On the axiomatic method of constructing a theory. Definition of a natural number. Study of axioms of the theory of integers Axiomatic construction of a system of integers

In the school mathematics course, real numbers were defined in a constructive way, based on the need to carry out measurements. This definition was not strict and often led researchers into dead ends. For example, the question of the continuity of real numbers, that is, are there voids in this set. Therefore, when conducting mathematical research, it is necessary to have a strict definition of the concepts being studied, at least within the framework of some intuitive assumptions (axioms) that are consistent with practice.

Definition: A set of elements x, y, z, …, consisting of more than one element, called a set R real numbers, if the following operations and relations are established for these objects:

I group of axioms– axioms of the operation of addition.

In abundance R the addition operation was introduced, that is, for any pair of elements a And b amount and designated a + b
I 1. a+b=b+a, a, b R .

I 2. a+(b+c)=(a+b)+c,a, b, c R .

I 3. There is such an element called zero and denoted by 0, which for any a R condition is met a+0=a.

I 4. For any element a R there is an element called it opposite and denoted by - a, for which a+(-a)=0. Element a+(-b), a, b R , called difference elements a And b and is designated a - b.

II – group of axioms - axioms of the multiplication operation. In abundance R operation entered multiplication, that is, for any pair of elements a And b a single element is defined, called them work and designated a b, so that the following conditions are satisfied:
II 1. ab=ba,a, b R .

II 2 a(bc)=(ab)c, a, b, c R .

II 3. There is an element called unit and denoted by 1, which for any a R condition is met a 1=a.

II 4. For anyone a 0 there is an element called it reverse and denoted by or 1/ a, for which a=1. Element a , b 0, called private from division a on b and is designated a:b or or a/b.

II 5. Relationship between addition and multiplication operations: for any a, b, c R condition is satisfied ( ac + b)c=ac+bc.

A collection of objects that satisfies the axioms of groups I and II is called a number field or simply a field. And the corresponding axioms are called field axioms.

III – the third group of axioms - axioms of order. For elements R order relation is defined. It is as follows. For any two different elements a And b one of two relationships holds: either a b(reads " a less or equal b"), or a b(reads " a more or equal b"). It is assumed that the following conditions are met:

III 1. a a for each a. From a b, b should a=b.

III 2. Transitivity. If a b And b c, That a c.

III 3. If a b, then for any element c occurs a+c b+c.

III 4. If a 0, b 0, That ab 0 .

Group IV of axioms consists of one axiom - the axiom of continuity. For any non-empty sets X And Y from R such that for each pair of elements x X And y Y inequality holds x < y, there is an element a R, satisfying the condition

Rice. 2

x < a < y, x X, y Y(Fig. 2). The listed properties completely define the set of real numbers in the sense that all its other properties follow from these properties. This definition uniquely defines the set of real numbers up to the specific nature of its elements. The caveat that a set contains more than one element is necessary because a set consisting of only zero obviously satisfies all the axioms. In what follows, we will call the elements of the set R numbers.

Let us now define the familiar concepts of natural, rational and irrational numbers. The numbers 1, 2 1+1, 3 2+1, ... are called natural numbers, and their set is denoted N . From the definition of the set of natural numbers it follows that it has the following characteristic property: If

1) A N ,

3) for each element x A the inclusion x+ 1 A, then A=N .

Indeed, according to condition 2) we have 1 A, therefore, by property 3) and 2 A, and then according to the same property we get 3 A. Since any natural number n is obtained from 1 by successively adding the same 1 to it, then n A, i.e. N A, and since by condition 1 the inclusion A N , That A=N .

The principle of proof is based on this property of natural numbers by mathematical induction. If there are many statements, each of which is assigned a natural number (its number) n=1, 2, ..., and if it is proven that:

1) statement number 1 is true;

2) from the validity of the statement with any number n N follows the validity of the statement with number n+1;

then the validity of all statements is thereby proven, i.e. any statement with an arbitrary number n N .

Numbers 0, + 1, + 2, ... is called integers, their set is denoted Z .

Numbers of the form m/n, Where m And n whole, and n 0, are called rational numbers. The set of all rational numbers is denoted by Q .

Real numbers that are not rational are called irrational, their set is denoted I .

The question arises that perhaps the rational numbers exhaust all the elements of the set R? The answer to this question is given by the axiom of continuity. Indeed, this axiom does not hold for rational numbers. For example, consider two sets:

It is easy to see that for any elements and the inequality . However rational there is no number separating these two sets. In fact, this number can only be , but it is not rational. This fact indicates that there are irrational numbers in the set R.

In addition to the four arithmetic operations on numbers, you can perform the operations of exponentiation and root extraction. For any number a R and natural n degree a n is defined as the product n factors equal a:

A-priory a 0 1, a>0, a- n 1/ a n, a 0, n- natural number.

Example. Bernoulli's inequality: ( 1+x)n> 1+nx Prove by induction.

Let a>0, n- natural number. Number b called root n th degree from among a, If b n =a. In this case it is written . Existence and uniqueness of a positive root of any degree n from any positive number will be proven below in Section 7.3.
Even root, a 0, has two meanings: if b = , k N , then -b= . Indeed, from b 2k = a follows that

(-b)2k = ((-b) 2 )k = (b 2)k = b 2k

A non-negative value is called its arithmetic value.
If r = p/q, Where p And q whole, q 0, i.e. r is a rational number, then for a > 0

(2.1)

Thus, the degree a r defined for any rational number r. From its definition it follows that for any rational r there is equality

a -r = 1/a r.

Degree a x(number x called exponent) for any real number x is obtained using continuous propagation of the degree with a rational exponent (see Section 8.2 for more information). For any number a R non-negative number

it's called absolute value or module. For absolute values ​​of numbers, the following inequalities are valid:

|a + b| < |a| + |b|,
||a - b|| < |a - b|, a, b R

They are proven using properties I-IV of real numbers.

