For mathematicians on Earth, being driven to mental collapse by a theorem at most leads to insanity. The number of mathematicians who have been in mental institutions is only slightly less than that of literati.

But for cultivators in Shenzhou, a broken Dao Heart can potentially lead to physical death.

Therefore, Godel's Incompleteness Theorem poses a great danger to Shenzhou cultivators.

Even for someone like Wang Qi, who understood this theorem in his past life and personally derived it in this life, encountered danger.

However, Wang Qi knows that this shouldn't be a mortal danger.

Just to be safe, he preemptively estimated ten possible states and devised twelve contingency plans.

Among them, "Condition B" refers to his current state.

Collapse indeed occurred, significantly, and no means could stop it. But at the last moment, the collapse automatically ceased, leaving Wang Qi with a sliver of life.

The viewpoint "All axiom systems are incomplete" is merely a common misunderstanding of Godel's Incompleteness Theorem. Actually, complete systems known to mankind exist. For example, Euclidean geometry can be axiomatized into a complete system. Moreover, Godel's Completeness Theorem that Wang Qi proposed several years ago [in Shenzhou, it's called "Wang's Completeness Law"] also demonstrates the completeness of all first-order logical systems.

The collapse is not endless. At least within Wang Qi's cultivation, there are things that "incompleteness" cannot doubt or overturn.

Wang Qi felt his situation. His Life Fire was barely flickering, his life teetering on the edge. Yet his pen still hadn't stopped.

Because he had prepared and left behind contingency plans.

The ring "Mathematics" on Wang Qi's hand and the shoulder guard "Discrete" on his shoulder lit up sequentially, then Jarvis input a new spell into Wang Qi's body.

——"Because of incompleteness, we can never prove an axiom system to be free of contradictions," this is also a misunderstanding of Godel's Incompleteness Theorem.

What Godel's Incompleteness Theorem denies is the possibility of "proving a system's consistency from within the system," a process described as complex but its core idea uses the so-called "self-reference."

——This statement refers to itself.

——This set contains itself.

This is also the source of many paradoxes in logic. Earth's "Barber Paradox" and Shenzhou's "Alchemist's Paradox" ultimately stem from the question of whether a set should include itself.

And this incompleteness law can be bypassed.

You cannot prove a system's completeness from within, but you can prove it from outside.

We cannot prove the system based on the system itself, but can prove it through another system.

Wang Qi conducted related theoretical research five or six years ago.

Indeed, the ideas of the Bourbaki School!

Proof from the outside!

The Bourbaki School, originally something derived from Earth's mathematical community after being heavily impacted by Godel's Incompleteness Theorem. By Shenzhou's classification, it also belongs to the Li Sect. And cultivation methods deduced according to Bourbaki School theories can just support the mana systems collapsing due to incompleteness law.

The support provided by Jarvis is recalibrating Wang Qi's collapsing mana within him.

Meanwhile, Wang Qi's paper is nearing its conclusion.

"Conclusion... Thus, I can prove that for any computational system, if it includes an arithmetic system, then this system cannot simultaneously satisfy completeness and consistency. For any arithmetic system, if it includes an arithmetic system, then we cannot prove its consistency from within the system..."

——Completeness and consistency cannot both be satisfied; this system is either self-contradictory or contains propositions that cannot be proven nor refuted!

Boom!

An electrical flash seemed to explode in Wang Qi's mind. He felt mentally invigorated, never having felt so clear-headed.

On another note, his mana system collapsed and then re-formed, almost as if he had experienced Nirvana. His cultivation level remained at early Sage Core Formation, but the essence of cultivation subtly transcended.

Pop pop pop pop...

A series of crisp sounds echoed in the air, like tearing silk. That was Wang Qi's unwittingly scattered mana disturbing the surroundings.

He stood up, then sat down again, pondering. He picked up a sheet of paper again, wrote an appendix, and reiterated the essence of Godel's Incompleteness Theorem, attaching papers he's authored intermittently over these years regarding "Transfinite Induction" and "Arithmetic Axiomatization."

Even he, while writing this paper, encountered a massive collapse of cultivation method. How would others fare?

So, I must add a bit of "insurance measure" to proceed.

Although Godel's Incompleteness Theorem occupies a very important place in mathematical history, the proof process is relatively simple, and the paper itself is quite short. With Wang Qi's speed of thought and writing, he finished the entire paper in just a couple of hours.

He pushed the door open to leave, and Chen Youjia suddenly lunged at him, grabbing his collar and reprimanding him: "What the hell are you doing?"

"Writing a paper..."

