Enter the Immortal Cultivation

In an incense burner, a few wisps of green smoke lingered. There was no light source in the whole room, but the light that came from nowhere illuminated every inch of the room. The best large pieces of rice paper were neatly stacked on the table. The owner of the study preferred soft brushes, so there was an inkstone on the desk. There was ink in the inkstone, which was evenly viscous, like oil and water. A brush was separated.

The ink seemed to be waiting, waiting for the moment when it would become a stunning article.

But in fact, these inks seemed to be newly ground, and they were just the magic effects of the inkstone. The pen had not moved for a long time.

The owner of the ink and the pen had been frozen here for a long time.

No matter what the master Xi Boche wrote, he would be regarded as a treasure outside. Although his handwriting was indeed excellent. However, his status was truly extremely high.

However, such a god-man also had a day of frowning.

"Alas...still no progress." He shook his head, his calm and distant eyes were filled with anxiety that could not be concealed.

These days, he has given his all. It cannot be said that there are no results, but they are not worth mentioning. Compared with his own ambitions, it is really insignificant.

"Teacher..." One of Xi Boche's disciples looked at him worriedly from behind: "You seem to be too worried recently..."

"Yes." The master of calculation sighed: "Master of Calculation... In terms of calculation, there is no one who can resist Master of Calculation. If I am not afraid of such a person, that place will not be mine."

The master of calculation is not a genius. In terms of his peers, he is a "hard-working genius". He cultivates himself diligently and perseveres. Naturally, the geniuses among his peers have fallen behind him.

He has not been defeated or afraid in his life. Therefore. He is not ashamed to talk about these. .

Naturally, he says whatever he thinks of.

"Teacher, maybe you can take a break like usual and relax. The level of completeness law is not something that must be conquered. You see, three or five years ago, we have proved the completeness of many branch systems of mathematics..."

"He Ke." Xi Boche called out the student's name: "Have you heard any news recently?"

This half-step free and unrestrained monk. He is one of his early disciples. The feelings are naturally different from the later disciples of the master of calculation. He was a little afraid to look at his teacher's back, and lowered his head and said: "Recently... I have been in contact with Junior Brother He..."

He Waier is the master of calculation's most accomplished disciple. Many of the master's disciples respectfully call him Senior Brother. But He Ke, who is older, is called Junior Brother.

Xi Boche showed a headache: "Wel... that guy changed after believing in Lian Zong. What did he say?"

"I have sounded out his mouth, and things seem to be really bad..."

"It seems that the master of calculation is still quite capable." The master of calculation nodded and didn't care: "There are many loopholes in my theory. If he can help me find one or two, I will be very grateful. But, he should not bother to think about this field, right? Humph."

He Ke gritted his teeth: "Also, Mr. Feng... Mr. Feng also knows. But, they are all worried that you can't make it..."

"I was wondering why my old friend has been sneaking around lately, so this is it?" Xi Boche's expression was a little solemn: "It seems that it is really a big problem. How basic can it be? Where on earth am I..."

He Ke's heart was cold: Up to now, the teacher's ideas have not leaned towards "completeness"... He did not doubt the completeness of mathematics at all...

"Teacher, we still have a way out. We can still prove the completeness of the system from outside the system through super-exhaustive induction..."

He Ke did not speak for too long. The master of calculation stared at him. The black pupils seemed to have boundless courage. Soon, he could not continue. After a long while, the master of calculation turned his gaze back to the proof that he had been stagnant for a long time. He seemed to want to follow this proof and "see" the other side of mathematics.

"We must know, we will know..."

He whispered softly.

"In short, my recent thoughts." Wang Qi reported his thoughts to Feng Luoyi with a calm face.

The so-called "thoughts" sometimes need to be hidden, but sometimes it doesn't matter to say them out.

Wang Qi's thoughts are like this for Feng Luoyi.

Feng Luoyi is already a top cultivator. Whether it is life or profit. He has no higher requirements. What this great mathematician thinks in his heart is nothing more than seeking the great way, and also getting rid of the so-called "ultimate picture". Apart from this, there is no other benefit that can move him.

It is better to say that in order to realize his ideal, the more talented juniors like Wang Qi, the better. From this perspective, Feng Luoyi's support for Wang Qi can really be said to be unsparing.

This is also the reason why those who have "reached the top" are not stingy in promoting the younger generation.

Feng Luoyi was thinking about Wang Qi's ideas in his mind. He thought: "Looking at your attitude, it seems that you are going to kill through these twenty-three questions."

