Some paradoxes – an anthology, that you don’t care about
I recently found a ton of these I think they’re pretty cool :]
I thought they were Interesting so maybe you will 
I’m gonna list a couple of my favorites.
“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.”
“Two closed boxes, Bl and B2, are on a table. Bl contains $1,000. B2 contains either nothing or $1 million. You do not know which. You have an irrevocable choice between two actions:
1. Take what is in both boxes.
2. Take only what is in B2.
At some time before the test a superior Being has made a prediction about what you will decide. It is not necessary to assume determinism. You only need be persuaded that the Being’s predictions are “almost certainly” correct. If you like, you can think of the Being as God, but the paradox is just as strong if you regard the Being as a superior intelligence from another planet or a supercomputer capable of probing your brain and making highly accurate predictions about your decisions. If the Being expects you to choose both boxes, he has left B2 empty. If he expects you to take only B2, he has put $1 million in it. (If he expects you to randomize your choice by, say, flipping a coin, he has left B2 empty.) In all cases Bl contains $1,000. You understand the situation fully, the Being knows you understand, you know that he knows, and so on.
What should you do? Clearly it is not to your advantage to flip a coin, so that you must decide on your own. The paradox lies in the disturbing fact that a strong argument can be made for either decision. Both arguments cannot be right. The problem is to explain why one is wrong.
Let us look first at the argument for taking only B2. You believe the Being is an excellent predictor. If you take both boxes, the Being almost certainly will have anticipated your action and have left B2 empty. You will get only the $1,000 in Bl. On the other hand, if you take only B2, the Being, expecting that, almost certainly will have placed $1 million in it. Clearly it is to your advantage to take only B2.
Convincing? Yes, but the Being made his prediction, say a week ago and then left. Either he put the $1 million in B2, or he did not. “If the money is already there, it will stay there whatever you choose. It is not going to disappear. If it is not already there, it is not going to suddenly appear if you choose only what is in the second box.” It is assumed that no “backward causality” is operating; that is, your present actions cannot influence what the Being did last week. So why not take both boxes and get everything that is there? If B2 is filled, you get $1,001,000. If it is empty, you get at least $1,000. If you are so foolish as to take only B2, you know you cannot get more than $1 million, and there is even a slight possibility of getting nothing. Clearly it is to your advantage to take both boxes!”
I don’t know if it was clear to you guys that the obvious choice would be to choose both boxes, but i saw that right away, this wasn’t a very good one but I did kind of like it.
“Proof that there exists a unicorn
I wish to prove to you that there exists a unicorn. To do this it obviously suffices to prove the (possibly) stronger statement that there exists an existing unicorn. (By an existing unicorn I of course mean one that exists.) Surely if there exists an existing unicorn, then there must exist a unicorn. So all I have to do is prove that an existing unicorn exists. Well, there are exactly two possibilities:
(1) An existing unicorn exists.
(2) An existing unicorn does not exist.
Possibility (2) is clearly contradictory: How could an existing unicorn not exist? Just as it is true that a blue unicorn is necessarily blue, an existing unicorn must necessarily be existing.”
This next one is good :]
“Poaching on the hunting preserves of a powerful prince was punishable by death, but the prince further decreed that anyone caught poaching was to be given the privilege of deciding whether he should be hanged or beheaded. The culprit was permitted to make a statement – if it were false, he was to be hanged; if it were true, he was to be beheaded. One logical rogue availed himself of this dubious prerogative – to be hanged if he didn’t and to be beheaded if he did – by stating: ‘I shall be hanged.’ Here was a dilemma not anticipated. For, as the poacher put it, ‘If you now hang me, you break the laws made by the prince, for my statement is true, and I ought to be beheaded, but if you behead me, you are also breaking the laws, for then what I said was false and I should therefore be hanged.’”
“Olbers’ paradox – why is the night sky dark?
… the apparent paradox, stated in 1826 and now explained by postulating a finite expanding universe, that the sky is dark at night although, as there are an infinite number of stars, it should be uniformly bright.”
“[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.”
“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.
What are we to say to the argument that goes to prove this unacceptable conclusion? Happily it rests on assumptions. 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 are confronted with nothing more than what logicians have been referring to for a couple of thousand years as a reductio ad absurdum . 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.”
“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.
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.”
All of these. and more, can be found at:
http://www.paradoxes.co.uk/
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