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Paradoxes
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 DraculaAssume : 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! Hilbert's hotel paradoxImagine 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. The barber paradoxIn 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? The unexpected hangingA 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." A judge tells a condemned prisoner two statements: 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. There is a simpler form of this paradox. The judge tells the condemned these two statements: 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 judge is tired of hanging people, so he just tells the prisoner one statement: 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'". 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. Arithmetic and algebraic paradoxes Proving that 2 = 1: Here is the version offered by Augustus De Morgan: Let x = 1. Then x² = x. So x² - 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, a² = ab. (2) Subtracting b² from both sides, a² - b² = ab - b² . (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. Dividing both sides by (A + B - C), that is, dividing by zero, we get A = - B, or A + B = 0, which is evidently absurd. 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: 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. |