Enter the Immortal Cultivation

The first of Hilbert's twenty-three problems is the continuum cardinality problem.

The continuum problem is the problem of "there is no other cardinality between the cardinality of countable sets and the cardinality of real number sets".

The so-called "cardinality" refers to the "absolute measure" of a set. If a set has one element, then the cardinality of this set is one. If there are two elements, the cardinality is two. And so on.

The cardinality of infinite countable sets such as "all integers" and "all natural numbers" is recorded as "Aleph Zero" - Shenzhou calls it "Tao Yuan Zero Number", the smallest infinite integer.

The ancients of Shenzhou once believed that the total number of numbers, infinite size, is the number of Tao.

Aleph Zero plus one is still Aleph Zero. Aleph Zero plus Aleph Zero is still Aleph Zero. Aleph Zero multiplied by Aleph Zero is still Aleph Zero.

Infinite, positive infinity. Ordinary operations are completely meaningless for this number.

So, is there a number in the world that is larger than this infinite number?

Actually, there is.

That is the cardinality of the "power set".

If a set has the element "1", then it has two power sets - "1" and the empty set?.

If a set has two elements "1, 2", then it has four power sets - the empty set?, set {1}, set {2}, set {1, 2}.

Similarly, when a set has three elements, then it has eight power sets. When the number of elements in the set increases to four, the number of power sets increases to sixteen.

The power set of a set is always more than the number of elements in the set. If a set has N elements, then it has 2 to the power of N power sets.

The power set of an infinite countable set, two to the power of Aleph Zero, is the second infinite number discovered by humans - Beth 1.

In addition to being the power set of the integer set, this "beth1" is also the cardinality of all real number sets.

The continuum problem can also be summarized as "Is there another cardinality between Aleph Zero and Beth 1?".

Is there a cardinality of a set that is clearly greater than one infinity and less than another infinity?

This is the first question among the twenty-three questions.

Among the twenty-three questions, the second and tenth questions are related to the foundation of mathematics and are considered extremely important. It is precisely because of the idea of ​​"completeness, consistency, and decidability" of the master of mathematics. Therefore, these two questions have always been mentioned together. But from the perspective of the "questioner", the relationship between the first and second questions is closer. The first and second questions, continuum and completeness, are fundamentally connected.

The question of the first question leads to the question of the second question, and the answer to the second question inspires the answer to the tenth question.

These questions can be regarded as a system.

Of course, each question in the twenty-three questions of Ximen is more or less related to the questions in the other twenty-three. The entire twenty-three questions are implicitly a whole. And this whole covers most aspects of mathematics. Once a question is solved, mathematics as a whole will show a huge progress. And the research of each mathematician is more or less related to one of the twenty-three questions.

No mathematician has ever been able to do this, not before, and unlikely to be in the future. For the history of mathematics, the twenty-three questions are an extremely magnificent leap.

And Wang Qi saw this point. He has already solved the second and tenth questions. Now he throws out the solution to the first question. In fact, it is not a particularly shocking thing.

In addition, the continuum hypothesis is similar to the completeness proof and the decidability proof. They are all of the kind that have extremely important status, but are relatively independent. They are like the first piece of a domino. They are not very good in themselves, but as long as they fall, they will trigger a chain reaction.

To solve these problems, you don’t need much accumulation. These problems are all very "clever".

On Earth. The solvers of the second and tenth questions are all very young genius scholars. And the solver of the first question, strictly speaking, does not even understand mathematical logic-P.J. Cohen's professional field is analysis, he was just attracted by this question, that's all.

P.J. Cohen, the solver of the first question, himself could not even understand the application of the proof method he invented in the field of logic.

In other words, this achievement can also be attributed to the "flash of genius inspiration".

However, the biggest problem is...

"I don't think I memorized this thing in my previous life..." Wang Qi felt a headache again.

The solution to a quadratic equation is now known to middle school students. However, how many people know how to prove that solution?

The distance between "knowing" and "proving" is probably equivalent to "cultivating the supreme mental method" and "creating the supreme mental method". The difficulty of the latter is countless times that of the former.

What's more, Wang Qi didn't even remember the solution to the first question, the "force method", but only remembered a general direction.

"How long will it take for me now to prove the first question by myself?"

Wang Qi couldn't help but start thinking about this question.

Until Zhen Chanzi came out and reminded: "Hey, kid."

"Thinking about something, don't bother me."

"I want to remind you that we're at the place." Zhen Chanzi sighed: "If you don't land, you'll fly over."

