Property of completeness of the set of real numbers. Axioms of real numbers. The role of the axiom of continuity in the construction of mathematical analysis
§ 7 . Foundation of Analysis, 4
Completeness of the set of real numbers.
7.1. Introduction.
Definition. By a real number a we mean the equivalence class a of fundamental sequences of rational numbers.
Definition. A bunch of R equivalence classes of fundamental sequences of rational numbers will be called the set of real numbers.
1) lim a n = a Û " 0< eÎR$ pО N(" nО N, n ³ p) Þ |a n - a| £ e
2) every sequence (a n) that is convergent is also fundamental
" 0 < eÎR$ pО N(("mО N, " nО N, m ³ p, n ³ p) Þ |a m - a n | £ e)
It is natural to try, by analogy with §6, to apply the factorization procedure to the set of fundamental sequences of real numbers. Wouldn't we get a set of equivalence classes of fundamental sequences of real numbers containing the set R as its own subset?
It turns out not.
In this §, a remarkable property will be established: the completeness property of the set of real numbers, which consists in the fact that any fundamental sequence of real numbers converges in R.
7.2. Approximation of real numbers by decimal fractions.
Definition. The sequence (q n) is bounded if $ 0< MÎQ, that (" nО N|q n | £ M)
Theorem 1. Every fundamental sequence of rational numbers is bounded.
Proof. Let (q n) be a fundamental sequence of rational numbers, then, due to the fundamental nature, for e=1 there is such a pн N, What:
$ pО N:((" m ³ p) Þ |q n -q m | £ 1)
m = p - fix, then " n ³ p |q n | £ |q p | + 1.
Indeed: |q n | = |qn -qp +qp | £ |q n -q p | + |q p | z |q n | £ 1 + |q p |.
Setting as M = max (|q 1 |, |q 2 |, … , |q p-1 |, …, 1+|q p |) we get: " nн N|q n | £ M.ð
In clause 6.3. the unary relation “to be positive” was given on the set. We agree to write “>0“. Then a ³ 0 w (a > 0 or a = 0).
Theorem 2 . Let the fundamental sequence (q n) of rational numbers represent a real number a, then:
a) ($ p 1 О N, $ MO Q(" nО N, " n ³ p 1) z |q n | £ M) z a £ M.
b) ($ p 2 О N, $ mО Q(" nО N, " n ³ p 2) Þ q n ³ m) Þ m £ a.
Proof. Since " n³p 1 q n -M £ 0, then the fundamental sequence q n -M - the difference between the fundamental sequence (q n) and the constant sequence M cannot be a positive sequence, since it is either zero or negative.
Therefore, the real number (a-M) represented by this sequence cannot be positive, i.e. a-M £ 0, i.e. a e M.
Similarly, b) is considered.
Theorem 3
. The fundamental sequence (q n) of rational numbers represents a real number a if and only if " 0
(q n)нa ы " 0< eÎR$ pО N(" nО N, n³p) Þ |q n -a| £ e.
Proof. Let's just prove the necessity. It is obvious that "eн R$ e 1 О Q(e 1 £e)
Let the fundamental sequence (q n) of rational numbers be a representative of the number a.
By assumption, it is fundamental, i.e. "0< eÎQ$ pО N(" nО N,"mО N, n³p, m³p) Þ |q n -q n | £e/2.
Fix n³p, then we get the fundamental sequence (q m -q n): (q 1 -q n; q 2 -q n; ...; q n-1 -q n; 0; q n+1 -q n; ...).
All terms of this sequence for m³p satisfy the inequality: |q m -q n |£ e/2.
By Theorem 2, the real number represented by this sequence | a-q n | £e/2.
| a-q n | £ e О R"n³p.
