Paradoxes are an interesting thing and have existed since the time of the ancient Greeks. However, they say that with the help of logic, one can quickly find a fatal flaw in the paradox, which shows why the seemingly impossible is possible or that the whole paradox is simply built on flaws in thinking.
Of course, I won't be able to refute the paradox, at least I would at least fully understand the essence of each. It's not always easy. Check it out …
12. Olbers paradox
In astrophysics and physical cosmology, Olbers' paradox is an argument that the darkness of the night sky conflicts with the assumption of an infinite and eternal static universe. This is one piece of evidence for a non-static universe, such as the current Big Bang model. This argument is often referred to as the “dark paradox of the night sky”, which states that from any angle from the ground, the line of sight will end when it reaches the star. To understand this, we will compare the paradox with finding a person in a forest among white trees. If, from any point of view, the line of sight ends at the treetops, does one still see only white? This belies the darkness of the night sky and leaves many people wondering why we don't only see light from the stars in the night sky.
11. The paradox of omnipotence
The paradox is that if a creature can perform any actions, then it can limit its ability to perform them, therefore, it cannot perform all actions, but, on the other hand, if it cannot limit its actions, then this is something that it cannot do. This seems to imply that the ability of an omnipotent being to limit itself necessarily means that it does indeed limit itself. This paradox is often expressed in the terminology of the Abrahamic religions, although this is not a requirement. One of the versions of the paradox of omnipotence is the so-called paradox about the stone: can an omnipotent being create such a heavy stone that even it will not be able to lift it? If this is so, then the being ceases to be omnipotent, and if not,that being was not omnipotent to begin with. The answer to the paradox is that the presence of a weakness, such as being unable to lift a heavy stone, does not fall under the category of omnipotence, although the definition of omnipotence implies the absence of weakness.
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10. Sorit's paradox
The paradox is this: consider a pile of sand, from which grains of sand are gradually removed. One can construct a reasoning using statements: - 1,000,000 grains of sand is a pile of sand - a pile of sand minus one grain of sand is still a pile of sand. If you continue the second action without stopping, then, ultimately, this will lead to the fact that the heap will consist of one grain of sand. At first glance, there are several ways to avoid this conclusion. You can counter the first premise by saying that a million grains of sand are not a heap. But instead of 1,000,000, there can be an arbitrarily large number, and the second statement will be true for any number with any number of zeros. So the answer is to outright deny the existence of things like a heap. In addition, one might object to the second premise by stating,that it is not true for all “grain collections” and that removing one grain or grain of sand still leaves a heap in a heap. Or it may declare that a pile of sand may consist of a single grain of sand.
9. The interesting numbers paradox
Statement: not such a thing as an uninteresting natural number. Proof by contradiction: suppose you have a non-empty set of natural numbers that are not interesting. Due to the properties of natural numbers, the list of uninteresting numbers will necessarily have the smallest number. Being the smallest number of a set, it could be defined as interesting in this set of uninteresting numbers. But since all the numbers in the set were initially defined as uninteresting, we came to a contradiction, since the smallest number cannot be both interesting and uninteresting. Therefore, the sets of uninteresting numbers must be empty, proving that there is no such thing as uninteresting numbers.
8. The flying arrow paradox
This paradox suggests that in order for movement to occur, the object must change the position it occupies. An example is the movement of an arrow. At any moment of time, a flying arrow remains motionless, because it is at rest, and since it is at rest at any time, it means that it is always motionless. That is, this paradox, put forward by Zeno back in the 6th century, speaks of the absence of movement as such, based on the fact that a moving body must reach halfway before completing the movement. But since it is motionless at every moment of time, it cannot reach half of it. This paradox is also known as the Fletcher paradox. It is worth noting that if the previous paradoxes spoke about space, then the next paradox is about dividing time not into segments, but into points.
7. The paradox of Achilles and the tortoise
In this paradox, Achilles runs after the turtle, having previously given it a head start of 30 meters. If we assume that each of the runners began to run at a certain constant speed (one very fast, the other very slowly), then after a while Achilles, having run 30 meters, will reach the point from which the turtle moved. During this time the turtle will “run” much less, say, 1 meter. Then Achilles will need some more time to cover this distance, for which the turtle will move even further. Having reached the third point, which the turtle visited, Achilles will advance further, but still will not catch up with it. This way, whenever Achilles reaches the turtle, it will still be ahead. Thus, since there are an infinite number of points that Achilles must reach, and which the turtle has already visited,he can never catch up with the turtle. Of course, logic tells us that Achilles can catch up with the turtle, which is why this is a paradox. The problem with this paradox is that in physical reality it is impossible to endlessly cross points - how can you get from one point of infinity to another without crossing the infinity of points? You cannot, that is, it is impossible. But in mathematics this is not the case. This paradox shows us how mathematics can prove something, but it doesn't really work. Thus, the problem of this paradox is that the application of mathematical rules for non-mathematical situations occurs, which makes it inoperative. The problem with this paradox is that in physical reality it is impossible to endlessly cross points across - how can you get from one point of infinity to another without crossing the infinity of points? You cannot, that is, it is impossible. But in mathematics this is not the case. This paradox shows us how mathematics can prove something, but it doesn't really work. Thus, the problem of this paradox is that the application of mathematical rules for non-mathematical situations occurs, which makes it inoperative. The problem with this paradox is that in physical reality it is impossible to endlessly cross points across - how can you get from one point of infinity to another without crossing the infinity of points? You cannot, that is, it is impossible. But in mathematics this is not the case. This paradox shows us how mathematics can prove something, but it doesn't really work. Thus, the problem of this paradox is that the application of mathematical rules for non-mathematical situations occurs, which makes it inoperative. This paradox shows us how mathematics can prove something, but it doesn't really work. Thus, the problem of this paradox is that the application of mathematical rules for non-mathematical situations occurs, which makes it inoperative. This paradox shows us how mathematics can prove something, but it doesn't really work. Thus, the problem of this paradox is that the application of mathematical rules for non-mathematical situations occurs, which makes it inoperative.
