Into the Immortal Cultivation

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In a censer, a few traces of green smoke lingered. There is no light source in the whole room, but the light coming from nowhere is to illuminate every inch of the room. The good large rice paper was neatly coded on the case. The owner of the study room prefers soft pens, so there is an inkstone on the desk. There is ink in the inkstone, which is even and sticky, like oil or water. A brush is across.

The ink seems to be waiting, waiting for the moment when he becomes a shocking article.

But in fact, these inks are like new grinds, and they are just the magic effects that come with the inkstone. The pen has not moved for a long time.

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

Fortuneteller Hebe Che, no matter what word he writes, he will be regarded as an ink treasure outside. Although his words are indeed excellent. However, his identity is truly extremely high.

However, such a man of God also has a day of frowning.

"Alas ... there is still no progress." He shook his head, his anxiety in his quiet and distant eyes concealed.

These days, he exhausted everything he had. The results cannot be said to be absent, but they are not worth mentioning. Compared with his own ambitions, it is really small.

"Teacher ..." One of Xi Baiche's disciples looked worriedly behind him: "You seem too worried recently ..."

"Yeah." The fortune teller sighed: "Fortune prince ...... Only for the sake of calculus, there is no resistance to the fortune prince. If such a character comes over, if I'm not afraid, it's not mine."

The master is not a genius. In the words of his peers, he is a "genius of hard work", diligently practicing himself and persevering. Naturally, the genius of his peers has fallen behind him.

In his life, he was not undefeated or frightened. Therefore, he is not shy about talking about these. .

Naturally, just say what you think.

"Teacher, you may be able to rest as usual, relax your mind, and complete the law. In fact, you don't have to overcome it. You see, three or five years ago, we have proved the completeness of many branch systems ..."

"He Ke." Xi Baiche called out the student's name in one breath: "Have you heard anything recently?"

This half-step monk was one of the disciples in his early years, and his feelings are different from the disciples' later disciples. He did not dare to look at the back of his teacher, and bowed his head: "Recently ... I have been in contact with Brother He ..."

He Waier is a disciple who has achievements. Many disciples of the fortune teller respect him as his brother. But the older He Ke is called a younger brother.

Xi Baiche showed a headache expression: "Waier ... That guy changed after he believed Lianzong. What did he say?"

"I checked his tone, things seemed really bad ..."

"It seems that Shujun still has some skills." The master nodded and didn't care: "I have a lot of loopholes in theory. If he can help me find one or two, I would be grateful. However, he should be disdain Think about this area? Huh. "

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

"I said how my old friend is sneaky recently, is this the case?" Xi Baiche's expression was a little dignified: "It seems, it is really a big problem. How can it be based? Where am I ... … "

He Ke's heart was cold: Up to now, the teacher's thinking has not relied on "completeness" ... He has no doubt about the completeness of mathematics ...

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

He Ke did not say too long. The foreman's eyes were fixed on him. The black eyes seem to have boundless courage. Soon, he could not continue. It took a while for the fortune teller to draw his gaze back to the proof that he had been stuck for a long time. He seemed to want to "see" the other side of mathematics along this proof.

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

He murmured softly.

"All in all, my recent thoughts." Wang Qi looked at Feng Luoyi plainly and reported his thoughts.

The so-called "thought" sometimes needs to be hidden, but sometimes it's okay to say it.

This idea of ​​Wang Qi is just like this for Feng Luoyi.

Feng Luoyi is already a top monk. Regardless of life or profit, he has no higher requirements. What the big calculator thinks is nothing more than seeking the road, and also getting rid of the so-called "ultimate picture". Other than that, there was no 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 perspective, Feng Luoyi's support for Wang Qi can be said to have spared no effort.

This is also the reason why those guys who have already "topped the mountain" are not stingy to promote their juniors.

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

"Dare not dare." Wang Qi wants it. Twenty-three asks how terrible? Even on the earth, there are still a few unanswered questions among these 23 questions. Even if he is confident to solve one or two of these questions independently, he dare not say that he can solve all of them.