Theorem 4
. Whatever the real number a, there will always be an integer M such that the inequality M £ a (" aО R$! MO Z(M £ a< M+1)) Proof. Step 1. Proof of existence. Let the fundamental sequence (q n) of rational numbers represent a real number a: ((q n)нa). By virtue of Theorem 1, $ Lн Z0, such that " nн N q n ³-L, q n £L: (-L£ q n £L). By Theorem 3 (q n)нa ы " e>0, eн R$ pО N: ((" nО N, n³p) z 1q n -a1 £ e). Then " n³p ½a1=½a- q n + q n ½ £½a- q n ½+½ q n ½ £ e + L. ½a1 £ e + L w -L-e £ a £ L+e. Because e is an arbitrary number >0, then –L £ a £ L. After that, it is obvious that -1-L<
a < L+1. Then among the finite set of integers: -L-1, -L, -L+1, ..., -1, 0, +1, ..., L, L+1, we find first number M+1 for which the condition a< M+1. Then the number M does not satisfy the inequality M £ a< M+1,
т.е. такое число M существует. Step 2. Proof of uniqueness.4 Mathematical theories, as a rule, find their way out in the fact that they allow one set of numbers (initial data) to be processed into another set of numbers, which constitutes an intermediate or final goal of calculations. For this reason, numerical functions occupy a special place in mathematics and its applications. They (more precisely, the so-called differentiable numerical functions) constitute the main object of study of classical analysis. But any description of the properties of these functions that is any complete from the point of view of modern mathematics, as you could already feel in school and as you will soon see, is impossible without a precise definition of the set of real numbers on which these functions act. A number in mathematics, like time in physics, is known to everyone, but is incomprehensible only to specialists. This is one of the main mathematical abstractions, which, apparently, has yet to undergo a significant evolution and the story of which can be devoted to an independent intensive course. Here we mean only to bring together what the reader basically knows about real numbers from high school, highlighting in the form of axioms the fundamental and independent properties of numbers. At the same time, our goal is to give an exact definition of real numbers suitable for subsequent mathematical use and pay special attention to their property of completeness, or continuity, which is the germ of the passage to the limit - the main non-arithmetic operation of analysis. Definition 1. The set E is called the set of real (real) numbers, and its elements are called real (real) numbers if the following set of conditions, called the axiomatics of real numbers, is satisfied: Mapping defined (addition operation) assigning to each ordered pair of elements from E some element called the sum of x and y. In this case, the following conditions are met: There is a neutral element 0 (called zero in the case of addition) such that for any For any element there is an element called opposite to such that Operation 4 is associative, i.e., for any elements from Operation 4 is commutative, i.e., for any elements of E, If an operation is defined on some set that satisfies the axioms, then it is said that the structure of a group is given on or that there is a group. If the operation is called addition, then the group is called additive. If, in addition, it is known that the operation is commutative, i.e., the condition is satisfied, then the group is called commutative or abelian. So the axioms say that E is an additive abelian group. Mapping defined (multiplication operation) assigning to each ordered pair of elements from E some element, called the product of x and y, and in such a way that the following conditions are satisfied: 1. There is a neutral element in the case of multiplication by one) such that 2. For any element there is an element called inverse, such that 3. The operation is associative, i.e. any of E 4. The operation is commutative, that is, for any Note that, with respect to the operation of multiplication, the set can be verified to be a (multiplicative) group. Multiplication is distributive with respect to addition, i.e. Note that, in view of the commutativity of multiplication, the last equality is preserved if the order of the factors in both its parts is reversed. If on some set there are two operations that satisfy all of the listed axioms, then it is called an algebraic field or simply a field. There is a relation between the elements of E, i.e., for elements from E it is established whether it is fulfilled or not. In this case, the following conditions must be satisfied: The relation is called the inequality relation. A set, between some of whose elements there is a relation that satisfies the axioms 0, 1, 2, is known to be called partially ordered, and if, in addition, axiom 3 is satisfied, i.e. any two elements of the set are comparable, then the set is called linearly ordered. Thus, the set of real numbers is linearly ordered by the inequality relation between its elements. If x, are elements of R, then If are elements of R, then If X and Y are non-empty subsets of E that have the property that for any elements, then there exists such that for any elements . This completes the list of axioms whose fulfillment on any set E makes it possible to consider this set as a concrete realization