The role of the axiom of continuity in the construction of mathematical analysis

The significance of the axiom of continuity is such that without it a rigorous construction of mathematical analysis is impossible. [ source not specified 1351 days] To illustrate, we present several fundamental statements of analysis, the proof of which is based on the continuity of real numbers:

· (Weierstrass's theorem). Every bounded monotonically increasing sequence converges

· (Bolzano-Cauchy theorem). A function continuous on a segment, taking values ​​of different signs at its ends, vanishes at some internal point of the segment

· (Existence of power, exponential, logarithmic and all trigonometric functions throughout the “natural” domain of definition). For example, it is proved that for everyone and the whole there exists , that is, a solution to the equation. This allows you to determine the value of the expression for all rationals:

Finally, again thanks to the continuity of the number line, it is possible to determine the value of the expression for an arbitrary one. Similarly, using the property of continuity, the existence of a number is proved for any .

For a long historical period of time, mathematicians proved theorems from analysis, in “subtle places” referring to geometric justification, and more often - skipping them altogether because it was obvious. The all-important concept of continuity was used without any clear definition. Only in the last third of the 19th century did the German mathematician Karl Weierstrass arithmetize analysis, constructing the first rigorous theory of real numbers as infinite decimal fractions. He proposed the classical definition of a limit in the language, proved a number of statements that had been considered “obvious” before him, and thereby completed the construction of the foundation of mathematical analysis.

Later, other approaches to determining a real number were proposed. In the axiomatic approach, the continuity of real numbers is explicitly highlighted as a separate axiom. In constructive approaches to the theory of real numbers, for example, when constructing real numbers using Dedekind sections, the property of continuity (in one form or another) is proven as a theorem.

Other formulations of the property of continuity and equivalent sentences[edit | edit wiki text]

There are several different statements expressing the property of continuity of real numbers. Each of these principles can be used 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.

Continuity according to Dedekind[edit | edit wiki text]

Main article:Theory of cuts in the field of rational numbers

Dedekind considers the question of the continuity of real numbers in his work “Continuity and Irrational Numbers”. In it, he compares rational numbers with points on a straight line. As is known, a correspondence can be established between rational numbers and points on a line when the starting point and the unit of measurement of the segments are chosen on the line. Using the latter, you can construct a corresponding segment for each rational number, and by putting it off to the right or left, depending on whether there is a positive or negative number, you can get a point corresponding to the number. Thus, for each rational number there corresponds one and only one point on the 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 constructed on a unit segment. Thus, the region 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 there is a certain point on a line, then all points on the line fall into two classes: points located to the left, and points located to the right. The point itself can be arbitrarily assigned to either the lower or upper class. Dedekind sees the essence of continuity in the reverse principle:

Geometrically, this principle seems obvious, but we are not able to prove it. Dedekind emphasizes that, in essence, this principle is a postulate that expresses the essence of that property attributed to the direct, 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 the numbers of one class lie on the number line to the left of all the numbers of the second. These classes are named accordingly lower And upper classes sections. In theory there are 4 possibilities:

1. The lower class has a maximum element, the upper class does not have a minimum

2. The lower class does not have a maximum element, but the upper class has a minimum

3. The lower class has the maximum and the upper class has the minimum elements

4. There is no maximum element in the lower class, and no minimum element in the upper class

In the first and second cases, the maximum element of the bottom or the minimum element of the top, respectively, produces this section. In the third case we have leap, and in the fourth - space. Thus, the continuity of the number line means that in the set of real numbers there are no jumps or gaps, that is, figuratively speaking, there are no voids.

If we introduce the concept of a section of a set of real numbers, then Dedekind’s principle of continuity can be formulated as follows.

Dedekind's principle of continuity (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 reality, these statements are equivalent, and are essentially different formulations of the same thing. Therefore, both of these statements are called Dedekind's principle of continuity of real numbers.

Lemma on nested segments (Cauchy-Cantor principle)[edit | edit wiki text]

Main article:Lemma on nested segments

Lemma on nested segments (Cauchy - Cantor). Any system of nested segments

has a non-empty intersection, that is, there is at least one number that belongs to all segments of a given system.

If, in addition, the length of segments of a given system tends to zero, that is

then the intersection of segments of this system consists of one point.

This property is called continuity of the set of real numbers in the sense of Cantor. Below we will show that for Archimedean ordered fields, Cantor continuity is equivalent to Dedekind continuity.

The supremum principle[edit | edit wiki text]

The supremum 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 essentially makes use of the continuity of the set of real numbers in some form. At the same time, one can, on the contrary, postulate the existence of a supremum for any non-empty set bounded above, and relying on this to prove, for example, the principle of continuity according to Dedekind. Thus, the supremum theorem is one of the equivalent formulations of the property of continuity of real numbers.

Comment. Instead of supremum, one can use the dual concept of infimum.

The principle of infimum. Every non-empty set of real numbers bounded from below has an infimum.

This proposal is also equivalent to Dedekind's continuity principle. Moreover, it can be shown that the statement of the supremum theorem directly follows from the statement of the infimum theorem, and vice versa (see below).

Finite covering lemma (Heine-Borel principle)[edit | edit wiki text]

Main article:Heine-Borel Lemma

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 principle)[edit | edit wiki text]

Main article:Bolzano-Weierstrass theorem

Limit point lemma (Bolzano - Weierstrass). Every infinite limited number set has at least one limit point.

Equivalence of sentences expressing the continuity of the set of real numbers[edit | edit wiki text]

Let us make some preliminary remarks. According to the axiomatic definition of a real number, the set of real numbers satisfies three groups of axioms. The first group is field axioms. The second group expresses the fact that the set 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 represents an ordered field. The third group of axioms consists of one axiom - the axiom of continuity (or completeness).

To show the equivalence of different formulations of the continuity of real numbers, it is necessary to prove that if one of these statements holds for an ordered field, then the validity of all the others follows from this.

Theorem. Let be an arbitrary linearly ordered set. The following statements are equivalent:

1. Whatever non-empty sets and such that for any two elements and the inequality holds, there exists an element such that for all and the relation holds

2. For every section in there is an element producing this section

3. Every non-empty set bounded above has a supremum

4. Every non-empty set bounded from below has an infimum

As can be seen from this theorem, these four sentences only use the fact that the linear order relation is introduced, and do not use the structure of the field. Thus, each of them expresses the property of being 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 the presence of a field structure.

