Axiomatic construction of a system of integers. Axiomatics of real numbers. Relationship between order and addition

Axiomatic method in mathematics.

Basic concepts and relations of the axiomatic theory of natural series. Definition natural number.

Addition of natural numbers.

Multiplication of natural numbers.

Properties of the set of natural numbers

Subtraction and division of natural numbers.

Axiomatic method in mathematics

In the axiomatic construction of any mathematical theory, the certain rules:

1. Some concepts of the theory are chosen as major and accepted without definition.

2. Formulated axioms, which in this theory are accepted without proof, they reveal the properties of the basic concepts.

3. Each concept of the theory, which is not contained in the list of basic ones, is given definition, it explains its meaning with the help of the main and preceding this concept.

4. Every sentence of the theory that is not contained in the list of axioms must be proved. Such proposals are called theorems and prove them on the basis of the axioms and theorems preceding the one under consideration.

The system of axioms should be:

a) consistent: we must be sure that, drawing all sorts of conclusions from a given system of axioms, we will never arrive at a contradiction;

b) independent: no axiom should be a consequence of other axioms of this system.

V) complete, if within its framework it is always possible to prove either the given statement or its negation.

The presentation of geometry by Euclid in his "Elements" (3rd century BC) can be considered the first experience of the axiomatic construction of a theory. A significant contribution to the development of the axiomatic method for constructing geometry and algebra was made by N.I. Lobachevsky and E. Galois. At the end of the 19th century Italian mathematician Peano developed a system of axioms for arithmetic.

Basic concepts and relations of the axiomatic theory of natural numbers. Definition of a natural number.

As a basic (undefined) concept in a certain set N is chosen attitude , as well as set-theoretic concepts, as well as the rules of logic.

The element immediately following the element A, designate A".

The "immediately follow" relationship satisfies the following axioms:

Peano's axioms:

Axiom 1. in multitude N there is an element, directly not next for any element of this set. Let's call him unit and symbolize 1 .

Axiom 2. For each element A from N there is only one element A" immediately following A .

Axiom 3. For each element A from N there is at most one element immediately followed by A .

Axiom 4. Any subset M sets N coincides with N , if it has the properties: 1) 1 contained in M ; 2) from what A contained in M , it follows that and A" contained in M.

Definition 1. A bunch of N , for whose elements the relationship is established "directly follow» that satisfies axioms 1-4 is called set of natural numbers, and its elements are natural numbers.

IN this definition nothing is said about the nature of the elements of the set N . So she can be anything. Choosing as a set N some particular set on which a particular "directly follow" relation is given that satisfies axioms 1-4, we get model of this system axioms.

standard model system of axioms Peano is emerged in the process historical development society series of numbers: 1,2,3,4,... The natural series begins with the number 1 (axiom 1); every natural number is immediately followed by a single natural number (axiom 2); each natural number immediately follows at most one natural number (axiom 3); starting from the number 1 and moving in order to the natural numbers immediately following each other, we obtain the entire set of these numbers (axiom 4).

So, we began the axiomatic construction of a system of natural numbers with the choice of the main "directly follow" relationship and axioms that describe its properties. Further construction of the theory involves consideration of the known properties of natural numbers and operations on them. They should be disclosed in definitions and theorems, i.e. derived in a purely logical way from the relation "immediately follow", and axioms 1-4.

The first concept that we introduce after the definition of a natural number is attitude "immediately precedes" , which is often used when considering the properties of the natural series.

Definition 2. If a natural number b directly follows natural number A, that number A called immediately preceding(or previous) number b .

The relation "before" has near properties.

Theorem 1. One has no preceding natural number.

Theorem 2. Every natural number A, other than 1, has a single preceding number b, such that b"= A.

