In an incense burner, a few wisps of green smoke filled. 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 rice paper is neatly stacked on the desk. The owner of the study prefers a soft pen, so there is also an inkstone on the desk. There is ink in the inkstone, which is uniform and viscous, like oil and water. A brush is separated.

Those inks, as if waiting, waiting for the moment when they become amazing articles.

But in fact, these inks seem to be freshly polished, and they are only the magic effects that come with the inkstone. The pen hasn't been touched for a long time.

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

The lord Xi Baiche, no matter what words he writes, will be enshrined as ink treasure outside. Although his handwriting is indeed excellent. However, his status is truly extremely high.

However, such a man of God also has a day when he is worried.

"Alas...... Still no progress. He shook his head, and there was anxiety in his quiet and distant eyes.

These days, he gave everything he had. It cannot be said that there are no results, but they are not worth mentioning. It was small compared to his own ambitions.

"Teacher......" One of Xi Baiche's disciples looked worriedly behind him, "You seem to have been too worried lately......

"yes. The lord sighed: "Count the king...... Only on the way of arithmetic, there is no opponent of the king. If such a character is killed, if I am not afraid, it is not mine over there. ”

The master is not a genius. In the words of his peers, he is a "genius of hard work" who cultivates himself diligently and persevers. Naturally. A genius among his contemporaries. He was left behind.

He lived all his life. It's not that there have been defeats, there have been no fears. Therefore, he is not ashamed to talk about these ......

Naturally, say whatever comes to mind.

"Teacher, you may be able to rest and rest as usual, scatter your mind, and complete the discipline level, but you don't have to overcome it. You see, three or five years ago, we have proven the completeness of many sub-systems of arithmetic......"

"He Ke. He called out the student's name, "Have you heard anything lately?"

This half-step-free monk. He was one of several disciples in his early years, and his feelings were different from those of the later disciples of the Counter. He was a little afraid to look at his teacher's back, and bowed his head and said, "Recently...... I've been in contact with Senior Brother He......"

He Waier is a disciple who has attained success. Many of the Lord's disciples respectfully called him Senior Brother. But He Ke, who is older, is called a junior brother.

Xi Baiche showed a headache expression: "Weyl...... After that guy believed Lianzong, he changed. What did he say?"

"I've probed his tone, and things seem really bad......"

"It seems that you still have some skills. The master nodded, disapproving: "There are many loopholes in my theory, and if he can help me find one or two, I would be grateful." It's just that he shouldn't bother to think about this field, right? ”

He Ke gritted his teeth: "There are also." Mr. Feng...... Mr. Feng also knows. It's just that they're worried that you're not going to make it......"

"I told me how my old friend had been sneaking around lately. I see, right?" Xi Baiche's expression was a little solemn: "It looks like it's really a big problem." How basic can I be? Where the hell am I ......?"

He Ke's heart was cold: until now, the teacher's thinking has not leaned on "completeness...... He had no doubts about the completeness of arithmetic......

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

He Ke didn't say it for too long. The eyes of the teller were fixed on him. Those black eyes seemed to have boundless strength. Soon, he couldn't talk anymore. After a long while, the reckoning lord drew his gaze back to the proof that he had been stagnant for a long time. It was as if he wanted to follow this proof and "see" the other side of arithmetic.

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

He muttered softly.

"All in all, I've been thinking about it lately. Wang Qi reported his thoughts to Feng Luoyi with a flat face.

The so-called "thinking" sometimes needs to be hidden, but sometimes it doesn't hurt to say it.

Wang Qi's idea is like this for Feng Luoyi.

Feng Luoyi is already a monk at the top. Whether it is life or profit, he has no higher requirements. What this great arithmetic has in mind is nothing more than to seek the avenue, and in addition, to get rid of the so-called "ultimate picture". Other than that, there was no more interest to impress him.

It is better to say that in order to realize his ideals, the more talented juniors like Wang Qi, the better. From this level, Feng Luoyi's support for Wang Qi can really be said to spare no effort.