6. The paradox of Buridan's donkey
This is a figurative description of human indecision. This refers to the paradoxical situation when a donkey, being between two absolutely identical in size and quality haystacks, will starve to death, since it will not be able to make a rational decision and start eating. The paradox is named after the 14th century French philosopher Jean Buridan, however, he was not the author of the paradox. He has been known since the time of Aristotle, who, in one of his works, talks about a man who was hungry and thirsty, but since both feelings were equally strong, and the man was between eating and drinking, he could not make a choice. Buridan, in turn, never spoke about this problem, but raised questions about moral determinism, which implied that a person, faced with the problem of choice, of course,should choose in the direction of the greater good, but Buridan allowed the possibility of slowing down the choice in order to assess all the possible advantages. Later, other writers satirized this view, referring to a donkey facing two identical haystacks and starving to make a decision.
5. The surprise execution paradox
The judge tells the convict that he will be hanged at noon on one of the working days next week, but the day of execution will be a surprise for the prisoner. He will not know the exact date until the executioner comes to his cell at noon. After a little reasoning, the offender comes to the conclusion that he can avoid execution. His reasoning can be divided into several parts. He begins by saying that he cannot be hanged on Friday, since if he is not hanged on Thursday, then Friday will no longer be a surprise. Thus, he ruled out Friday. But then, since Friday had already been struck off the list, he came to the conclusion that he could not be hanged on Thursday, because if he was not hanged on Wednesday, then Thursday would not be a surprise either. Reasoning in a similar way, he consistently eliminated all the remaining days of the week. Joyful, he goes to bed with the certainty that the execution will not happen at all. The executioner came to his cell at noon Wednesday the following week, so, despite all his reasoning, he was extremely surprised. Everything the judge said came true.
4. The hairdresser's paradox
Suppose there is a city with one male hairdresser, and that every man in the city shaves his head, some on his own, some with the help of a hairdresser. It seems reasonable to assume that the process obeys the following rule: the hairdresser shaves all men and only those who do not shave themselves. In this scenario, we can ask the following question: Does the barber shave himself? However, asking this, we understand that it is impossible to answer it correctly: - if the hairdresser does not shave himself, he must follow the rules and shave himself; - if he shaves himself, then according to the same rules he should not shave himself.
3. The Epimenides paradox
This paradox stems from a statement in which Epimenides, contrary to the general belief of Crete, suggested that Zeus was immortal, as in the following poem: They created a tomb for you, High Holy Cretans, eternal liars, evil beasts, slaves of the belly! But you are not dead: you are alive and you will always be alive, For you live in us, and we exist. However, he did not realize that by calling all Cretans liars, he involuntarily called himself a deceiver, although he "implied" that all Cretans, except him. Thus, if you believe his statement, and all Cretans are liars in fact, he is also a liar, and if he is a liar, then all Cretans are telling the truth. So, if all Cretans speak the truth, then he is included, which means, based on his verse, that all Cretans are liars. So the line of reasoning goes back to the beginning.
2. The Evatla paradox
This is a very old problem in logic, stemming from Ancient Greece. It is said that the famous sophist Protagoras took Evattla to his teachings, while he clearly understood that the student could pay the teacher only after he won his first case in court. Some experts argue that Protagoras demanded money for tuition immediately after Evatl finished his studies, others say that Protagoras waited for a while until it became obvious that the student was not making any effort to find clients, still others we are sure that Evatl tried very hard, but he never found clients. In any case, Protagoras decided to sue Evatl to repay the debt. Protagoras argued that if he won the case, he would be paid his money. If Evattl won the case,then Protagoras still had to receive his money in accordance with the original agreement, because this would be Evatl's first winning deal. Evatl, however, insisted that if he won, then by court order he would not have to pay Protagoras. If, on the other hand, Protagoras wins, then Evatl loses his first case, and therefore does not have to pay anything. So which man is right?
1. The paradox of force majeure
The Force Majeure Paradox is a classic paradox formulated as "what happens when an irresistible force meets a stationary object?" The paradox should be seen as a logical exercise, not as a postulation of a possible reality. According to modern scientific understanding, no force is completely irresistible, and there is and cannot be completely immovable objects, since even a slight force will cause a slight acceleration of an object of any mass. An immovable object must have infinite inertia, and, therefore, infinite mass. Such an object will be compressed by its own gravity. An irresistible force will require infinite energy that does not exist in a finite universe.