"Oh." Wang Qi looked down. It was an endless plain. The flat ground extended all the way to the horizon, and only in the west was there a faint shadow of the mountains. That was the northern section of Kunlun Mountain. To the south of Wang Qi, there was a river, which was flat on the ground like satin, almost straight. That was the second largest river in Shenzhou, the Dajiang River. There were houses gathered, with the Dajiang River as the center, spreading all the way to the north and south. Outside the town, there were those fields like handkerchiefs. Now the new season of wheat has just sprouted, and it looks green and fluffy from the sky.

Green and brown are the colors of this land.

"Jiangbei City, we're here." Wang Qi sighed.

This kind of scene is not so easy to see north of Kunlun. The dragon clan nailed the plate of the West Sea, and also made the movement of the plate of the West Land of Shenzhou weird. The terrain west of Kunlun is far more complicated than that east of Kunlun.

Wang Qi flew all the way from the west of Kunlun, and he had already seen the scenery of the mountains. Seeing this plain now, he felt relaxed and refreshed, and the melancholy feeling he had when thinking about the problem just now completely disappeared.

Speaking of which, although he has been cultivating immortals for more than ten years, he has never found time to see this world. In the first few years, he concentrated on his studies in the Immortal Academy, and then he was "under house arrest" in Shenjing because of the suspicion of "exiled immortals". Except for going to Xiaodong with Chen Feng once, and going to Tianlingling with Chen Feng and Chen Youjia, he basically never went out. After arriving in Leiyang, he stayed on the sea and almost never took a good look at the world where he lived in this life.

This universe is infinite, but there is only this planet at this moment, which is quite special for him.

Zhen Chanzi also sighed: "The bloodline of the Luo family should...should be here?"

Liwu Luo family, where the descendants of Zhen Chanzi's adoptive father's family may exist.

Wang Qi heard this and patted his ring: "Yeah, I flew for more than half a month just to help you find your roots!"

Zhen Chanzi was angry: "Why does this sound so much like the dead returning to their roots? And why is it so dirty?"

"No, no, no, other people look for their ancestors, and you look for your descendants." Wang Qi looked innocent: "As for the dirty... Am I not helping you find your bloodline roots? Is there something else in it?"

That day, after saying goodbye to Chen Feng, Wang Qi also began his "journey to help the old man in the ring find his roots." He first went to find Mao Zimiao to say hello. After having a simple meal, Wang Qi first went north and entered the scope of Lie Dian Island.

Lie Dian Island is very small, not comparable to Fusang Continent. It is located at the junction of the West Sea and the North Sea. There is no overly large biological community here, and the number of monsters is much less than that of the West Sea. And after the sea monsters lose their most terrifying number, they are not a concern. On the other hand, it is also an unusual place. The first cultivator of the modern law, Yuanli Shangren, was born on this island and attained enlightenment. For ordinary modern practitioners, Lieding Island can even be called half a holy land. The mortals near Lieding Island can also be said to live and work in peace and contentment.

When Wang Qi came to Lieding Island, it was the first day of the new year. Lieding Island was covered with snow, and it was covered with silver, a northern scenery. In addition to enjoying the snow scenery, Wang Qi also visited Ai Changyuan who went home for the New Year.

Wang Qi and Ai Changyuan had a friendship when they were young. However, Wang Qi had a cheating device, and he was about three or four years ahead of Ai Changyuan in cultivation. Before breaking through the realm, Ai Changyuan, who was in the foundation-building period, could not beat Wang Qi, who had already formed a pill. This made the young master of the Ai family quite depressed.

After leaving Lieding Island, Wang Qi went against the Luo River all the way to the west foot of Kunlun. Then, at the foot of Kunlun Mountain, Wang Qi abandoned the method of flying and escape, and did not even use the protective qi, and relied purely on his hands and feet to go to the world's first famous mountain.

This is not a "training" or "increasing the difficulty". Even if Wang Qi does not use the flying method or the protective qi, he is still a Jindan cultivator, and his physiological functions are far superior to any ordinary creature. With only his hands and feet, he can easily climb this mountain.

However, the feeling of letting the strong wind blow across his face and the feeling of the suddenly widened vision at the moment of reaching the top are really interesting.

At the top of Kunlun Mountain, Wang Qi changed direction, went north to find the upper reaches of the river, and then went all the way down.

He is now a Jindan cultivator. In the absence of enemies, he can fly around the world in more than ten hours-of course, the environment of the West Sea where there are enemies everywhere is another matter. Along the way, he spent much more time playing than flying.

"Let me see... well, this local specialty food in Jiangbei... well, bran rice cake, and noodle tea... and then the scenic spots... there are no underworld gods, and the most famous scenery is still in spring and autumn." Wang Qi fluttered in a travel journal in his hand, as if he was planning to play first.

However, Zhen Chanzi burst out with unimaginable stubbornness: "Let me go to Luojia Village in the north of the city first!" (To be continued ~^~)