11 Mind-Bending Paradoxes

Paradoxes have existed since the days of Ancient Greece, with modern logicians credited for their widespread popularity. By applying logic, one can typically identify a critical flaw in the paradox, explaining why the seemingly impossible is either achievable or built on flawed reasoning. Can you solve the issues presented in the 11 paradoxes here? If so, share your solutions or highlight the fallacies in the comments.

11. The Paradox of Omnipotence

This paradox suggests that if a being possesses the power to perform any action, it must also have the ability to limit its own actions, which would then imply it cannot perform all actions. However, if it cannot limit its own actions, that becomes something it cannot do. This creates the paradox that an omnipotent being's ability to limit itself means it will ultimately limit itself. The paradox is often discussed in relation to the God of the Abrahamic religions, though it is not exclusive to that context. A common version is the paradox of the stone: 'Could an omnipotent being create a stone so heavy that even it cannot lift it?' If the answer is yes, the being seems no longer omnipotent; if no, it seems the being was never omnipotent in the first place. One solution to the paradox is that a weakness, like a stone the being cannot lift, does not fall under omnipotence, as omnipotence implies the absence of weaknesses.

10. The Paradox of Sorites

The paradox is as follows: imagine a heap of sand from which individual grains are removed. One could construct an argument using premises as follows:

1,000,000 grains of sand make up a heap of sand. (Premise 1) A heap of sand minus one grain is still considered a heap. (Premise 2) Repeatedly applying Premise 2 (each time removing one grain) eventually leads to the conclusion that a heap could consist of just a single grain of sand.

At first glance, there seem to be ways to avoid this conclusion. One might challenge the first premise by rejecting that 1,000,000 grains of sand constitute a heap. However, 1,000,000 is simply an arbitrarily large number, and the argument will hold for any such number. Therefore, the response must entirely deny the existence of heaps. Peter Unger supports this resolution. Alternatively, one might object to the second premise by asserting that not every collection of grains will still form a heap if one grain is removed. Or, one could accept the conclusion by arguing that a heap can indeed consist of just a single grain of sand.

9. The Paradox of the Interesting Number

Claim: No natural number is uninteresting.

Proof by Contradiction: Suppose you have a set of natural numbers that are all not interesting. Due to the well-ordered nature of natural numbers, there must be a smallest number in this set. But being the smallest number in a set of non-interesting numbers makes it interesting. Since the set was defined as containing only non-interesting numbers, this creates a contradiction, as this smallest number cannot simultaneously be interesting and uninteresting. Thus, the set of uninteresting numbers must be empty, proving that no natural number is uninteresting.

8. The Paradox of the Arrow

In the arrow paradox, Zeno argues that for motion to occur, an object must change the position it occupies. He uses the example of an arrow in flight. He states that at any single moment in time, for the arrow to be moving, it must either move to the position where it currently is, or to a position where it is not. It cannot move to where it is not, because that is just one moment, and it cannot move to where it is, as it is already there. In other words, no motion occurs at a single instant because that moment is static. Thus, if the arrow cannot move in a single instant, it cannot move at all, making any motion impossible. This paradox is also known as the Fletcher's paradox, with a fletcher being a maker of arrows. While the first two paradoxes dealt with dividing space, this one divides time—not into intervals, but into discrete points.

7. The Paradox of Achilles and the Tortoise

In the paradox of Achilles and the Tortoise, Achilles races against the tortoise, who is given a 100-foot head start. Assuming both start running at a constant speed, with Achilles being much faster, after some time Achilles will have covered the 100 feet, reaching the tortoise’s starting point. However, during this time, the tortoise has only moved a much shorter distance, say 10 feet. It will take Achilles additional time to cover that 10 feet, during which the tortoise moves further ahead. Every time Achilles reaches a point where the tortoise has been, the tortoise moves farther, so it seems like Achilles can never overtake the tortoise. The paradox arises from the fact that there are infinitely many points Achilles must pass before overtaking the tortoise. Despite this, experience shows us that Achilles will indeed catch up, making this a paradox.

[JFrater: To highlight the issue with this paradox and give insight into the flaws of similar ones: in reality, it’s impossible to traverse infinity – how can one move from one point in infinity to another without crossing an infinite number of points? You can’t, and that’s why it's impossible. But in mathematics, this is not the case. This paradox demonstrates how mathematics may suggest a conclusion that is actually false in the physical world. The problem here is that mathematical concepts are being applied to a non-mathematical situation, rendering the paradox invalid.]

