When Newton created calculus, it triggered the second mathematical crisis. For example, Newton's definition of derivative is not too strict.
The derivative of x2, first take an increment δx that is not 0, then get 2xδx+(δx)2 from (x+δx)2, and then get it from δ X..
Divide by 2x+Δ x, and finally suddenly make Δ δx = 0, and the derivative is 2x.
. We know that this result is correct, but there are obvious mistakes in the process of derivation: in the first part of the argument, it is assumed that δ x is not 0, and in the second part of the argument, it is taken as 0. So is it 0 after all?
In addition, Newton's calculus regards infinitesimal as a finite value that is not zero, which is eliminated from both ends of the equation, but sometimes it is ignored as zero, which leads to the problem that infinitesimal must be both zero and not zero in terms of its practical application at that time. But as far as formal logic is concerned, this is undoubtedly a contradiction. Newton later failed to prove himself.
The essence of the two major mathematical crises is actually caused by the imperfection of the real number system. So Wilstras and others launched the "arithmetic of analysis" movement.
Wilstrass believes that real numbers are the origin of all analysis. To make the analysis strict, we must first make the real number system itself strict. The most reliable method is to convert real numbers into integers (rational numbers) according to strict reasoning. In this way, all the concepts of analysis can be deduced from integers, thus filling the previous loopholes and defects. This is the so-called "analysis arithmetic" program.
Under the guidance of Wilstrass's "Analytical Arithmetic" movement, Dai Dejin, Cantor and Wilstrass all put forward their own real number theory.
1872, German mathematician Dai Dejin defined irrational numbers by dividing rational numbers from the requirements of continuity, and established the theory of real numbers on a strict scientific basis. He divides the set of all rational numbers into two non-empty and disjoint subsets A and A', so that each element in set A is smaller than that in set A'. Set a is called the lower group of division, and set a' is called the upper group of division, and this division is marked as A|A'. Dai Dejin defined this division as the division of rational numbers, in which Dai Dejin expanded from rational numbers to real numbers, and established the theory of irrational numbers and the definition of continuous pure arithmetic.
Calculation process of Dydykin's division theorem
Cantor also achieved the same goal through the theory of rational number sequence. Both Cantor and Dai Dejin defined real numbers as some type of "set" of rational numbers. Dai Dejin method can be called sequential completion method and Cantor method can be called measurement completion method. These methods have become typical structural methods in modern mathematics, and have been continuously popularized and developed by later generations as powerful tools in mathematical theory.
Cantor's Theory of Rational Number Sequence
Wilstrass published the theory of bounded monotone sequences. The basic sequence of rational numbers is to assume the completeness of real numbers, and then define irrational numbers of rational numbers according to the limits of rational number sequences. There are many rational number columns, which are themselves basic columns, but there are no restrictions in the rational number system, so there is a definition: if a basic column converges to a rational number, it is called a rational number basic column; If a basic column does not converge to any rational number or converges to null, it is called an irrational basic column. The basic column of rational number defines rational number, and the basic column of irrational number defines irrational number.
Verification process of bounded monotone sequence theory
These three real number theories prove the completeness of the real number system. The definition of real number and its completeness mark the completion of analytical arithmetic movement advocated by Wilstrass. In this way, the logical cycle around the concept of real numbers for a long time has been completely eliminated, and the establishment of real number system also marks that algebra has completely got rid of the geometric haze.
Because of the establishment of real number system, mathematics and even the whole scientific community are shrouded in a happy and peaceful atmosphere. Scientists generally believe that the systematicness and rigor of mathematics have been achieved, and the science building has been basically completed. However, this statement was finally hit in the face.
Although Wilstrass's "Analytic Arithmetic" movement once solved two major crises in the history of mathematics, it also triggered the third mathematical crisis, which continues to this day, making the whole mathematical building in jeopardy.
In this movement, Cantor said in a letter to Dai Dejin on 1873 165438129 October that the problem that led to set theory was finally clearly put forward: whether the set of positive integers (n) and the set of real numbers (x) can correspond to each other. On February 7, 65438, Cantor wrote to Dai Dejin, saying that he had successfully proved that the "set" of real numbers was uncountable, that is, it could not correspond to the "set" of positive integers one by one. This day should be regarded as the birth day of set theory.
Simple set knowledge
Set theory founded by Cantor can be said to be a basic branch of mathematics, and its research object is general set. Set theory occupies a unique position in mathematics, and its basic concepts have penetrated into all fields of mathematics. Set theory or set theory is a mathematical theory to study a set (a whole composed of a bunch of abstract objects), including the most basic mathematical concepts such as set, element and subordinate relationship. We have been exposed to simple collection knowledge since high school, and you can simply recall it.
Set theory begins with the binary relationship between an object O and a set A: if O is an element of A, it can be expressed as O ∈ A. Since a set is also an object, the above relationship can also be used in the relationship between sets and sets. Another relationship between two sets is called inclusion relationship. If all the elements in set A are elements in set B, then set A is called a subset of b with the symbol A? B. For example {1, 2}
Is a subset of {1, 2,3}, but {1, 4} is not {1, 2,3}.
A subset of. By definition, any set is also a subset of itself, and a subset without considering itself is called proper subset. Set a is the proper subset of set b if and only if set a is a subset of set b and set b is not a subset of set a.
There are many unary and binary operations in the arithmetic of numbers, and there are also many unary and binary operations on sets in set theory.
