Achilles is the fastest runner in ancient Greek mythology, but if there is a turtle in front of him (climbing up from point A), he will never catch up with it. The reason is as follows: He had to pass the place A where the tortoise started, but when he caught up with this place, the tortoise climbed some distance forward. At point B, he had to pass point B, but when he caught up with point B, the tortoise climbed to point C again.

Achilles and the tortoise

One day, Achilles ran a race with the tortoise. Achilles thought he was faster than the tortoise, so he told the tortoise to run a little less. Their agreement is that Achilles will start somewhere at d 1, and the tortoise will start at d 2, which is closer to the finish line. But imagine that when Achilles runs to d 2, the tortoise will run to d 3, another place. By the time Achilles caught up with d 3, the tortoise had reached d 4. By analogy, every time Achilles ran to the place where the tortoise had been before, the tortoise ran a distance forward. So, how can Achilles catch up with the tortoise?

Dune paradox

When sand grains pile up together, many of them become a pile. For example,100000 grains of sand are piled together to form a sand dune. The sand dune is so big that if you take away a grain of sand casually, the sand dune will still exist, because a grain of sand is really insignificant. Similarly, if a grain of sand is taken from a sand dune consisting of 99,999 grains of sand, the sand dune will not disappear. In short, if you take a grain of sand from a sand dune, the sand dune will continue to exist. But if it does happen,

Paradox of not claiming oneself

If a predicate cannot be applied to itself, we call it "no assertion". On the contrary, we call it "advocacy". For example, the predicate "composed of Chinese characters" is a self-predicate predicate. "A red fruit" can only describe the fruit, but not itself, so it is not a claim.

So "not advocating" itself is not advocating? If so, it should not be applied to yourself, that is, it should be applied to yourself. But if it is not, it should not be applied to yourself. In other words, if you apply it to yourself, you should not apply it to yourself!

Lawyers and apprentices

Student A is an apprentice to a barrister. When he was still training, he promised his teacher that he would pay the tuition after the training and won the first lawsuit. But after graduation, student A never took over any lawsuit, so the teacher decided to sue him for arrears of tuition.

The teacher's statement is that if the teacher wins the lawsuit himself, student A must pay the tuition fee immediately; If student A wins the case, student A should pay the tuition according to the original agreement, so students should pay the tuition anyway.

But a's statement is that if the court wins his case, he won't have to pay tuition; If the teacher wins, he never wins, so there is no need to pay tuition according to the agreement.

Whose argument makes sense?

We are liars.

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

So, is what this person said true or false? If what he said is true, then he is lying, that is to say, what he said is false. But if what he said is false, isn't it right that he said he was lying? But how can a word be both true and false? Some people may think that what he said is not true, but if what he said is not true, but he says he is lying, isn't he really lying?

Newcomb paradox

Imagine there are two boxes in front of you. One is transparent, with 10 thousand yuan in it, and the other is opaque. There may be/kloc-0,000,000 in it, or there may be no money. You have two choices: take the opaque box or both. All the money in the box you take is yours.

However, a very accurate (close to 100% accurate) prophet will be present to predict your choice. Before you make a decision, he will predict your choice first. If he calculates that you will only take away the opaque box, he will put 1 million into it. If he thinks you will take two boxes, he will give you an empty opaque box.

Now, he has made his prediction and arranged the appropriate box. From your point of view, whether there is money in the opaque box is a foregone conclusion. It is also reasonable to take 1 10,000 yuan more than one box. But most people who decide to take two boxes only get 1 ten thousand yuan, not10/ten thousand yuan. How do you think to choose rationally?

Prisoner's dilemma

Suppose you and I broke the law and were put in prison together. According to our lawyer:

If one of us pleads guilty and the other one doesn't, the one who pleads guilty will be released, and the one who doesn't will be sentenced to ten years in prison.

If we all plead guilty, everyone will be sentenced to seven years in prison.

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

Suppose we are all smart and think that the shorter the sentence, the better. Now, you and I are separated, unable to communicate, and we have to decide whether to plead guilty or not.

I don't know if you will plead guilty. But if you do, I should also plead guilty, because it will only lead to seven years in prison instead of ten years. If you don't plead guilty, I should plead guilty, because then I will be released. So I should confess anyway.

But if you do the same reasoning and finally decide to plead guilty, we will be sentenced to seven years in prison. This is far worse than a year in which both of us plead not guilty. Why does rational reasoning lead to such consequences?

Russell's paradox

We usually divide things and people into different groups. For example, 2, 16 is part of an even set, but most sets are not part of the set themselves. An even set contains 2, 16, but the set itself is not even, so it is not a part of itself, just as an alliance composed of several countries is not a country itself, but it is not an animal.

Ok, so the "not your own" refers to your own?

A sudden test

Is a pop quiz possible? A teacher told her students that there would be a pop quiz next week. she

Some students infer that the date of the exam will not be Friday, because if the exam is not held on Thursday, then all students will know that the exam will be held on Friday, so this exam cannot be regarded as a surprise exam. Since the possibility of holding a test on Friday has been ruled out, it is impossible to conduct a surprise test on Thursday for the same reason. By analogy, surprise testing is impossible. But next Monday, the teacher will really take a surprise exam, and all the students will be surprised.

A barber who cuts his own hair.

There is a barber in the village who can only cut hair for people who can't cut their own hair. Do you think he will cut his own hair?

There is no almighty god in the world.

According to the truth, "omnipotence" means the ability to do anything possible. So, can Almighty God make a work? A stone you can't lift? If you can, then there is one thing that God can't do, and that is to lift it? The stone he created himself. If God can't make such a stone, then God is not omnipotent, because we can make a stone that we can't lift. Therefore, there is no almighty God in the world.