6. The Paradox of Buridan's Ass

This is a symbolic illustration of indecision. It describes a situation where a donkey, placed exactly midway between two equally sized and good-quality haystacks, is unable to choose between them and thus starves, as it cannot make up its mind about which to eat. The paradox is named after the 14th-century French philosopher Jean Buridan, although he did not create it. It actually traces back to Aristotle’s *De Caelo*, where he describes a man frozen in indecision because he is equally hungry and thirsty, caught between food and drink. This notion was later parodied by thinkers who imagined an ass unable to decide between two equal and tempting bales of hay, ultimately starving while stuck in contemplation.

5. The Paradox of the Unexpected Hanging

A judge tells a condemned prisoner that he will be hanged at noon on a weekday during the following week, but that the hanging will be a surprise to him. The prisoner will not know the day of his execution until the executioner knocks on his cell door at noon on the chosen day. After considering his situation, the prisoner concludes that the hanging cannot happen. He begins by reasoning that the hanging cannot be on Friday, because if he hasn’t been hanged by Thursday, then Friday would be the only remaining option, and thus it would no longer be a surprise. Since the judge’s statement requires the hanging to be a surprise, he rules out Friday. He then applies the same logic to Thursday, reasoning that if he isn’t hanged by Wednesday, Thursday becomes the only option, eliminating the surprise again. Following similar logic, he eliminates Wednesday, Tuesday, and Monday. Confident that the hanging will not occur at all, he rests in his cell. However, the next week, at noon on Wednesday, the executioner knocks on his door, and despite all his reasoning, the hanging is still an unexpected surprise. Everything the judge predicted has come true.

4. The Barber's Paradox

Imagine a town with only one male barber, and every man in the town keeps himself clean-shaven, either by shaving himself or by going to the barber. It appears logical to assume that the barber follows a simple rule: he shaves all the men who do not shave themselves, and only those men.

In this scenario, a curious question arises: Does the barber shave himself? However, upon considering this, we realize that the situation is paradoxical.

- If the barber does not shave himself, he must follow the rule and shave himself. - If the barber shaves himself, according to the rule, he should not shave himself.

3. The Paradox of the Unstoppable Force

The Irresistible Force Paradox, also known as the Unstoppable Force Paradox, is a classic logical dilemma phrased as: "What happens when an irresistible force meets an immovable object?" This paradox serves as a thought experiment rather than a claim about reality. According to modern science, no force is truly irresistible, and immovable objects do not exist. Even the smallest force can cause any object to accelerate. An immovable object would have to possess infinite mass and inertia, causing it to collapse under its own gravity and form a singularity. An unstoppable force would require infinite energy, which is not possible in a finite universe.

2. The Paradox of the Court

The Paradox of the Court is an ancient logical problem from Greece. It is said that the renowned sophist Protagoras took on a pupil named Euathlus with the understanding that the student would pay Protagoras for his teachings after winning his first case (in some versions: only if Euathlus wins his first case). Some versions suggest Protagoras demanded payment immediately after Euathlus finished his education, while others claim Protagoras waited, observing that Euathlus made no effort to seek clients. Some versions also claim that although Euathlus made an effort, no clients ever came. Eventually, Protagoras decided to sue Euathlus for the amount owed. Protagoras argued that if he won the case, he would be paid. If Euathlus won the case, Protagoras would still be paid as per the contract, because Euathlus would have won his first case.

Euathlus, however, argued that if he were to win the case, the court's ruling would absolve him from any obligation to pay Protagoras. On the other hand, if Protagoras were victorious, Euathlus still would not have won his first case, thus avoiding the need to pay. The question arises: who among the two is correct?

1. Epimenides’ Paradox

This paradox stems from the statement made by Epimenides, who, contrary to the common belief in Crete, declared that Zeus was immortal, as expressed in the following verse:

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 unaware that by declaring all Cretans as liars, he had inadvertently labeled himself as one as well, even though he intended to exclude himself. This creates a paradox: if all Cretans are liars, then he must also be a liar. But if he is a liar, it implies that all Cretans are truthful. If all Cretans are truthful, then he is telling the truth, and if he is telling the truth, then all Cretans are liars. This leads to an endless cycle of contradictions.

+ Olbers’ Paradox

In the realms of astrophysics and physical cosmology, Olbers’ paradox challenges the idea of a static, infinite universe. Also known as the 'dark night sky paradox,' it questions why the night sky remains dark if the universe is infinite and eternal. According to the paradox, from any point on Earth, the line of sight should eventually reach the surface of a star. To illustrate, imagine standing in a forest of white trees—if your line of sight always ended at the surface of a tree, you would only see white. This contradicts the darkness of the night sky and raises the question of why we don’t see only starlight.

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