"What I am going to talk about now is the most important concept that I have learned from the work of Lord Hemen and from the work of the Gotting! ”

As soon as Wang Qi said this, there was an extra furrow on the brow of the counter.

And behind him, more Shaoli faction arithmetic almost out of anger.

The Lord Hepache had countless achievements in his life. However, his views on the nature of arithmetic are widely criticized and difficult for most arithmetic scholars to accept.

In the view of the arithmetic, the essence of arithmetic is "form". Symbols alone do not have any meaning. The "form" of those symbolic arrangements is arithmetic.

This is like saying that in the original formula of "1+1=2", the symbols "1", "+", "=", and "2" are meaningless, and only the complete equation has mathematical significance.

On the other hand, many of the economists of the Geting faction are also in mixed moods. They had a sinister look on their faces, mixed with confusion and anger.

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

And it is Wang Qi himself who denies this ideal.

Wang Qi used the idea of the master to veto the pursuit of the master and the efforts of the Geting faction for decades.

However, this kind of complicated mood did not affect Wang Qi.

Wang Qi understands that now he represents not himself, but "correct".

Self-referential is not a mathematical description - it is indeed logic, but whether it is mathematics or not is debatable, and not all mathematicians admit it. Whether it is the earth or China, there are such signs.

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

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

The earth calls it "Gödel Digitization".

――Maybe in China, it will be called "Wangqi Digital"?

"When the average person understands this incompleteness, it is easy to fall into a fog of self-reference and find it difficult to extricate themselves. I suspect that this is probably related to Kang's predecessors' diagonal proofs, infinite cardinality violating our innate instincts, and so on. That's why. That's why for thousands of years, our predecessors turned a blind eye to this important truth. ”

"Rather than unraveling this fog, we have to use this formalized approach. ”

Wang Qi's hands emitted golden light, and he conjured up a curtain of light, on which countless operators flew and arranged, listing a great proof.

The second and greatest part of Gödel's proof of incompleteness theorem is here.

"Any axiom system. There is a limit to the operators that can be used, and the axioms that can exist. Therefore, the statements that these axioms, these operators, can enumerate, must also be countable—infinitely countable, dod-element zero, and natural numbers. That's how it's described. ”

And the length of these possible statements is necessarily countable. And since it is still within the infinite range of counts, we can number it with natural numbers. Each number is unique. ”

"Then, we can build a collection 'Zhongtian'. This collection, 'Zhongtian', contains all possible 'numbers'. within an axiomatic system. All possible statements must be within this number. ”

......

By the time the sermon had reached this point, it had begun to fall out of the realm of what most people could understand. What is "countable infinite", what is "cardinality", "ordinal". These are beyond their comprehension.

Even the carefree monks who showed up to listen to Wang Qi's sermon showed a look of confusion.

Every step that the teenager said, they could understand. But. What is the mathematical significance of these things when they come together?

I don't understand at all!

More people who watched this "live broadcast" with the help of the Ten Thousand Immortals Fantasy Realm were dizzy. This seemingly simple proof seems to contain infinite magic, as if to drag their minds into an abyss.

Feng Luoyi had to sigh. In the form of "subtitles", it was explained to all the monks who watched the live broadcast.

This process of digitization is, to put it bluntly, "mapping".

Symbols, expressions, and sequences of expressions in the arithmetic system were mapped into numbers - and the "Gödel number" was introduced to realize the procedure of digitizing objects. The result of this processing provides a digital tool for mathematical logic and other related branches in terms of research methods, 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 relevant problems.

Transform a quasi-mathematical statement into a mathematically meaningful statement.

This is the meaning of this "digitization".

And when this proof entered the second half of the second stage, in the Geting faction, Ike Man sighed softly: "Primitive recursion......"

His expression was full of regret and remorse.

――I've studied this field as well...... If I had gone a little deeper, would I have been able to avoid today's catastrophe?

A few monks from the Geting Sect immediately sent a private chat: "Brother Ai, you have research in this field, can you stumble over Wang Qi at this step?"

Ike shook his head with a wry smile. Wang Qi did not make a mistake in this step. Can he say what is right as wrong?

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

Of course, those present did not know that this digital proof has more meaning than its own.

It is also the source of recursion.

Recursion 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 one last time. The operator arrangement becomes a proof of the incomplete theorem.

"That's the whole process. ”

The audience was silent.

Arithmetic is dying at this moment.

"Consistency and completeness cannot be both, and arithmetic is incomplete without contradictions...... Many Wanfa disciples who watched the live broadcast almost cried.

After Wang Qi finished explaining the proof of the incomplete theorem, he went to the edge of the pulpit and sat quietly for a while. Today's sermon is divided into four parts. The first part deals with the incompleteness theorem, and the next part deals with the indecidability theorem.

Then, it's time to ask questions and summarize.

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

He closed his eyes and recuperated, as if he didn't feel the malice of the overhauls around him.

What if you look at the enemy? This is the realm of seekers, not the occasion for fighting. No matter what you say, it is impossible to reverse right and wrong.

After a short break, Wang Qi walked back to the center of the pulpit. The whispers that had been made up for a short period of time had also faded away. The attention of countless monks was focused on Wang Qi.

Like one poisonous snake after another, looking for the vital point of the prey, trying to kill with one blow.

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

"The first part of the indeterminability theorem, like the incomplete theorem, is 'self-referential', and I will not repeat it here. I'm going to move on to the second part of the undecidable theorem proof. ”

"First of all, I need to thank Turing for his help with this issue, and for the proof of equivalence. ”

"First of all, the first proof method I want to introduce is my own proposal, which is the mechanism of the narrow Turing arithmetic...... And the second part, which Turing calls 'all things that are all things to come about'. It is constructed according to the incompleteness theorem, a complete and inconsistent algorithm. ”

This is also the part that Wang Qi and Turing Zhenren discussed.

The narrow Turing Calculator and the Panacea, i.e., the Turing machine and the lambda algorithm, have the same potential to surpass the proof itself, as does the second stage of the second half of the incomplete theorem proof. In later generations, they each developed their own leafy paths.

Turing was also very happy that his little eye-catching theorem could be preached. Wang Qi also needs the support of some carefree monks, so that he will not be so weak, so that more arithmetic scholars can follow him and create a new world of arithmetic.

The two can be said to hit it off.

The process of proving an indefinable theorem is much easier to deal with than it is incomplete. Especially the first part, Turing machine proof. This story is almost in the same vein as the "puppet inquisitor" told by Wang Qi before, as well as the mirage "Silver Wing Assassin".

Many monks who do not belong to the Ten Thousand Dharmas, especially those small sects that take "literature and art" and "novelists' words" as their practice methods, have listened to them with relish. They can't understand Wang Qi's theory, and they can't understand the mathematical meaning behind Wang Qi's theory. But they can understand the story! There are almost endless stories in their minds.

It is foreseeable that in the coming period, "puppet people" will become a popular subject among storytellers.

And the disciples of the Ten Thousand Methods Sect, who had not yet collapsed, showed greater interest in the second method.

Bo Xiaofeng's whole person was in a state of confusion. "Complete but paradoxical systems"? Does such a thing really exist? Is there any arithmetic significance in this?

Wang Qi's narration began at noon, and from the time of the hour. When the bell rang in the land of China, this sermon also came to an end.

"To sum up, 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 of the twenty-three questions of Simon have been satisfactorily resolved. Among the arithmetic foundations, the most important proofs of consistency, completeness, and decidability have also been completed. ”

"Arithmetic is about to enter a new era. ”

No one applauded.

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