Theorem. Let be an arbitrary ordered field. The following sentences are equivalent:

1. (as a linearly ordered set) is Dedekind complete

2. To fulfill Archimedes' principle And principle of nested segments

3. For the Heine-Borel principle is satisfied

4. The Bolzano-Weierstrass principle is fulfilled

Comment. As can be seen from the theorem, the principle of nested segments itself not equivalent Dedekind's principle of continuity. From Dedekind's principle of continuity the principle of nested segments follows, but for the converse it is necessary to additionally require that the ordered field satisfy the Archimedes axiom

The proof of the above theorems can be found in the books from the reference list below.

· Kudryavtsev, L. D. Course of mathematical analysis. - 5th ed. - M.: “Drofa”, 2003. - T. 1. - 704 p. - ISBN 5-7107-4119-1.

· Fikhtengolts, G. M. Fundamentals of mathematical analysis. - 7th ed. - M.: “FIZMATLIT”, 2002. - T. 1. - 416 p. - ISBN 5-9221-0196-X.

· Dedekind, R. Continuity and irrational numbers = Stetigkeit und irrationale Zahlen. - 4th revised edition. - Odessa: Mathesis, 1923. - 44 p.

· Zorich, V. A. Mathematical analysis. Part I. - Ed. 4th, corrected. - M.: "MCNMO", 2002. - 657 p. - ISBN 5-94057-056-9.

· Continuity of functions and numerical domains: B. Bolzano, L. O. Cauchy, R. Dedekind, G. Cantor. - 3rd ed. - Novosibirsk: ANT, 2005. - 64 p.

4.5. Axiom of continuity

Whatever are the two non-empty sets of real numbers A and

B , for which for any elements a ∈ A and b ∈ B the inequality

a ≤ b, there is a number λ such that for all a ∈ A, b ∈ B the following holds:

equality a ≤ λ ≤ b.

The property of continuity of real numbers means that on real

there are no “voids” in the vein line, that is, the points representing numbers fill

the entire real axis.

Let us give another formulation of the axiom of continuity. To do this, we introduce

Definition 1.4.5. We will call two sets A and B a section

set of real numbers, if

1) sets A and B are not empty;

2) the union of sets A and B constitutes the set of all real

numbers;

3) every number in set A is less than a number in set B.

That is, every set forming a section contains at least one

element, these sets do not contain common elements and, if a ∈ A and b ∈ B, then

We will call set A the lower class, and set B the upper class.

section class. We will denote the section by A B.

The simplest examples of sections are the sections obtained following

blowing way. Let's take some number α and put

A = ( x x< α } , B = { x x ≥ α } . Легко видеть, что эти множества не пусты, не пере-

are cut and if a ∈ A and b ∈ B, then a< b , поэтому множества A и B образуют

section. Similarly, you can form a section by sets

A =(x x ≤ α ) , B =(x x > α ) .

We will call such sections sections generated by the number α or

we will say that the number α produces this section. This can be written as

Sections generated by any number have two interesting

properties:

Property 1. Either the upper class contains the smallest number, and the lower one

class does not have the largest number, or the lower class contains the largest number

lo, and in the upper class there is no least.

Property 2. The number generating a given section is unique.

It turns out that the axiom of continuity formulated above is equivalent to

is consistent with the statement called Dedekind's principle:

Dedekind's principle. For each section there is a number generating

this is a section.

Let us prove the equivalence of these statements.

Let the axiom of continuity be true, and some se-

reading A B . Then, since classes A and B satisfy the conditions, the formula

stated in the axiom, there is a number λ such that a ≤ λ ≤ b for any numbers

a ∈ A and b ∈ B. But the number λ must belong to one and only one of

classes A or B, therefore one of the inequalities a ≤ λ will be satisfied< b или

a< λ ≤ b . Таким образом, число λ либо является наибольшим в нижнем классе,

or the smallest in the upper class and generates the given section.

Conversely, let Dedekind’s principle be satisfied and two non-empty

sets A and B such that for all a ∈ A and b ∈ B the inequality

a ≤ b. Let us denote by B the set of numbers b such that a ≤ b for any

b ∈ B and all a ∈ A. Then B ⊂ B. For the set A we take the set of all numbers

villages not included in B.

Let us prove that the sets A and B form a section.

Indeed, it is obvious that the set B is not empty, since it contains

non-empty set B. The set A is also not empty, since if a number a ∈ A,

then the number a − 1∉ B, since any number included in B must be at least

numbers a, therefore, a − 1∈ A.

the set of all real numbers, due to the choice of sets.

And finally, if a ∈ A and b ∈ B, then a ≤ b. Indeed, if any

number c will satisfy the inequality c > b, where b ∈ B, then the incorrect

equality c > a (a is an arbitrary element of the set A) and c ∈ B.

So, A and B form a section, and by virtue of Dedekind’s principle, there is a number

lo λ generating this section, that is, being either the largest in the class

Let us prove that this number cannot belong to class A. Valid

but, if λ ∈ A, then there is a number a* ∈ A such that λ< a* . Тогда существует

the number a′ lying between the numbers λ and a*. From the inequality a′< a* следует, что

a′ ∈ A , then from the inequality λ< a′ следует, что λ не является наибольшим в

class A, which contradicts Dedekind's principle. Therefore, the number λ will be

is the smallest in class B and for all a ∈ A and the inequality will hold

a ≤ λ ≤ b , which is what needed to be proven.◄

Thus, the property formulated in the axiom and the property

formulated in Dedekind's principle are equivalent. In the future these

properties of the set of real numbers we will call continuity

according to Dedekind.

From the continuity of the set of real numbers according to Dedekind it follows

two important theorems.

Theorem 1.4.3. (Archimedes' principle) Whatever the real number is

a, there is a natural number n such that a< n .