The axiomatic construction of the theory of natural numbers is not considered either in the initial or in high school. However, those properties of the "directly follow" relation, which are reflected in Peano's axioms, are the subject of study in primary course mathematics. Already in the first grade, when considering the numbers of the first ten, it turns out how each number can be obtained. The terms “follow” and “before” are used. Each new number acts as a continuation of the studied segment of the natural series of numbers. Students are convinced that each number is followed by the next, and moreover, only one, that the natural series of numbers is infinite.

Addition of natural numbers

According to the rules for constructing an axiomatic theory, the definition of addition of natural numbers must be introduced using only the relation "directly follow", and concepts "natural number" And "prior number".

Let us preface the definition of addition with the following considerations. If for any natural number A add 1, we get the number A", immediately following A, i.e. A+ 1= a" and hence we get the rule of adding 1 to any natural number. But how to add to the number A natural number b, different from 1? Let us use the following fact: if it is known that 2 + 3 = 5, then the sum 2 + 4 = 6, which immediately follows the number 5. This happens because in the sum 2 + 4 the second term is the number immediately following the number 3. So 2 + 4 = 2+3 " =(2+3)". IN general view we have , .

These facts underlie the definition of addition of natural numbers in axiomatic theory.

Definition 3. Addition of natural numbers is an algebraic operation that has the following properties:

Number a + b called sum of numbers A And b , and the numbers themselves A And b - terms.

In the axiomatic construction of any mathematical theory, certain rules:

some concepts of the theory are chosen as the main ones and are accepted without definition;

each concept of the theory, which is not contained in the list of basic ones, is given a definition;

axioms are formulated - sentences that are accepted in this theory without proof; they reveal the properties of the basic concepts;

· each sentence of the theory that is not contained in the list of axioms must be proved; such propositions are called theorems and are proved on the basis of axioms and terems.

In the axiomatic construction of a theory, all statements are derived from the axioms by way of proof.

Therefore, the system of axioms is subject to special requirements:

Consistency (a system of axioms is called consistent if it is impossible to logically derive two mutually exclusive sentences from it);

independence (a system of axioms is called independent if none of the axioms of this system is a consequence of other axioms).

A set with a relation given in it is called a model of a given system of axioms if all the axioms of this system are satisfied in it.

There are many ways to construct a system of axioms for the set of natural numbers. For the basic concept, one can take, for example, the sum of numbers or the order relation. In any case, it is necessary to specify a system of axioms that describe the properties of the basic concepts.

Let us give a system of axioms, adopting the basic concept of the operation of addition.

Non-empty set N is called the set of natural numbers if the operation (a; b) → a + b, called addition and having the properties:

1. addition is commutative, i.e. a + b = b + a.

2. addition is associative, i.e. (a + b) + c = a + (b + c).

4. in any set A, which is a subset of the set N, Where A there is a number a such that all Ha, are equal a+b, Where bN.

Axioms 1 - 4 are enough to construct the whole arithmetic of natural numbers. But with such a construction, it is no longer possible to rely on the properties of finite sets that are not reflected in these axioms.

Let us take as the basic concept the relation “directly follow…” defined on a non-empty set N. Then the natural series of numbers will be the set N, in which the relation "directly follow" is defined, and all elements of N will be called natural numbers, and the following hold: Peano's axioms:

AXIOM 1.

in multitudeNthere is an element that does not immediately follow any element of this set. We will call it a unit, and denote it by the symbol 1.

AXIOM 2.

For each element a ofNthere is a single element a immediately following a.

AXIOM 3.

For each element a ofNthere is at most one element immediately followed by a.

AXOIM 4.

Any subset M of the setNcoincides withN, if it has the properties: 1) 1 is contained in M; 2) from the fact that a is contained in M, it follows that a is also contained in M.

A bunch of N, for the elements of which the relation "immediately follow ..." is established, satisfying axioms 1 - 4, is called set of natural numbers , and its elements are natural numbers.

If as a set N choose some specific set on which a specific relation "directly follow ..." is given, satisfying axioms 1 - 4, then we get different interpretations (models) given axiom systems.

