After matching the answers, Yi Cheng found that the answers of the two people were exactly the same.

In other words, the method of solving the problem is different.

If nothing else, it should be two perfect scores.

This obviously can't tell the winner or loser, and I can only hope that the second test can open up the gap slightly.

At 9:40, the second test officially began.

The questions in the second test can be described as simple and crude, with a total of 4 answers or proof questions.

The score is also super violent:

The first two questions are worth 40 points each, and the last two questions are worth 50 points each, and the full score of the whole paper is 180 points.

A few students who participated in the high school for the first time saw such a score and were so frightened that even their hands holding pens began to tremble.

"Mom...... 40 points for a question, just go away. ”

"I've never seen such an exaggerated score."

……

Yi Cheng took a deep breath, calmed his mind, and opened the test paper.

"Mom, what the hell is this?"

A young man's soft call came from the side.

"Be quiet in the exam room." The invigilator reminded.

It's no wonder that he sighed, because there are many people who are as confused and uncomfortable as him.

It's just that others don't show it.

The first question is as follows:

[Horses, so the name is also; The white one, so the name is also. The name is not the name of the color. Therefore, it is said: white horses are not horses. Ask for horses, yellow and black horses can be caused. Ask for a white horse, and a yellow and black horse cannot be done. …… Therefore, the yellow and black horses are also the same, and there should be horses, but not white horses, which is the non-horse judgment of white horses. Horses, there is no color to take, so yellow and black are so appropriate. The white horse has to take the color, and the yellow and black horses are all so colored, so only the white horse can respond to the ear. If there is no one to go, there is no one to go. Therefore, it is said: white horses are not horses. Horses are colored, so there are white horses. He makes the horse colorless, and the horse is like his own ears. Antori Hakuba? Therefore, the white one is not a horse. The white horse, the horse and the white, the white and the horse. Therefore, it is said: white horses are not horses.

(1) Test: White horse non-horse (5 points)

(2) If there is a horse, it has to find food for all the horses that do not find food for themselves, test that this horse is not this horse, and give an example of the existence of "this horse is not this horse" (35 points)]

Yi Cheng couldn't help but let out a soft sigh.

Now I can't even do math problems if I don't speak well.

This is an allusion to an ancient sophist named Gongsun Long:

Once Gongsun Long passed the customs, the customs official said: "According to the custom, the people who pass the customs can pass, but the horses can't." Gongsun Long said that the white horse is not a horse, and after some argumentation, the official nodded again and again after hearing it, and said: "What you said is very reasonable, please pay for the horse."

Now this problem requires you to translate classical Chinese in mathematical language, and prove that "white horses are not horses"

It can be said that the previous words are all nonsense, and it is somewhat useful to say that it is useful, and it is not much use to say that it is useless.

It can only be said that the person who wrote the question is an avid lover of ancient culture.

The first question is obviously a send-off question.

Yi Cheng shook his head and began to prove it:

Suppose the horse is set A and the white horse is element B.

Then there are B∈A

B ≠A

In other words, Gongsun Long had to define the relationship between the two before he could discuss the results.

If, according to the first case, B∈A, the white horse is an element in the set of horses, then the white horse is a horse, this is a false proposition.

If we follow the second case, B ≠ A, the white horse is just one element of the set of horses, so the white horse is not equal to the horse, this is a true proposition.

After the first question was successfully confirmed, it came to the second question.

Yi Cheng stood for three seconds.

This horse is not this horse.

No way?

This question obviously shouldn't be placed here.

Because this is a typical Russell paradox question.

What is Russell's paradox?

This is a terrible story that caused an uproar in the mathematical community, and has not yet been perfectly answered:

The German mathematician Cantor founded the famous set theory, which became the cornerstone of modern mathematics. Mathematicians were fascinated by the discovery that "all mathematical achievements can be based on set theory".

In 1903, a shocking news came out that set theory was flawed! This is the famous Russell paradox proposed by the British mathematician Russell.

Russell gives a very simple example to illustrate this vulnerability in set theory:

There is a barber in a certain city who only shaves the faces of people who [don't shave their own faces].

But one day, the barber saw in the mirror that his beard had grown, and he instinctively grabbed the razor.

So should this barber shave his face? 】

The paradox is obvious.

If he shaves his face, then he violates the principle of the one who shaves his face.

If he doesn't shave his face, then he has to shave his face for [those who don't shave themselves].

That's where the contradiction lies.

This paradox triggered the third crisis in the history of mathematics.

It would be too embarrassing for high school students to prove it here.

So Yi Cheng thinks that this question should not appear here.

It's over.

The first question is so difficult, this time Gaolian obviously doesn't want anyone to live.

"Teacher!"

It was at this moment that a student in the classroom raised his right hand.

The invigilator looked back.

"What's wrong?"

"This question is wrong." The student said stiffly.

Everyone looked up at her in unison.

This student is Yan Ziqi from Yi Cheng's table.

It was clear that she had also noticed that the test questions were out of the syllabus.

"The first and second questions are obviously a Russell paradox, this question is obviously out of the outline, even the top mathematicians today cannot perfectly solve Russell's paradox, it should not appear here." Yan Ziqi said loudly.

She is the gold medalist of last year's Mathematics Olympiad, she is the first in mathematics in the school's grade, she is the pride of mathematics in the province, and she is the mathematics talent that the country will focus on cultivating in the future.

She is qualified to challenge.

The invigilator walked over and looked at Yan Ziqi's exam card.

Then he double-checked the paper.

The invigilator looked at it for about half a minute, turned around, faced the candidates in the entire classroom, and said lightly, "There is no mistake in this question, let's continue to answer the question." ”

……

Impossible.

Yan Ziqi and Yi Cheng fell silent in unison.

As for the others, even if they understood Ziqi's words and knew that this was a problem related to Russell's paradox, they didn't know what to do.

Some people have given up on answering the second question and have begun to turn to the later questions.

According to the teacher's earnest teaching, don't attack a topic, put it first, and come back after solving the easy problem.

The result is –

The further back you go, the more you won't do it.

"Damn, who came up with this topic?!"

"This is already an Olympiad problem, right?"

"No, it's already beyond the Olympiad questions, right?!"

Only a few are still patiently answering.

Among them are Yi Cheng and Yan Ziqi.

They don't plan to give up yet.

Yi Cheng was puzzled until he saw the two words in front of the second question:

[Proof.] 】

Cunning questioner!

Actually playing this kind of word game.

Proof questions generally use the word test, and the result may be to prove that the proposition is false.

[Try], this word is very spiritual.

According to the current situation, the test is to try.

This is something that does not need to be proven, and cannot be proven.

All you need to do is describe the proof idea in mathematical language.

As for whether it can be proven, it is not the point of this question.

The latter example is the point, which examines your understanding of the paradoxical proposition.

To solve Russell's paradox, even the best mathematicians have to take a detour.

But to describe Russell's paradox through the language of general mathematics, this is something that junior high school students can do.

The corners of Yi Cheng's mouth rose slightly, and a relaxed smile surfaced.

Once you figure out this relationship, it's all easy.

He took up his pen and wrote:

Let the property P(x) mean "x does not belong to x".

Suppose a class A is determined by the property P

That is, "A={x|x∉A}".

First, if A belongs to A, then A is an element of A, then A has the property P, and from the property P knows that A does not belong to A;

Second, if A does not belong to A, that is, A has the property P, and A is made up of all classes with the property P, so A belongs to A.

……

Okay, now that you've written your idea, here's an example:

Yi Cheng wrote on the test paper:

"What I wrote is false."

40 points in hand.

……