The paradox is an absurd truth based on seemingly correct premisses and inference but leads to an anti-intuitive and unacceptable conclusion. They have existed since ancient Greek times, but the credit of popularizing goes to modern logicians. By using logic one can usually find a fatal error in a paradox, proving that it is real or, on the contrary, absolute nonsense based on false assumptions. Confused? You should be.
The Paradox of Omnipotence
The idea of the paradox is that if there is a being that is capable of everything, it is also capable of limiting its capabilities, which in turn prevents doing something. So this being is not omnipotent because the term itself includes the idea of “being able to do everything”.
The most common version of this paradox is the “stone version”. “Can this creature create a stone so heavy that it cannot lift it by itself?”. If so, at that point, he stops be omnipotent, but if not, the being was not omnipotent from the beginning. The answer is that you still have a weakness, such as being unable to lift that stone, then this is not omnipotency, because the very definition of omnipotency excludes the possibility of weakness
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The Digital Paradox
There is no such concept as a boring (uninteresting, dull) natural number. Let’s say you have a set of numbers that are considered boring. By the order of numbers, there will always be some smallest number. This number, by its very nature, is defined as the most uninteresting, which makes it interesting. A number cannot be interesting and uninteresting at the same time. Thus, the set of numbers should be empty, as a result of which it is proved that there is no uninteresting natural number.
Surprise Hang-Man Paradox
The judge informs the sentenced person that he will be hanged on a working day next week. But the execution will be unexpected. When the convict starts to think about the judge’s words, he comes to the conclusion that he won’t get hanged.
It all starts with the convict’s handling the days with the exclusion method. It cannot be Friday, because if nothing happened all week, Friday is the last day and it is no longer a surprise. Continuing with this method is also can’t be Thursday because Friday is already off and the last day is Thursday. Thus, he also excludes Wednesday, Tuesday, and Monday for consideration. As a result, very relieved, he settles on his straw bag and, in full confidence that he will not get hanged, goes to sleep.
Next week, at noon on Wednesday, the hangman opens the door and the death penalty is executed, to the surprise of the convict. Everything the referee said has come true.
Although confronted with an irrefutable fact, a person is able to use his logical thinking to protect himself from something by using absolutely unrelated, logical conclusions. For the most part, it is just self-deception.
The Barber Paradox
Let’s imagine that there is a small town with only one barber. And all the men in the city always keep a smooth shaved face. Some of them go to the barbershop and the other part shave themselves. This creates the premise: The barber shaves all those who do not shave themselves.
But there are logical contradictions in this scenario: 1) If the barber doesn’t shave himself, he shaves himself. 2) If the barber shaves, according to the premise, he does not shave himself.
The Bootstrap Paradox
The Bootstrap Paradox is a paradox of time travel that questions how something that is taken from the future and placed in the past could ever come into being in the first place.
Imagine that a time traveler buys a copy of Hamlet from a bookstore, travels back in time to Elizabethan London, and hands the book to Shakespeare, who then copies it out and claims it as his own work. Over the centuries that follow, Hamlet is reprinted and reproduced countless times until finally, a copy of it ends up back in the same original bookstore, where the time traveler finds it, buys it, and takes it back to Shakespeare. Who, then, wrote Hamlet?
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Achilles and The Tortoise
The Paradox of Achilles and the Tortoise is number one of theoretical discussions of movement, put forward by the Greek philosopher Zeno of Elea in the 5th century BC. It begins with the great hero Achilles challenging a tortoise to a footrace. To keep things fair, he agrees to give the tortoise a head start of, say, 500m.
When the race begins, Achilles unsurprisingly starts running at a speed much faster than the tortoise, so that by the time he has reached the 500m mark, the tortoise has only walked 50m further than him. But by the time Achilles has reached the 550m mark, the tortoise has walked another 5m. And by the time he has reached the 555m mark, the tortoise has walked another 0.5m, then 0.25m, then 0.125m, and so on. This process continues again and again over an infinite series of smaller and smaller distances, with the tortoise always moving forwards while Achilles always plays catch up.
Logically, this seems to prove that Achilles can never overtake the tortoise—whenever he reaches somewhere the tortoise has been, he will always have some distance still left to go no matter how small it might be. Except, of course, we know intuitively that he can overtake the tortoise. The trick here is not to think of Zeno’s Achilles Paradox in terms of distances and races, but rather as an example of how any finite value can always be divided an infinite number of times, no matter how small its divisions might become.
The Boy or Girl Paradox
Imagine that a family has two children, one of whom we know to be a boy. What then is the probability that the other child is a boy? The obvious answer is to say that the probability is 1/2—after all, the other child can only be either a boy or a girl, and the chances of a baby being born a boy or a girl are (essentially) equal.
In a two-child family, however, there are actually four possible combinations of children: two boys (MM), two girls (FF), an older boy and a younger girl (MF), and an older girl and a younger boy (FM). We already know that one of the children is a boy, meaning we can eliminate the combination FF, but that leaves us with three equally possible combinations of children in which at least one is a boy—namely MM, MF, and FM. This means that the probability that the other child is a boy—MM—must be 1/3, not 1/2.
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The Card Paradox
Imagine you’re holding a postcard in your hand, on one side of which is written, “The statement on the other side of this card is true.” We’ll call that Statement A. Turn the card over, and the opposite side reads, “The statement on the other side of this card is false” (Statement B). Trying to assign any truth to either Statement A or B, however, leads to a paradox: if A is true then B must be as well, but for B to be true, A has to be false. Oppositely, if A is false then B must be false too, which must ultimately make A true.