The standard model of the system of Peano's axioms is a series of numbers that arose in the process of the historical development of society: 1, 2, 3, 4, 5, ...

Any countable set can be a model of the Peano axioms.

For example, I, II, III, III, ...

oh oh oh oh oh...

one two three four, …

Consider a sequence of sets in which the set (oo) is the initial element, and each subsequent set is obtained from the previous one by assigning one more circle (Fig. 15).

Then N is a set consisting of sets of the described form, and it is a model of the system of Peano's axioms.

Indeed, in many N there is an element (oo) that does not immediately follow any element of the given set, i.e. axiom 1 holds. For each set A of the set under consideration, there is a unique set that is obtained from A by adding one circle, i.e. Axiom 2 holds. For each set A there is at most one set from which the set is formed A by adding one circle, i.e. Axiom 3 holds. If MN and it is known that the set A contained in M, it follows that the set in which there is one circle more than in the set A, is also contained in M, That M =N, which means that Axiom 4 is satisfied.

In the definition of a natural number, none of the axioms can be omitted.

Let us establish which of the sets shown in Fig. 16 are a model of Peano's axioms.

1 a b d a

G) Fig.16

Solution. Figure 16 a) shows a set in which axioms 2 and 3 are satisfied. Indeed, for each element there is a unique element that immediately follows it, and there is a unique element that it follows. But axiom 1 does not hold in this set (axiom 4 does not make sense, because there is no element in the set that does not immediately follow any other). Therefore, this set is not a model of Peano's axioms.

Figure 16 b) shows the set in which axioms 1, 3 and 4 are satisfied, but behind the element A two elements immediately follow, and not one, as required in axiom 2. Therefore, this set is not a model of Peano's axioms.

On fig. 16 c) shows a set in which axioms 1, 2, 4 are satisfied, but the element With immediately follows two elements. Therefore, this set is not a model of Peano's axioms.

On fig. 16 d) shows a set that satisfies axioms 2, 3, and if we take the number 5 as the initial element, then this set will satisfy axioms 1 and 4. That is, in this set for each element there is a single one immediately following it, and there is a single element that it follows. There is also an element that does not immediately follow any element of this set, this is 5 , those. Axiom 1 holds. Correspondingly, Axiom 4 also holds. Therefore, this set is a model of Peano's axioms.

Using the Peano axioms, we can prove a number of statements. For example, we prove that for all natural numbers the inequality x x.

Proof. Denote by A set of natural numbers for which a a. Number 1 belongs A, since it does not follow any number from N, and therefore does not follow by itself: 1 1. Let aa, Then a a. Denote A through b. By virtue of axiom 3, Ab, those. bb And bA.

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

Axioms of multiplication

The operation of multiplication ("·") is introduced, that is, 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

The order relation "" (less than or equal to) is given on, that is, for any pair x, y of 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 refer to the set X those numbers whose square is less than 2, and the rest - to Y. Then between X And Y cannot be pasted rational number(is not a rational number).

This key axiom provides density and thus makes the construction of calculus possible. To illustrate its importance, we point out two fundamental consequences of it.

Consequences of the axioms

It follows directly from the axioms that some important properties real numbers, for example,

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

Literature

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

Links

Wikimedia Foundation. 2010 .

See what "Axiomatics of Real Numbers" is in other dictionaries:

A real or real number is a mathematical abstraction that arose from the need to measure the geometric and physical quantities of the world around us, as well as to perform operations such as taking a root, calculating logarithms, solving ... ... Wikipedia

Real or real numbers are a mathematical abstraction that serves, in particular, to represent and compare the values of physical quantities. Such a number can be intuitively represented as describing the position of a point on a straight line. ... ... Wikipedia

Wiktionary has an article on "axiom" Axiom (dr. Greek ... Wikipedia

An axiom that occurs in various axiomatic systems. Axiomatics of real numbers Hilbert's axiomatics of Euclidean geometry Kolmogorov's axiomatics of probability theory ... Wikipedia

The given system of axioms of the theory of integers is not independent, as noted in Exercise 3.1.4.

