Exploring Immortality Cultivation-Chapter 910 - 239: No Applause

"Next, what I'm going to talk about is the most important concept I've learned from Ximen Master and the work of the Genting Sect! Formalism."

As soon as Wang Qi said this, the Master of Calculation's brow furrowed deeply.

Behind him, more Arithmeticians from the Shaoli Sect became almost outrageously angry.

In his lifetime, Xi Baiche achieved countless accomplishments. However, his views on the essence of mathematics were widely criticized and hard for most Arithmeticians to accept.

In his view, the essence of mathematics is "form". Isolated symbols have no meaning. It is the "form" of these symbol arrangements that constitutes mathematics.

For example, in the fundamental equation "1+1=2", the symbols "1", "+", "=", and "2" are meaningless by themselves; only the complete equation holds mathematical significance.

Meanwhile, Arithmeticians from the Genting Sect wore complex expressions on their faces, mixed with bewilderment and anger.

The idea of "form" includes consistency, completeness, and decidability. It could be said that this was the greatest pursuit and ideal of Xi Baiche's life.

And the one who negated this ideal was Wang Qi himself.

Using Xi Baiche's line of thinking, Wang Qi rejected Xi Baiche's pursuit and also the decades of effort by the Genting Sect.

However, all these complicated emotions did not affect Wang Qi.

Wang Qi understood that at this moment, he represented not himself, but "correctness".

Self-reference is not a mathematical description—while it indeed belongs to logic, whether it is part of mathematics is debatable, and not all mathematicians agree. Such signs are evident on Earth as well as in Shenzhou.

"This statement cannot be proven" is not strictly a mathematical statement.

The second step in Godel's proof method is to transform this non-mathematical or quasi-mathematical statement into a mathematical statement.

On Earth, this is called—"Godel Numbering".

—perhaps in Shenzhou, it might be called "Wang Qi Numbering"?

"When people understand this incompleteness, it's easy to get trapped in layers of self-referential fog, unable to extricate themselves. I suspect this is related to Senior Kang's diagonal argument and infinite cardinals defying our natural intuition. It is precisely for this reason that our predecessors have ignored this principle for centuries."

"To unravel this fog, one must use this formalized method."

Waving his hands, Wang Qi released golden light, conjuring layers of light screens, countless operators dancing upon them, arranging a grand proof.

The second part of Godel's Incompleteness Theorem, and its most profound section, is right here.

"In any axiomatic system, the operators used, and the axioms that can exist, are finite. Hence, the statements that can be enumerated by these axioms and operators must also be countable—infinitely countable, cardinal of zero, the number of natural numbers. That's how it's described."

"And the potential statements, their length must also be countable. And since it is within the domain of countable infinity, we can enumerate them with natural numbers. Each number is unique."

"We can then establish a set called 'Central Heaven', this set 'Central Heaven' includes all potential statement 'numbers'. Within an axiomatic system, all possible statements must fall within this numbering."

...

As the lecture reached this point, it began to move beyond what the majority could understand. Terms like "countable infinity", "cardinal", and "ordinal" were beyond their grasp.

Even among the Carefree Practitioners present to hear Wang Qi's lecture, a few displayed bewildered expressions.

Every step the young man described was understandable to them. But when all these steps came together, what mathematical significance did they hold?

Completely incomprehensible!

Many Modern Cultivators watching this "live broadcast" through the Ten Thousand Immortals Illusion Realm experienced a throbbing headache. What seemed like a simple proof seemed infused with endless magic, pulling their minds towards an abyss.

Feng Luoyi had no choice but to sigh and, through "subtitles", explain to all the cultivators watching via the broadcast.

This numbering process, in essence, is "mapping".

Mapping symbols, expressions, and expression sequences in the arithmetic system into numbers—achieving the numbering process through the introduction of "Godel Numbers". This method provides a digital tool for research methods in mathematical logic and other related branches, facilitating the conversion of some discussion objects into natural numbers or functions of natural numbers, allowing related issues to be explored using the theory of natural numbers.

Converting a quasi-mathematical statement into one with mathematical significance.

This is the significance of this "numbering".

As the proof entered the latter half of the second phase, Ai Kemeng from the Genting Sect sighed softly: "Primitive recursion..."

His expression was filled with deep regret and remorse.

—I have also studied this field... would deeper insights back then have spared today's calamity?

Some cultivators from the Genting Sect immediately sent a private message: "Brother Ai, do you have research in this domain? Can you trip up Wang Qi at this step?"

