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11 Brain-Twisting Paradoxes

Paradoxes have been around since the time of Ancient Greeks & the credit of popularizing them goes to recent logicians. Using logic you can usually find a fatal flaw in the paradox which shows why the seemingly impossible is either possible or the entire paradox is built on flawed thinking. Can you all work out the problems in each of the 11 paradoxes shown here? If you do, post your solutions or the fallacies in the comments.

11

The Omnipotence Paradox

The paradox states that if the being can perform such actions, then it can limit its own ability to perform actions and hence it cannot perform all actions, yet, on the other hand, if it cannot limit its own actions, then that is—straight off—something it cannot do. This seems to imply that an omnipotent being’s ability to limit itself necessarily means that it will, indeed, limit itself. This paradox is often formulated in terms of the God of the Abrahamic religions, though this is not a requirement. One version of the omnipotence paradox is the so-called paradox of the stone: “Could an omnipotent being create a stone so heavy that even that being could not lift it?” If so, then it seems that the being could cease to be omnipotent; if not, it seems that the being was not omnipotent to begin with. An answer to the paradox is that having a weakness, such as a stone he cannot lift, does not fall under omnipotence, since the definition of omnipotence implies having no weaknesses.

10

The Sorites’ Paradox

The paradox goes as follows: consider a heap of sand from which grains are individually removed. One might construct the argument, using premises, as follows:

1,000,000 grains of sand is a heap of sand. (Premise 1)
A heap of sand minus one grain is still a heap. (Premise 2)
Repeated applications of Premise 2 (each time starting with one less grain), eventually forces one to accept the conclusion that a heap may be composed of just one grain of sand.

On the face of it, there are some ways to avoid this conclusion. One may object to the first premise by denying 1,000,000 grains of sand makes a heap. But 1,000,000 is just an arbitrarily large number, and the argument will go through with any such number. So the response must deny outright that there are such things as heaps. Peter Unger defends this solution. Alternatively, one may object to the second premise by stating that it is not true for all collections of grains that removing one grain from it still makes a heap. Or one may accept the conclusion by insisting that a heap of sand can be composed of just one grain.

9

The Interesting number paradox

Claim: There is no such thing as an uninteresting natural number.

Proof by Contradiction: Assume that you have a non-empty set of natural numbers that are not interesting. Due to the well-ordered property of the natural numbers, there must be some smallest number in the set of not interesting numbers. Being the smallest number of a set one might consider not interesting makes that number interesting. Since the numbers in this set were defined as not interesting, we have reached a contradiction because this smallest number cannot be both interesting and uninteresting. Therefore the set of uninteresting numbers must be empty, proving there is no such thing as an uninteresting number.

8

The arrow paradox

In the arrow paradox, Zeno states that for motion to be occurring, an object must change the position which it occupies. He gives an example of an arrow in flight. He states that in any one instant of time, for the arrow to be moving it must either move to where it is, or it must move to where it is not. It cannot move to where it is not, because this is a single instant, and it cannot move to where it is because it is already there. In other words, in any instant of time there is no motion occurring, because an instant is a snapshot. Therefore, if it cannot move in a single instant it cannot move in any instant, making any motion impossible. This paradox is also known as the fletcher’s paradox—a fletcher being a maker of arrows.
Whereas the first two paradoxes presented divide space, this paradox starts by dividing time – and not into segments, but into points.

7

Achilles & the tortoise paradox

In the paradox of Achilles and the Tortoise, Achilles is in a footrace with the tortoise. Achilles allows the tortoise a head start of 100 feet. If we suppose that each racer starts running at some constant speed (one very fast and one very slow), then after some finite time, Achilles will have run 100 feet, bringing him to the tortoise’s starting point. During this time, the tortoise has run a much shorter distance, say, 10 feet. It will then take Achilles some further time to run that distance, by which time the tortoise will have advanced farther; and then more time still to reach this third point, while the tortoise moves ahead. Thus, whenever Achilles reaches somewhere the tortoise has been, he still has farther to go. Therefore, because there are an infinite number of points Achilles must reach where the tortoise has already been, he can never overtake the tortoise. Of course, simple experience tells us that Achilles will be able to overtake the tortoise, which is why this is a paradox.

