As we all know, people can't set flags.

The cow that was blown out on a whim won't be long before it turns into a slap and slapped on his face again.

In the history of science written in Chinese, the act of scientists setting up flags is called "building buildings".

Two of the most famous of these, one in 1900 and the other in 1900.

The protagonist of the first time is Lord Kelvin, who has just helped Chen Muwu and provided him with a fund for the development of a particle accelerator.

In his New Year's address to the Royal Society in 1900, he gave an outlook on the development of physics in the coming new century, and then uttered the famous sentence:

"The edifice of physics has been completed, and all that remains is some hammering and commemorating work, and there are only two small dark clouds floating in the beautiful and clear sky."

Mr. Lu Xun has said more than once: "I have never said such a thing. ”

But this time Lord Kelvin also said, "I haven't said anything like that either." ”

In fact, Kelvin did mention the concept of dark clouds back then, but he never mentioned buildings.

The place of the lecture was not the Royal Society, but the Royal Institute, and the time was not the first day of the new year, but April 27.

Kelvin gave a lecture on that day entitled "The Clouds of the Nineteenth Century Covering the Kinetic Theory of Heat and Optics."

What he said in his speech was not as light as in the Chinese expression: "The theory of dynamics asserts that both heat and optics are forms of motion, and now the beauty and clarity of this theory are overshadowed by two dark clouds." ”

The word eclipsed reflects the seriousness of these two dark clouds, which are not at all as light and light as in the first expression, as if the two small dark clouds were insignificant.

Chen Muwu always feels that the use of the word "building" in Chinese is to describe a sense of crisis in which the foundation is not strong and crumbling.

Then two fierce men, Planck and Einstein, descended from the sky to "support the building to fall into the future", opening up two new paths for the development of physics: quantum theory and relativity.

The second non-existent edifice took place in 1900 at the Second International Congress of Mathematicians in Paris, the capital of France.

I don't know whether it was at the opening or closing ceremony, but the convener of the conference, the French mathematician Henri Poincaré, is said to have said: "...... With the concept of set theory, we can build the entire mathematical edifice...... Today, we can say that the absolute rigor of mathematics has been reached! ”

Poincaré said that he did not say the above paragraph, but only appeared in the history of Chinese mathematics about the building, which is somewhat doubtful.

It's just that the edifice of mathematics at that time, like the edifice of physics, was equally crumbling.

After that, mathematicians came up with a bunch of paradoxes, among which Russell, the philosopher who recruited Chen Muwu to the Cambridge Apostolic Society, proposed the "Russell paradox".

In some popular science books, Russell's paradox is reduced to the barber's paradox.

In a city, there was a barber.

He declares that he will shave the faces of all the people in the city who don't shave their faces, and he only shaves their faces.

One day, the barber saw in the mirror that his beard had grown, and he subconsciously grabbed the razor, but before he could do it, he suddenly remembered what he had said.

If he doesn't shave his face, then he belongs to the "people in the city who don't shave his face", so he has to shave himself.

But if he shaves himself, he is again a "person who shaves himself", so he should not shave himself.

In addition to the barber's paradox, Russell's paradox has another form of popular science that is easy to understand.

A library has a book name dictionary that contains all the books in the library that don't list their names.

It doesn't matter if the dictionary lists its own name or not, and the principle is similar to the barber's paradox above.

Russell's paradox was put forward in the face of the mathematicians who said that "all mathematical results can be based on set theory".

Gottlob Frege, a German logician, wrote a book on the basic theory of sets.

Just as the book was about to be handed over to the printing press, Frege received a letter from Russell about Russell's paradox.

He immediately found himself in a mess with Russell's paradox, and could only add at the end of the book: "The most unfortunate thing that can happen to a scientist is that when his work is about to be completed, he finds that the foundation of his work has collapsed." ”

Russell's paradox was followed by a series of paradoxes: Richard's paradox, Perry's paradox, Grelyn's and Nelson's paradox......

These paradoxes, known as semantic paradoxes, shook the foundations of the mathematical edifice and triggered the third mathematical crisis.

The first of the first two mathematical crises occurred in ancient Greece.

Pythagoras' student Hippasus discovered that the length of the diagonal line of a square with one side is neither an integer nor a ratio of two integers.

At that time, the ancient Greek mathematicians did not know the root number two, let alone the existence of irrational numbers in the world.

Unable to solve this problem, they finally chose to solve the problem of the person who raised the problem:

They threw Hippasos into the Aegean Sea and fed sharks.

The second mathematical crisis, Zeno's paradox, which sprouted in ancient Greece, can Achilles catch up with the turtle, and whether the moving arrow will move or not?

The ancient Greeks first came into contact with the problem of infinitesimality, and the real outbreak of this mathematical crisis was in the time of Newton and Leibniz.

The two of them invented calculus, which is very convenient to use, but there is only one question, is the infinitesimal quantity in calculus zero?

The infinitesimal may appear on the denominator, so it shouldn't be zero.

But if you take the infinitesimal quantity as zero, and remove the terms that contain it, the resulting formula can be proved correct in mechanics and geometry.

At the time, calculus was criticized as a "devil's trick" and "a scientific but incorrect result by chance with a double error."

It was not until the 19th century, when mathematicians led by Cauchy perfected the specific concept of limits, that the crisis was finally resolved.

As for the third mathematical crisis caused by these paradoxes, it is the one that has been solved the fastest.

German mathematicians Ernst Zermelo and Abraham Frankel proposed two sets of theories in 1908 and 1922 respectively, and together they became the Z(ermelo)-F (raenkel) axiom system.

This axiomatic system axiomatizes the construction of sets to exclude the existence of sets like Russell's paradox, which can be regarded as solving this mathematical crisis.