or, as they say, a model of real numbers. This definition formally does not presuppose any preliminary information about numbers, and from it, "including mathematical thought", again, formally, we must already obtain the rest of the properties of real numbers as theorems. We would like to make a few informal remarks about this axiomatic formalism. Imagine that you have not gone from adding apples, cubes, or other named quantities to adding abstract natural numbers; that you did not measure segments and did not come to rational numbers; that you do not know the great discovery of the ancients that the diagonal of a square is incommensurable with its side and therefore its length cannot be a rational number, that is, irrational numbers are needed; that you do not have the notion of “more” that arises in the process of measurement, that you do not illustrate order to yourself, for example, by the image of a number line. If all this had not happened beforehand, then the enumerated set of axioms would not only not be perceived as a definite result of spiritual development, but would rather seem at least strange and in any case an arbitrary figment of fantasy. With regard to any abstract system of axioms, at least two questions immediately arise. First, are these axioms compatible, i.e., is there a set that satisfies all the above conditions? This is the question of the consistency of axiomatics. Secondly, does the given system of axioms uniquely determine the mathematical object, i.e., as logicians would say, is the system of axioms categorical. The unambiguity here should be understood as follows. If persons A and B, independently, have built their own models, for example, of numerical systems that satisfy the axiomatics, then a bijective correspondence can be established between the sets, even if it preserves arithmetic operations and the order relation, i.e. From a mathematical point of view, in this case, they are just different (completely equal) implementations (models) of real numbers (for example, infinite decimal fractions, and - points on the number line). Such realizations are called isomorphic, and the mapping is called an isomorphism. The results of mathematical activity, therefore, do not refer to an individual implementation, but to each model from the class of isomorphic models of a given axiomatics. We will not discuss the above questions here and confine ourselves to informative answers to them. A positive answer to the question about the consistency of axiomatics is always conditional. With regard to numbers, it looks like this: based on the axiomatics of set theory adopted by us (see Ch. I, § 4, item 2), we can construct a set of natural numbers, then a set of rational ones, and, finally, a set E of all real numbers that satisfies all of the above properties. 15. If non-empty sets A and B of real numbers are such that for any and the inequality a< b, то найдется такое действительное число с, что a < с < b. The axiom of completeness is valid only in R. It can be proved that between any unequal rational numbers it is always possible to insert a rational number unequal to them. From the axioms given above, one can deduce the uniqueness of zero and one, the existence and uniqueness of the difference and the quotient. Note, in addition, the properties of inequalities that are widely used in various transformations: 1. If a< b, с < d , то a+c < b+d. 2. If a< b, то –a >-b. 3. If a > 0, b< 0, то ab < 0, а если a < 0, b < 0, то ab >0. (The latter is also true for a > 0, b > 0.) 4. If 0< a < b, 0 < c < d, то 0 < ac < bd . 5. If a< b, c >0, then ac< bc , а если a < b, c < 0, то bc < ac . 6. If 0< a < b, то . 7. 0 < 1, то –
1 < 0. 8. For any positive numbers a and b, there is a number nО N such that na > b (axiom Archimedes, for segments of length a, b, na). The following notation for numerical sets is used: N –
set of natural numbers; Z –
set of integers; Q –
set of rational numbers; I –
set of irrational numbers; R –
set of real numbers; R + is the set of real positive numbers; R_ –
the set of real negative numbers; R 0 is the set of real non-negative numbers; C is the set of complex numbers (the definition and properties of this set are discussed in Section 1.1). Let us introduce the concept of boundedness on the set of real numbers. It will be actively used in the discussion below. We will call a set UPPER (BOTTOM) Bounded if there exists such a real number M ( m )
that any element satisfies the inequality: The number M is called the UPPER BOUND OF THE SET A, and the number m –
LOWER Bound of this set. A set bounded above and below is called bounded. A bunch of N
natural numbers is bounded below, but not bounded above. Set of integers Z
not bounded from above or below. If we consider the set of areas of arbitrary triangles inscribed in a circle of diameter D
, then it is bounded by zero from below, and from above –
the area of any polygon that includes a circle (in particular, the area of the circumscribed square, equal to D
2
). Any set bounded from above (from below) has infinitely many upper (lower) faces. Then, is there a smallest of all upper bounds and a largest of all lower bounds? Let's call the number
the least upper bound of a set bounded above AÌ R
, If: 1.