[JFrater: I will point out the problem with this paradox to give you all an idea of how the others might be wrong: in physical reality it is impossible to transverse the infinite - how can you get from one point in infinity to another without crossing an infinity of points? You can't - thus it is impossible. But in mathematics it is not. This paradox shows us how mathematics may appear to prove something - but in reality, it fails. So the problem with this paradox is that it is applying mathematical rules to a non-mathematical situation. This makes it invalid.]

6

The Buridan’s ass paradox

This is a figurative description of a man of indecision. It refers to a paradoxical situation wherein an ass, placed exactly in the middle between two stacks of hay of equal size and quality, will starve to death since it cannot make any rational decision to start eating one rather than the other. The paradox is named after the 14th century French philosopher Jean Buridan. The paradox was not originated by Buridan himself. It is first found in Aristotle’s De Caelo, where Aristotle mentions an example of a man who remains unmoved because he is as hungry as he is thirsty and is positioned exactly between food and drink. Later writers satirised this view in terms of an ass who, confronted by two equally desirable and accessible bales of hay, must necessarily starve while pondering a decision.

5

The unexpected hanging paradox

A judge tells a condemned prisoner that he will be hanged at noon on one weekday in the following week, but that the execution will be a surprise to the prisoner. He will not know the day of the hanging until the executioner knocks on his cell door at noon that day. Having reflected on his sentence, the prisoner draws the conclusion that he will escape from the hanging. His reasoning is in several parts. He begins by concluding that the “surprise hanging” can’t be on a Friday, as if he hasn’t been hanged by Thursday, there is only one day left – and so it won’t be a surprise if he’s hanged on a Friday. Since the judge’s sentence stipulated that the hanging would be a surprise to him, he concludes it cannot occur on Friday. He then reasons that the surprise hanging cannot be on Thursday either, because Friday has already been eliminated and if he hasn’t been hanged by Wednesday night, the hanging must occur on Thursday, making a Thursday hanging not a surprise either. By similar reasoning he concludes that the hanging can also not occur on Wednesday, Tuesday or Monday. Joyfully he retires to his cell confident that the hanging will not occur at all. The next week, the executioner knocks on the prisoner’s door at noon on Wednesday — which, despite all the above, will still be an utter surprise to him. Everything the judge said has come true.

4

The barber’s Paradox

Suppose there is a town with just one male barber; and that every man in the town keeps himself clean-shaven: some by shaving themselves, some by attending the barber. It seems reasonable to imagine that the barber obeys the following rule: He shaves all and only those men in town who do not shave themselves.

Under this scenario, we can ask the following question: Does the barber shave himself?
Asking this, however, we discover that the situation presented is in fact impossible:

- If the barber does not shave himself, he must abide by the rule and shave himself.
- If he does shave himself, according to the rule he will not shave himself

3

Epimenides’ Paradox

This paradox arises from the statement in which Epimenides, against the general sentiment of Crete, proposed that Zeus was immortal, as in the following poem:

They fashioned a tomb for thee, O holy and high one
The Cretans, always liars, evil beasts, idle bellies!
But thou art not dead: thou livest and abidest forever,
For in thee we live and move and have our being.

He was, however, unaware that, by calling all Cretens liars, he had, unintentionally, called himself one, even though what he ‘meant’ was all Cretens except himself. Thus arises the paradox that if all Cretens are liars, he is also one, & if he is a liar, then all Cretens are truthful. So, if all Cretens are truthful, then he himself is speaking the truth & if he is speaking the truth, all Cretens are liars. Thus continues the infinite regression.

2

The paradox of the court

The Paradox of the Court is a very old problem in logic stemming from ancient Greece. It is said that the famous sophist Protagoras took on a pupil, Euathlus, on the understanding that the student pay Protagoras for his instruction after he had won his first case (in some versions: if and only if Euathlus wins his first court case). Some accounts claim that Protagoras demanded his money as soon as Euathlus completed his education; others say that Protagoras waited until it was obvious that Euathlus was making no effort to take on clients and still others assert that Euathlus made a genuine attempt but that no clients ever came. In any case, Protagoras decided to sue Euathlus for the amount owed.
Protagoras argued that if he won the case he would be paid his money. If Euathlus won the case, Protagoras would still be paid according to the original contract, because Euathlus would have won his first case.

Euathlus, however, claimed that if he won then by the court’s decision he would not have to pay Protagoras. If on the other hand Protagoras won then Euathlus would still not have won a case and therefore not be obliged to pay. The question is: which of the two men is in the right?

