The first of Hilbert's twenty-three questions is the cardinal problem of continuity.

The continuum problem, i.e., the problem of "there is no other cardinality between the cardinality of the countable set and the cardinality of the real set".

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

The cardinality of an infinite set of "all integers" and "all real numbers" is recorded as "Alev zero", which is called "Dow zero number" in China, the smallest infinite integer.

The ancients of China once believed that the total number of numbers, infinitely large, was the number of the Tao.

Alev zero plus one or Alev zero. Alev zero plus Alev zero or Alev zero. Aleph zero multiplied by Aleph zero or Aleph zero.

Infinity, infinity. The ordinary way of operating is completely meaningless for this number.

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

Actually, there is.

That is the cardinality of the "power set".

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

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

And so on, when a set has three elements, then it has eight power sets. When the set element is increased by four, the power set is increased to sixteen.

The power set of a set will always have more elements than the set. If a set has N elements, then it has a power set of 2 to the Nth power.

The power set of infinitely countable sets, the Alev to the zero power of two, is the second infinitely large number discovered by mankind - Alev one.

And the continuum problem. It can also be summarized as "Is there another cardinal between Alev zero and Alev one?".

Is there a cardinality of a set that is explicitly greater than one infinity and less than the other infinite?

This is the first of the twenty-three questions.

Among the 23 questions, the second and tenth questions are related to the foundation of arithmetic and are considered to be extremely important. It is precisely because of the idea of "completeness, consistency, and decidability" of the counter. So these two questions have always been compared. However, from the perspective of the "questioner", the relationship between the first question and the second question is closer. The first question and the second question, continuum and completeness, are fundamentally connected.

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

These problems can be regarded as a system.

Of course, each of the twenty-three questions of Simon is more or less related to the other twenty-three questions. The whole twenty-three questions are faintly a whole. And this whole, covering almost every aspect of arithmetic, once the problem is solved, the arithmetic as a whole will show a huge progress. And the research of each arithmetic is more or less related to one of the twenty-three questions.

No arithmetic has ever been able to do this, not before, and not in the future. For the history of arithmetic. Twenty-three questions is a magnificent leap.

And Wang Qi also took a fancy to this. He has already solved the second and tenth questions. Now the solution to the first question is actually not a particularly shocking thing.

Other than that. The continuum hypothesis is similar to the proof of completeness and the proof of decidability, which are the kind that have an extremely important position, but are relatively independent in themselves. They're like the first piece of a domino that isn't good in itself, but as soon as they fall, they set off a chain reaction.

If you want to solve these problems, you don't need to have a lot of deep accumulation. These are all problems that focus on "ingenuity".

on Earth. The second and tenth questions were answered by young genius scholars. And the person who answered the first question did not even strictly understand mathematical logic - P.J. Cohen's specialty was analysis, and he was only attracted to this one question, nothing more.

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

That is, this one outcome. The same can be pushed into the "flash of genius inspiration".

The biggest problem, though......

"I don't seem to have memorized this thing in my previous life......" Wang Qi felt a little headache again.

The solution of binary equations is now a middle school student. But how many people know how to prove that solution?

The distance between "knowing" and "proving" is roughly equivalent to "cultivating the supreme mind" and "creating the supreme mind". The latter is countless times more difficult than the former.

What's more, Wang Qi doesn't even remember the solution to the first question, the "force forcing method" itself, only remembers a general direction.

"How long will it take for me to be able to put aside the proof process of the first question on my own?"

Wang Qi couldn't help but think about this problem.

Until the real son came out and reminded: "Hey, boy." ”

"Don't worry about things. ”

"I wanted to remind you that it's time to get there. Zhen Yanzi sighed: "If you don't land, you will fly over." ”

"Oooh. Wang Qi looked down. It was an endless plain. The flat ground stretches all the way to the skyline, and only to the west is the shadow of the mountains. That is the northern section of Kunlun Mountain. To the south of Wangqi, there is a river, which is spread flat on the ground like satin, almost straight. That is the second largest river in China. There are private houses gathered, with the river as the center, all the way to the north and south. Beyond the town, there are fields like handkerchiefs. Now the new season of wheat has just sprouted, and it looks green and fluffy from above.

Green and brown are the colors of the ground.