Feng Luoyi shook his head: "Who knows what will happen next? I didn't see it five or six years ago. You can follow that completeness and break through the second and tenth questions at once-um, Maybe the first question will be added. "

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

"detour?"

Wang Qi was stunned. Is forced method the perfect choice to solve the problem of continuum hypothesis? Why is it a detour?

"I saw an idea recently, which is very peculiar and very interesting." Feng Luoyi stared at Wang Qi's eyes and said, "Make all the buildable sets into one system and compare the infinite size in the infinite ..."

Wang Qi was stunned: "Someone figured it out?"

This is the idea of ​​Gödel's proof of the continuum hypothesis!

Infer within the range allowed by the ZF axiom, prove that if the ZF axiom system has consistency, the continuum assumption is true. Within the ZF axiom system, the continuum assumption cannot be falsified.

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

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

Following this line of thought, one can also prove that within the ZF axiom system, the continuum assumption cannot be verified.

In other words, within the category of set theory, the continuum problem is an undecidable problem. If this "cannot be confirmed nor falsified" conclusion is made earlier, then there is no need to strictly prove the tenth question, which is an absolute counterexample to "decidability".

It can only be proved outside the existing axiom system.

The forced method is to break the shackles of the original axiom system, open the system, and set the way.

However, if the set theory and the existing axiom system are the cornerstones, how should this "self-made" system be counted? Beyond the cornerstone? Counting it?

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

Upon hearing Wang Qi's exclaimation, Feng Luoyi's eyes flashed: "You have thought about this idea, have you?"

"I vaguely thought about it, but ... I don't like it very much." Wang Qi could only say so. In terms of the simplicity and fluency of the theoretical argumentation process, Cohen's forced method is far less than Gödel's ideas. Unless it is silly and crazy, ordinary people will not go out of this road when there is still ahead.

"Is there another unfounded 'intuition'?" Feng Luoyi shook his head, and rarely blamed Wang Qi, but just sighed: "Several years ago, it seems that this kind of thing happened ... you avoid the normal people's feeling On the road, I walked the 'twisted road'. It wasn't until the incomplete laws were thrown out in the past days that I broke the complete thought. I found out that the long road you detoured a few years ago was the only right way. "

What Feng Luoyi is referring to is the way to prove the consistency of the arithmetic system before and after Wang Qi received the Dao reward. According to the thinking of ordinary people, he should proceed from the first order to the higher order according to the "complete first-order predicate logic system" he proved.

However, Wang Qi chose at the time a very far road-prove that there is no contradiction in the system outside the system.

Everyone will be surprised, there is obviously a smooth road, you do n’t want to go, you have to go around the system, is this not stupid?

At the time, only Wang Qi knew that this system could not "do not ask outsiders". The circle he went around was necessary.

However, this time is not the same. Gödel's idea of ​​proof is just not as meaningful as the forced method, nor the irrefutable strength of the forced method. In fact, this is still the right path.

He can only say: "This ... is really just a matter of personal preference. There should be nothing wrong with that idea."

"Oh." Feng Luoyi nodded, not knowing whether to listen.

"Right, teacher." Wang Qi was suddenly curious: "Who is the author of that idea?"

Where is the wonderful flower, can you think independently of the same idea as God Gordel?

Feng Luoyi smiled and said: "You should also know. The True Story of Ten Thousand Methods, Su Junyu."

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

In the impression, the guy Su Junyu studied in a very complicated field, it seems to have everything, mainly probability and geometry ... um, if you think of it this way, it seems that he does not have the possibility of touching logic? If he ran out of his mind that day to learn logic-very likely!

However, this is really a wonderful overlap. Wang Qi thought: Cohen, the founder of the forced method, is not in the field of mathematics. At first, he was only interested in the first question, so he wanted to cross-discipline. He himself cannot actually understand the immense significance of the forced law.

Su Junyu thinks so too ... I want to get involved in the field of the first question, but this leg has made great achievements.

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