"You almost died..." Chen Youjia's previously indifferent expression became noticeably marked by anger and grievance in her eyes. She used mana to keep the tears from spilling over her eyes, gritting her teeth, "You almost died... You almost died! What kind of paper are you writing?"

Only then did Wang Qi remember, while he had sealed the doors and windows, he had forgotten to block his aura.

"Back then, Senior Pu, who pioneered the Mysterious Path, dissipated his power when he discovered discontinuity." Wang Qi patted Chen Youjia's shoulder: "It's more or less the same principle..."

Chen Youjia held her breath for a moment, then stabilized: "Really..." She buried her head in Wang Qi's shoulder: "Is that so..."

If anyone else claimed they were about to overturn the existing academic system, Chen Youjia would most likely think he was crazy, or fundamentally incompetent. But for some reason, when Wang Qi said it, she believed him.

Very strange, although Wang Qi had a glorious past, it was only within the realm of "genius"; to say he could now produce the results of the Carefree Level was not very credible. Yet, Chen Youjia just believed him.

Although Wang Qi was quite insane, he was not insane about this matter.

"Hey, Junior Brother." Mi suddenly appeared and said to Wang Qi: "Master is looking for you."

"Huh?" Wang Qi was a bit surprised. Chen Youjia asked in confusion: "Could it be that Mr. Feng is looking for you? Is it about the paper you just wrote?"

"Absolutely not." Wang Qi was still conscious of the situation. Before writing the paper, he installed several firewalls on his calculator to prevent Feng Luoyi's attention. Moreover, he used mana to shield the surroundings; the dataized cultivation method was also a nemesis of the Spiritual Sense simulated by the calculator. Feng Luoyi could not possibly know anything about this.

If forced to say, maybe... maybe it's because of some "ideas" he previously expressed?

Wang Qi felt some uncertainty. He quickly used his calculator ring to complete the paper's integration, then placed the manuscript paper in his storage bag, sealing it with strong cultivation to ensure that no one could open it.

Chen Youjia was right here. Letting her see it would be really dangerous.

With a bit of pressure and excitement, Wang Qi once again ventured into the Ten Thousand Immortals Illusion Realm.

Feng Luoyi was already waiting for Wang Qi in one of the illusions. The place still looked like a library. Feng Luoyi and Wang Qi sat opposite each other. The elder's face had a faint tinge of anger as he reprimanded: "Wang Qi... what have you been thinking about lately?"

"Recently, I've been trying to describe mathematics in straightforward language. This process, since it doesn't involve overly complicated mathematical concepts, is quite suitable for dissemination." Wang Qi respectfully said: "I plan to print it into a book and spread it worldwide, to propagate our Wanfa Sect's teachings, hoping that everyone will know about our Wanfa Sect's principles, inspiring more people to join."

Popularizing knowledge is also a great way to gain fame.

Hawking is the best example. This is also one of Wang Qi's long-term plans.

Feng Luoyi accepted Wang Qi's explanation but snorted coldly: "It's good that you have this intention. But don't you think some 'principles' you want to propagate are rather reckless?"

Wang Qi feigned ignorance: "What do you say, teacher?"

"'Numbers themselves are transcendent objects beyond our thinking.' 'Logically, we cannot deny the possibility of one plus one equalling three.' 'No one worries about the matter of one plus one equalling three.'" Feng Luoyi glared at Wang Qi: "What do you mean to say? You surely don't mean that everyone relies on an ultra-logical intuition to complete the initial accumulation of mathematics, right?"

"I just demonstrated a possibility." Wang Qi maintained a low profile: "Actually, before the Master of Calculation proves the completeness of set theory, we can only say 'it's possible.' We certainly can't deny the logical possibility of proving 'one plus one equals three.' As for the other two points, I was merely stating facts."

Feng Luoyi shook his head helplessly: "When exactly did you fall towards the Mathematics Lord's sect?"

"I have never leaned towards the Mathematics Lord's sect." Wang Qi respectfully said. Perhaps in terms of "completeness," the Li Sect's thoughts were utterly defeated, but their research approaches and working methods were still commendable. At least, Wang Qi thinks so.

"But your ideas are becoming somewhat dangerous."

Wang Qi respectfully said: "Because, in my view, existing logic is truly unreliable."

"Unreliable?" Feng Luoyi laughed. Wang Qi felt large waves of aggressive data streams almost overwhelming him.

"You should explain how it's unreliable? What's unreliable?"

"Can you find anything in the existing system that it cannot explain?"

Wang Qi replied: "Only what's currently existing."

"Could you possibly see what's not existent now? Did you create a new mathematical branch and then prove it can't be axiomatized?"

"No, teacher." Wang Qi shook his head: "I didn't create a branch; I calculated the future."