Enter the Immortal Cultivation

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

This unexpected question left Austria stunned for a moment. 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..."

The words came to my lips but I couldn't say them anymore.

What is long?

I'm afraid everyone knows this question. Even newly born monsters or children who have just opened their spirits can have many concepts such as the size, length, etc. of babbling gestures.

However, the simpler the concept, the harder it is to describe it. Because in most people's minds, the concept of "length" is already basic enough. Ask them to use more basic concepts to express such a concept...

It's really difficult.

Seeing his look, Wang Qi was slightly amused, but he still said seriously: "Since you can't answer this question, just keep it simple. Just ask you, what is the length?"

Aoliu could no longer say anything like "Don't underestimate the demon".

He thought about the problem very seriously, and finally came to the conclusion cautiously: "The distance between two points..."

"Haha." Wang Qi smiled.

Aoliu was a little irritated. He stood up and said: "Who doesn't understand this self-evident truth? Since everyone understands it, why study it? If you ask me, you and other human beings have already entered the obsession barrier and only focus on Looking at my own little piece of land, I can’t see the vast sky.”

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

“The name of this kind of illusion is ‘I am born with knowledge’, or ‘I am very smart’ or ‘I am short-sighted’.”

Wang Qi stood up again: "Perhaps the spiritual wisdom acquired the day after tomorrow gives you a feeling of 'being born with spiritual wisdom'. We really can't understand this kind of feeling. For us humans, 'linghui' 'It's really common. That's why we feel that Linghui is not that great - at least the wisdom of our human race is not that great. In fact, everything we know is the shackles set by wisdom. "

"Only by recognizing obsession can we get rid of it. Only by knowing the insignificance of the sky can we calculate the vastness of the sea of ​​stars."

Aoliu frowned: "Are you going to sit down and discuss the truth? It's just nice to say it."

He was actually a little happy in his heart. This battle of concepts and ideas can go on for a long time, and may eventually turn into a situation where no one can convince anyone. If he entered such a situation, he would have a chance to erase the embarrassing scene just now.

Unfortunately, Wang Qi would not give him such a chance.

The world is objective after all. What is true is true, what is false is false.

Wang Qi turned back, as if 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'. You must also know the 'fifth condition', right?"

Aoliu nodded: "Not absolutely."

"Being able to call the Sky Painting Technique a 'condition' means you have some wisdom in updating the Monster Clan. What a pity..." Wang Qi shook his head. He didn't know whether it was a pity that the Demon Clan was not updated more deeply, or that he was pitying a certain junior. Not good at academics. He said: "Since you know this postulate, which we humans call the method of drawing the sky, you should also know the definition of points, lines and planes?"

Olyu nodded. They are indeed aware of this self-evident problem.

“A line is the sum of an infinite number of points, right?”

Olyu nodded again.

Wang Qi continued to ask: "Then have you ever thought that a point has no length? In other words, the length of a point is zero. A line is made up of countless points, so where does the length of the line come from?"

"Well..." This question made all the students stunned for a moment.

Everyone, including the two dragon clansmen Yueluo Liuli and Bai Xiansu Zheng, were stunned.

——This... no one has really thought about it!

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

——You guys, I can’t cure you anymore?

——Let me tell you, my place looks simple, but the most difficult question is how much you want.

The question Wang Qi just asked is actually an important part of the "measure theory". And its solution is called "Lebesgue integral" on Earth. In this world, it serves as a deviant classic and was completed by the Geting sect.

Although this is just a small footnote to the master's glorious life, most people know that if all disciples of Wan Dharma sect understand this thoroughly, there will be no obstacles in rising to the Void Refining Stage.

How long does it take for a genius who has never been exposed to mainstream mathematics to completely figure out this problem?

Grothendieck spent three years in high school.

"If you want to explain this problem, you have to start with set theory." Wang Qi said, and began to explain the initial content of the set. Sets, elements, unions, intersections, complements, etc. ,

All of these contents are geared toward ordinary high school students. In about three or five classes, the average high school student on Earth can master this content. Wang Qi's current students are at least high-level monsters in the transformation stage, and their memories are even stronger. It only took him an hour to ensure that they mastered 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 very complex. I know that outside of arithmetic, this concept has also been discussed by some people. Maybe some of you I have thought about this question - what is infinity. Now, throw away those mysterious definitions in your mind. In arithmetic, there are only two known types of infinity: countable infinity and uncountable infinity. It is called 'countable infinity'. Uncountable infinity is greater than countable infinity, and there is no infinity smaller than countable infinity..."

"The set of all real numbers is an uncountable infinity. And all sets with the same potential as the 'set of all real numbers' are called 'continuums.'" At this point in his speech, Wang Qi smiled: "By the way, There are quite a few things in this system that I have proven. In other words, this is something I have learned.”

Aoliu's face was ashen.

"I will tell you my story about myself and the 'continuum hypothesis' when I have time. All in all, you just need to know this concept. If you want to ask about the difference between 'countable infinity' and 'uncountable infinity', then Let’s just say it briefly – this may be the secret between point, line and surface without any size.”

"Now back to the topic. What do we usually mean by the words length, area, and volume?" Wang Qi put his hands on the desk: "In order to clarify this topic more clearly, let us only focus on the simplest The one-dimensional case of The word class can be understood similarly."

"First of all, we can make this definition. A straight line is a huge set of points."

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

"The union of two subsets that are disjoint 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. Simply put, two disjoint and non-overlapping sets The total length of the line segments can be regarded as the sum of their respective lengths."

"Furthermore, the sum of the measures of three disjoint subsets should also be equal to the measure of the set of these three subsets, whether four, five, and so on, for infinite measures of disjoint subsets The sum should also be equal to the measure of the set obtained by adding them together.”

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

"First, the measure corresponding to the empty set is zero. Second, several disjoint subsets, and the measure of the subset obtained by combining them is exactly equal to the sum of the measures of these subsets. Third, if a straight line is regarded as real number axis, then the corresponding measure of the line segment from the sub-point to the ugly point on the number axis should be equal to the sub-point.”

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

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

Of course, this sentence is slightly exaggerated. The Lebesgue measurement that Wang Qi just described [in this world, it is called the ‘Singing Measurement’] is just one type of measurement. In fact, there will be other measurement systems in this world. Mathematics also recognizes other measures than this one.

For example, another type of measurement involved in geophysics is the Dirac measurement [also called "infinite measurement" in this world]

This has led many mathematicians to believe that there is actually a more universal measure, but humans have not yet understood that step.

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

For a moment, Aoliu was really shocked. He marveled at this rigorous wisdom. But he still asked: "What's the use of this? This is just a word game. I can play this kind of game all day..."

"Yeah, what's the use." Wang Qi flashed and appeared in front of Aoliu again: "This definition just makes you realize your own insignificance." (To be continued ~^~)