In the same year, Hilbert came up with the idea of finding a universal solution to the mathematical crisis that had already erupted three times.

He came up with an idea called the Hilbert Project, proposing to base all existing theories on a limited set of complete axioms and giving proof that these axioms are consistent.

Hilbert wanted mathematics to be complete and decidable, to be based on rigorous logic, to be the most unassailable truth in the world.

There is such a clause in the Hilbert plan, which is called completeness, and one can deduce all the theorems from the axioms.

If it can't be deduced, it is not the completeness of the above that is the problem, but the personal ability.

Axioms are the basic mathematical knowledge that people summarize in long-term practice, and serve as the basis for judging the truth or falsity of other propositions, which cannot be proved and do not need to be proved.

Theorems, on the other hand, are true propositions obtained by reasoning from axioms.

Hilbert is the greatest mathematician in the world today, and his words are eloquent and compelling.

From the moment he proposed this plan, mathematicians have always believed that this plan is correct and have been trying to prove it.

It's just that many years have passed, and none of the mathematicians have been able to get this proof.

It was not until 1931 that Gödel proved another point, in an axiom system, there is always at least one proposition that cannot be proved true or false, and in order to prove or falsify these propositions, it is necessary to use new axioms outside the system.

This is Gödel's first theorem of incompleteness, and the emergence of this theorem can be regarded as a complete rejection of the Hilbert plan and the crushing of the dreams of Hilbert and all mathematicians.

Hilbert's original intention was to completely solve the mathematical crisis, but unexpectedly, he almost toppled the foundation of the mathematical edifice.

This Gödel is the one who solved Einstein's gravitational field equation and solved the Gödel universe that supports time travel.

Chen Muwu understood this person because of the Gödel universe, so he naturally knew the two incompleteness theorems he proposed.

When he heard the incompleteness of Bohr's words, he thought of this theorem and of Hilbert, a mathematician.

Chen Muwu did not have much prejudice against Hilbert, but he clearly remembered that Hilbert once said, "Physics is too difficult for physicists."

His original intention was to say that modern physics, although highly dependent on higher mathematics, has always been less rigorous.

But when this sentence comes out of the mouth of a mathematician, it is still very uncomfortable for a person who studies physics to hear it.

Anyway, now that the particle accelerator is being manufactured step by step, Chen Muwu has nothing else to do except supervise the work.

So now that we have thought of this, it is better to figure out this incompleteness theorem, which can be regarded as a small shock from the physicist to Master Hilbert.

After "denouncing" the Germans, Chen Muwu once again fell into a long distraction.

Rutherford had long been accustomed to his beloved apprentice coming to such an out-of-body experience from time to time, so he simply pulled his other beloved apprentice, Bohr, to discuss and impart the management of laboratories and research institutes.

After a long time, Chen Muwu woke up from his meditation, opened his mouth and blinked his eyes.

"You're finally alive, did you think of any good ideas just now?"

Rutherford asked with a smile.

Chen Muwu touched his head embarrassedly: "I really had some immature ideas just now. ”

"Which aspect of the particle...... Experiment? ”

Bohr followed Rutherford: "Or quantum mechanics? ”

"Well, it's neither, it's just that after Professor Bohr's reminder, I suddenly seem to have some ideas about mathematics."

He scratched his head more often.

Although Rutherford was accustomed to Chen Muwu changing his research direction at any time, he still could not imagine that a good student would study mathematics.

He subconsciously stretched out his hand to the pipe on the desk, and then remembered that Chen Muwu didn't like the taste of this tobacco.

Bohr had a look of disappointment on his face.

"However, Professor Bohr, I also have a few new ideas on quantum mechanics, and I may write a paper or two in the near future, and I will ask for more advice from you when the time comes."

"It's easy to say, it's easy to say, I can't talk about advice, just discuss with each other."

The office began to fill with a cheerful air again.

After the day's chat, the guests and hosts enjoyed themselves.

Rutherford acquiesced to the purpose of Chen Muwu's trip and did not recruit students in Cavendis's laboratory.

Bohr also got a fairly satisfactory result from Chen Muwu, since he wanted to study theoretical knowledge and said that he wanted to consult with himself, then it was inevitable to exchange letters and telegrams.

Maybe the relationship between the two of them will become closer and closer, so that you can slowly figure him out, and you must dig him up, oh no, please go to Copenhagen.

Bohr spent a few more days at Cambridge University, then headed north to Manchester, where he continued to visit friends and family, and finally boarded the ship from Norwich and returned to Copenhagen, Denmark.

Chen Muwu walked out of the leisurely state he had maintained for more than a year, and once again started the mode of liver papers.

Although I know the law of incompleteness and how Gödel proved it, it is not easy to reproduce that paper.

Fortunately, in addition to Chen Muwu's own plug-in, he also has a humanoid plug-in at Cambridge University.

Many of the Cambridge Apostolic Society were mathematicians and logicians, all of whom were students of Russell, and even Russell himself.

There is an intrinsic connection between the incompleteness theorem and Russell's paradox, both of which involve negation of self-referential and diagonal methods.

It would be a fool to have this kind of ready-made thighs and study them on their own, so they took advantage of the opportunity to meet in the Apostolic Society every Saturday night.

The other sons and buddies were all chatting with wine glasses, and Chen Muwu was eating the precious ingredients prepared on the table, while lowering his posture and asking others about mathematics and logic.

He also found the opportunity to go to Russell's office several times to ask for advice, just so that he could write this paper on the incompleteness theorem and get recognized by the mathematical community.

It is rumored that Dr. Chan of Trinity College has become desireless in physics after winning the Nobel Prize in Physics.

He has recently become very close to Russell, and may be about to develop in the direction of philosophy.

(End of chapter)