The elements in set theory also have three characteristics: certainty, mutual difference and disorder. First, the elements in the set must be definite. For example, {the handsome guy in our company} is not the same, because the definition of handsome is different. Some people think that Wei Meng is handsome, while others think that he is weak and handsome, so the elements are uncertain. The elements in the collection must be different from each other.
For example, {5,6} is a set, but it cannot be expressed as {5,6,5}, which is anisotropy; {1, 2,4} and {4,2, 1} are the same set, which is the disorder of the set, because the elements in the set have no order.
A poet and lead singer
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.
At the 1900 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 …". This discovery fascinated mathematicians.
Unfortunately, it was not until three years later, that is, 1903, that Russell discovered the problem of set theory. Russell is a rare all-rounder in the west, a famous British philosopher, mathematician, logician, historian and writer. He had a debate with Hilbert, the leader of Gottingen School, on the philosophical basis of mathematics.
Russell thinks that "mathematics is logic", while Hilbert puts forward formalism, thinking that the object of mathematical thinking is the mathematical symbol itself. The debate involving two people includes set theory.
Russell found a BUG in Cantor's set theory from three characteristics of set elements. The set S consists of all sets that do not belong to itself. 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.
The vernacular version of Russell's paradox is the famous barber's paradox: there is a barber in a city, and his advertisement reads: "My hairdressing skills are superb and the whole city is famous. I will shave all the people in this city who don't shave themselves. I will only shave these people. I would like to extend a warm welcome to everyone! " When people come to him to shave, they naturally don't shave themselves. One day, however, the barber saw in the mirror that his beard had grown. He instinctively grabbed the razor. Do you think he can shave himself? If he doesn't shave himself, then he belongs to the "person who doesn't shave himself" and he has to shave himself. What if he shaved himself? He belongs to the "person who shaves himself" and should not shave himself.
This is the famous saying in the history of mathematics that "a barber rushed into the building, turned the whole building upside down, and even directly shook the foundation of the whole mathematics building." So far, no one has invited the barber out.
If the first and second mathematical crises only affected the construction of the whole mathematical building, then the third mathematical building directly shook the whole foundation because it involved the basic problems of mathematics.
Because Russell's paradox only involves the most basic concepts of set theory: set, element, attribution and generalization principle, its composition is very clear. The appearance of this paradox shows that the previous naive set theory contains contradictions, so the whole mathematics based on set theory cannot be without contradictions. This paradox also shows that the logic used in mathematics is not without problems. The third crisis of mathematics has made the mathematical and logical circles feel the seriousness of the problem.
Many paradoxes caused by this.
Russell's paradox shows that the principle of generalization cannot be unconditionally recognized, but the change of the principle of generalization will greatly change the set theory, so the influence on the whole mathematics is enormous. To put it simply, it seems that the infinite set and infinite radix can eliminate the paradox and solve the contradiction, but it gradually loses the mathematical certainty. This is the contradiction of the problem.
Russell's problems directly ruined the life work of many mathematicians. Frege, a famous German logician, received a letter from Russell about this paradox when he finished printing his Basis of Set Theory. He immediately found that a series of achievements he had been busy with for a long time were all messed up by this paradox. He can only write at the end of the book: "The worst thing for a scientist is to find that the foundation of his work has collapsed when his work is about to be completed." This is really helpless. Even though we have spent so much energy on the mathematical construction of logic, many of our conclusions are still not rigorous enough and may have loopholes.
Of course, the restoration work is also going on vigorously. If we want to solve this crisis, we must establish a more rigorous solution to unify these contradictions.
The most famous is the Zemello-frankl axiom system. On 1908
In, Ernst Zemello put forward the first axiomatic set theory-Zemello set theory. This axiomatic theory does not allow the construction of ordinal numbers; However, the development of most "general mathematics" is inseparable from ordinal number, which is the fundamental tool in most set theory research. In addition, an axiom of Zermelo involves the concept of "clarity", and its operational meaning is vague.
So it was later improved by frankl and called the Zemello-frankl axiomatic system. In this axiomatic system, due to the classification axiom that P(x) is a property of X, for any known set A, there exists a set B such that for all elements x∈B is if and only if x∈A and P (x); Therefore {x∣x is a set} cannot be written as a set in this system because it is not a subset of any known set; Through this axiom, the existence set A={x∣x is a set} can be proved to be contradictory in ZF system.
In a word, it is the Zemello-frankl axiom system that strictly stipulates the conditions for the existence of sets (in short, there is an empty set axiom; Every set has a power set axiom; The union of all sets in each set also forms the axiom of set union; Elements in each set that meet certain conditions constitute subset axioms; Functions whose domain is a "have the axiom of" range "substitution, and so on. ), so you can't define a set in the paradox. So Russell paradox is avoided in this system.
However, it does not solve the problem from the effectiveness of the whole basic structure of mathematics, so as to repair the whole mathematics building on the basis of mathematics. Many important topics of mathematical foundation and mathematical logic have not been fundamentally solved, so there are still some defects. /kloc-more than 0/00 years have passed, and the crisis continues. When will the foundation of the math building be compacted? Now it seems that there is still a long way to go.
However, the third mathematical crisis undoubtedly played a great role in promoting the development of the whole mathematical field, promoting the research of basic mathematical theories, promoting the birth of Godel's incompleteness theorem and the development of mathematical logic. It can be said that every crisis is like the birth of a cornucopia, bringing new contents, new progress and even revolutionary changes to mathematics.