Let us assume that the statement of the theorem is false, that is, there is such a

some number b0 such that the inequality n ≤ b0 holds for all natural numbers

n. Let's divide the set of real numbers into two classes: into class B we include

all numbers b satisfying the inequality n ≤ b for any natural n.

This class is not empty because it contains the number b0. We'll put everything in class A

the remaining numbers. This class is also not empty, since any natural number

included in A. Classes A and B do not intersect and their union is

the set of all real numbers.

If we take arbitrary numbers a ∈ A and b ∈ B, then there is a natural number

number n0 such that a< n0 ≤ b , откуда следует, что a < b . Следовательно, классы

A and B satisfy Dedekind's principle and there is a number α that

generates a section A B, that is, α is either the largest in class A or

or the smallest in class B. If we assume that α is in class A, then

one can find a natural number n1 for which the inequality α< n1 .

Since n1 is also included in A, the number α will not be the largest in this class,

therefore, our assumption is incorrect and α is the smallest in

class B.

On the other hand, take the number α − 1, which is included in the class A. Sledova-

Therefore, there is a natural number n2 such that α − 1< n2 , откуда получим

α < n2 + 1 . Так как n2 + 1 - натуральное число, то из последнего неравенства

it follows that α ∈ A. The resulting contradiction proves the theorem.◄

Consequence. Whatever numbers a and b are such that 0< a < b , существует

a natural number n for which the inequality na > b holds.

To prove it, it is enough to apply Archimedes’ principle to the number

and use the property of inequalities.◄

The corollary has a simple geometric meaning: Whatever the two

segment, if on the larger of them, from one of its ends successively

put the smaller one, then in a finite number of steps you can go beyond

larger segment.

Example 1. Prove that for every non-negative number a there exists

the only non-negative real number t such that

t n = a, n ∈ , n ≥ 2 .

This theorem about the existence of an arithmetic root of the nth degree

from a non-negative number in a school algebra course is accepted without proof

deeds.

☺If a = 0, then x = 0, so the proof of the existence of arithmetic

The real root of a is required only for a > 0.

Let us assume that a > 0 and divide the set of all real numbers

for two classes. In class B we include all positive numbers x that satisfy

create the inequality x n > a, in class A, everyone else.

According to Archimedes' axiom, there are natural numbers k and m such that

< a < k . Тогда k 2 ≥ k >a and 2 ≤< a , т.е. оба класса непусты, причем класс

A contains positive numbers.

Obviously, A ∪ B = and if x1 ∈ A and x2 ∈ B, then x1< x2 .

Thus, classes A and B form a cross-section. The number that makes up this

section, denoted by t. Then t is either the largest number in the class

ce A, or the smallest in class B.

Let us assume that t ∈ A and t n< a . Возьмем число h , удовлетворяющее нера-

sovereignty 0< h < 1 . Тогда

(t + h)n = t n + Cnt n−1h + Cn t n−2h2 + ... + Cnn hn< t n + Cnt n−1h + Cn t n−2h + ... + Cn h =

T n + h (Cnt n−1 + Cn t n−2 + ... + Cn + Cn t n) − hCn t n = t n + h (t + 1) − ht n =

T n + h (t + 1) − t n

Then we get (t + h)< a . Это означает,

Hence, if we take h<

that t + h ∈ A, which contradicts the fact that t is the largest element in the class A.

Similarly, if we assume that t is the smallest element of class B,

then, taking a number h satisfying the inequalities 0< h < 1 и h < ,

we get (t − h) = t n − Cnt n−1h + Cn t n−2 h 2 − ... + (−1) Cn h n >

> t n − Cnt n−1h + Cn t n−2h + ... + Cn h = t n − h (t + 1) − t n > a .

This means that t − h ∈ B and t cannot be the smallest element

class B. Therefore, t n = a.

Uniqueness follows from the fact that if t1< t2 , то t1n < t2 .☻ n

Example 2. Prove that if a< b , то всегда найдется рациональное число r

such that a< r < b .

☺If the numbers a and b are rational, then the number is rational and satisfactory

satisfies the required conditions. Let us assume that at least one of the numbers a or b

irrational, for example, let's say that the number b is irrational. Presumably

We also assume that a ≥ 0, then b > 0. Let us write the representations of numbers a and b in the form

decimal fractions: a = α 0,α1α 2α 3.... and b = β 0, β1β 2 β3..., where the second fraction is infinite

intermittent and non-periodic. As for the representation of the number a, we will consider

It should be noted that if a number a is rational, then its notation is either finite or it is not

a periodic fraction whose period is not equal to 9.

Since b > a, then β 0 ≥ α 0; if β 0 = α 0, then β1 ≥ α1; if β1 = α1, then β 2 ≥ α 2

etc., and there is a value of i at which for the first time there will be

the strict inequality βi > α i is satisfied. Then the number β 0, β1β 2 ...βi will be rational

nal and will lie between the numbers a and b.

If a< 0 , то приведенное рассуждение надо применить к числам a + n и

b + n, where n is a natural number such that n ≥ a. The existence of such a number

follows from Archimedes' axiom. ☻

Definition 1.4.6. Let a sequence of segments of the number line be given

([ an ; bn ]), an< bn . Эту последовательность будем называть системой вло-

of segments if for any n the inequalities an ≤ an+1 and

For such a system, inclusions are made

[a1; b1 ] ⊃ [ a2 ; b2 ] ⊃ [ a3 ; b3 ] ⊃ ... ⊃ [ an ; bn ] ⊃ ... ,

that is, each subsequent segment is contained in the previous one.

Theorem 1.4.4. For any system of nested segments there is

at least one point that is included in each of these segments.

Let's take two sets A = (an) and B = (bn). They are not empty and for any

n and m the inequality an< bm . Докажем это.

If n ≥ m, then an< bn ≤ bm . Если n < m , то an ≤ am < bm .

Thus, classes A and B satisfy the axiom of continuity and,

therefore, there is a number λ such that an ≤ λ ≤ bn for any n, i.e. This

the number belongs to any segment [ an ; bn ] .◄

In what follows (Theorem 2.1.8) we will refine this theorem.

