"After hearing this story, you all should know the content of my thesis. Regarding this, I find it difficult to say whether it's a blessing or a misfortune."

Wang Qi's tone was calm and steady, as if the astonishing theory just now wasn't spoken by him, but someone else.

"Blessing?" Ai Kemeng looked up at Wang Qi, slightly smiled bitterly, then shook his head: "I can't see anything lucky in this."

Wang Qi faintly smiled: "If one day, we can truly establish a set of logic, like the Puppet Judge determining the Mechanical Man itself in my story, to determine the truth of a mathematical proposition, then what's the purpose of us arithmeticians existing? I'm afraid by then, Shenzhou will no longer need the Wanfa Sect. An ultimate computing device would outweigh all of us."

All the cultivators of the Wanfa Sect hung their heads, divine light flickering in their eyes.

If it really comes to that point, then the Wanfa Sect indeed has no reason to exist. An ultimate computing device — an ultimate magic artifact that can explain any issue in mathematics in a short time. That's practically a thing close to the Dao.

Once reaching that step, mathematics will have reached its end.

So, has mathematics reached its end now?

I'm afraid not yet.

On Earth, countless scholars once clamored "Physics has reached the end," "Humankind is about to grasp the ultimate truth." That era was the pinnacle of classical physics—yes, merely classical physics.

People of that era even declared, "The sky of physics is bright and sunny, with only two small dark clouds."

This was simply because they couldn't see the next step.

Unable to see the next step, they believed they had reached the limit. In truth, they hadn't even barely reached the initial realm.

How similar is Shenzhou's current mathematics to that time?

Now it seems, the Heavenly Material Circulation of Shenzhou is as far from the true Heavenly Dao as Mathematics Dao is from the true Heavenly Dao.

After hearing Wang Qi's words, all the arithmeticians finally regained some enthusiasm. Yet, there were still sighs: "That may be true...but who dares to pursue the Mathematics Dao in the future?"

Wang Qi's incompleteness and undecidability are almost like slapping a tag of "nihility" on the entire Mathematics Dao.

Under such circumstances, who would dare to explore the Mathematics Dao? You might not even dare to believe that the Dao you seek is meaningful.

This is not a problem of a weak Heart of Seeking. Because Wang Qi's incompleteness and undecidability are derived from mathematics itself, the proof process is impeccable.

This is like saying, if Buddha suddenly engages in a massacre at a dharma assembly, devouring human flesh, transforming into a tusked demon, even the most devout Buddhist followers may not withstand such a shock.

Wang Qi, this is stripping away the gentle facade of mathematics, exposing its terrifying essence.

Seeing the expressions of the Carefree Practitioners, Wang Qi smiled bitterly in his heart.

The mathematics world on Earth, after going through this turmoil, became despondent for twenty or thirty years. Indeed, there were still mathematical achievements during this period, but it couldn't match the prosperity of before.

To ask these Carefree Practitioners to immediately accept the principles of incompleteness and undecidability is, indeed, demanding too much.

However, regarding the question of "how to research," Wang Qi still has some thoughts.

"Actually, as for how mathematics should proceed in the future, I do have some ideas."

Wang Qi's words caused a slight disturbance among the listeners.

Envisioning the future is not something everyone is qualified to do. Those here are the top arithmeticians of Shenzhou. Besides the Mathematics Lord and Master of Calculation, who else in this world can point fingers at them and say, "This is what you should do next"?

This isn't directed at the masses or low-level cultivators, but at these top mathematicians.

What he is doing is not some irresponsible prediction, but akin to a "Five-Year Plan," a programmatic thing. Those capable of doing this are all people who can lead an era and stand at the forefront. Even if he's not a school leader or at the helm of the Mathematics Sect, he wouldn't be far off.

Geting Sect's former leader, the once Divine Venerable, Cloud Prince Ke Lanyin, also used a program to unify near ancient geometry and transformation groups, ushering in the era of Wanfa Sect's "One Law Leads to Liberation." It was precisely because of this that he was considered someone who could rival the Mathematics Lord. [This refers to the El Langen Program]

And what he summarized was only the domains of geometry and group theory.

What Wang Qi spoke of, however, was the entire mathematics world.

If an ordinary person said this, even assuming he was a Carefree Practitioner, these top-level cultivators would probably reprimand him.

However, there was no one who dared to criticize Wang Qi.

The second and tenth questions of the Ximen Twenty-three were too significant. Many Carefree Practitioners from the Genting Sect had tried to solve them without seeing a glimmer of hope. Yet, this young man solved them.