is one of the upper bounds of the set A
; 2.
is the smallest of the upper bounds of the set A
. In other words, the real number
is the least upper bound of the set AÌ R
, If: Accepted designation Enter in the same way: –
least infimum of a set bounded below A
and corresponding designations In Latin: supremum - the highest, infimum - the lowest. The exact faces of a set may or may not belong to it. THEOREM. Bounded from above (from below) non-empty set of real numbers the exact upper (lower) bound. We accept this theorem without proof. For example, if , then the upper limit can be considered the number 100, the lower -10, and . If , then . In the second example, exact boundaries do not belong to this set. On the set of real numbers, two non-intersecting subsets of algebraic and transcendental numbers can be distinguished. ALGEBRAIC NUMBERS are numbers that are the roots of a polynomial whose coefficients –
whole numbers. In higher algebra, it is proved that the set of complex roots of a polynomial is finite and equal to n. (Complex numbers are a generalization of real numbers). The set of algebraic numbers is countable .
It includes all rational numbers, since numbers of the form satisfy the equation It is also proved that there are algebraic numbers that are not radicals from rational numbers. This very important result stopped fruitless attempts to find solutions of equations of degree higher than the fourth in radicals. The centuries-old search for algebraists who studied this problem was able to generalize the French mathematician E. Galois, who absurdly died at the age of 21. His scientific works are only 60 pages, but they were a brilliant contribution to the development of mathematics. A young man who passionately and uncontrollably loved this science, twice tried to enter the most prestigious educational institution in France at that time. –
Polytechnic School –
unsuccessfully. Started studying at a privileged high school –
expelled due to a conflict with the director. Having become a political prisoner after speaking out against Louis Philippe, he handed over from prison to the Paris Academy of Sciences a manuscript with a study of solving an equation in radicals. The Academy rejected this work. An absurd death in a duel ended the life of this outstanding man. The set that is the difference between the sets of real and algebraic numbers is called the set of TRANSCENDENT NUMBERS .
Obviously, every transcendental number cannot be a root of a polynomial with integer coefficients. At the same time, the proof of the transcendence of any individual numbers caused enormous difficulties. Only in 1882, Professor of the University of Koenigsberg F. Lindemann managed to prove the transcendence of the number, from which it became clear that it was impossible to solve the problem of squaring a circle (to construct a square with the area of a given circle using a compass and a ruler). We see that the ideas of algebra, analysis, geometry mutually penetrate each other. The axiomatic introduction of real numbers is far from the only one. These numbers can be introduced by combining the set of rational and irrational numbers, or as infinite decimals, or by using sections on the set of rational numbers. *1) This material is taken from the 7th chapter of the book: L.I. Lurie FOUNDATIONS OF HIGHER MATHEMATICS / Textbook / M .: Publishing and Trade Corporation "Dashkov and Co", - 2003, - 517 S. 1
/
5 ✪ Axiomatics of real numbers ✪ Introduction. Real numbers | matan #001 | Boris Trushin + ✪ The principle of nested segments | matan #003 | Boris Trushin! ✪ Various principles of continuity | matan #004 | Boris Trushin! ✪ Axiom of continuity. Cantor's principle of nested cuts The following proposition is perhaps the simplest and most convenient for applications formulation of the continuity property of real numbers. In the axiomatic construction of the theory of the real number, this statement, or equivalent to it, is certainly included in the number of axioms of the real number. Axiom of continuity (completeness). A ⊂ R (\displaystyle A\subset \mathbb (R) ) And B ⊂ R (\displaystyle B\subset \mathbb (R) ) and the inequality is satisfied, there is such a real number ξ (\displaystyle \xi ) that for everyone a ∈ A (\displaystyle a\in A) And b ∈ B (\displaystyle b\in B) there is a relation Geometrically, if we treat real numbers as points on a straight line, this statement seems obvious. If two sets A (\displaystyle A) And B (\displaystyle B) are such that on the number line all elements of one of them lie to the left of all elements of the second, then there is a number ξ (\displaystyle \xi ), separating these two sets, that is, lying to the right of all elements A (\displaystyle A)(except perhaps the ξ (\displaystyle \xi )) and to the left of all elements B (\displaystyle B)(same clause). It should be noted here that despite the "obviousness" of this property, for rational numbers it is not always satisfied. For example, consider two sets: It is easy to see that for any elements a ∈ A (\displaystyle a\in A) And b ∈ B (\displaystyle b\in B) the inequality a<