Theorem 1. The axiomatic theory of integers is consistent.

Proof. We will prove the consistency of the axiomatic theory of integers, starting from the assumption that the axiomatic theory of natural numbers is consistent. To do this, we construct a model on which all the axioms of our theory are satisfied.

Let's build a ring first. Consider the set

N´ N = {(a, b)ï a, bÎ N}.

a, b) natural numbers. By such a pair we mean the difference of natural numbers a-b. But until the existence of a system of integers in which such a difference exists has not been proved, we have no right to use such a designation. At the same time, this understanding gives us the opportunity to set the properties of the pairs as we need.

We know that different differences of natural numbers can be equal to the same integer. Accordingly, we introduce on the set N´ N equality relation:

(a, b) = (c, d) Û a + d = b + c.

It is easy to see that this relation is reflexive, symmetric, and transitive. Therefore, it is an equivalence relation and has the right to be called an equality. Factor set of sets N´ N Z. Its elements will be called integers. They are equivalence classes on a set of pairs. The class containing the pair
(a, b), denoted by [ a, b].

Z a, b] how about the difference a-b

[a, b] + [c, d] = [a+c, b+d];

[a, b] × [ c, d] = [ac+bd, ad+bc].

It should be borne in mind that, strictly speaking, the use of operation symbols is not entirely correct here. The same symbol + denotes the addition of natural numbers and pairs. But since it is always clear in which set a given operation is performed, we will not introduce separate notation for these operations here.

It is required to check the correctness of the definitions of these operations, namely, that the results do not depend on the choice of elements a And b defining the pair [ a, b]. Indeed, let

[a, b] = [a 1 ,b 1 ], [c, d] = [With 1 , d 1 ].

It means that a+b 1 = b+a 1 , c + d 1 =d + With 1 . Adding these equalities, we get

a+b 1 + c + d 1 = b+a 1 +d + With 1 Þ[ a + b, c + d] = [a 1 +With 1 ,b 1 + d 1 ]

Þ [ a, b] + [c, d] = [a 1 ,b 1 ] + [c 1 , d 1 ].

The correctness of the definition of multiplication is defined similarly. But here we must first check that [ a, b] × [ c, d] = [a 1 ,b 1]×[ c, d].

Now we should check that the resulting algebra is a ring, that is, the axioms (Z1) - (Z6).

Let us check, for example, the commutativity of addition, that is, the axiom (Z2). We have

[c, d] + [a, b] = = [a+c, b+d] = [a, b] + [c, d].

The commutativity of addition for integers is derived from the commutativity of addition for natural numbers, which is assumed to be already known.

Axioms (Z1), (Z5), (Z6) are verified similarly.

The role of zero is played by a couple. Let's denote it by 0 . Really,

[a, b] + 0 = [a, b] + = [a+ 1,b+ 1] = [a, b].

Finally, -[ a, b] = [b, a]. Really,

[a, b] + [b, a] = [a+b, b+a] = = 0 .

Now let's check the extension axioms. It should be borne in mind that in the constructed ring there are no natural numbers as such, since the elements of the ring are classes of pairs of natural numbers. Therefore, it is required to find a subalgebra isomorphic to the semiring of natural numbers. Here again the notion of the pair [ a, b] how about the difference a-b. Natural number n can be represented as the difference of two natural numbers, for example, in the following way: n = (n+ 1) - 1. Hence the proposal to establish a correspondence f: N ® Z according to the rule

f(n) = [n + 1, 1].

This correspondence is injective:

f(n) = f(m) Þ [ n + 1, 1]= [m+ 1, 1] Þ ( n + 1) + 1= 1 + (m+ 1) n=m.

