What is the origin of mathematical paradox?

19 19, the famous British mathematician Russell made up a very interesting "joke".

There is a barber who likes to brag in town. One day, the barber boasted, "I shave all the people in town who don't shave themselves. Only such people shave."

Everyone laughed at this. Someone asked him, "Mr. Barber, can you shave yourself?"

"This, this, …" The barber was tongue-tied and couldn't say a word for a long time.

It turned out that this boastful barber was caught in a dilemma. If he shaves himself, which doesn't conform to the first half of his sentence, he shouldn't shave; However, if he doesn't shave himself, which doesn't conform to his second sentence, he should shave himself again. Whether you shave or not, it's not right anyway.

A logically contradictory statement like a barber is called paradox. This joke made up by Russell is the famous Barber Paradox in the history of mathematics.

The barber's appearance is funny, but mathematicians can't laugh at it, because they themselves, like barbers who like to brag, are caught in a contradictory embarrassing situation.

In fact, mathematicians in the early 20th century were even more embarrassed than that boastful barber. As long as the barber cancels the original statement and laughs brazenly, nothing will happen; Mathematicians are not as lucky as he is, because they have encountered an unavoidable mathematical paradox. If the original "declaration" is revoked, most valuable knowledge in modern mathematics will cease to exist.

This mathematical paradox was also put forward by Russell. 1902, Russell constructed this mathematical paradox "strictly" from the set theory that has been recognized as the basic theory of mathematics according to the logic method commonly used by mathematicians. Popularization is the barber paradox.

Set theory is a mathematical theory developed at the end of 19, which quickly penetrated into every corner of mathematics until the middle school mathematics textbook. It has greatly changed the whole face of mathematics. Just as mathematicians have just laid mathematics on the basis of set theory, Russell's paradox has emerged. It points out with irrefutable facts that whoever agrees with set theory will become a "barber who likes to brag" and thus fall into a contradictory dilemma. Mathematicians are extremely embarrassed. If we continue to recognize set theory, mathematics, which claims to be absolutely rigorous, will not be justified by such a monster as Russell Paradox. If set theory is not recognized, many important mathematical inventions will cease to exist.

Russell's paradox shocked the mathematical world and led to a crisis involving the basis of mathematics. It has been found that there is a huge crack in the foundation of this glorious building of mathematics. If it is not repaired, the whole building is in danger of collapse at any time.

Mathematicians bravely accepted the challenge. They carefully studied the causes of Russell's paradox. It turns out that Russell's paradox is such a behemoth because in set theory, the phrase "set of sets" can't be said casually. Therefore, mathematicians began to explore under what circumstances mathematical conclusions are true and mathematical reasoning is effective ..., thus creating a new branch of mathematics-mathematical basic theory.

In this field, due to the different views of mathematicians, three famous schools have emerged. Mathematicians represented by Russell are called logicians. They believe that Russell paradox will not happen as long as the illogical language of "set of sets" is not allowed. Mathematicians represented by Brouwer are called intuitionists. They think that the "set of sets" is incomprehensible. If we don't admit its rationality, Russell's paradox will naturally not arise. Mathematicians represented by Hilbert are called formalism school. They think that paradox is the manifestation of incompatibility.

All three schools have put forward plans to mend the foundation of mathematics, and a big debate broke out because of their own opinions. This great debate has had a far-reaching impact on the development of modern mathematics, and also led to the birth of many new branches of mathematics.

At present, the work of mending the foundation of mathematics has not achieved completely satisfactory results, and mathematicians are still struggling tenaciously.