b
{\displaystyle a The significance of the axiom of continuity is such that without it a rigorous construction of mathematical analysis is impossible. To illustrate, we present several fundamental statements of analysis, the proof of which is based on the continuity of real numbers: A m / n = (a n) m (\displaystyle a^(m/n)=\left((\sqrt[(n)](a))\right)^(m)) Finally, again due to the continuity of the number line, one can determine the value of the expression a x (\displaystyle a^(x)) already for arbitrary x ∈ R (\displaystyle x\in \mathbb (R) ). Similarly, using the continuity property, we prove the existence of the number log a b (\displaystyle \log _(a)(b)) for any a , b > 0 , a ≠ 1 (\displaystyle a,b>0,a\neq 1).
For a long historical period of time, mathematicians proved theorems from analysis, in “thin places” referring to the geometric justification, and more often skipping them altogether, since it was obvious. The essential concept of continuity was used without any clear definition. It was only in the last third of the 19th century that the German mathematician Karl Weierstrass produced the arithmetization of analysis, constructing the first rigorous theory of real numbers as infinite decimal fractions. He proposed the classical definition of the limit in the language ε − δ (\displaystyle \varepsilon -\delta ), proved a number of statements that were considered “obvious” before him, and thus completed the construction of the foundation of mathematical analysis. Later, other approaches to the definition of a real number were proposed. In the axiomatic approach, the continuity of real numbers is explicitly singled out as a separate axiom. In constructive approaches to the theory of a real number, for example, when constructing real numbers using Dedekind sections, the continuity property (in one formulation or another) is proved as a theorem. There are several different statements expressing the continuity property of real numbers. Each of these principles can be taken as the basis for constructing the theory of the real number as an axiom of continuity, and all the others can be derived from it. This issue is discussed in more detail in the next section. The question of the continuity of real numbers Dedekind considers in his work "Continuity and irrational numbers" . In it, he compares rational numbers with points on a straight line. As you know, between rational numbers and points of a straight line, you can establish a correspondence when the starting point and unit of measurement of the segments are chosen on the straight line. With the help of the latter, for every rational number a (\displaystyle a) construct the corresponding segment, and putting it aside to the right or to the left, depending on whether there is a (\displaystyle a) positive or negative number, get point p (\displaystyle p) corresponding to the number a (\displaystyle a). So every rational number a (\displaystyle a) matches one and only one point p (\displaystyle p) on a straight line. It turns out that there are infinitely many points on the line that do not correspond to any rational number. For example, a point obtained by plotting the length of the diagonal of a square built on a unit segment. Thus, the realm of rational numbers does not have that completeness, or continuity, which is inherent in a straight line. To find out what this continuity consists of, Dedekind makes the following remark. If p (\displaystyle p) is a certain point of the line, then all points of the line fall into two classes: points located to the left p (\displaystyle p), and points to the right p (\displaystyle p). The very point p (\displaystyle p) can be arbitrarily assigned to either the lower or the upper class. Dedekind sees the essence of continuity in the reverse principle: Geometrically, this principle seems obvious, but we are not in a position to prove it. Dedekind emphasizes that, in essence, this principle is a postulate, which expresses the essence of that property attributed to the direct line, which we call continuity. To better understand the essence of the continuity of the number line in the sense of Dedekind, consider an arbitrary section of the set of real numbers, that is, the division of all real numbers into two non-empty classes, so that all numbers of one class lie on the number line to the left of all numbers of the second. These classes are named respectively