1

The unstoppable force paradox

The Irresistible force paradox, also the unstoppable force paradox, is a classic paradox formulated as “What happens when an irresistible force meets an immovable object?” The paradox should be understood as an exercise in logic, not as the postulation of a possible reality. According to modern scientific understanding, no force is completely irresistible, and there are no immovable objects and cannot be any, as even a minuscule force will cause a slight acceleration on an object of any mass. An immovable object would have to have an inertia that was infinite and therefore infinite mass. Such an object would collapse under its own gravity and create a singularity. An unstoppable force would require infinite energy, which does not exist in a finite universe.

Bonus

Olbers’ Paradox

In astrophysics and physical cosmology, Olbers’ paradox is the argument that the darkness of the night sky conflicts with the assumption of an infinite and eternal static universe. It is one of the pieces of evidence for a non-static universe such as the current Big Bang model. The argument is also referred to as the “dark night sky paradox” The paradox states that at any angle from the earth the sight line will end at the surface of a star. To understand this we compare it to standing in a forest of white trees. If at any point the vision of the observer ended at the surface of a tree, wouldn’t the observer only see white? This contradicts the darkness of the night sky and leads many to wonder why we do not see only light from stars in the night sky.

Text is available under the Creative Commons Attribution-ShareAlike License; additional terms may apply. Text is derived from Wikipedia.

10 places you don't want to visit

10

Great Pacific Garbage Patch

Pacific Ocean

The Great Pacific Garbage Patch, also described as the Pacific Trash Vortex, is a gyre of marine litter in the central North Pacific Ocean located roughly between 135° to 155°W and 35° to 42°N. Most current estimates state that it is larger than the U.S. state of Texas, with some estimates claiming that it is larger than the continental United States, however the exact size is not known for sure. The Patch is characterized by exceptionally high concentrations of pelagic plastics, chemical sludge, and other debris that have been trapped by the currents of the North Pacific Gyre. The patch is not easily visible because it consists of very small pieces, almost invisible to the naked eye, most of its contents are suspended beneath the surface of the ocean. This is not a place the average Joe would want to visit.

9

Izu Islands

Japan

The Izu Islands are a group of volcanic islands stretching south and east from the Izu Peninsula of Honshū, Japan. Administratively, they form two towns and six villages; all part of Tokyo. The largest is Izu Ōshima, usually called simply Ōshima. Because of their volcanic nature, the islands are constantly filled with the stench of sulfur (extremely similar to the smell of thousands of farts). Residents were evacuated from the islands in 1953 and 2000 due to volcanic activity and dangerously high levels of gas. The people returned in 2005 but are now required to carry gas masks with them at all times in case gas levels rise unexpectedly.

8

The Door to Hell

Turkmenistan

Address: Derweze, Turkmenistan

This has featured on listverse before, but it would be remiss of us to exclude it from this list. While drilling in Derweze in Turkmenistan in 1971, geologists accidentally found an underground cavern filled with natural gas. The ground beneath the drilling rig collapsed, leaving a large hole with a diameter of about 50-100 meters. To avoid poisonous gas discharge, scientists decided to set fire to the hole. Geologists had hoped the fire would go out in a few days but it has been burning ever since. Locals have named the cavern The Door to Hell. As you can see from the picture above, it is one hell of an amazing place, but certainly one you wouldn’t want to visit.

7

Alnwick Poison Gardens

England

Address: Denwick Lane, Alnwick, NE66 1YU, England

Inspired by the Botanical Gardens in Padua, Italy (the first botanical garden which was created to grow medicinal and poisonous plants in the 1500s), the Alnwick Poison Garden is a garden devoted entirely to plants that can kill. It features many plants grown unwittingly in back gardens, and those that grow in the British countryside, as well as many more unusual varieties. Flame-shaped beds contain belladonna, tobacco and mandrake. The Alnwick Garden has a Home Office license to grow some very special plants; namely, cannabis and coca which are found behind bars in giant cages – for obvious reasons.

6

Asbestos Mine

Canada

Address: Thetford-Mines, Quebec, Canada

Asbestos is a set of six naturally occurring silicate minerals highly prized for their resistance to fire and sound absorption abilities. On the downside, exposure to this stuff causes cancer and a variety of other diseases. It is so dangerous that the European Union has banned all mining and use of asbestos in Europe. But, for those curious enough to want to get close to the stuff, all is not lost. In Canada at the Thetford Mines, you can visit an enormous open pit asbestos mine which is still fully operational. The workers in the mines aren’t required to wear any sort of respiratory protection, and in some sections of the nearby town, residential areas are butted right next up against piles of asbestos waste. The mine offers bus tours of the deadly environment during the summer months. Tickets are free (would you expect it to be any other way?). If you decide to visit, don’t forget your full body bio-hazard suit.

