What are some famous paradoxes?

Zeno's Paradox:

Achilles is the fastest man in ancient Greek mythology, but if there is a tortoise in front of him (crawling forward from point A) , he can never catch up with the turtle. The reason is as follows: if he wants to catch up with the turtle, he must pass through the place A where the turtle started, but when he catches up to this place, the turtle crawls forward some distance and reaches point B. , he had to pass through point B to catch up with the tortoise, but when he caught up with point B, the tortoise climbed to point C... So Achilles could never catch up with the tortoise! ​​

One day Achilles and the Tortoise had a race. Because Achilles thought he was faster than the Tortoise, he asked the Tortoise to run a shorter distance. Their agreement is that Achilles will start at a certain point d 1, and the tortoise will start from a place d 2 closer to the end. But just think about it, when Achilles ran to d 2, the tortoise would run to another place d 3. When Achilles caught up to d 3, the tortoise had already reached d 4. By analogy, every time Achilles ran to the place where the tortoise had been before, the tortoise had already run some distance further. From this point of view, how can Achilles catch up with the tortoise?

Dune Paradox

Sand grains pile up together, and a small amount becomes a large number, forming a sand dune. For example, one hundred thousand grains of sand piled together form a sand dune. The sand dunes are so big that if you take away just one grain of sand, the dunes will still exist, because a grain of sand is really insignificant. Likewise, if you take away one grain of sand from a dune consisting of ninety-nine thousand nine hundred and ninety-nine grains of sand, the dune will not disappear. In summary, take away a grain of sand from a dune and the dune will continue to exist. But if this is the case, continue to remove the sand grains one by one until the last grain of sand is left, and the dune will continue to exist. But how can a grain of sand form a dune?

The paradox of not claiming to be true

If a predicate cannot be applied to itself, we call it "not claiming to be true". On the contrary, we call it "self-proclaimed". For example, the predicate "composed of Chinese characters" is composed of Chinese characters, so it is a self-proclaimed predicate. "It's a red fruit" can only describe the fruit, not yourself, so I don't call myself one.

So "it does not claim to be" itself does not mean that it does not claim to itself? If so, it should not be applied to oneself, that is, it should be applied to oneself. But if not, it applies to oneself, which means that it should not apply to oneself. In other words, if it applies to yourself, it doesn’t apply to yourself!

Lawyer and Apprentice

Student A is the apprentice of a barrister. When he was still in training, he promised his teacher that he would pay his tuition after he completed his training and won his first lawsuit. However, after graduation, Student A never took up any lawsuit, so the teacher decided to sue him for defaulting on tuition fees.

The teacher’s argument is that if the teacher wins the lawsuit, student A must pay the tuition immediately; if student A wins, student A should pay the tuition according to the original agreement. So students should pay tuition fees anyway.

But A’s argument is that if the court rules that he wins, he does not need to pay tuition; if the teacher wins, he himself has never won, so according to the agreement, he does not need to pay tuition fee.

Whose argument is reasonable?

Liar

Someone said: "What I am saying now is false."

So, what exactly is this person saying? True or false? If what he says is true, then he is telling lies, that is, what he says is false. But if what he said was false, then wouldn’t he be right to say that he was telling a lie? But how can a statement be both true and false? Some people may think that what he said is neither true nor false, but if what he said is actually neither true nor false, but he says he is telling a lie, then isn't he really telling a lie?

Newcomb's Paradox

Just imagine, there are two boxes in front of you, one is transparent with ten thousand dollars inside, and the other is opaque. There might be a million dollars in it, or there might not be any money at all. You have two choices: you can take the opaque box, or you can take both boxes, and all the bills in the box you take are yours.

However, there is a very accurate (nearly 100% accurate) oracle who will be there to predict your choices. He will predict your choices before you make a decision. If he calculates that you will only take away the opaque box, he will put a million dollars into the box. If he thinks you will take both boxes, he will give you an empty opaque box.

Now he has made his prediction and arranged the appropriate boxes. From your point of view, it is a foregone conclusion whether the opaque box contains the money or not. Logically speaking, if you take away two boxes, you will get 10,000 yuan more than if you take one box. But the vast majority of people who decided to take away two boxes only got $10,000 instead of $1.01 million. How do you think we should choose rationally?

Prisoner's Dilemma

Suppose you and I break the law and are imprisoned together. According to our lawyers:

If one of us pleads guilty and the other If a person does not plead guilty, the one who pleads guilty will be released, and the one who does not plead guilty will be sentenced to ten years in prison.

If we all plead guilty, each of us will go to prison for seven years.

If we all plead not guilty, we will only be sentenced to one year in prison.

Suppose we are both very smart and feel that the shorter the sentence, the better. Now, you and I are separated and unable to communicate. We each have to decide whether to plead guilty.

I don’t know if you will plead guilty. But if you plead guilty, I should plead guilty too, because that would only result in seven years in prison instead of ten. If you don't plead guilty, I should plead guilty, because then I will be released. So I should plead guilty anyway.

But if you reason the same way and finally decide to plead guilty, we will be sentenced to seven years in prison. This is much worse than neither of them pleading guilty and being sentenced to one year in prison. How can rational inference lead to such consequences?

Russell’s Paradox

We are accustomed to classifying things and people into different sets. For example, 2, 16, etc. are part of the even number set. But most sets are not themselves members of the set. The set of even numbers contains numbers such as 2, 16, etc., but the set itself is not an even number, so it is not its own member, just as a union composed of several countries is not itself a country. But the set that "is not an animal" refers to is a part of itself. Because the set contains things like pencils and trees, it is naturally not an animal.

Okay, so the group referred to by "not one's own members" is one's own members?

Sudden test

Is a surprise test possible? A teacher told her students that there would be a pop quiz next week. Her students reasoned that the test date must not be on Friday, because if the test has not been held by Thursday, then all students will know that the test will occur on Friday, so the test cannot be regarded as a surprise test. Tested. Now that the possibility of a test taking place on a Friday has been eliminated, a surprise test cannot take place on a Thursday by the same token. By analogy, pop quizzes are simply out of the question. But on the next Monday, the teacher actually gave a surprise test. All the students were surprised. What was wrong with their inferences?

The barber who cuts his own hair

In a certain village there is a barber who only cuts the hair of people who cannot cut their own hair. So tell me, will he cut his own hair?

There is no almighty God in the world

Logically, "omnipotence" means the ability to do anything possible. So, can an almighty God create a stone that he cannot lift? If so, then there is one thing that God cannot do, and that is lift the stone of His own creation. If God cannot create such a stone, then God is not almighty, because we can also create a stone that we cannot lift. Therefore, there is no almighty God in the world.