"Mathematics... incomplete?" Upon hearing this sentence, Bao Xiaoya's first reaction was—Are you freaking kidding me?

Mathematics incomplete?

How could mathematics possibly be incomplete?

Mathematics must be complete!

Even those whose mathematical views are completely different from the Master of Calculation agree with his famous saying, "We must know, we shall know."

For the Master of Calculation and his followers, mathematics must be complete and consistent. In the eyes of these cultivators, the essence of mathematics is 'form' made up of operators. Every statement formed by operators can be judged as "correct" or "incorrect." A correct statement can always be proved using another combination of operators. Moreover, a statement can only be judged as either "correct" or "incorrect," there can never be a situation where something is "both correct and incorrect" or "neither correct nor incorrect."

Completeness, consistency, and decidability are the foundations of the beliefs of the Master of Calculation and the Geting Sect.

For general mathematicians, although this is inevitable, they instinctively feel that mathematics must be complete.

Those cultivators who exhaustively study the laws of material flow have repeatedly found, to their amazement, that when they plan to attack a particular domain, the arithmeticians of the Wanfa Sect have already prepared the 'seeking Dao artifact'—mathematical formulas. From the early Elemental Power Sect to the Guiyi Alliance and then to today's Mysterious Palace, these sects that empirically explore the heavens and earth have continually provided the Wanfa Sect with more and more formulas—ingenious formulas derived from all things under heaven and the Daoist Natural Law. These formulas have fed back into the development of mathematics.

In the eyes of the cultivators of the Wanfa Sect, everything in the universe moves respectfully according to mathematical laws.

By saying mathematics is incomplete, are you saying that this world is flawed, that the Heavenly Dao has problems?

Bao Xiaofeng was steadier than his younger sister. He pressed down on Su Junyu's shoulder, transferring a stream of inner qi: "Su Junyu, stabilize your spirit, guard against the growth of heart demons..."

On the other side, a Grandmaster from the Yangshen Pavilion couldn't sit still and came over to ask: "Need any help?"

Su Junyu smiled, waved off Bao Xiaofeng's hand, and saluted the Primordial Spirit Master, saying: "I am fine, thank you, Uncle Master."

After dismissing those from the Yangshen Pavilion, Su Junyu's eyes were blazing as he said, "I'm not crazy. I'm just stating a possibility."

"A possibility? You call this a 'possibility'?" Bao Xiaofeng frowned: "Are you crazy? Is that possible?"

"Until completeness is proven, no one can deny this possibility," Su Junyu said earnestly: "We've never truly known..."

"This is the result of believing in the Li Sect," Suo Wenhong said disdainfully, frowning: "This is purely... purely..."

For disciples of the Lian Sect, some things are self-evident and not to be denied.

For the Li Sect disciples, some things are inevitable, just not yet proven, and can't be denied.

Only on this point have the views of the Li Sect and Lian Sect been aligned.

Qin Chuanhe also frowned: "Proof?"

Su Junyu shrugged: "For now, there is none."

"Without it, you dare to say such things..."

"Well, it's not a formal occasion anyway," Su Junyu was indifferent.

Bao Xiaofeng was curious: "Su Junyu, what on earth is your thought process?"

"I think this study right now is definitely to pave the way for or buffer some new theory. However, what we are learning now is all about 'axiomatization' or 'proof of axiomatic system completeness.' These things, not to speak of a hundred or two hundred years ago, have had new breakthroughs several years ago. It definitely can't be called revolutionary." Su Junyu slowly said, "Since what we are studying now is not the revolutionary content itself, it must be something preceding that revolutionary content or something that can counteract the shock of that revolutionary content."

Bao Xiaofeng nodded, indicating that Su Junyu's words made sense.

"Since that revolutionary theory is definitely related to what we've been learning these past few days, then starting from what we've been learning, it's relatively simple to think about."

"And have you noticed that everything we've learned has a characteristic—they are all products of 'a certain idea,' ways to cleverly circumvent 'a certain high wall.'

Bao Xiaoya immediately reacted: "Senior Brother Wang Qi's idea from seven years ago... this clever approach to... circumventing the proof of higher-order logical completeness!"

—Again with this Wang Qi...

Qin Chuanhe's brow involuntarily furrowed: "The things we are learning are ways to circumvent the proof of higher-order logical completeness, so you speculate that mathematics is incomplete? Why don't you say that completeness has been proven?"

"If completeness had been proven, then these things would be meaningless!" Su Junyu retorted: "Moreover, complete mathematics doesn't contradict any of our intuitive senses! That wouldn't count as 'revolutionary' then!"

Suo Wenhong shook his head: "Absurd."

Qin Chuanhe shouted angrily: "Proof! For this idea, you must provide proof!"