The statement formulated in Theorem 1.4.4 is called the principle

Cantor, and a set that satisfies this condition will be called non-

discontinuous according to Cantor.

We have proven that if an ordered set is Dede-continuous

kindu, then the principle of Archimedes is fulfilled in it and it is continuous according to Cantor.

It can be proven that an ordered set in which the principles are satisfied

cipes of Archimedes and Cantor, will be continuous according to Dedekind. Proof

This fact is contained, for example, in.

Archimedes' principle allows each line segment to compare non-

which is the only positive number satisfying the conditions:

1. equal segments correspond to equal numbers;

2. If the point B of the segment AC and the segments AB and BC correspond to the numbers a and

b, then the segment AC corresponds to the number a + b;

3. The number 1 corresponds to a certain segment.

The number corresponding to each segment and satisfying the conditions 1-3 on-

is called the length of this segment.

Cantor's principle allows us to prove that for every positive

number, you can find a segment whose length is equal to this number. Thus,

between the set of positive real numbers and the set of segments

kovs, which are laid off from a certain point on a straight line along a given side

from this point, a one-to-one correspondence can be established.

This allows us to define the numerical axis and introduce correspondence between

I'm waiting for real numbers and points on a line. To do this, let's take some

first line and select point O on it, which will divide this line into two

beam. We will call one of these rays positive, and the second negative.

nom. Then we will say that we have chosen the direction on this straight line.

Definition 1.4.7. We will call the number axis the straight line on which

a) point O, called the origin or origin of coordinates;

b) direction;

c) a segment of unit length.

Now for each real number a we associate a point M with a number

howl straight so that

a) the number 0 corresponded to the origin of coordinates;

b) OM = a - the length of the segment from the origin to point M was equal to

modulo number;

c) if a is positive, then the point is taken on the positive ray and, if

If it is negative, then it is negative.

This rule establishes a one-to-one correspondence between

a set of real numbers and a set of points on a line.

We will also call the number line (axis) the real line

This also implies the geometric meaning of the modulus of a real number.

la: the modulus of a number is equal to the distance from the origin to the point depicted

pressing this number on the number line.

Now we can give a geometric interpretation to properties 6 and 7

modulus of a real number. For positive C of the number x, I satisfy

satisfying property 6, fill the interval (−C, C), and the numbers x satisfying

property 7, lie on the rays (−∞,C) or (C, +∞).

Let us note one more remarkable geometric property of the module of matter:

real number.

The modulus of the difference between two numbers is equal to the distance between the points, corresponding to

corresponding to these numbers on the real axis.

ry standard number sets.

Set of natural numbers;

Set of integers;

Set of rational numbers;

Set of real numbers;

Sets, respectively, of integers, rational and real

real non-negative numbers;

Set of complex numbers.

In addition, the set of real numbers is denoted as (−∞, +∞) .

Subsets of this set:

(a, b) = ( x | x ∈ R, a< x < b} - интервал;

[ a, b] = ( x | x ∈ R, a ≤ x ≤ b) - segment;

(a, b] = ( x | x ∈ R, a< x ≤ b} или [ a, b) = { x | x ∈ R, a ≤ x < b} - полуинтерва-

ly or half-segments;

(a, +∞) = ( x | x ∈ R, a< x} или (−∞, b) = { x | x ∈ R, x < b} - открытые лучи;

[ a, +∞) = ( x | x ∈ R, a ≤ x) or (−∞, b] = ( x | x ∈ R, x ≤ b) - closed rays.

Finally, sometimes we will need gaps in which we will not care

whether its ends belong to this interval or not. We will have such a period

denote a, b.

§ 5 Boundedness of numerical sets

Definition 1.5.1. A numerical set X is called bounded

from above, if there is a number M such that x ≤ M for every element x from

set X.

Definition 1.5.2. A numerical set X is called bounded

below, if there is a number m such that x ≥ m for every element x from

set X.

Definition 1.5.3. A numerical set X is called bounded,

if it is limited above and below.

In symbolic notation, these definitions would look like this:

a set X is bounded from above if ∃M ∀x ∈ X: x ≤ M,

is bounded below if ∃m ∀x ∈ X: x ≥ m and

is limited if ∃m, M ∀x ∈ X: m ≤ x ≤ M .

Theorem 1.5.1. A numerical set X is bounded if and only if

when there is a number C such that for all elements x from this set

The inequality x ≤ C holds.

Let the set X be bounded. Let's put C = max (m, M) - the most

the greater of the numbers m and M. Then, using the properties of the module of reals

numbers, we obtain the inequalities x ≤ M ≤ M ≤ C and x ≥ m ≥ − m ≥ −C , from which it follows

It is true that x ≤ C.

Conversely, if the inequality x ≤ C is satisfied, then −C ≤ x ≤ C. This is the three-

expected if we put M = C and m = −C .◄

The number M that bounds the set X from above is called the upper

boundary of the set. If M is the upper bound of a set X, then any

a number M ′ that is greater than M will also be the upper bound of this set.

Thus, we can talk about the set of upper bounds for the set

X. Let us denote the set of upper bounds by M. Then, ∀x ∈ X and ∀M ∈ M

the inequality x ≤ M will be satisfied, therefore, according to the axiom, continuously

There exists a number M 0 such that x ≤ M 0 ≤ M . This number is called the exact

no upper bound of a numerical set X or the upper bound of this

set or the supremum of a set X and is denoted by M 0 = sup X .

Thus, we have proven that every non-empty number set,

bounded above always has an exact upper bound.

It is obvious that the equality M 0 = sup X is equivalent to two conditions:

1) ∀x ∈ X the inequality x ≤ M 0 holds, i.e. M 0 - upper limit of the multiplicity

2) ∀ε > 0 ∃xε ∈ X so that the inequality xε > M 0 − ε holds, i.e. this game

The price cannot be improved (reduced).

Example 1. Consider the set X = ⎨1 − ⎬ . Let us prove that sup X = 1.

