Enter the Immortal Cultivation

"Next, I will talk about the most important concept I learned from Master Xi and the work of the Goting School! Formalization."

As soon as Wang Qi said this, a furrow appeared on Master Xu's brow.

Behind him, more Shaoli School mathematicians were almost furious.

Master Xu Boche had countless achievements in his life. However, his views on the nature of mathematics were widely criticized, and most mathematicians could not accept it.

In Master Xu's view, the essence of mathematics is "form". Individual symbols do not have any meaning. The "form" of the arrangement of those symbols is mathematics.

It's like saying that in the original formula "1+1=2", the symbols "1", "+", "=", and "2" are all meaningless, and only the complete equation has mathematical meaning.

On the other hand, many mathematicians of the Goting School also had mixed feelings. They showed strange expressions on their faces, mixed with confusion and anger.

The concept of "form" includes consistency, completeness, and decidability. It can be said that this is the greatest pursuit of the master of calculation, Xi Boche, and the ideal of his life.

And it was Wang Qi himself who denied this ideal.

Wang Qi used the master of calculation's thinking to veto the master's pursuit and the efforts of the Goting School for decades.

However, these complex emotions did not affect Wang Qi.

Wang Qi understood that what he represented now was not himself, but "correctness".

Self-reference is not a mathematical description - it does belong to logic. But whether it belongs to mathematics is still debatable, and not all mathematicians admit this. There are such signs on Earth and in Shenzhou.

"This statement cannot be proved" is not a mathematical statement in the narrow sense.

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

Earth calls it "Gödel's numeration".

- Maybe in Shenzhou, it will be called "Wang Qi's numeration"?

"When ordinary people understand this incompleteness, they are easily trapped in a fog of self-reference and cannot extricate themselves. I guess this is probably related to the fact that Kang's diagonal proof and infinite cardinality violate our natural intuition. It is for this reason that our predecessors have turned a blind eye to this truth for thousands of years."

"If you want to solve this fog, you must use this formal method."

Wang Qi's hands emitted golden light, which transformed into layers of light curtains, with countless operators flying on them, arranging and listing a great proof.

The second part of Gödel's proof of the incompleteness theorem, and also its greatest part, is here.

"Any axiom system. The operators used, the axioms that can exist, are all finite. Therefore, the statements that these axioms and operators can list must also be countable - infinite countable, Dao Yuan number zero, the number of natural numbers. That's how it is described."

"And the length of these possible statements must also be countable. And since it is still within the scope of countable infinity, we can number it with natural numbers. Each number is unique."

"Then, we can establish a set 'Zhongtian', and this set 'Zhongtian' includes the 'numbers' of all possible statements. Within an axiom system. All possible statements must be within this number."

...

When the sermon reached this point, it had begun to deviate from the scope that most people could understand. What "countable infinity", what "cardinality", "ordinal number". These have exceeded their scope of understanding.

Even the free and easy monks who appeared on the scene to listen to Wang Qi's sermon showed confusion.

They could understand every step that this young man said. However, after these things were put together. What mathematical meaning does it have?

I don't understand it at all!

More modern practitioners who watched this "live broadcast" with the help of the Fantasy of Ten Thousand Immortals were all confused. This seemingly simple proof seemed to contain infinite magic, as if it was going to drag their minds into an abyss.

Feng Luoyi had to sigh and explain to all the monks who watched the live broadcast in the form of "subtitles".

This process of digitization is simply "mapping".

The symbols, expressions, and sequences of expressions in the arithmetic system are mapped to numbers - the digitization procedure of objects is realized by introducing "Gödel numbers". The result of this treatment provides a digital tool in terms of research methods for mathematical logic and other related branches, which can easily convert some discussion objects into natural numbers or functions of natural numbers, and can use the theory of natural numbers to discuss related issues.

Convert a quasi-mathematical statement into a statement with mathematical meaning.

This is the meaning of this "digitization".

When the proof entered the second half of the second stage, in the Ge Ting faction, Ai Keman sighed softly: "Primitive recursion..."

His expression contained a lot of regret and remorse.

——I have also studied this field... If I could have gone deeper back then, would I have been able to avoid today's disaster?