Therefore, we have a one-to-one correspondence between N and some subset Z, which we denote by N*. Let's check that it saves operations:

f(n) + f(m) = [n + 1, 1]+ [m + 1, 1] = [n + m + 2, 2]= [n + m+ 1, 1] = f(n+m);

f(n) × f(m) = [n+ 1, 1]× [ m + 1, 1] = [nm+n + m + 2, n+m+ 2]= [nm+ 1, 1] = f(nm).

Thus, it has been established that N* forms in Z under the operations of addition and multiplication, a subalgebra isomorphic to N

Denote a pair [ n+ 1, 1] from N* n, through n a, b] we have

[a, b] = [a + 1, 1] + = [a + 1, 1] – [b + 1, 1] = a – b .

Thus, finally, the concept of the pair [ a, b] as a difference of natural numbers. At the same time, it was established that each element from the constructed set Z represented as the difference of two natural numbers. This will help to test the axiom of minimality.

Let M - subset Z, containing N* and together with any elements A And b their difference a - b. Let us prove that in this case M =Z. Indeed, any element of Z represented as the difference of two natural numbers, which by condition belong to M along with its difference.

Theorem 2. The axiomatic theory of integers is categorical.

Proof. Let us prove that any two models on which all the axioms of the given theory hold are isomorphic.

Let a Z 1 , +, ×, N 1 c and b Z 2 , +, ×, N 2 ñ are two models of our theory. Strictly speaking, the operations in them must be denoted by different symbols. We will deviate from this requirement so as not to clutter up the calculations: it is clear every time which operation is in question. The elements belonging to the considered models will be provided with the corresponding indexes 1 or 2.

We are going to define an isomorphic mapping from the first model to the second. Because N 1 and N 2 are semirings of natural numbers, then there exists an isomorphic mapping j of the first semiring onto the second. Let us define the mapping f: Z 1® Z 2. Every whole number X 1 О Z 1 is represented as the difference of two natural numbers:
X 1 = a 1 – b 1 . We believe

f (x 1) = j( a 1) – j( b 1).

Let's prove that f is an isomorphism. The mapping is well defined: if X 1 = at 1 , where y 1 = c 1 – d 1 , then

a 1 – b 1 = c 1 – d 1 a 1 +d 1 = b 1 + c 1 Þ j( a 1 +d 1) = j( b 1 + c 1)

Þ j( a 1) + j( d 1) = j( b 1) + j( c 1) Þ j( a 1)–j( b 1)=j( c 1) – j( d 1) f(x 1) =f (y 1).

Hence it follows that f- unambiguous mapping Z 1 in Z 2. But for anyone X 2 of Z 2 can find natural elements a 2 and b 2 such that X 2 = a 2 – b 2. Since j is an isomorphism, these elements have inverse images a 1 and b 1 . Means, x 2 = j( a 1) – j( b 1) =
= f (a 1 – b 1), and each element from Z 2 is a prototype. Hence the correspondence f mutually unambiguous. Let's check that it saves operations.

If X 1 = a 1 – b 1 , y 1 = c 1 – d 1 , then

X 1 + y 1 = (a 1 + c 1) – (b 1 +d 1),

f(X 1 + y 1) = j( a 1 + c 1) – j( b 1 +d 1) =j( a 1)+ j( c 1) – j( b 1) – j( d 1) =

J( a 1) – j( b 1)+ j( c 1)– j( d 1) =f(X 1) + f(y 1).

Similarly, we check that the multiplication is preserved. Thus, it has been established that f is an isomorphism, and the theorem is proved.

Exercises

1. Prove that any ring containing the system of natural numbers also includes the ring of integers.

2. Prove that every minimal ordered commutative ring with unity is isomorphic to the ring of integers.

3. Prove that every ordered ring with unity and no zero divisors contains only one subring isomorphic to the ring of integers.