☺Indeed, firstly, inequality 1 −< 1 выполняется для любого

n ∈ ; secondly, if we take an arbitrary positive number ε, then by

Using Archimedes' principle, one can find a natural number nε such that nε > . That-

where the inequality 1 − > 1 − ε is satisfied, i.e. found element xnε multi-

of X, greater than 1 − ε, which means that 1 is the least upper bound

Similarly, one can prove that if a set is bounded below, then

it has an exact lower bound, which is also called the lower bound

new or infimum of the set X and is denoted by inf X.

The equality m0 = inf X is equivalent to the conditions:

1) ∀x ∈ X the inequality x ≥ m0 holds;

2) ∀ε > 0 ∃xε ∈ X so that the inequality xε holds< m0 + ε .

If a set X has the largest element x0, then we will call it

the maximum element of the set X and denote x0 = max X . Then

sup X = x0 . Similarly, if there is a smallest element in a set, then

we will call it minimal, denote min X and it will be an in-

fimum of the set X.

For example, the set of natural numbers has the smallest element -

unit, which is also the infimum of the set. Supre-

This set has no muma, since it is not bounded from above.

The definitions of precise upper and lower bounds can be extended to

sets that are unbounded above or below, assuming sup X = +∞ or, correspondingly,

Accordingly, inf X = −∞ .

In conclusion, we formulate several properties of the upper and lower boundaries.

Property 1. Let X be some number set. Let us denote by

− X set (− x | x ∈ X ) . Then sup (− X) = − inf X and inf (− X) = − sup X .

Property 2. Let X be some number set λ be real

number. Let us denote by λ X the set (λ x | x ∈ X ) . Then if λ ≥ 0, then

sup (λ X) = λ sup X , inf (λ X) = λ inf X and, if λ< 0, то

sup (λ X) = λ inf X , inf (λ X) = λ sup X .

Property 3. Let X1 and X2 be number sets. Let us denote by

X1 + X 2 is the set ( x1 + x2 | x1 ∈ X 1, x2 ∈ X 2 ) and through X1 − X 2 the set

( x1 − x2 | x1 ∈ X1, x2 ∈ X 2) . Then sup (X 1 + X 2) = sup X 1 + sup X 2 ,

inf (X1 + X 2) = inf X1 + inf X 2 , sup (X 1 − X 2) = sup X 1 − inf X 2 and

inf (X1 − X 2) = inf X1 − sup X 2 .

Property 4. Let X1 and X2 be numerical sets, all elements of which

ryh are non-negative. Then

sup (X1 X 2) = sup X1 ⋅ sup X 2 , inf (X1 X 2) = inf X 1 ⋅ inf X 2 .

Let us prove, for example, the first equality in Property 3.

Let x1 ∈ X1, x2 ∈ X 2 and x = x1 + x2. Then x1 ≤ sup X1, x2 ≤ sup X 2 and

x ≤ sup X1 + sup X 2 , whence sup (X1 + X 2) ≤ sup X1 + sup X 2 .

To prove the opposite inequality, take the number

y< sup X 1 + sup X 2 . Тогда можно найти элементы x1 ∈ X1 и x2 ∈ X 2 такие,

that x1< sup X1 и x2 < sup X 2 , и выполняется неравенство

y< x1 + x2 < sup X1 + sup X 2 . Это означает, что существует элемент

x = +x1 x2 ∈ X1+ X2, which is greater than the number y and

sup X1 + sup X 2 = sup (X1 + X 2) .◄

The proofs of the remaining properties are carried out similarly and provide

are revealed to the reader.

§ 6 Countable and uncountable sets

Definition 1.6.1. Consider the set of the first n natural numbers

n = (1,2,..., n) and some set A. If it is possible to establish mutual

one-to-one correspondence between A and n, then the set A will be called

final.

Definition 1.6.2. Let some set A be given. If I may

establish a one-to-one correspondence between the set A and

set of natural numbers, then the set A will be called a count-

Definition 1.6.3. If the set A is finite or countable, then we will

believe that it is no more than countable.

Thus, a set will be countable if its elements can be counted

put in a sequence.

Example 1. The set of even numbers is countable, since the mapping n ↔ 2n

is a one-to-one correspondence between the set of natural

numbers and many even numbers.

Obviously, such a correspondence can be established not only in

zom. For example, you can establish a correspondence between set and multi-

gestion (of integers), establishing correspondence in this way

When constructing an axiomatic theory of natural numbers, the primary terms will be “element” or “number” (which in the context of this manual we can consider as synonyms) and “set”, the main relations: “belonging” (the element belongs to the set), “equality” and " follow up”, denoted a / (reads “the number a stroke follows the number a”, for example, a two is followed by a three, that is, 2 / = 3, the number 10 is followed by the number 11, that is, 10 / = 11, etc.).

The set of natural numbers(natural series, positive integers) is a set N with the introduced “follow after” relation, in which the following 4 axioms are satisfied:

A 1. In the set N there is an element called unit, which does not follow any other number.

A 2. For each element of the natural series, there is only one next to it.

A 3. Each element of N follows at most one element of the natural series.

A 4.( Axiom of induction) If a subset M of a set N contains one, and also, together with each of its elements a, also contains the following element a / , then M coincides with N.

The same axioms can be written briefly using mathematical symbols:

A 1 ( 1  N) ( a  N) a / ≠ 1

A 2 ( a  N) ( a /  N) a = b => a / = b /

A 3 a / = b / => a = b

If element b follows element a (b = a /), then we will say that element a is prior to element b (or precedes b). This system of axioms is called Peano axiom systems(since it was introduced in the 19th century by the Italian mathematician Giuseppe Peano). This is just one of the possible sets of axioms that allow us to define the set of natural numbers; There are other equivalent approaches.

The simplest properties of natural numbers

Property 1. If the elements are different, then the ones following them are different, that is

a  b => a /  b / .

Proof is carried out by contradiction: suppose that a / = b /, then (by A 3) a = b, which contradicts the conditions of the theorem.

