Paradoxes are an interesting thing and they have existed since the time of the ancient Greeks. However, it is said 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 the flaws in thinking.
Of course, I will not pull the paradox, I would at least fully understand the essence of each. It is not always easy. Check it out ...
12. The Olbers Paradox
In astrophysics and physical cosmology, the Olbers paradox is the argument that the darkness of the night sky conflicts with the assumption of an infinite and eternal static Universe. This is one evidence of 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 says that from any angle from the ground the line of sight will end when it reaches the star. To understand this, we compare the paradox with finding a man in the forest among white trees. If from any point of view the line of sight ends at the treetops,does a person continue to see only white? This contradicts the darkness of the night sky and makes many people wonder why we do not see only the 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 restrict its actions, then that something that it cannot do. This, to all appearances, implies that the ability of an omnipotent being to limit oneself necessarily means that it does limit itself. This paradox is often formulated in the terminology of the Abrahamic religions, although this is not a mandatory requirement. One of the versions of the omnipotence paradox is the so-called stone paradox: can an omnipotent being create such a heavy stone that it will not even be able to lift it? If so, the creature ceases to be omnipotent, and if not, the creature was not omnipotent from the very beginning. The answer to the paradox is this: the presence of weakness,such as the impossibility of raising a heavy stone, does not fall under the category of omnipotence, although the definition of omnipotence implies the absence of weaknesses.
10. The Paradox of Soryt
The paradox is this: consider a heap of sand from which grains of sand are gradually removed. You can build a reasoning using the statements: - 1000000 grains of sand are a bunch of sand - a bunch of sand minus one grain of sand is still a bunch of sand. If we 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. One can argue with the first premise, saying that a million grains of sand is not a pile. 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. Thus, the answer must directly deny the existence of such things as a bunch. In addition, someone may object to the second premise, saying that it is not true for all “grain collections” and that removing one grain or grain of sand still leaves a bunch. Or it may declare that a pile of sand may consist of a single grain of sand.
9. The paradox of interesting numbers
Statement: not such a thing as an uninteresting natural number.Proof by contradiction: Suppose that you have a non-empty set of natural numbers that are uninteresting. Due to the properties of natural numbers, the list of uninteresting numbers is sure to be the smallest number. Being the smallest number in a set, it could be defined as interesting in this set of uninteresting numbers. But since initially all the numbers of the set were 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 paradox of a flying arrow
This paradox says that in order for a movement to occur, an object must change the position it occupies. An example is the movement of an arrow. At any time, the flying arrow remains motionless, because it is at rest, and since it rests at any time, it means that it is always motionless. That is, this paradox, advanced by Zeno as early as the 6th century, speaks of the absence of movement as such, based on the fact that the moving body must reach half,before you complete the move. But since it is motionless at every moment of time, it cannot reach half. 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 a turtle, after giving it a head start in 30 meters. If we assume that each of the runners began to run at a certain constant speed (one very quickly, the second very slowly), then after a while Achilles, having run 30 meters, will reach the point from which the tortoise moved. During this time, the turtle will run much less, say, 1 meter. Then it will take Achilles some more time to cover this distance, over which the tortoise will go even further. Reaching the third point, in which the tortoise visited, Achilles will advance further, but still will not catch up with her. Thus, whenever Achilles reaches the tortoise, it will still be ahead. Thus, since there are an infinite number of points that Achilles has to reach, and in which the tortoise has already been, he will never be able to catch up with the tortoise.Of course, logic tells us that Achilles can overtake the tortoise, because this is a paradox. The problem with this paradox is that in physical reality it is impossible to cross points indefinitely - how can you get from one point of infinity to another without intersecting the infinity of points? You cannot, that is, it is impossible. But in mathematics it is not. This paradox shows us how mathematics can prove something, but in reality it does not 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 ass
This is a figurative description of human indecision. This refers to a paradoxical situation where an ass, staying between two haystacks of exactly the same size and quality, will starve to death, since they 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 labors 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 a problem of choice, certainly had to choose more good, but Buridan admitted the possibility of slowing down the choice in order to assess all possible benefits. Later, other authors responded with satire to this point of view, speaking of the donkey, who, faced with two identical haystacks, would starve, making a decision.
5. The paradox of unexpected execution
The judge tells the convict that he will be hanged at noon on one of the working days next week, but the day of the execution will be a surprise for the prisoner. He will not know the exact date until the executioner at noon comes to his cell. After a bit of speculation, the offender concludes that he will be able to avoid execution. His reasoning can be divided into several parts. He begins with the fact that he can not hang on Friday, because if it is not hung on Thursday, then Friday will not be a surprise. Thus, Friday he ruled out.But then, since Friday was already removed from 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 also not be a surprise. Reasoning in a similar way, he consistently excluded all the remaining days of the week. Joyful, he goes to bed with the certainty that there will be no execution at all. The following week, at noon on Wednesday, an executioner came to his cell, so, despite all his arguments, he was extremely surprised. Everything that the judge said came true.
4. The paradox of the barber
Suppose that there is a city with one male hairdresser, and that every man in the city shaves bald: some independently, some with the help of a hairdresser. It seems reasonable to assume that the process is subject to the following rule: the barber shaves all men and only those who do not shave themselves. According to this scenario, we can ask the following question: does the hairdresser 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 follows from a statement in which Epimenides, contrary to the general conviction of Crete, suggested that Zeus was immortal, as in the following poem: They created a tomb for you, the highest holy Cretans, eternal liars, evil beasts, slaves of the stomach! But you did not die: you are alive and you will live forever, For you live in us, and we exist. However, he did not realize that by calling all Cretans a liar, he involuntarily and himself called a deceiver, although he “meant” 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 tell the truth. So, if all Cretans tell the truth, then he is including, and this means, based on his verse, that all Cretans are liars. Thus, the chain of reasoning returns to the beginning.
2. Paradox Evatla
This is a very old logic problem stemming from ancient Greece. It is said that the famous sophist Protagoras took Evatl’s teaching to him, and he clearly understood that the student would be able to pay the teacher only after he won his first case in court. Some experts claimthat Protagoras demanded tuition money immediately after Evatle finished his studies, others say that Protagoras waited some time until it became clear that the student was not making any effort to find clients, still others were convinced Evatle tried very hard, but he never found any clients. In any case, Protagoras decided to sue Evatla to return the debt. Protagoras argued that if he won the case, he would be paid his money. If Evatl won the case, Protagoras still had to receive their money in accordance with the original contract, because this would be Evatla's first winning case. Evatl, however, said that if he won, then by court decision he would not have to pay Protagoras. If, on the other hand, Protagoras wins, Evatl loses his first business, therefore he does not have to pay anything. So which man is right?
1. The paradox of force majeure
The paradox of force majeure is a classic paradox, formulated as “what happens when an irresistible force encounters a fixed object?” The paradox should be taken as a logical exercise, and not as a postulation of possible reality.According to modern scientific understanding, no force is completely irresistible, and there are no and cannot be completely immovable objects, since even a small force will cause a slight acceleration of an object of any mass. A fixed object must have infinite inertia, and, consequently, infinite mass. Such an object will be compressed by its own gravity. The irresistible force will require infinite energy that does not exist in the finite universe.