By the end of this conversation, almost everyone left with resentment. Meanwhile, Su Junyu's brow furrowed deeply once again.

Yes... proof, I must provide proof...

—But, Wang Qi, if you've truly achieved such an incredible feat, how did you prove it?

For the first time, Su Junyu felt somewhat powerless. He contemplated the principle of "incompleteness," which was inspired by Wang Qi's notes stating that "even the simplest numbers have meanings beyond human comprehension." He could not independently figure out how to prove it...

Due to cognitive reasons, Su Junyu did not realize that Wang Qi and Feng Luoyi had already hinted at the key to this proof.

"Silver-Winged Assassin," "Self-reference."

In the end, Su Junyu sighed deeply.

—Damn, I can't keep up with that kid's thinking anymore.

..................................................................

In the Ten Thousand Immortals Illusion Realm, Feng Luoyi and Fairy Ruoche witnessed the entire scene.

Fairy Ruoche's tone carried a touch of mockery, ninety percent bitterness: "It seems your and Wang Qi's efforts were in vain. Not a single person realized that the Mirage Game was the key to solving this puzzle... Self-reference, self-reference... haha."

Feng Luoyi's tone was calm: "It's ultimately a good phenomenon."

Fairy Ruoche asked, "What should we do next? Should we give a more explicit hint?"

Feng Luoyi shook his head: "No need for now. Our main goal is to guide and observe the changes in their thought processes." He paused and then said, "Let's observe for another seven days. If this doubt isn't ignored or forgotten, then let the first group of Primordial Spirit Cultivators join this training."

Fairy Ruoche smiled bitterly: "Watching them, each one clearly holds the key yet fails to enter the gate, feels so familiar... For centuries, I've been like a headless fly, unaware that the treasure mountain is right beside me. 'The Alchemist's Paradox,' haha, such an ancient theory. Self-reference... self-reference..."

In the end, all her reflections turned into a sigh: "Why didn't I think of it?"

"The average person pondering this question only gets trapped in a maze of self, hard to escape. However, there's a type of genius who can cut through layers of illusions and see the core." Feng Luoyi's tone was calm.

Godel's Incompleteness Theorem—the proof of Wang's Incompleteness Law, is not achieved overnight. It also requires a winding and arduous process of thought.

"Is Wang Qi that genius?"

"He is undoubtedly a genius." Seeing Fairy Ruoche's low spirits, Feng Luoyi changed the topic: "Speaking of, how is Ximen Master doing?"

"As for Master..." Fairy Ruoche smiled bitterly: "Perhaps the Knowledge Barrier is too daunting. He hasn't realized the problem in this area at all. And... and..."

Feng Luoyi frowned: "And what? Is there another issue?"

"In fact, within the Genting Sect, a certain ideology has emerged." Fairy Ruoche felt a bit ashamed to begin: "They say, the statements and proofs of the Incompleteness Law are too arbitrary. 'Mathematical Incompleteness' is only a short phrase, but with operators... it's almost inexpressible. They believe that as long as they refuse to acknowledge this as mathematics and perceive it as a 'word game,' they can... they can avoid Dao Heart fluctuations."

"Ha." Feng Luoyi showed a look of mockery: "So, the Mathematics Lord criticized us this way, and now we're using it to criticize others?"

Fairy Ruoche lamented: "But... it really isn't a mathematical statement... Maybe, this ideology can be used to protect the teacher..."

Feng Luoyi angrily said: "Do you want to make Ximen Master a cowardly mouse?"

Fairy Ruoche sighed: "Incompleteness Law, its persuasive power is still somewhat lacking."

.........................................................

"It seems the persuasive power of the Incompleteness Theorem is indeed somewhat lacking..."

This was what Wang Qi thought after his seventh failed attempt to send a Spirit Message to Chen Youjia.

Godel's Incompleteness Theorem has displayed extraordinary value in Earth's philosophical domain. Along with the uncertainty principle of quantum mechanics, it forms the basis of modern human cognitive concepts. Yet, comparatively, it holds less significance in the mathematical domain. It's a mathematical milestone, but its traces are rarely seen in later territories.

Because, in the eyes of practical people, Godel's Incompleteness Theorem is meaningless. And in the eyes of theoretical people, Godel's Incompleteness Theorem is not very mathematical.

"Should I throw more fuel on the fire..."

If I were to add more fuel, perhaps the resistance my teacher faces in promoting this theorem would lessen significantly... Youjia wouldn't have to stay in Wanfa Sect for so long... and my plan could be implemented faster...

Wang Qi's thoughts gradually steered in this direction.

If something is irrefutably persuasive...

"Should I solve Ximen 23 Questions' first question?"

Wang Qi just happened to know how to increase the persuasiveness of his theoretical system.