4. Prove that the second-order matrix ring over the field of real numbers contains infinitely many subrings isomorphic to the ring of integers.

Field of rational numbers

The definition and construction of a system of rational numbers are carried out in the same way as it is done for a system of integers.

Definition. A system of rational numbers is a minimal field that is an extension of the ring of integers.

In accordance with this definition, we obtain the following axiomatic construction of the system of rational numbers.

Primary terms:

Q is the set of rational numbers;

0, 1 are constants;

+, × are binary operations on Q;

Z- subset Q, the set of integers;

Å, Ä are binary operations on Z.

Axioms:

I. Field axioms.

(Q1) a+ (b+c) = (a+b) + c.

(Q2) a + b = b + a.

(Q3)(" a) a + 0 = a.

(Q4)(" a)($(–a)) a + (–a) = 0.

(Q5) a× ( b× c) = (a× b) × c.

(Q6) a× b = b× a.

(Q7) A× 1 = A.

(Q8)(" a¹ 0)($ a –1) a × a –1 = 1.

(Q9) ( a+b) × c = a × c + b× c.

II. Extension axioms.

(Q10) a Z, M, L, 0, 1ñ be the ring of natural numbers.

(Q11) Z Í Q.

(Q12)(" a,bÎ Z) a+b=aÅ b.

(Q13)(" a,bÎ Z) a× b = aÄ b.

III. Axiom of minimality.

(Q14) MÍ Q, ZÍ M, ("a, bÎ M)(b ¹ 0 ® a× b–1 О M)Þ M = Q.

Number a× b-1 is called a quotient A And b, denoted a/b or .

Theorem 1. Every rational number is represented as a quotient of two integers.

Proof. Let M is the set of rational numbers representable as a quotient of two integers. If n is an integer, then n = n/1 belongs M, hence, ZÍ M. If a, bÎ M, That a = k/l, b = m/n, Where k, l, m, nÎ Z. Hence, a/b=
= (kn) / (lm)Î M. By axiom (Q14) M= Q, and the theorem is proved.

Theorem 2. The field of rational numbers can be linearly and strictly ordered, and in a unique way. The order in the field of rational numbers is Archimedean and continues the order in the ring of integers.

Proof. Denote by Q+ a set of numbers representable as a fraction, where kl> 0. It is easy to see that this condition does not depend on the type of fraction representing the number.

Let's check that Q + – positive part of the field Q. Since for an integer kl three cases are possible: kl = 0, klÎ N, –kl Î N, then for a = we obtain one of three possibilities: a = 0, aн Q+ , –aО Q + . Further, if a = , b = belong Q+ , then kl > 0, mn> 0. Then a + b = , and ( kn+ml)ln = kln 2 + mnl 2 > 0. Hence, a + bн Q + . It can be verified similarly that abн Q + . Thus, Q + is the positive part of the field Q.

Let Q++ is some positive part of this field. We have

l =.l 2 н Q ++ .

From here NÍ Q++ . By Theorem 2.3.4, the reciprocals of natural numbers also belong to Q++ . Then Q + Í Q++ . By Theorem 2.3.6 Q + =Q++ . Therefore, the orders defined by the positive parts also coincide. Q+ and Q ++ .

Because Z + = NÍ Q+ , then the order in Q continues the order Z.

Let now a => 0, b => 0. Since the order in the ring of integers is Archimedean, for positive kn And ml there is a natural With such that With× kn>ml. From here With a = With>= b. Hence, the order in the field of rational numbers is Archimedean.

Exercises

1. Prove that the field of rational numbers is dense, that is, for any rational numbers a < b there is a rational r such that a < r < b.

2. Prove that the equation X 2 = 2 has no solutions in Q.

3. Prove that the set Q countable.

Theorem 3. The axiomatic theory of rational numbers is consistent.

Proof. The consistency of the axiomatic theory of rational numbers is proved in the same way as for integers. To do this, a model is built on which all the axioms of the theory are fulfilled.