5

Ramree Island

Burma

Ramree Island in Burma is a huge swamp home to 1000s of salt water enormous salt water crocodiles, the deadliest in the world. It is also home to malaria carrying mosquitos, and venomous scorpions. During the Second World War, the island was the site of a six week battle in the Burma campaign. Here is a description of one of those horrifying nights: “That night [of the 19 February 1945] was the most horrible that any member of the M.L. [motor launch] crews ever experienced. The scattered rifle shots in the pitch black swamp punctured by the screams of wounded men crushed in the jaws of huge reptiles, and the blurred worrying sound of spinning crocodiles made a cacophony of hell that has rarely been duplicated on earth. At dawn the vultures arrived to clean up what the crocodiles had left…Of about 1,000 Japanese soldiers that entered the swamps of Ramree, only about 20 were found alive.”

4

Yungas Road

Bolivia

The North Yungas Road (Road of Death or Death Road) is a 61 kilometres (38 mi) or 69 kilometres (43 mi) road leading from La Paz to Coroico, 56 kilometres (35 mi) northeast of La Paz in the Yungas region of Bolivia. It is legendary for its extreme danger with estimates stating that 200 to 300 travelers are killed yearly along it. The road includes crosses marking many of the spots where vehicles have fallen. The road was built in the 1930s during the Chaco War by Paraguayan prisoners. It is one of the few routes that connects the Amazon rainforest region of northern Bolivia, or Yungas, to its capital city. Because of the extreme dropoffs of at least 600 metres (2,000 ft), single-lane width – most of the road no wider than 3.2 metres (10 ft) and lack of guard rails, the road is extremely dangerous. Further still, rain, fog and dust can make visibility precarious. In many places the road surface is muddy, and can loosen rocks from the road.

3

Mud Volcanoes of Azerbaijan

Azerbaijan

In the Spring of 2001, volcanic activity under the Caspian Sea off the Azeri coast created a whole new island. In October 2001 there was an impressive volcanic eruption in Azerbaijan at Lokbatan, but there were no casualties or evacuation warnings. But Azerbaijan does not have a single active volcano, at least not in the usual sense of the word. What Azerbaijan does have is mud volcanoes – hundreds of them. Mud volcanoes are the little-known relatives of the more common magmatic variety. They do erupt occasionally with spectacular results, but are generally not considered to be dangerous – unless you happen to be there at the wrong time: every twenty years or so, a mud volcano explodes with great force, shooting flames hundreds of meters into the sky, and depositing tonnes of mud on the surrounding area. In one eruption, the flames could easily be seen from 15 kilometers away on the day of the explosion, and were still burning, although at a lower level, three days later.

2

The Zone of Alienation

Eastern Europe

The Zone of Alienation is the 30 km/19 mi exclusion zone around the site of the Chernobyl nuclear reactor disaster and is administrated by a special administration under the Ukrainian Ministry of Extraordinary Situations (Emergencies). Thousands of residents refused to be evacuated from the zone or illegally returned there later. Over the decades this primarily elderly population has dwindled, falling below 400 in 2009. Approximately half of these resettlers live in the town of Chernobyl; others are spread in villages across the zone. After recurrent attempts at expulsion, the authorities became reconciled to their presence and even allowed limited supporting services for them. Because of looting, there is a strong police presence – so be warned, if you visit, you may either be shot or get radiation poisoning – and we all know how awful that can be.

1

Ilha de Queimada Grande

Brazil

Off the shore of Brazil, almost due south of the heart of São Paulo, is a Ilha de Queimada Grande (Snake Island). The island is untouched by human developers, and for very good reason. Researchers estimate that on the island live between one and five snakes per square meter. That figure might not be so terrible if the snakes were, say, 2 inches long and nonvenomous. The snakes on Queimada Grande, however, are a unique species of pit viper, the golden lancehead. The lancehead genus of snakes is responsible for 90% of Brazilian snakebite-related fatalities. The golden lanceheads that occupy Snake Island grow to well over half a meter long, and they possess a powerful fast-acting poison that melts the flesh around their bites. This place is so dangerous that a permit is required to visit.