In terms of mana, he might still lack some accumulation. However, in the realm of mathematical logic, he is at a peak level.

"In fact, I neither can nor intend to point out a path for the future development of mathematics—I don't even have the qualification for that, and I know it would certainly be in vain. It is actually a ridiculous thing. I have always believed that future arithmeticians, just like past ones, will never tread the old paths. They will see the problems with new perspectives through relationships we can't conceive of, solving the big problems we leave them—though due to our limited insight, we don't know that now."

"But at least I know what I can do."

"Incompleteness and undecidability are not terrible things. The Mysterious Path has proven the loss of certainty in the microcosm. However, this has no impact on the macroscopic world. We are still victorious in a world of cause and effect," Wang Qi said. "Although I proved the lack of completeness and decidability in mathematics, all I did was prove their existence; I still don't know where they occur. We are far from touching the edges of mathematics. In the known field of mathematics, incompleteness and undecidability don't hold much significance."

"At least for now, the mathematics we know has not conflicted with the objective facts of this world."

"Perhaps, one day we will face all of this, just like we need to confront 'uncertainty' when exploring the microcosm. But that day is not today."

"Such research hasn't been conducted yet, and it's still too early for it. At least the existing mathematics isn't sufficient to touch that area. I believe that when we reach that realm, the need for proof will have reached the level of consciousness, and the technology to meet this demand will have developed—but not now."

"And outside of this domain, we still have many meaningful problems to study. For example, the fifth calculation of the Ximen Twenty-three—the Heavenly Transformation Formula and transformation groups; Li's Conjecture—when we gave up using the Heavenly Transformation Formula as the tool to conquer it, new hope was in sight; the senior Calculation King's Law of Quadratic Reciprocity... We can see that mathematics is still progressing. We can still move forward."

"In fact, for Daoist fellows not pursuing metamathematics, incompleteness and undecidability are either non-existent or have no impact on their work."

"As for Daoist fellows in the Dao of Metamathematics, our journey has just begun! We still have many things to do!"

Someone from the Li Sect was disgruntled: "What else can we do?"

—What we pursue has already been destroyed by you.

"Just because I completed Senior Xi's problem, do you think there are no more problems to solve?" Wang Qi shook his head.

In the academic world, solving a problem is essentially "taking away others' jobs." When a conjecture is verified, countless mathematicians who depend on this conjecture for their livelihood become unemployed. For these Carefree Practitioners here, not only was their Dao Heart shaken, but their goals for which they fought were taken away.

It's just that Wang did it in a way entirely contrary to their ideal approaches, so they couldn't accept it for a moment.

But...

"The Master of Calculation once said that if there are problems, it means the discipline is developing." Wang Qi sighed, "Incompleteness and undecidability are the huge problems I've exposed. As for how to bypass this problem—actually, I mentioned it six years ago."

"Six years ago..."

Some arithmeticians showed a complex expression. Indeed, at the Banquet of Completion six years ago, Wang Qi did mention some "ideas." But at the time, no one cared. They all dismissed it as some personal insight from this junior—just an individual's research direction. Even if fully developed, what significance would it have?

But no one anticipated that today, after proving the fallacy of the Master of Calculation's ideas, the thought that no one paid attention to would become the development direction of the Li Sect in the future.

Looking at this youth again now.

Li Sect's rebel?

The future leader of the Li Sect?

Many Carefree Practitioners vaguely saw this point. It's just that emotionally, they couldn't accept it immediately.

Feng Luoyi lightly coughed twice: "I believe you all have gained a general understanding of Wang Qi's thought process and ideals by now. Wang Qi, you can now explain your paper to everyone."

Wang Qi nodded and, with a seventy percent standard, began to articulate his paper "On Computable Numbers and Their Application in Decision Problems."

The Mathematics Lord was still not very interested, shaking his head and saying, "A hundred useless things."

This paper "On Computable Numbers and Their Application in Decision Problems" was similarly not very long. Like "A Tentative Discussion on Unprovable Statements in Volume One of the Arithmetic Chapter of 'All Laws Mathematical Collection' and Related Systems," its proof process was truly simple.

Simple, yet filled with wisdom.

All the Carefree were unconsciously immersed in the piece of work, which itself exudes an alluring brilliance of wisdom. Even knowing that this river of logic would flow toward a thousand-foot abyss, the arithmeticians couldn't help but listen.

Soon, a practitioner stood up, entranced, loudly applauding: "Well done!"

The one who stood was the Pavilion Master of the Thousand Mechanism Pavilion, Master Turing.