As a basis, we take the set

Z´ Z* = {(a, b)ï a, bÎ Z, b ¹ 0}.

The elements of this set are pairs ( a, b) integers. By such a pair we mean the quotient of integers a/b. In accordance with this, we set the properties of the pairs.

We introduce on the set Z´ Z* equality relation:

(a, b) = (c, d) Û ad = bc.

We note that it is an equivalence relation and has the right to be called an equality. Factor set of sets Z´ Z* with respect to this equality relation, we denote by Q. Its elements will be called rational numbers. A class containing a pair ( a, b), denoted by [ a, b].

We introduce in the constructed set Q operations of addition and multiplication. It will help us to make an idea about the element [ a, b] how about private a/b. In accordance with this, we assume by definition:

[a, b] + [c, d] = [ad+bc, bd];

[a, b] × [ c, d] = [ac, bd].

We check the correctness of the definitions of these operations, namely, that the results do not depend on the choice of elements a And b defining the pair [ a, b]. This is done in the same way as in the proof of Theorem 3.2.1.

The role of zero is played by a couple. Let's denote it by 0 . Really,

[a, b] + 0 = [a, b] + = [a× 1+0× b, b× 1] = [a, b].

Opposite to [ a, b] is the pair –[ a, b] = [–a, b]. Really,

[a, b] + [–a, b]= [ab-ab, bb] = = 0 .

The unit is a pair = 1 . Inverse to pair [ a, b] - pair [ b, a].

Now let's check the extension axioms. Let's establish a correspondence
f: Z ® Q according to the rule

f(n) = [n, 1].

We verify that this is a one-to-one correspondence between Z and some subset Q, which we denote by Z*. We check further that it preserves operations, hence it establishes an isomorphism between Z and subring Z* V Q. Hence, the extension axioms have been verified.

Denote a pair [ n, 1] from Z* corresponding to natural number n, through n . Then for an arbitrary pair [ a, b] we have

[a, b] = [a, 1] × = [ a, 1] / [b, 1] = a /b .

This substantiates the concept of the pair [ a, b] as about the quotient of integers. At the same time, it was established that each element from the constructed set Q represented as a quotient of two integers. This will help to test the axiom of minimality. The verification is carried out as in Theorem 3.2.1.

Thus, for the constructed system Q all the axioms of the theory of integers are satisfied, that is, we have built a model of this theory. The theorem has been proven.

Theorem 4. The axiomatic theory of rational numbers is categorical.

The proof is similar to the proof of Theorem 3.2.2.

Theorem 5. The Archimedean ordered field is an extension of the field of rational numbers.

The proof is as an exercise.

Theorem 6. Let F is an Archimedean ordered field, a > b, Where a, bÎ F. There is a rational number н F such that a > > b.

Proof. Let a > b³ 0. Then a-b> 0, and ( a-b) –1 > 0. There is a natural T such that m×1 > ( a-b) –1 , whence m –1 < a-b £ A. Further, there is a natural k such that k× m-1³ a. Let k – smallest number for which this inequality holds. Because k> 1, then we can put k = n + 1, n Î N. Wherein
(n+ 1)× m-1³ a, n× m –1 < a. If n× m-1 £ b, That a = b + (a-b) > b+m-1³ n× m –1 + m –1 =
= (n+ 1)× m-1 . Contradiction. Means, a >n× m –1 > b.

Exercises

4. Prove that any field containing the ring of integers also includes the field of rational numbers.

5. Prove that every minimal ordered field is isomorphic to the field of rational numbers.

Real numbers

Axiomatic construction of a system of integers. Axiomatics of real numbers. Relationship between order and addition

Axioms of multiplication

Relationship between addition and multiplication

Axioms of order

Relationship between order and addition

Relationship between order and multiplication

Axiom of continuity

A comment

Consequences of the axioms

Literature

see also

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