A few monks from the Ge Ting faction immediately sent a private message: "Brother Ai, do you have research in this field? Can you trip Wang Qi up at this step?"

Ai Keman smiled bitterly and shook his head. Wang Qi did not make a mistake in this step. Can he say that the right thing is wrong?

At the same time, he also made up his mind to go back and study this field that was not valued in the past.

Of course, the people present did not know that this numerical proof had more significance than itself.

It is also the source of recursion theory.

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

At this point, Wang Qi's proof has also come to an end.

Wang Qi waved his hands for the last time. The operators were arranged into a proof of the incomplete theorem.

"This is the whole process."

The whole audience was silent.

Mathematics went to death at this moment.

"Consistency and completeness cannot be achieved at the same time. Mathematics is incomplete without contradictions..." Many Wanfamen disciples who watched the live broadcast almost cried.

After Wang Qi finished the proof of the incomplete theorem, he sat quietly at the edge of the podium for a while. Today's sermon is divided into four parts. The first part talks about the incomplete theorem, and the next part talks about the undecidable theorem.

Then, it is time for questions and summary.

Those free-spirited monks also need a certain amount of time to absorb and understand these things.

He closed his eyes and rested, seemingly unable to feel the malice of the top monks around him.

Even if everyone is an enemy, so what? This is the realm of seekers, not a place for fighting. No matter how much you talk, it is impossible to reverse right and wrong.

After a short rest, Wang Qi walked back to the center of the podium. The whispers that appeared because of the short rest gradually disappeared. Countless monks focused their attention on Wang Qi.

Like one poisonous snake after another, they are looking for the vital points of their prey, trying to kill them with one blow.

Wang Qi continued to speak: "After talking about 'incomplete', let's talk about 'nothing more' - that is, the undecidable theorem."

"The first part of the undecidable theorem is 'self-referential', just like the incomplete theorem, so I won't go into details here. I will go directly to the second part of the proof of the undecidable theorem."

"First of all, here I need to thank Master Turing for his help in this issue and for providing an equivalent proof method."

"First of all, the first proof method I want to introduce is proposed by myself, which is based on the mechanism of the narrow Turing calculator... and the second part, Master Turing called it 'the universal method'. It is a complete but inconsistent algorithm constructed based on the incomplete theorem."

This is also the part that Wang Qi and Master Turing agreed on.

The narrow Turing calculator and the universal method - that is, the Turing machine and the λ algorithm, like the second stage of the second half of the proof of the incomplete theorem, have the potential to surpass the proof itself. In later generations, they each developed a flourishing path.

Turing Zhenren was also very happy that his theorem, which he did not think highly of, could be spread. Wang Qi also needed some free and easy monks to show their support, so that he would not be so weak, so that more mathematicians could follow him and create a new world of mathematics.

The two of them hit it off immediately.

The proof process of the undecidable theorem seemed much easier to handle than that of the incompleteness. Especially the first part, the Turing machine proof. This story is almost the same as the "puppet judge" told by Wang Qi before, and the mirage "Silver Wing Assassin".

Many monks who did not belong to the Wanfa Sect, especially those small sects that used "literary arts" and "novelists' words" as their methods of cultivation, listened with relish. They could not understand Wang Qi's theory, nor could they understand the mathematical meaning behind Wang Qi's theory. However, they could understand the story! Almost endless stories emerged in their minds.

It can be predicted that in the future, "puppet people" will become a hot topic among storytellers.

The Wanfamen disciples who had not collapsed showed greater interest in the second method.

Bo Xiaofeng was in a state of confusion. "Complete but self-contradictory system"? Such a thing really exists? Is there any mathematical significance in it?

Wang Qi's narration started at noon and ended at You. When the You bell rang in the land of Shenzhou, this sermon also came to an end.

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

"I think we can proudly announce that the second and tenth questions of Ximen's twenty-three questions have been successfully solved. The most important proofs of consistency, completeness, and decidability in the foundation of mathematical reasoning have also been completed."

"Mathematics is about to enter a new era."

No one applauded.

However, everyone felt that a revolutionary change might really be coming... (To be continued ~^~)