"What is the length, what is the area, what is the volume?"

This unexpected question made Oro linger for a moment. Pen ~ fun ~ Pavilion www.biquge.info he didn't expect Wang Qi to ask such a question. After thinking for a moment, he opened his mouth and said, "The length is ......."

When the words came to his lips, he couldn't say them again.

What is long?

I'm afraid everyone knows this question. Even if it is a newborn demon clan or a child who has just opened the spirit, they can have many concepts such as babbling and gesizing, size, length, etc.

However, the simpler the concept, the more difficult it is to narrate. Because in the minds of most people, the concept of "length" is basic enough. Ask them to express such a concept in a more basic way......

It's really hard.

Seeing his appearance, Wang Qi was slightly amused, but still said seriously: "Since you can't answer this question, it's better to keep it simple." I'll just ask you, what's the length?"

Oroyu can't say anything like "don't underestimate the demon" anymore.

He thought about the problem very carefully, and finally carefully said the conclusion: "The distance between the two points ......"

"Huh. Wang Qi Fu smiled.

Oriyu was somewhat enraged. He stood up and said, "Who doesn't understand this self-evident truth? Since everyone understands it, why should they study it? If you want me to say, you and the other human race have already entered the obsession obsession, staring only at their own one-third of an acre, but not seeing the vast sky. ”

"Self-explanatory?" Wang Qi's tone was full of ridicule. He flipped off the lectern with his hands and feet, pulled a somersault in the air, and landed in front of Aoliu's desk, staring straight at Aoliu. His smile was a little grim: "Boy, I have to tell you, nothing is self-explanatory. The so-called 'self-evident' is nothing but an illusion, an illusion given to you by your brain, your soul, and your wisdom. ”

"The name of this illusion is 'I was born to know,' or 'I'm smart,' or 'I'm near-minded.' ”

Wang Qi stood up again: "Maybe the acquired spiritual wisdom gives you a feeling of 'being born only when you have spiritual intelligence'." We really can't understand this kind of feeling. For us Terrans, 'Linghui' is really commonplace. That's why we feel that Linghui is actually not that great—at least the wisdom of our human race at this level is not so great. Everything we know is actually a shackle that wisdom has set on itself. ”

"Only by recognizing obsessions can we break obsessions. Only by knowing the insignificance of the sky can we calculate the vastness of the sea of stars. ”

Ao Liu frowned: "Do you want to sit down and talk about it? ”

There was actually some joy in his heart. This kind of battle of ideas and ideas can be said for a long time, and it may eventually become a situation where no one can convince anyone. If he enters such a situation, he has a chance to erase the embarrassing scene just now.

It's a pity that Wang Qi won't give him this opportunity.

The world is objective after all. Truth is truth, and falsehood is falsehood.

Wang Qi rolled backwards, as if he was about to fall to the ground. The next second, he was standing on the podium. He said: "Just now you mentioned a set of geometry supported by the 'four conditions', so you must also know the 'fifth condition', right?"

Ao Liu nodded: "Not absolutely. ”

"To be able to call the Heaven Painting Method a 'condition', you also have some wisdom in updating the demon clan. It's a pity......" Wang Qi shook his head, and he didn't know if it was a pity that the demon clan was not deepened, or it was a pity that a junior was not proficient. He said, "Since you know about this public establishment that our human race calls the law of painting heaven, you should also know the definition of points, lines, and surfaces?"

Oroyu nodded. They are aware of this self-evident problem.

"Lines are the sum of countless dots, right?"

Oroyu nodded again.

Wang Qi continued to ask: "So have you ever thought that dots have no length. In other words, the length of the dot is zero. Lines are made up of countless points, so where does the length of the line come from?"

The question of "Hmm......" made all the students stunned for a moment.

Including the two dragon clans, Moonfall Liuli and Baixian Suzheng, everyone was stunned.

- This ...... No one really thought about it!

Wang Qi looked at the expressions of the group of students below, and finally nodded with satisfaction.

- Look, I can't cure you anymore?

- I'm telling you, I look simple here, but the question of how much to die for is how much and how much.

The question Wang Qi just asked is actually an important part of the "measurement theory". And its solution is called the "Lebegus integral" on Earth. In this world, it was completed by the Geting school as a classic of renunciation.