lower And upper classes sections. Theoretically, there are 4 possibilities: In the first and second cases, the maximum element of the lower or the minimum element of the upper, respectively, produces this section. In the third case we have jump, and in the fourth space. Thus, the continuity of the number line means that there are no jumps or gaps in the set of real numbers, that is, figuratively speaking, there are no voids. This proposition is also equivalent to Dedekind's continuity principle. Moreover, it can be shown that the statement of the infimum theorem directly follows from the assertion of the supremum theorem, and vice versa (see below). Finite Cover Lemma (Heine - Borel). In any system of intervals covering a segment, there is a finite subsystem covering this segment. Limit Point Lemma (Bolzano - Weierstrass). Every infinite bounded number set has at least one limit point.. The second group expresses the fact that the set of real numbers is , and the order relation is consistent with the basic operations of the field. Thus, the first and second groups of axioms mean that the set of real numbers is an ordered field. The third group of axioms consists of one axiom - the axiom of continuity (or completeness). To show the equivalence of various formulations of the continuity of the real numbers, it must be proved that if one of these propositions holds for an ordered field, then all the others are true. Theorem. Let be an arbitrary linear ordered set . The following statements are equivalent: As can be seen from this theorem, these four sentences use only what is on R (\displaystyle (\mathsf (R))) introduced a linear order relation, and do not use the field structure. Thus, each of them expresses the property R (\displaystyle (\mathsf (R))) as a linearly ordered set. This property (of an arbitrary linearly ordered set, not necessarily the set of real numbers) is called continuity, or completeness, according to Dedekind. Proving the equivalence of other sentences already requires a field structure. Theorem. Let R (\displaystyle (\mathsf (R)))- an arbitrary ordered field. The following sentences are equivalent: Comment. As can be seen from the theorem, the principle of nested segments in itself is not equivalent Dedekind's continuity principle. The principle of nested segments follows from the Dedekind continuity principle, but for the converse it is required to additionally require that the ordered field . Continuity of real numbers- a property of the system of real numbers, which the set of rational numbers does not have. Sometimes, instead of continuity, they talk about completeness of the system of real numbers. There are several different formulations of the continuity property, the best known of which are: Dedekind's principle of continuity of real numbers, principle of nested segments Cauchy - Cantor, supremum theorem. Depending on the accepted definition of a real number, the continuity property can either be postulated as an axiom - in one formulation or another, or proven as a theorem. The following proposition is perhaps the simplest and most convenient for applications formulation of the continuity property of real numbers. In the axiomatic construction of the theory of a real number, this statement, or equivalent to it, is certainly among the axioms of a real number. Geometric illustration of the axiom of continuity Axiom of continuity (completeness). Whatever the non-empty sets and , such that for any two elements and the inequality holds, there exists a number ξ such that for all and the relation holds Geometrically, if we treat real numbers as points on a straight line, this statement seems obvious. If two sets A And B are such that on the number line all elements of one of them lie to the left of all elements of the second, then there is a number ξ, separating these two sets, that is, lying to the right of all elements A(except, perhaps, ξ itself) and to the left of all elements B(same clause). It should be noted here that despite the "obviousness" of this property, for rational numbers it is not always satisfied. For example, consider two sets: It is easy to see that for any elements and the inequality a < b. However rational there is no number ξ separating these two sets. Indeed, this number can only be , but it is not rational. The significance of the axiom of continuity is such that without it a rigorous construction of mathematical analysis is impossible. To illustrate, we present several fundamental statements of analysis, the proof of which is based on the continuity of real numbers: Finally, again due to the continuity of the number line, one can determine the value of the expression a x already for arbitrary . Similarly, using the continuity property, we