So is the axiomatic method of the so-called real number system. We list the independent properties that real numbers should have as few as possible as axioms, so as to deduce other properties from axioms and construct an axiom system ("real number axiom").
Hilbert's axiomatic method describes what we need the real number system, which solves many problems left over from middle school mathematics, such as what is the addition and multiplication of real numbers, why the addition of real numbers satisfies the commutative law and associative law, and the multiplication also satisfies the commutative law and associative law, which can be understood as axiomatic provisions. In fact, if more basic assumptions are provided (for example, on the basis of rational numbers), these operational laws can be proved. It also ensures the establishment of the basic theorem of real number system and provides a necessary stage for the development of limit theory in mathematical analysis. Whether the real number system satisfying these axioms exists is solved by the following construction method, that is, the concrete generation method of the real number system is given, and it is proved that all the axioms listed in the axiomatic method are satisfied. For the axiomatic method, please refer to zorich's Mathematical Analysis (Volume I). Introducing the existence of real number system by construction method. The following are three ways to construct a real number system (mainly defining irrational numbers from rational numbers).
1.Dedeking segmentation method
Dydykin's segmentation method has been introduced in many mathematical analysis works. The most classic narrative is Landau's short book The Basis of Analysis. The subtitle of this book is "Operation of Integer, Rational Number, Real Number and Complex Number", which starts from natural number and defines it to complex number, showing the complete definition of number system.
Dai Dejin's method is fully described in the mathematical analysis textbooks of the former Soviet Union, and a classic textbook consisting of three volumes is introduced for the first time: A Course in Calculus by Fehin Geerds. The introduction gives a complete description of Dydykin's segmentation method, which lays a solid foundation for the book. In addition, we can also see the Initial Order of Generalization of Sets and Functions by Alexander Rove, the first lecture of Eight Lectures on Mathematical Analysis by Qin Xin, Appendix 1 of Theory of Real Variability by Jin Lu, and Appendix 2 of Mathematical Analysis (3rd Edition) by the Department of Mathematics of East China Normal University.
In western textbooks, spivak's Calculus introduced the axioms of number system in detail in two chapters at the beginning, and at the end of the book, he used three chapters to talk about how to construct real numbers. The first chapter and appendix of Rudin's Principles of Mathematical Analysis have a brief description of real number theory. These two textbooks changed Dydykin's segmentation method. From the history of mathematics (Boyer's book History of Calculus Concepts, Historical Review of Derivatives and Integrals), we know that this is basically Russell's definition method of real numbers.
Among various methods of introducing real number system, Dydykin's segmentation method has always been highly respected, and it is called the creation of intuitive human wisdom completely independent of space and time.
2. Cantor's basic column (that is, Cauchy column) method
In this respect, we can refer to Chapter 4 of A Concise Course of Mathematical Analysis, Section 68 of Algebra edited by Xu Shaopu and Song, Chapter 5 of Mathematical Analysis edited by Xu Shaopu and Song, Chapter 2 of Mathematical Analysis edited by the Department of Mathematics of East China Normal University, and Appendix 2 of Mathematical Analysis (first edition).
3. Wilstrass method based on decimal representation
This method is different from the first two methods. It does not need to introduce new mathematical objects as irrational numbers, but starts from the existing definition in middle school, that is, it recognizes that decimal finite decimals and infinite cyclic decimals are rational numbers, while decimal infinite cyclic decimals are irrational numbers. This is more acceptable to middle school students. So it is also called middle school students' real number theory.
But why do decimals cycle infinitely? The limit problem is inevitably involved here. With Cauchy criterion, we can understand decimal infinite acyclic decimal from the limit of sequence or the sum of infinite series. But we can't understand it like this before establishing the real number system, otherwise we will make the same mistakes as Cauchy in history.
Therefore, in order to avoid the logical circular definition, when decimal infinite acyclic decimal is defined as irrational number, it can not be regarded as the sum of infinite series at the beginning, but only as a pure symbol and a mathematical object with unclear meaning. Then, the addition and multiplication operations are introduced into the set of all decimal decimals, and the order between any two decimals is specified, and it is verified that they meet the four axioms of domain axiom, order axiom, Archimedes axiom and continuity axiom. Of course, there are many inference steps here. In fact, it is also a mathematical abstraction to think that such symbols represent real numbers, and it is also another equivalent form of continuity axiom. Historically, Wallis equated rational numbers with cyclic decimals in 1696. Stoertz put forward the definition of decimal infinite acyclic decimal as irrational number in 1886, but he still has not established a satisfactory real number theory.
There are many textbooks about real numbers starting from decimals, such as Lecture Notes on Mathematical Analysis by Ahebov, Advanced Mathematics Course by Guan, and Introduction to Advanced Mathematics by Hua. In the first chapter of Zhang Zhusheng's New Lecture Notes on Mathematical Analysis, the strict method of introducing four operations into decimals is explained in detail.
Another method that can be classified into this method is to introduce the principle of closed interval set with rational number as the end point as a substitute for the continuity axiom. It is not only intuitive, but also avoids the infinite cycle of decimals that are difficult to explain at first. It is also a good method. First of all, we should understand the exact meaning of uniqueness here, which means uniqueness in the sense of isomorphism. Specifically, it is proved that all real number system models that satisfy the axiom of real numbers are isomorphic.
The discussion on the uniqueness of the real number system in isomorphism after it is established according to Dai Dejin's method can be found in the last chapter of spivak's Calculus. For the discussion on the uniqueness of establishing real number system according to Cantor Cauchy method, please refer to the proof of the last part of Chapter 5 of Mathematical Analysis edited by Xu Shaopu and Song.