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

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

"Don't dare, don't dare. "Wang Qi wants to have. How terrible are the twenty-three questions? Even on Earth, there are still many of these twenty-three questions that remain unanswered. Even if he is confident that he can solve one or two of these problems independently, he does not dare to 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 were able to follow that completeness and penetrate the second and tenth questions in one fell swoop. ”

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

"Detours?"

Wang Qi was stunned. Is the force force method the best way to solve the problem of the continuum hypothesis?

"I recently saw an idea that was very peculiar and interesting. Feng Luoyi stared at Wang Qi's eyes and said, "Combine all the buildable set contracts into a system, and compare the infinite size in the infinite......"

Wang Qi was stunned: "Someone came up with it?"

This is the idea of Gödel's continuum hypothesis!

Extrapolating to the extent that the ZF axiom allows, it is proved that if the ZF axiom system is consistent, then the continuum is assumed to be true. Within the ZF axiom system, the continuum hypothesis cannot be falsified.

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

However, "unfalsifiable" does not mean "proven".

Along these lines, one can also show that the continuum hypothesis cannot be proved within the ZF axiom system.

In other words, the continuum problem is an undecidable problem within the realm of set theory. If this conclusion of "cannot be verified and falsifiable" is a little earlier, then there is no need to strictly prove the tenth question, which is an absolute counterexample of "decidability".

It can only be proved outside the existing axiomatic system.

The law of force and coercion is to break out of the shackles of the original axiom system, open up the system by itself, and set its own path.

However, if set theory and the existing axiomatic system are the cornerstones, how should this "self-created" system be calculated?

This is also the source of the great significance of the coercion law.

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

"I vaguely thought of it, but ...... I don't really like it. That's all Wang Qi can say. In terms of the conciseness and fluency of the theoretical argumentation process, Cohen's method of force is far inferior to Gödel's line of thought. Unless you're stupid and crazy, most people won't break out of this path when there is still a time ahead.

"Is it an unfounded 'intuition' again?" Feng Luoyi shook his head, rarely did not reprimand Wang Qi, but sighed: "It seems that this kind of thing happened a few years ago...... You avoided what ordinary people think is the normal path and took the 'crooked path', until a few days ago you threw out the law of incompleteness and broke the idea of perfection, and I realized that the long road you took a few years ago was the only right way. ”

What Feng Luoyi refers to is the path of Wang Qi before and after he received the Dao Seed Reward, and used the superlimit induction method to prove the consistency of the arithmetic system. According to the common people's thinking, he should start from the "completeness of the first-order predicate logic system" that he has proved, and push from the first order to the higher order.

However, what Wang Qi chose at that time was an incomparably far path - to prove that there were no contradictions in the system outside the system.

Everyone will be surprised, there is obviously a smooth road to the sky, you don't want to go, you have to go around the outside of the system, isn't it stupid?

At that time, only Wang Qi understood that this system could not do "without pretense". The circle he went around was what he had to do.

This time, though, it's different. Gödel's idea of proof is just not as significant as the method of coercion, and it does not have the irrefutable force of the method of coercion. In fact, it's still the right way.

All he could say was, "This ...... It's really just a matter of personal preference. That line of thinking should be not wrong. ”

"Oh. Feng Luoyi nodded, but he didn't know if he heard it or not.

"That's right, teacher. Wang Qi suddenly became a little curious: "Who is the person who proposed that idea?"

Where did the strange thing come from, that he could independently think of the same ideas as the Gödel God?

Feng Luoyi smiled: "You should also know each other." The true biography of Wanfamen, Su Junyu. ”

"Senior Brother Su, does his domain include logic?" Wang Qi was stunned.

In my impression, Su Junyu's field of study is very miscellaneous, as if he has everything, mainly probability and geometry...... Well, if he thinks about it this way, it seems that he is not without the possibility of getting in touch with logic? If he runs away to learn logic that day, it is very likely!

But it's a wonderful coincidence. Wang Qi thought: Cohen, the founder of the force forcing method, is not in the field of arithmetic logic. At first, he was only interested in the first question, so he wanted to get involved in the cross-field. He himself could not understand the enormous significance of the Law of Coercion.

Su Junyu probably thinks the same way...... I wanted to get a kick in the field of the first question, and it turned out that this leg was a big achievement. (To be continued.) )

PS: Still QAQ in Cartext