said turned out to be true, even though the prisoner had "proved" that the judge was contradicting himself. What was wrong with his reasoning?

    The Simplest Form of the Paradox

    The judge is tired of hanging people, so he just tells the prisoner one statement:

  • The prisoner cannot know that this statement is true.

    The prisoner thinks about that statement: "I'll need to reason logically about this statement. Suppose I logically prove that it's true. Then I will know it is true. But then it must be wrong when it said I couldn't know that. So then it must not be true. If I logically decide that it isn't true, then the prediction in that statement was accurate, so it must be true. Either way I see contradictions. Therefore, I conclude that this statement is as self-contradictory and meaningless as the claim 'this statement is false'".

    The judge and the other people in the courtroom listen to this speech. They see the prisoner's confusion, and see that he doesn't end up knowing the statement is true. That's exactly what the statement predicted, so they all see that the judge was correct once again. There is no contradiction, despite the prisoner's conclusion that there is. Moved by pity for the poor man's confusion, the judge grants a full pardon.

    Discussion

    This paradox is unsettling because the prisoner seems to show that the judge is being self-contradictory, yet in the end the judge ends up being perfectly correct in every statement. Several solutions have been suggested for this paradox.

    One solution has been to take into account human nature. In the first form of the paradox, the prisoner would probably wake up every morning convinced that this is the day. Therefore, he wouldn't be surprised, no matter what day the hanging happened to be. When the judge claimed that he would be surprised, the judge was simply wrong.

    Using that definition of "surprise", the paradox is hardly very interesting. A more interesting definition of surprise would be that the prisoner can't logically and consistently prove what will happen, using the judge's statements as axioms. In that case, the prisoner truly is surprised by the hanging. Although the prisoner couldn't prove what would happen, everyone else could. The contradictions only arose when the prisoner tried to prove something from the axioms.

    Using that definition of "surprise", the paradox is hardly very interesting. A more interesting definition of surprise would be that the prisoner can't logically and consistently prove what will happen, using the judge's statements as axioms. In that case, the prisoner truly is surprised by the hanging. Although the prisoner couldn't prove what would happen, everyone else could. The contradictions only arose when the prisoner tried to prove something from the axioms.

  • Arithmetic and algebraic paradoxes

    Proving that 2 = 1:
    Here is the version offered by Augustus De Morgan: Let x = 1. Then x = x. So x - 1 = x -1. Dividing both sides by x -1, we conclude that x + 1 = 1; that is, since x = 1, 2 = 1.

    Assume that
    a = b. (1)

    Multiplying both sides by a,

    a = ab. (2)

    Subtracting b from both sides,

    a - b = ab - b . (3)

    Factorizing both sides,

    (a + b)(a - b) = b(a - b). (4)

    Dividing both sides by (a - b),

    a + b = b. (5)

    If now we take a = b = 1, we conclude that 2 = 1. Or we can subtract b from both sides and conclude that a, which can be taken as any number, must be equal to zero. Or we can substitute b for a and conclude that any number is double itself. Our result can thus be interpreted in a number of ways, all equally ridiculous.

    The paradox arises from a disguised breach of the arithmetical prohibition on division by zero, occurring at (5): since a = b, dividing both sides by (a - b) is dividing by zero, which renders the equation meaningless. As Northrop goes on to show, the same trick can be used to prove, e.g., that any two unequal numbers are equal, or that all positive whole numbers are equal.

    Here is another example:
    Proving that 3 + 2 = 0

    Assume A + B = C, and assume A = 3 and B = 2.

    Multiply both sides of the equation A + B = C by (A + B).

    We obtain A + 2AB + B = C(A + B)

    Rearranging the terms we have

    A + AB - AC = - AB - B + BC

    Factoring out (A + B - C), we have

    A(A + B - C) = - B(A + B - C)

    Dividing both sides by (A + B - C), that is, dividing by zero, we get A = - B, or A + B = 0, which is evidently absurd.

    Proving that n = n + 1

    (a) (n + 1) = n + 2n + 1

    (b) (n + 1) - (2n + 1) = n

    (c) Subtracting n(2n + 1) from both sides and factoring, we have

    (d) (n + 1) - (n + 1)(2n + 1) = n - n(2n +1)

    (e) Adding (2n + 1) to both sides of (d) yields

    (n + 1) - (n + 1)(2n + 1) + (2n + 1) = n - n(2n + 1) + (2n + 1)

    This may be written:

    (f) [(n + 1) - (2n + 1)] = [(n - (2n + 1)]

    Taking square roots of both sides,

    (g) n + 1 - (2n + 1) = n - (2n + 1)

    and, therefore,

    (h) n = n + 1

    The trick here is to ignore the fact that there are two square roots for any positive number, one positive and one negative: the square roots of 4 are 2 and -2, which can be written as 2. So (g) should properly read:

    (n + 1 - (2n + 1)) = (n - (2n + 1))

  • Walking forever

    Ever wondered if you could walk forever? Literally? Suppose you are given a distance of 100 meters to walk. 2 points, A & B, 100M apart. A man walks from point A to B. He'll reach point B eventually right? But if he walks according to the following rule, he will not reach point B. The rule is: always walk half of the remaining distance. You start out with a distance of 100M remaining. Walk half the remaining distance and you would have walked 50M. Then walk another half of the remaining 50M and you would have walked a total of 75M with 25M left. Keep walking like this and you'll never reach the end. You'll be walking for an infinite distance. You will come close to 100M but you will never cover that distance. Pretty neat eh? Thanks Max, my bunk mate for providing this paradox.