Ai Kemeng shook his head with a bitter smile. Wang Qi hadn't done anything wrong at this step. Could he say something right was wrong?

At the same time, he also made up his mind to go back and study this previously neglected field in detail.

Of course, the people present did not know that this proof of numbering had a significance beyond itself.

It is also the source of recursive theory.

And recursive theory is precisely one of the most important branches of modern logic.

At this point, Wang Qi's proof was also coming to a close.

Wang Qi waved his hands one last time. The operators arranged into the proof of the Incompleteness Theorem.

"This is the entire process."

The entire arena was silent.

At this moment, mathematics was heading towards death.

"Consistency and completeness cannot both be attained; mathematics is non-contradictory but incomplete..." Many disciples of the Wanfa Sect watching the live broadcast almost burst into tears.

After Wang Qi finished proving the Incompleteness Theorem, he sat quietly at the edge of the podium for a moment. Today's lecture was divided into four parts. The first part was about the Incompleteness Theorem; the next part was about the Undecidability Theorem.

Then came the time for questions and summaries.

Even the Carefree Practitioners needed some time to absorb and understand these things.

He closed his eyes to rest, seemingly oblivious to the malice imposed on him by the peak cultivators around him.

What does it matter if everyone is an enemy? This is the domain of the Seeking Dao Seeker, not a place for combat. No matter how you argue, you cannot reverse right and wrong.

After a brief rest, Wang Qi returned to the center of the podium. The whispered conversations that arose due to the short break gradually faded. The attention of countless cultivators was focused on Wang Qi.

Like one poisonous snake after another, seeking the vital point of their prey, aiming for a fatal strike.

Wang Qi continued the lecture: "After discussing 'incompletion', let's talk about 'nothing but'—that is, the Undecidability Theorem."

"The first part of the Undecidability Theorem is the same as the Incompleteness Theorem, both are 'self-reference.' I won't elaborate on this again. I will directly move on to the second part of the Undecidability Theorem proof."

"First, I must thank Master Turing here, thank him for his help with this issue, and for providing an equivalent proof method."

"First, the first proof method I want to introduce is one I proposed myself, utilizing the mechanism of the Narrow-Turing Calculator... and the second part, which Master Turing called the 'Wanfa Wanxiang Style.' It is an algorithm constructed based on the Incompleteness Theorem, complete but inconsistent."

This was also the part Wang Qi and Master Turing had discussed.

The Narrow-Turing Calculator and the Wanfa Wanxiang Style—also known as the Turing Machine and λ-Algorithm—like the second stage of the latter part of the Incompleteness Theorem proof, they have potential that exceeds the proof itself. They each developed into thriving roads in later generations.

Master Turing was also pleased that his not-so-highly-regarded theorem could be propagated. Wang Qi also needed some Carefree Practitioners to show support, making himself appear less isolated and weak, so more arithmeticians could follow him and create new territories in mathematics.

The two could be said to be a perfect match.

The proof process of the Undecidability Theorem, compared to that of the Incompleteness Theorem, seemed to be much easier to handle. Especially the first part, the Turing Machine proof. This story is almost a continuation of the "Puppet Judge" Wang Qi previously lectured on, and the Mirage Game "Silver-Winged Assassin."

Many cultivators not belonging to the Wanfa Sect, especially those from small sects that practice through "art" and "novelist comments," listened with great interest. They could not understand Wang Qi's theories or comprehend the mathematical significance behind them. But they could understand the stories! Their minds were filled with endless stories.

It can be predicted that in a period to come, "Puppet People" will become a popular theme in the mouths of storytellers.

And those Wanfa Sect disciples who were not yet in collapse showed greater interest in the second method.

Bao Xiaofeng was entirely in a state of confusion. "A complete but self-contradictory system"? Does such a thing really exist? Is there any mathematical significance in this?

Wang Qi's lecture started at noon and ended at the stroke of the clock at the evening hour. When the clock at 5 p.m. to 7 p.m. struck in the Shenzhou Land, this lecture was also drawing to a close.

"In summary, we can make such a proof..." Wang Qi took a deep breath and spoke the last two sentences.

"I think we can proudly announce that the most critical second question of the twenty-three Ximen questions, the tenth question, has already been perfectly resolved. The most important consistency, completeness, and decidability proofs in the foundation of mathematical reasoning have also been completed."

"Mathematics is about to enter a new era."

No one applauded.

However, everyone felt—as if a reformative overhaul might truly be coming...