"I dare not, I dare not." Wang Qi was afraid. How terrible are the twenty-three questions? Even on Earth, there are still several questions that are unresolved among these twenty-three questions. Even if he has the confidence to solve one or two of them independently, he dare not say that he can solve all of them.

Feng Luoyi shook his head: "Who knows what will happen in the future? Five or six years ago, I didn't see that you could follow that completeness and break through the second and tenth questions in one fell swoop - well, maybe you will have to add the first question."

"But..." Feng Luoyi's eyes suddenly became sharp: "You seem to always like to take detours."

"Detour?"

Wang Qi was stunned. The forcing method is the only choice to perfectly solve the continuum hypothesis problem? Why is it a detour?

"I recently saw an idea, which is very strange and interesting." Feng Luoyi stared at Wang Qi's eyes and said: "All the sets that can be established are combined into one system, and the size of infinity is compared in infinity..."

Wang Qi was stunned: "Has anyone figured it out?"

This is the idea that Gödel used to prove the continuum hypothesis!

Infer within the scope allowed by the government axioms, and prove that if the government axiom system is consistent, the continuum hypothesis is true. Within the government axiom system, the continuum hypothesis cannot be falsified.

For the twenty-three questions, this should be a perfect answer.

However, "cannot be falsified" does not mean "proven".

Following this idea, people can also prove that within the government axiom system, the continuum hypothesis cannot be proved.

In other words, the continuum problem is an undecidable problem within the scope of set theory. If this conclusion of "cannot be confirmed or falsified" was reached earlier, then there would be no need to strictly prove the tenth question, and this would be an absolute counterexample to "decidability".

It can only be proved outside the existing axiom system.

The method of forcing is to break out of the constraints of the original axiom system, open a system and set a path by yourself.

However, if we use set theory and the existing axiom system as the cornerstone, then how should this "self-created" system be calculated? Outside the cornerstone? Within mathematics?

This is also the source of the great significance of the method of forcing.

Hearing Wang Qi's exclamation, Feng Luoyi's eyes flashed: "You have thought of this idea, right?"

"I vaguely thought of it, but... I don't like it very much." Wang Qi could only say this. In terms of the simplicity and fluency of the theoretical argumentation process, Cohen's forcing method is far inferior to Gödel's idea. Unless you are stupid or crazy, most people will not break out of this path when there is still time ahead.

"Another unfounded 'intuition'?" Feng Luoyi shook his head. He rarely scolded Wang Qi, but sighed: "Several years ago, this kind of thing seemed to have happened... You avoided the path that most people think is normal and took the 'wrong path'. It was not until a few days ago that you threw out the law of incompleteness and broke the idea of ​​perfection that I realized that the long way you took in the past few years was the only right way."

Feng Luoyi was referring to the path that Wang Qi took before and after winning the Daozhong Award to prove the consistency of the arithmetic system by transfinite induction. According to what most people think, he should start from the "completeness of the first-order predicate logic system" he proved, and push from the first order to the higher order.

However, what Wang Qi chose at that time was an extremely long way - proving that there is no contradiction in the system outside the system.

Everyone would be surprised that there is a straight path to heaven, but you don't want to go, and you have to go around the system. Isn't this stupid?

At that time, only Wang Qi understood that this system could not be "self-sought". The circle he went around was necessary.

However, this time it was different. Gödel's proof was not as meaningful as the forcing method, nor as irrefutable as the forcing method. In fact, it was still the right way.

He could only say: "This... is really just a matter of personal preference. That idea should be correct."

"Oh." Feng Luoyi nodded, not knowing whether he had heard it or not.

"By the way, teacher." Wang Qi suddenly became curious: "Who is the originator of that idea?"

Where did this weirdo come from, who could think independently and come up with the same idea as Gödel?

Feng Luoyi smiled and said: "You should know him too. The true disciple of Wanfamen, Su Junyu."

"Brother Su? Does his field include logic?" Wang Qi was stunned.

In his impression, Su Junyu studied a lot of fields, it seemed that he had everything, mainly probability and geometry... Well, if you think about it this way, it seems that he is not without the possibility of contacting logic? If he suddenly went to study logic that day - it is very possible!

However, this is really a wonderful overlap. Wang Qi thought: Cohen, the founder of the force method, did not study mathematical logic. He was only interested in the first question at first, so he wanted to cross the field to get involved. He himself could not understand the great significance of the force method.

Su Junyu probably thought the same way... He wanted to get involved in the field of the first question, and ended up making great achievements. (To be continued ~^~)

PS: Still stuck in the text QAQ