Second, the rationality of calculus is seriously questioned, which almost subverts the whole calculus theory.
Third, Russell's paradox: S is composed of all elements that do not belong to itself. Does s contain S? In layman's terms, one day Xiaoming said, "I'm lying!" "Ask xiao Ming is lying or telling the truth. The terrible thing about Russell's paradox is that it doesn't involve the profound knowledge of sets like the maximum ordinal paradox or the maximum cardinal paradox. It is simple, but it can easily destroy set theory!
Chinese name
Three crises in mathematics
Foreign name
Three crises in mathematics
first time
Found the root number 2 and overthrew "everything is counted"
second time
The rationality of the concept of calculus is seriously questioned.
the third time
Russell Paradox in Set Theory
The First Mathematical Crisis
Pythagoras was a famous mathematician and philosopher in ancient Greece in the fifth century BC. He once founded a school of mysticism: Pythagoras School, which integrates politics, scholarship and religion. Pythagoras' famous proposition "Everything is a number" is the philosophical cornerstone of this school. Pythagoras school refers only to integers. "All numbers can be expressed as integers or the ratio of integers" is the mathematical belief of this school. Dramatically, however, the Pythagorean theorem established by Pythagoras has become the "grave digger" of Pythagoras' mathematical belief.
Pythagoras, ancient Greek philosopher
After the Pythagorean theorem was put forward, hippasus, a member of his school, considered a question: What is the diagonal length of a square with a side length of 1? He found that this length can not be expressed by integer or fraction, but only by a new number. Hippasus's discovery led to the birth of the first irrational number in the history of mathematics. Small? However, the appearance of the "new world" set off a huge storm in the field of mathematics at that time. It directly shook the Pythagorean school's mathematical belief and made the Pythagorean school panic. In fact, this great discovery was not only a fatal blow to the Pythagorean school, but also a great impact on the thoughts of all ancient Greeks at that time. The paradox of this conclusion lies in its conflict with common sense: any quantity can be expressed as a rational number within any precision range. This is a widely accepted belief not only in Greece at that time, but also in today's highly developed measurement technology. But what we believe by experience and are completely in line with common sense are actually small things? The overthrown existence! How contrary to common sense and ridiculous this should be! It just subverts the previous understanding. To make matters worse, people are powerless in the face of this absurdity. This directly led to the crisis of people's understanding at that time, which led to a big storm in the history of western mathematics, known as the "first mathematical crisis."
The second mathematical crisis
appear
The second mathematical crisis stems from the use of calculus tools. With the improvement of people's understanding of scientific theory and practice, calculus, a sharp mathematical tool, was discovered by Newton and Leibniz almost simultaneously in the17th century. As soon as this tool came out, it showed its extraordinary power. After using this tool, many difficult problems have become easy. But Newton and Leibniz's calculus theory is not strict. Their theories are all based on infinitesimal analysis, but their understanding and application of the basic concept of infinitesimal is confusing. Therefore, calculus has been opposed and attacked by some people since its birth. Among them, the most violent attack was British Archbishop Becquerel.
solve
After Cauchy defined infinitesimal by limit method, the theory of calculus was developed and perfected, thus making the mathematics building more brilliant and beautiful!
the third mathematical crisis
appear
/kloc-In the second half of the 9th century, Cantor founded the famous set theory, which was severely criticized by many people when it was first produced. But soon this groundbreaking achievement was accepted by mathematicians and won wide and high praise. Mathematicians found that starting from natural numbers and Cantor's set theory, the whole mathematical building could be established. Therefore, set theory has become the cornerstone of modern mathematics. The discovery that "all mathematical achievements can be based on set theory" intoxicated mathematicians. 1900, at the international congress of mathematicians, poincare, a famous French mathematician, declared cheerfully: "... with the help of the concept of set theory, we can build the whole mathematical building ... Today, we can say that we have reached absolute strictness ..."
However, the good times did not last long. 1903, a shocking news came out: set theory is flawed! This is the famous Russell paradox put forward by British mathematician Russell.
Russell built a set S: S is made up of all elements that don't belong to him. Then Russell asked: Does S belong to S? According to law of excluded middle, an element belongs to a set or not. Therefore, for a given set, it is meaningful to ask whether it belongs to itself. But this seemingly reasonable question, the answer will be in a dilemma. If s belongs to s, according to the definition of s, s does not belong to s; On the other hand, if S does not belong to S, then S also belongs to S by definition. It is contradictory in any case.
In fact, this paradox was discovered in the set theory before Russell. For example, in 1897, Burali and Folthy put forward the paradox of maximum ordinal number. 1899, Cantor himself discovered the paradox of maximum cardinality. However, because these two paradoxes involve many complicated theories in the set, they have only produced small ripples in the field of mathematics and failed to attract much attention. Russell paradox is different. Very simple and easy to understand, only involving the most basic things in set theory. So Russell's paradox caused a great shock in mathematics and logic at that time when it was put forward. For example, after receiving a letter from Russell introducing this paradox, G Frege said sadly, "The most unpleasant thing that a scientist encounters is that his foundation collapses at the end of his work. A letter from Mr. Russell put me in this position. " Dai Dejin therefore postponed the second edition of his article "What is the Nature and Function of Numbers". It can be said that this paradox is like throwing a boulder on the calm water of mathematics, which caused great repercussions and led to the third mathematical crisis.
solve
Eliminate paradox
After the crisis, mathematicians put forward their own solutions. I hope to reform Cantor's set theory and eliminate the paradox by limiting the definition of set, which requires the establishment of new principles. "These principles must be narrow enough to ensure that all contradictions are eliminated; On the other hand, it must be broad enough so that all valuable contents in Cantor's set theory can be preserved. " 1908, Tzemero put forward the first axiomatic set theory system according to his own principles, which was later improved by other mathematicians and called ZF system. This axiomatic set theory system makes up for the defects of Cantor's naive set theory to a great extent. Besides ZF system, there are many axiomatic systems in set theory, such as NBG system proposed by Neumann et al.
Axiomatic set system
The paradox in set theory was successfully eliminated, thus the third mathematical crisis was successfully solved. On the other hand, Russell's paradox has a far-reaching influence on mathematics. It puts the basic problems of mathematics in front of mathematicians for the first time with the most urgent needs, and guides mathematicians to study the basic problems of mathematics. The further development of this aspect has profoundly affected the whole mathematics. For example, the debate on the basis of mathematics has formed three famous schools of mathematics in the history of modern mathematics, and the work of each school has promoted the great development of mathematics.