Property 2. If the elements are different, then the ones preceding them (if they exist) are different, that is

a /  b / => a  b.

Proof: suppose that a = b, then, according to A 2, we have a / = b /, which contradicts the conditions of the theorem.

Property 3. No natural number is equal to the next one.

Proof: Let us introduce into consideration the set M, consisting of such natural numbers for which this condition is satisfied

M = (a  N | a  a / ).

We will carry out the proof based on the induction axiom. By definition of the set M, it is a subset of the set of natural numbers. Next 1M, since one does not follow any natural number (A 1), which means that also for a = 1 we have: 1  1 / . Let us now assume that some a  M. This means that a  a / (by definition of M), whence a /  (a /) / (property 1), that is, a /  M. From all of the above, based on Using the axioms of induction, we can conclude that M = N, that is, our theorem is true for all natural numbers.

Theorem 4. For any natural number other than 1, there is a number preceding it.

Proof: Consider the set

M = (1)  (c N | ( a  N) c = a / ).

This M is a subset of the set of natural numbers, one clearly belongs to this set. The second part of this set is the elements for which there are predecessors, therefore, if a  M, then a / also belongs to M (its second part, since a / has a predecessor - this is a). Thus, based on the axiom of induction, M coincides with the set of all natural numbers, which means that all natural numbers are either 1 or those for which there is a preceding element. The theorem has been proven.

Consistency of the axiomatic theory of natural numbers

As an intuitive model of the set of natural numbers, we can consider sets of lines: the number 1 will correspond to |, the number 2 ||, etc., that is, the natural series will look like:

|, ||, |||, ||||, ||||| ….

These rows of lines can serve as a model of natural numbers if “attributing one line to a number” is used as the “follow after” relation. The validity of all axioms is intuitively obvious. Of course, this model is not strictly logical. To build a rigorous model, you need to have another obviously consistent axiomatic theory. But we do not have such a theory at our disposal, as noted above. Thus, either we are forced to rely on intuition, or not to resort to the method of models, but to refer to the fact that for more than 6 thousand years, during which the study of natural numbers has been carried out, no contradictions with these axioms have been discovered.

Independence of the Peano axiom system

To prove the independence of the first axiom, it is enough to construct a model in which axiom A 1 is false, and axioms A 2, A 3, A 4 are true. Let us consider the numbers 1, 2, 3 as primary terms (elements), and define the “follow” relation by the relations: 1 / = 2, 2 / = 3, 3 / = 1.

There is no element in this model that does not follow any other (axiom 1 is false), but all other axioms are satisfied. Thus, the first axiom does not depend on the others.

The second axiom consists of two parts - existence and uniqueness. The independence of this axiom (in terms of existence) can be illustrated by a model of two numbers (1, 2) with the “follow” relation defined by a single relation: 1 / = 2:

For two, the next element is missing, but axioms A 1, A 3, A 4 are true.

The independence of this axiom, in terms of uniqueness, is illustrated by a model in which the set N will be the set of all ordinary natural numbers, as well as all kinds of words (sets of letters that do not necessarily have meaning) made up of letters of the Latin alphabet (after the letter z the next one will be aa, then ab ... az, then ba ...; all possible two-letter words, the last of which is zz, will be followed by the word aaa, and so on). We introduce the “follow” relation as shown in the figure:

Here axioms A 1, A 3, A 4 are also true, but 1 is immediately followed by two elements 2 and a. Thus, axiom 2 does not depend on the others.

The independence of Axiom 3 is illustrated by the model:

in which A 1, A 2, A 4 are true, but the number 2 follows both the number 4 and the number 1.

To prove the independence of the induction axiom, we use the set N, consisting of all natural numbers, as well as three letters (a, b, c). The following relation in this model can be introduced as shown in the following figure:

Here, for natural numbers, the usual follow relation is used, and for letters, the follow relation is defined by the following formulas: a / = b, b / = c, c / = a. It is obvious that 1 does not follow any natural number, for each there is a next, and only one, each element follows at most one element. However, if we consider a set M consisting of ordinary natural numbers, then this will be a subset of this set containing one, as well as the next element for each element from M. However, this subset will not coincide with the entire model under consideration, since it will not contain letters a, b, c. Thus, the induction axiom is not satisfied in this model, and, therefore, the induction axiom does not depend on the other axioms.

The axiomatic theory of natural numbers is categorical(complete in the narrow sense).

 (n /) =( (n)) / .

Principle of complete mathematical induction.

Induction theorem. Let some statement P(n) be formulated for all natural numbers, and let a) P(1) be true, b) from the fact that P(k) is true, it follows that P(k /) is also true. Then the statement P(n) is true for all natural numbers.

To prove this, let us introduce a set M of natural numbers n (M  N) for which the statement P(n) is true. Let's use axiom A 4, that is, we'll try to prove that:

1. k  M => k /  M.

If we succeed, then, according to axiom A 4, we can conclude that M = N, that is, P(n) is true for all natural numbers.

1) According to condition a) of the theorem, P(1) is true, therefore, 1  M.

2) If some k  M, then (by construction of M) P(k) is true. According to condition b) of the theorem, this entails the truth of P(k /), which means k /  M.

Thus, by the induction axiom (A 4) M = N, which means P(n) is true for all natural numbers.

Thus, the axiom of induction allows us to create a method for proving theorems “by induction.” This method plays a key role in proving the basic theorems of arithmetic concerning natural numbers. It consists of the following:

1) the validity of the statement is checked forn=1 (induction base) ,

2) the validity of this statement is assumed forn= k, Wherek– arbitrary natural number(inductive hypothesis) , and taking this assumption into account, the validity of the statement is established forn= k / (induction step ).

A proof based on a given algorithm is called a proof by mathematical induction .

