The first question among Hilbert's twenty-three problems, the cardinality of the continuum.

The continuum problem, namely the question of whether there are any cardinal numbers between the cardinality of countable sets and that of the real number set.

The so-called "cardinal number" refers to the "absolute measure" of a set. If a set has one element, then its cardinality is one; if it has two elements, the cardinality is two. And so on.

The cardinality of infinite countable sets like "all integers" or "all natural numbers" is denoted as "Aleph Zero"—known in Shenzhou as "Taoyuan Zero Number," the smallest infinite integer.

The ancients of Shenzhou once believed that the total number of numbers, or the infinitely large, is the number of the way (Tao).

Aleph Zero plus one is still Aleph Zero. Aleph Zero plus Aleph Zero is still Aleph Zero. Aleph Zero times Aleph Zero is still Aleph Zero.

Infinity, positive infinity. Ordinary operations have no meaning for this number at all.

So, are there numbers larger than this infinite number in the world?

In fact, there are.

That is the cardinality of the "power set."

If a set has one element "1," then its power set has two elements—"1" and the empty set ∅.

If a set has two elements "1, 2," then it has four power set elements—the empty set ∅, set {1}, set {2}, and set {1, 2}.

By analogy, if a set has three elements, it has eight power set elements. When a set has four elements, the power set increases to sixteen.

The power set of a set always has more elements than the set itself. If a set has N elements, its power set has 2 to the power of N elements.

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

And this "beth1" aside from being the power set of the integer set, is also the cardinality of all real number sets.

The continuum problem can also be summarized as "whether there exists another cardinal number between Aleph Zero and Beth One?"

Is there a cardinality of a set that is clearly greater than one infinite large and smaller than another infinite large?

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

The second and tenth questions among the twenty-three questions are related to the foundations of mathematics, considered extremely important. It is precisely because of the Master's ideas on "completeness, consistency, and decidability" that these two questions are always juxtaposed. But from the questioner's line of thought, the relationship between the first and second questions is actually closer. The first and second questions, the continuum and completeness, are fundamentally connected.

The first question leads to the problem of the second question, and the solution to the second question inspired the solution to the tenth question.

These questions can be seen as a system.

Of course, each question among the twenty-three questions of Ximen is related more or less to the other twenty-three questions, forming a subtle whole. This whole encompasses almost all aspects of mathematics, where solving one question will cause the entire mathematics to show great progress. Each arithmetician's research is more or less related to one of these questions.

No arithmetician has ever achieved this point; it has never been done before and is unlikely to happen in the future. In the history of mathematics, the twenty-three questions represent an extremely grand leap.

And Wang Qi has precisely noticed this point. He has already solved the second and tenth questions. Now, presenting the solution to the first question is not particularly earth-shattering.

Additionally, the continuum hypothesis and the proofs of completeness and decidability all hold extremely important positions yet are relatively independent on their own. They are like the first domino in a set, not remarkable in themselves but capable of causing a chain reaction when they fall.

Solving these problems doesn't require very deep accumulation. These problems tend to emphasize "ingenious thoughts."

On Earth, the solvers of the second and tenth questions were young genius scholars, and the solver of the first question strictly speaking didn't even understand mathematical logic—P.J. Cohen's field was analysis, he was merely attracted by this question, that's all.

The solver of the first question, P.J. Cohen himself, couldn't even understand the application of his invented proof method in the field of logic.

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

However, the biggest problem is...

"I don't think I particularly memorized this thing in my past life..." Wang Qi felt a bit of a headache again.

The solution to a quadratic equation is something any middle school student can do nowadays. But how many people know how to prove that solution?

The gap between "knowing" and "proving" is roughly equivalent to the difference between "cultivating the Supreme Heart Method" and "creating your own Supreme Heart Method." The difficulty of the latter multiplies the former countless times.

Moreover, Wang Qi doesn't even remember the "Forcing Method" itself for the first question, only the general direction.

"How long will it take for me to prove the first question on my own now?"

Unconsciously, Wang Qi began pondering this issue.

Until Zhen Chan's reminder pulled him back: "Hey, kid."

"I'm thinking, don't bother me."

"I just wanted to remind you that we've arrived." Zhen Chan sighed, "If you don't descend now, you'll fly right past it."

"Oh, oh." Wang Qi looked below. It was an endless plain. The flat ground stretched to the horizon, with only faint shadows of mountains to the west. That was the northern section of Kunlun Mountain. To Wang Qi's south, a river flowed, spread out like satin on the ground, almost straight. That was the second-largest river in Shenzhou, the Great River. There were clusters of houses extending from the river north and south. Beyond the towns were fields like patches of cloth. The new season's wheat had just sprouted and looked lush and green from high above.