Although this is only a small footnote to the glorious life of the Lord, most people know that if there are Ten Thousand Fa Sect disciples who have eaten this thoroughly, then there will be no obstacle to ascending to the refining stage.

How long does it take for a genius who has not been exposed to mainstream mathematics to fully understand this question?

Grothendieck's high school years took three years.

To explain this, we have to start with set theory. Wang Qi said, and began to explain the original content of the collection. Sets, zis, elements, unions, intersections, complements, and so on. 、

All of this content is intended for regular high school students. In about three or five lessons, the average high school student on Earth will be able to master this content. And Wang Qi's current students are at least high-level demon clans in the transformation stage, and their memory is even stronger, and it only took him an hour to ensure that they can master the basic concepts of those collections.

Of course, it's really just a basic concept.

"Next, we can introduce some more complex concepts, such as 'infinity'. The concept of 'infinity' is also complex. I know that outside of arithmetic, this concept is also discussed by some people. Perhaps some of you have pondered the question of infinity. And now, throw away all the mystical definitions in your minds. In arithmetic, there are only two known infinity. The countable and the uncountable. The set of all integers is called 'countable infinity'. The uncountable infinity is greater than the countable infinite, and there is no infinity smaller than the countable infinite......"

"The set of all real numbers is an uncountable infinity. And all sets of potentials equal to the 'set of all real numbers' are called 'continuums'. Speaking of this, Wang Qi smiled: "By the way, there are quite a few things in this system that I have proved. In other words, it's something I've realized. This must be emphasized. ”

Aoliu's face turned pale.

"I'll tell you the story of myself and the 'continuum hypothesis' when I have time. All in all, you just know the concept. If you want to ask the difference between 'countable infinity' and 'uncountable infinity', let's just say it – this may be the mystery between the infinite dots and lines and planes. ”

"Now back to the topic. What do we usually mean by the words length, area, volume?" Wang Qi propped his hands on the lectern: "In order to clarify this topic more clearly, let's focus on the simplest one-dimensional situation, that is, we will only consider the word 'length'. We hope that when we take out a part of the straight line, there is a 'length'. If this can be done, then similarly, words like area and volume can be understood similarly. ”

"First of all, we can make such a definition. A straight line is a huge set of dots. ”

"Every subset of this set of points, including itself, has a 'potential'. This potential is a measure. ”

"The union of two subsets that do not intersect with each other—that is, another subset of this large point set—also has a measure, and this measure should be equal to the sum of the two. To put it simply, the total length of two line segments that do not intersect or overlap can be regarded as the sum of their respective lengths. ”

Further, the sum of the measures of the three disjoint subsets should also be equal to the measures of the set of the three subsets combined, four or five, and so on, the sum of the measures of the infinite disjoint subsets should also be equal to the measure of the set obtained by combining them. ”

Speaking of this, Wang Qi said solemnly: "Next, we can make the final definition. ”

"One, the measure corresponding to the empty set is zero. Second, a number of disjoint subsets, which are combined together to obtain a measure of the subset that is exactly equal to the sum of the measures of each of these subsets. Third, if the straight line is regarded as the real number axis, then the measure corresponding to the line segment from the subpoint to the ugly point on the number line should be equal to the ugly minus the sub. ”

Then, Wang Qi closed his eyes with a solemn expression.

"It's a strict definition of what people usually call 'length', and it's the only correct one. ”

Of course, this statement is a slight exaggeration. The Lebegus measure that Wang Qi just described [in this world, it is called the 'Geting measure'] is just one of the measures. In fact, there are other measurement systems in this world as well. Other measures that differ from this measure are also recognized mathematically.

For example, another measure that is involved in geophysics, the Dirac measure [also called "immeasurable measure" in this world

This has led many mathematicians to believe that there are actually more universal measures that humans have not yet understood.

However, at this stage, the word "length" has been strictly defined, and human cognition of arithmetic has reached a deeper level.

For a moment, Oroyu was really shocked. He marveled at this rigorous wisdom. But he still asked, "What's the use of that? It's just a play on words." This kind of game that I can play on a day ......"

"yes, what's the use. With a flash of figure, Wang Qi appeared in front of Ao Liu again: "This definition just makes you realize your own insignificance." (To be continued.) )