prove the existence of the number log a b for any . For a long historical period of time, mathematicians proved theorems from analysis, in “thin places” referring to the geometric justification, and more often skipping them altogether because it was obvious. The essential concept of continuity was used without any clear definition. Only in the last third of the 19th century did the German mathematician Karl Weierstrass produce the arithmetization of analysis, constructing the first rigorous theory of real numbers as infinite decimal fractions. He proposed a classical definition of the limit in the language, proved a number of statements that were considered "obvious" before him, and thus completed the foundation of mathematical analysis. Later, other approaches to the definition of a real number were proposed. In the axiomatic approach, the continuity of real numbers is explicitly singled out as a separate axiom. In constructive approaches to the theory of a real number, for example, when constructing real numbers using Dedekind sections, the continuity property (in one formulation or another) is proved as a theorem. There are several different statements expressing the continuity property of real numbers. Each of these principles can be taken as the basis for constructing the theory of the real number as an axiom of continuity, and all the others can be derived from it. This issue is discussed in more detail in the next section. The question of the continuity of real numbers is considered by Dedekind in his work Continuity and Irrational Numbers. In it, he compares the rational numbers with the points of a straight line. As you know, a correspondence can be established between rational numbers and points of a straight line when a starting point and a unit of measurement of segments are chosen on a straight line. With the help of the latter, for every rational number a construct the corresponding segment, and putting it aside to the right or to the left, depending on whether there is a positive or negative number, get point p corresponding to the number a. So every rational number a matches one and only one point p on a straight line. It turns out that there are infinitely many points on the line that do not correspond to any rational number. For example, a point obtained by plotting the length of the diagonal of a square built on a unit segment. Thus, the realm of rational numbers does not have that completeness, or continuity, which is inherent in a straight line. To find out what this continuity consists of, Dedekind makes the following remark. If p is a certain point of the line, then all points of the line fall into two classes: points located to the left p, and points to the right p. The very point p can be arbitrarily assigned to either the lower or the upper class. Dedekind sees the essence of continuity in the reverse principle: Geometrically, this principle seems obvious, but we are not in a position to prove it. Dedekind emphasizes that, in essence, this principle is a postulate, which expresses the essence of that property attributed to the direct line, which we call continuity. To better understand the essence of the continuity of the number line in the sense of Dedekind, consider an arbitrary section of the set of real numbers, that is, the division of all real numbers into two non-empty classes, so that all numbers of one class lie on the number line to the left of all numbers of the second. These classes are named respectively lower And upper classes sections. Theoretically, there are 4 possibilities: In the first and second cases, the maximum element of the lower or the minimum element of the upper, respectively, produces this section. In the third case we have jump, and in the fourth space. Thus, the continuity of the number line means that there are no jumps or gaps in the set of real numbers, that is, figuratively speaking, there are no voids. If we introduce the concept of a section of the set of real numbers, then the Dedekind continuity principle can be formulated as follows. Dedekind's continuity principle (completeness). For each section of the set of real numbers, there is a number that produces this section. Comment. The formulation of the Axiom of Continuity about the existence of a point separating two sets is very reminiscent of the formulation of Dedekind's principle of continuity. In fact, these statements are equivalent, and, in essence, are different formulations of the same thing. Therefore, both of these statements are called the principle of continuity of real numbers according to Dedekind. Lemma on nested segments (Cauchy - Kantor). Any system of nested segments has a non-empty intersection, that is, there is at least one number that belongs to all