Green and brown were the colors of this land.

"Jiangbei City, we've arrived." Wang Qi marveled.

Such scenery is not easy to see north of Kunlun. The Dragon Clan nailed the West Sea plate in place, causing mysterious movements on the Shenzhou West Continent plate. The terrain west of Kunlun is far more complex than to the east of it.

Wang Qi flew all the way from west of Kunlun, having long seen the layered mountain scenery. Seeing this plain now, he felt refreshed, and the bit of gloomy mood brought by pondering difficult questions completely dissipated.

Come to think of it, though he had cultivated for over ten years, he never had the time to see this world. In the initial years, he was focused on his studies at the Immortal Institute; then, due to suspicion of being a "Banished Immortal," he was "softly confined" in the Divine Capital. Apart from trips to Xiaodong with Chen Feng and to Tianling Ridge with both Chen Feng and Chen Youjia, he hardly left home. After reaching Leiyang, he mostly stayed at sea, barely having the chance to properly see this earthly realm he lived in.

The universe is infinitely vast, but only this planet holds special meaning for him.

Zhen Chan also sighed, "The Luo Family's bloodlines should... should be here, right?"

Luo Family from Liwu, possibly the place where Zhen Chan's godfather's family's descendants exist.

Upon hearing this, Wang Qi patted his ring: "Yeah, yeah, I flew for over half a month just to help you find your roots!"

Zhen Chan was exasperated: "Why does this sound like dead leaves returning to their roots? And why is it so filthy?"

"No, no, others seek roots to find ancestors, you're seeking 'roots' to find descendants." Wang Qi looked innocent, "As for filthy... Am I not helping you find your bloodline root? Is there any other meaning in this?"

That day, after bidding farewell to Chen Feng, Wang Qi also began his "journey to help the old man in the ring find his roots." He first went to see Mao Zimiao, as a New Year blessing. After a simple meal, Wang Qi headed north, into the territory of Split Peak Island.

Split Peak Island is small and cannot be compared to the Fusang Continent. It is located at the junction of the West Sea and the North Sea. There aren't any overly large biological communities here, and the number of the Demon Clan is much less than in the West Sea. Without the terrifying numbers, the Sea Demons aren't much of a threat. On the other hand, it's also an extraordinary place. The first Yuanli Master of Modern Cultivation emerged on this island, receiving enlightenment, for the typical Modern Cultivator, Split Peak Island could even be called a semi-Holy Land. The mortals near Split Peak Island also live contently and in harmony.

When Wang Qi arrived at Split Peak Island, it was the first day of the year. Snow was swirling on the island, wearing a silver-white garb, displaying Northern Country's scenery. Besides enjoying the snowy views, Wang Qi also visited Ai Changyuan, who went home for the New Year.

Wang Qi and Ai Changyuan had forged a friendship in their youth. However, Wang Qi had an advantage with his cheat device, leading in cultivation by about three or four years over Ai Changyuan. Before breaking through realms, the Foundation Establishment Stage Ai Changyuan couldn't beat the already Core Formation Wang Qi, leaving the young master of the Ai Family somewhat melancholic.

After leaving Split Peak Island, Wang Qi went against the flow of the Luo River, all the way to the western foot of Kunlun. Then, at the foothills of Kunlun Mountain, Wang Qi abandoned the Flying Escape Technique and even the Protective Gang Qi, relying purely on his physical prowess to climb the world's premier mountain.

This wasn't some kind of "training" or "raising difficulty." Even without the Flying Escape Technique or the Protective Gang Qi, as a Golden Core Cultivator, Wang Qi's physiological capabilities far surpassed any ordinary creature. Using just his hands and feet, he could easily climb over this mountain.

However, that feeling of letting the strong wind blow against his face, and the sudden expansion of his view at the summit, was indeed very interesting.

Atop Kunlun Mountain, Wang Qi changed direction, heading north to find the upstream of the Great River, then continued along the way down.

He is now a Golden Core Cultivator. In the absence of enemies, he could fly around the world in over ten hours—of course, in enemy-dense environments like the West Sea, that's another matter. Along the way, he spent more time enjoying the sights than actually flying.

"Let's see... Hmm, this Jiangbei's local specialty food... Hmm, husk rice cakes, also flour tea... And then the scenic spots... No Netherworld God, the most famous scenes are only in spring and autumn." Wang Qi skims the travel guide in his hand, seemingly planning to explore around first.

However, Zhen Chan erupted with unimaginable stubbornness: "Take the old man to the north of the city to see the former Luo Family Village first!"