Tasks for independent solution

No. 1.1. Find out which of the listed systems satisfy the Peano axioms (they are models of the set of natural numbers), determine which axioms are satisfied and which are not.

a) N =(3, 4, 5...), n / = n + 1;

b) N =(n  6, n  N), n / = n + 1;

c) N =(n  – 2, n  Z), n / = n + 1;

d) N =(n  – 2, n  Z), n / = n + 2;

e) odd natural numbers, n / = n +1;

f) odd natural numbers, n / = n +2;

g) Natural numbers with the ratio n / = n + 2;

h) N =(1, 2, 3), 1 / = 3, 2 / = 3, 3 / = 2;

i) N =(1, 2, 3, 4, 5), 1 / = 2, 2 / = 3, 3 / = 4, 4 / = 5, 5 / = 1;

j) Natural numbers, multiples of 3 with the ratio n / = n + 3

k) Even natural numbers with the ratio n / = n + 2

m) Integers,
.

Integer system

Let us remember that the natural series appeared to list objects. But if we want to perform some actions with objects, then we will need arithmetic operations on numbers. That is, if we want to stack apples or divide a cake, we need to translate these actions into the language of numbers.

Please note that to introduce the operations + and * into the language of natural numbers, it is necessary to add axioms that define the properties of these operations. But then the set of natural numbers itself is also expanding.

Let's see how the set of natural numbers expands. The simplest operation, which was one of the first to be required, is addition. If we want to define the operation of addition, we must define its inverse - subtraction. In fact, if we know what will be the result of addition, for example, 5 and 2, then we should be able to solve problems like: what should be added to 4 to get 11. That is, problems related to addition will definitely require ability to perform the reverse action - subtraction. But if adding natural numbers gives a natural number again, then subtracting natural numbers gives a result that does not fit into N. Some other numbers were required. By analogy with the understandable subtraction of a smaller number from a larger number, the rule of subtracting a larger number from a smaller number was introduced - this is how negative integer numbers appeared.

By supplementing the natural series with the operations + and -, we arrive at the set of integers.

Z=N+operations(+-)

The system of rational numbers as a language of arithmetic

Let us now consider the next most complex action - multiplication. In essence, this is repeated addition. And the product of integers remains an integer.

But the inverse operation to multiplication is division. But it does not always give the best results. And again we are faced with a dilemma - either to accept as given that the result of division may “not exist”, or to come up with numbers of some new type. This is how rational numbers appeared.

Let's take a system of integers and supplement it with axioms that define the operations of multiplication and division. We obtain a system of rational numbers.

Q=Z+operations(*/)

So, the language of rational numbers allows us to produce all arithmetic operations over the numbers. The language of natural numbers was not enough for this.

Let us give an axiomatic definition of the system of rational numbers.

Definition. A set Q is called a set of rational numbers, and its elements are called rational numbers, if the following set of conditions, called the axiomatics of rational numbers, is satisfied:

Axioms of the operation of addition. For every ordered pair x,y elements from Q some element is defined x+yОQ, called sum X And at. In this case, the following conditions are met:

1. (Existence of zero) There is an element 0 (zero) such that for any XÎQ

X+0=0+X=X.

2. For any element XО Q there is an element - XО Q (opposite X) such that

X+ (-X) = (-X) + X = 0.

3. (Commutativity) For any x,yО Q

4. (Associativity) For any x,y,zО Q

x + (y + z) = (x + y) + z

Axioms of the multiplication operation.

For every ordered pair x, y elements from Q some element is defined xyО Q, called the product X And u. In this case, the following conditions are met:

5. (Existence of a unit element) There is an element 1 О Q such that for any XО Q

X . 1 = 1. x = x

6. For any element XО Q , ( X≠ 0) there is an inverse element X-1 ≠0 such that

X. x -1 = x -1. x = 1

7. (Associativity) For any x, y, zО Q

X . (y . z) = (x . y) . z

8. (Commutativity) For any x, yО Q

Axiom of the connection between addition and multiplication.

9. (Distributivity) For any x, y, zО Q

(x+y) . z = x . z+y . z

Axioms of order.

Any two elements x, y,О Q enter into a comparison relation ≤. In this case, the following conditions are met:

10. (X≤at)L ( at≤x) ó x=y

11. (X≤y) L ( y≤ z) => x≤z

12. For anyone x, yО Q or x< у, либо у < x .

Attitude< называется строгим неравенством,

The relation = is called the equality of elements from Q.

Axiom of the connection between addition and order.

13. For any x, y, z ОQ, (x £ y) Þ x+z £ y+z

Axiom of the connection between multiplication and order.

14. (0 £ x)Ç(0 £ y) Þ (0 £ x´y)

Archimedes' axiom of continuity.

15. For any a > b > 0, there exist m О N and n О Q such that m ³ 1, n< b и a= mb+n.

*****************************************

Thus, the system of rational numbers is the language of arithmetic.

However, this language is not enough to solve practical computing problems.

For real numbers, denoted by (the so-called R chopped), the operation of addition (“+”) is introduced, that is, for each pair of elements ( x,y) from the set of real numbers the element is assigned x + y from the same set, called the sum x And y .

Axioms of multiplication

The multiplication operation (“·”) is introduced, that is, for each pair of elements ( x,y) from the set of real numbers, an element is assigned (or, in short, xy) from the same set, called the product x And y .

Relationship between addition and multiplication

Axioms of order

On a given relation of order "" (less than or equal to), that is, for any pair x, y from at least one of the conditions or .

Relationship between order and addition

Relationship between order and multiplication

Axiom of continuity

A comment

This axiom means that if X And Y- two non-empty sets of real numbers such that any element from X does not exceed any element from Y, then a real number can be inserted between these sets. For rational numbers this axiom does not hold; classic example: consider positive rational numbers and assign them to the set X those numbers whose square is less than 2, and the others - to Y. Then between X And Y You cannot insert a rational number (it is not a rational number).

This key axiom provides density and thereby makes the construction of mathematical analysis possible. To illustrate its importance, let us point out two fundamental consequences from it.

Corollaries of the axioms

Some important properties of real numbers follow directly from the axioms, for example,

• the uniqueness of zero,
• the uniqueness of the opposite and inverse elements.

Literature

• Zorich V. A. Mathematical analysis. Volume I. M.: Phasis, 1997, chapter 2.

see also

Links

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