"Jiangbei City, it's here. Wang Qi sighed.

This kind of scene is not so easy to see in the north of Kunlun. The dragon clan nailed the plate of the West Sea, which also made the movement of the West Continent plate of Shenzhou strange, and the terrain west of Kunlun was far more complex 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 and mountains. Now that I saw this to the plain, I suddenly felt relaxed and happy, and the depression I brought when I was thinking about the problem just now disappeared completely.

Speaking of which, although he has cultivated immortals for more than ten years, he has never been able to take the time to look at this world. In the first few years, he concentrated on his studies in the Immortal Academy, and then was "placed under house arrest" in Shenjing because of the suspicion of "Immortals". Except for going to Xiaodong once with Chenfeng, and going to Tianlingling once with Chenfeng Chen Youjia, they basically haven't been out. After arriving in Leiyang, he stayed at sea again, and almost never had a good look at the world where he lived in this life.

This universe is infinite, but there is only one moment of planet, which is special to him.

Zhen Yanzi also sighed: "The bloodline of the Luo family should be ...... It should be right here, right?"

The Liwu Luo clan is a place where the descendants of the family of the righteous father of the true son may exist.

When Wang Qi heard this, he patted his ring: "Well, I've been flying for more than half a month, just to help you find your roots!"

"Why does this sound so much like a dead man's fallen leaves, and why is it so filthy?"

"No, no, no, others are looking for their ancestors, and you're looking for 'roots' to find descendants. Wang Qi looked innocent: "As for the filthy...... Am I not trying to help you find the root of your bloodline? Is there something else in this?"

On that day, after saying goodbye to Chenfeng, Wang Qi also began his "journey to help the old man of the ring find his roots". He first went to find Mao Zimiao, which was regarded as an early visit. After eating a light meal, Wang Qi first went all the way north and entered the category of Rift Peak Island.

Rift Island is very small, not comparable to the Fuso mainland. It is located at the junction of the West Sea and the North Sea. There are no overly large biomes here, and the number of demon clans is much smaller than that of the West Sea. And after the Kraken is going to lose the most terrible number, there is nothing to worry about. On the other hand, it is also an unusual place. The Venerable Master of Yuanli, the first cultivator of the Present Dharma, was born on this island and became enlightened, and for ordinary Present Dharma cultivators, the Rift Peak Island can even be called half of the Holy Land. The mortals near Rift Island can also be called to live and work in peace and contentment.

When Wang Qi came to Rift Island, it was the first day of the new year. The snow is falling heavily on the cracked island, covered in silver, and it is a northern scenery. While enjoying the snow scene, Wang Qi also went to visit Ai Changyuan, who went home for the New Year.

Wang Qi and Ai Changyuan made a friendship together when they were young. It's just that Wang Qi has opened the cheat, and he is about three or four years ahead of Ai Changyuan in practice. Before breaking through the realm, Ai Changyuan in the foundation building period couldn't beat Wang Qi, who had already formed a Dan. This made the young master of the Ai family a little depressed.

After leaving Rift Island, Wang Qi went against the Luo River and went all the way to the western foot of Kunlun. Then, at the foot of Kunlun Mountain, Wang Qi abandoned the method of flying away, and did not even use the protective qi, and walked towards the first mountain in the world purely with his hands and feet.

This is not an "experience" or a "difficulty". Even if Wang Qi doesn't need the method of flying away or the protective qi, he is also a Jindan monk, and his physiology is far superior to that of any mortal creature. With just his hands and feet, he was able to climb the mountain with ease.

It's just that the feeling of letting the strong wind blow across your face, and the feeling of suddenly widening your field of view at the moment of reaching the top, is 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 monk of Jindan. In the absence of an enemy, it can fly all over the world in just over 10 hours -- of course, the environment in the West Sea where enemies are everywhere is another matter. Along the way, he spent much more time playing than flying.

"I'll take a look...... Well, this is a local specialty of Jiangbei...... Well, the bran rice baba, and the noodle tea...... And then the places of interest...... There is no underworld, and the most famous scenery is still in spring and autumn. Wang Qi had a travelogue in his hand, and he seemed to be planning to play it first.

However, Zhen Yanzi broke out with unimaginable stubbornness: "Let the old man go to Luojia Village in the north of the city first!" (To be continued.) )