segments of the given system. If, in addition, the length of the segments of the given system tends to zero, that is, then the intersection of the segments of this system consists of one point. This property is called continuity of the set of real numbers in the sense of Cantor. It will be shown below that for the Archimedean ordered fields the continuity according to Cantor is equivalent to the continuity according to Dedekind. The supremacy principle. Every non-empty set of real numbers bounded above has a supremum. In calculus courses, this proposition is usually a theorem, and its proof makes significant use of the continuity of the set of real numbers in one form or another. At the same time, on the contrary, it is possible to postulate the existence of a supremum for any non-empty set bounded from above, and relying on this to prove, for example, the Dedekind continuity principle. Thus, the supremum theorem is one of the equivalent formulations of the continuity property of real numbers. Comment. Instead of the supremum, one can use the dual concept of the infimum. The infimum principle. Every non-empty set of real numbers bounded below has an infimum. This proposition is also equivalent to Dedekind's continuity principle. Moreover, it can be shown that the statement of the infimum theorem directly follows from the assertion of the supremum theorem, and vice versa (see below). Finite Cover Lemma (Heine - Borel). In any system of intervals covering a segment, there is a finite subsystem covering this segment. Limit Point Lemma (Bolzano - Weierstrass). Every infinite bounded number set has at least one limit point. Let's make some preliminary remarks. In accordance with the axiomatic definition of a real number, the set of real numbers satisfies three groups of axioms. The first group is the field axioms. The second group expresses the fact that the collection of real numbers is a linearly ordered set, and the order relation is consistent with the basic operations of the field. Thus, the first and second groups of axioms mean that the set of real numbers is an ordered field. The third group of axioms consists of one axiom - the axiom of continuity (or completeness). To show the equivalence of various formulations of the continuity of the real numbers, it must be proved that if one of these propositions holds for an ordered field, then all the others are true. Theorem. Let be an arbitrary linearly ordered set. The following statements are equivalent: As can be seen from this theorem, these four propositions only use what the linear order relation has introduced and do not use the field structure. Thus, each of them expresses a property as a linearly ordered set. This property (of an arbitrary linearly ordered set, not necessarily the set of real numbers) is called continuity, or completeness, according to Dedekind. Proving the equivalence of other sentences already requires a field structure. Theorem. Let be an arbitrary ordered field. The following sentences are equivalent: Comment. As can be seen from the theorem, the principle of nested segments in itself is not equivalent Dedekind's continuity principle. The principle of nested segments follows from the Dedekind continuity principle, but for the converse it is required to additionally require that the ordered field satisfies the Archimedes axiom The proof of the above theorems can be found in the books from the bibliography given below.§ 1. Axiomatics and some general properties of the set of real numbers
1. Definition of the set of real numbers
(I) Axioms of addition
(II) Axioms of multiplication
(I, II) Relation between addition and multiplication
(III) Axioms of order
(I, III) Relation between addition and order in R
(II, III) Relation between multiplication and order in R
(IV) Axiom of completeness (continuity)
Encyclopedic YouTube
Subtitles
Axiom of continuity
The role of the axiom of continuity in the construction of mathematical analysis
Other Statements of the Continuity Property and Equivalent Propositions
Continuity according to Dedekind
Finite cover lemma (Heine-Borel principle)
Limit point lemma (Bolzano-Weierstrass principle)
Plan:
Introduction
Literature Introduction
1. Axiom of continuity
2. The role of the axiom of continuity in the construction of mathematical analysis
3. Other formulations of the continuity property and equivalent propositions
3.1. Continuity according to Dedekind
3.2. Lemma on nested segments (Cauchy-Cantor principle)
3.3. The supremum principle
3.4. Finite cover lemma (Heine-Borel principle)
3.5. Limit point lemma (Bolzano-Weierstrass principle)
4. Equivalence of sentences expressing the continuity of the set of real numbers
Notes
Literature
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