Surfing the Paradox A trip into the flipside of your mind Infinity 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.     Infinite income-tax     Now if a man had an unlimited income, it is an immediate inference that, however low income-tax may be, he would have to pay annually to the Exchequer of his nation a sum equal in value to his whole income. Further, if his income was derived from a capital invested at a finite rate of interest (as is usual), the annual payments of income-tax would each be equal in value to the man's whole capital. If, then, the man with an unlimited income chose to be discontented, he would be sure of a sympathetic audience among philosophers and business acquaintances; but discontent could not last long, for the thought of the difficulties he would put in the way of the Chancellor of the Exchequer, who would find the drawing up of his budget most puzzling, would be amusing. Again, the discovery that, after paying an infinite income-tax, the income would be quite undiminished, would obviously afford satisfaction, though perhaps the satisfaction might be mixed with a slight uneasiness as to any action the Commissioners of Income Tax might take in view of this fact. 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!" Hempel's ravens (the confirmation paradox) While taking a group of benefactors on a tour through the new aviary they had just helped to build, a noted ornithologist commented, "And here we have two of the finest examples of ravens that I have ever seen. Notice the lustrous black plumage for which all ravens are famous." The ornithologist continued his lecture, commenting on the corvine feeding and nesting habits as well as on the birds' legendary role as harbingers of ill fortune. When the ornithologist had finished, a young man said, "Sir, excuse me, but did you say that 'All ravens are black'?"  "I don't know if I said exactly that, but it's true.All ravens are black." "But, how do you know that - for certain, I mean?" asked the young man. "Well, I've seen a few hundred ravens in my day and every one of them has been black." "Yes, but a few hundred are not all. How many ravens are there, anyway?"  "I would guess several million. As for your question, many other scientists, and non-scientists for that matter, have observed ravens over thousands of years and so far the birds have all been black. At least, I don't know of a single instance in which someone has produced a non-black raven." "That's true, but it's still not all - just most." "True, but there is other evidence. For example, take all these lovely multicolored birds we have seen today - the parrots, toucans, the peacocks -"  "They're lovely, but what do they have to do with your claim that all ravens are black?"  "Don't you see?" asked the ornithologist.  "No, I don't see. Please explain." "Well, you accept the idea that every new instance of another black raven that is observed adds to the support of the generalization that all ravens are black?"  "Yes, of course."  "Well then, the statement 'All ravens are black' is logically equivalent to the statement 'All non-black things are non-ravens.' This being so and because whatever confirms a statement also confirms any logically equivalent statement, it's clear that any non-black non-raven supports the generalization 'All ravens are black.' Hence, all these colorful, non-black non-ravens also support the generalization." "That's ridiculous," chided the young man. "In that case you might as well say that your blue jacket and gray pants also confirm the statement 'All ravens are black.' After all, they're also non-black non-ravens."  "That's correct," said the ornithologist. "Now you're beginning to think like a true scientist."  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. Two of Zeno's paradoxes Achilles and the tortoise Zeno's second paradox of motion, of Achilles and the tortoise, is probably the best known of his four paradoxes of motion. In this problem, the fleet Greek warrior runs a race against a slow-moving tortoise. Assume Achilles runs at ten times the speed of the tortoise (1 meter per second to 0.1 meter per second). The tortoise is given a 100-meter handicap in a race that is 1,000 meters. By the time Achilles reaches the tortoise's starting point T0, the tortoise will have moved on to point T1. Soon, Achilles will reach point T1, but by then the tortoise would have moved on to T2, and so on, ad infinitum. Every time Achilles reaches a point where the tortoise has just been, the tortoise has moved on a bit.  Although the distances between the two runners will diminish rapidly,  Achilles can never catch up with the tortoise, or so it would seem. Paradox of the arrow In the paradox of the arrow, Zeno asks us to consider an arrow in flight and argues that, in fact, the arrow must always be at rest. At each instant the arrow occupies a space equal to itself. Movement is impossible, because an instant by definition has no parts. If the arrow were capable of moving during an instant, we would contradict the definition of an instant, for the arrow would be in one position during the first part of the instant and in another position in the other part of the instant. The ship of Theseus The ship wherein Theseus and the youth of Athens returned had thirty oars, and was preserved by the Athenians down even to the time of Demetrius Phalereus, for they took away the old planks as they decayed, putting in new and stronger timber in their place, insomuch that this ship became a standing example among the philosophers, for the logical question of things that grow; one side holding that the ship remained the same, and the other contending that it was not the same. Protagoras's pupil Another paradox which has its foundation - real or legendary - in antiquity concerns the sophist Protagoras, who lived and taught in the fifth century BC. It is said that Protagoras made an arrangement with one of his pupils whereby the pupil was to pay for his instruction after he had won his first case. The young man completed his course, hung up the traditional shingle, and waited for clients. None appeared. Protagoras grew impatient and decided to sue his former pupil for the amount owed him. 'For,' argued Protagoras, 'either I win this suit, or you win it. If I win, you pay me according to the judgement of the court. If you win, you pay me according to our agreement. In either case I am bound to be paid.' 'Not so,' replied the young man. 'If I win, then by the judgement of the court I need not pay you. If you win, then by our agreement I need not pay you. In either case I am bound not to have to pay you.' Sorites paradox (paradox of the heap) Suppose you have a heap of sand. If you take away one grain of sand, what remains is still a heap: removing a single grain cannot turn a heap into something that is not a heap. If two collections of grains of sand differ in number by just one grain, then both or neither are heaps. This apparently obvious and uncontroversial supposition appears to lead to the paradoxical conclusion that all collections of grains of sand, even one-membered collections, are heaps. Hanging or beheading  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." [back to Writings] Digg This!  Stumble It!