Axioms of real numbers. Follow-up of the axioms of the theory of numbers

Speech numbers, which are indicated through (so called R ruban), the operation of adding (“+”) is introduced, so that the skin pair of elements ( x,y) with impersonal speech numbers put at the vіdpovіdnіst element x + y z tsієї w multiplier, titles sumo xі y .

Axioms of plurality

The multiplication operation (“·”) is introduced, so the skin pair of elements ( x,y) for impersonal speech numbers, put an element (otherwise, abbreviated, xy) s tsієї w multiplier, titles of creation xі y .

Zvyazok dodavannya that plural

Axioms to order

On the task of the order "" (less than one), then for bet x, y vykonuєtsya wanting to be one of the minds abo.

Zv'yazok in order that folding

Zvyazok vіdnoshennia order that plural

Axiom of continuity

Commentary

This axiom means that Xі Y- two empty multipliers of real numbers such that there is any element of X do not overturn any element Y, then you can insert a speech number between them. For rational numbers this axiom is not victorious; classic butt: recognizably positive rational numbers and visibly to impersonality X those numbers, the square of which is less than 2, and the other - up to Y. Todi mizh Xі Y cannot insert a rational number (not a rational number).

This is the key axiom that secures the security and thereby allows for mathematical analysis. For illustration of its importance, let me point out two fundamental implications from it.

Heritage of axioms

Without intermediary axiom, deacons are important to the power of today's numbers, for example,

unity of zero,
the unity of the proliferative and the virulence elements.

Literature

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

Div. also

Posilannya

Wikimedia Foundation. 2010 .

See also "Axiomatics of real numbers" in other dictionaries:

Speech, which is a real number, is a mathematical abstraction, which vinikla z requires the use of geometric and physical quantities of the necessary world, as well as carrying out such operations as root extraction, calculation of logarithms, solutions.

Speech, chi actual numbers is a mathematical abstraction, what to serve, zokrema, the manifestation of that similarity of the value of physical quantities. Such a number can be intuitively represented as describing the position of a point on a straight line.

Wiktionary has the article "axiom" Axiom (in. Greek ... Wikipedia

An axiom, as it is used in various axiomatic systems. Axiomatics of real numbers Hilbert's axiomatics of Euclidean geometry Axiomatics of Kolmogorov's theory of imovirnosti ... Wikipedia

Number system

Let's guess that the natural series has appeared for the transfer of objects. But if we want to work with objects, then we need arithmetic operations on numbers. Tobto, if we want to fold an apple or divide a cake, we need to translate the number of numbers.

It is a shameful respect that after the introduction of operations + і * in the language of natural numbers it is necessary to add axioms that signify the power of these operations. Aletodes and impersonal natural numbers tezh expanding.

We marvel at how the impersonal natural numbers expand. The simplest operation, as it was necessary for one of the first - ce dodavannya. If we want to appoint an additional operation, it is necessary to designate a return to it - a decision. It’s true, as we know, that as a result of adding, for example, 5 and 2, then we are guilty of adding to the order of the type: what needs to be added to 4, to take 11. vimagatimut vminnya viroblyat i zvorotnu diyu - vіdnіmannya. Ale, yakscho dodavannya natural numbers give again natural number, then looking at natural numbers gives a result that does not fit into N. We need more numbers. By analogy of a sensible vision of greater number lesser boulo introduced the rule of vidnіmannya z lesser greater - so the number of negative numbers appeared.

Complementing the natural series with operations + і - mi, we arrive at impersonal integers.

Z=N+operations(+-)

System of rational numbers yak mov arithmetic

Now let's look at this for folding diu - plural. As a matter of fact, this is a bagatarase addition. І additional number of integers is filled with a whole number.

Ale, a reverse operation to a multiple - tse podіl. But it is far from always giving a good result. And again we are faced with a dilemma - or else to accept as if the result could not be “understood”, or to guess the number of a new type. So they blamed rational numbers.

Let's take a system of integers and supplement it with axioms, which determine the operation of multiplication and the bottom. We take away the system of rational numbers.

Q=Z+operations(*/)

Father, the language of rational numbers allows you to work all arithmetic operations over numbers. The language of natural numbers was not enough.

Let us introduce axiomatically the system of rational numbers.

Appointment. The impersonal Q is called the impersonality of rational numbers, like the elements - rational numbers, as the advancing complex of minds, titles is called the axiomatics of rational numbers:

Axioms of folding operation. For be-like-ordered bet x,y elements Q deyaky element x+yÎQ, ranks in sum Xі at. When you win, think like this:

1. (Isnuvannya zero) Iznuє element 0 (zero) such that for any XОQ

X+0=0+X=X.

2. For any element X Q Q main element - XО Q (opposite X) such that

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

3. (Commutativity) For whatever x,yО Q

4. (Associativity) For any x, y, z Q

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

Axioms of the multiplication operation.

For be-like-ordered bet x, y elements of Q assigned to the actual element huÎ Q, titles of creation Xі y. When you win, think like this:

5. (Isnuvannya single element) Iznuє element 1 Q such that for whatever XО Q

X . 1 = 1. x = x

6. For any element X Q Q , ( X≠ 0) main element X-1 ≠0 such that

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

7. (Associativity) For be-things x, y, zО Q

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

8. (Commutativity) For whatever x, yО Q

Axiom zv'azku folded and multiplied.

9. (Distributive) For whatever x, y, zО Q

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

Axioms are in order.

Be like two elements x, y, Q Q start at the end of the line ≤. When you win, think like this:

10. (X≤at)L ( at≤x) ó x=y

11. (X≤y) L ( y≤ z) => x≤z

12. For be-yakah x, yО Q or x< у, либо у < x .

Setting< называется строгим неравенством,

Ratio = called equalness of Q elements.

Axiom zv'yazku dodavannya that order.

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

Axiom zv'yazku mnozhennya that order.

14. (0 £ x)Ç(0 £ y) z (0 £ x´y)

The axiom of the perpetuity of Archimedes.

15. For whether a > b > 0, we have m N and n Q so that m ³ 1, n< b и a= mb+n.

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

Thus, the system of rational numbers is Zem's arithmetic.

Prote, on the top of practical counting tasks, the movie is not enough.

Axiomatic method in mathematics.

Basic understanding and understanding of the axiomatic theory of the natural series. Appointment of a natural number.

Addition of natural numbers.

An increase in natural numbers.

Power of the multiplier of natural numbers

Vіdnіmannya raspodіl natural numbers.

Axiomatic method in mathematics

With axiomatic prompting, some kind of mathematical theory is supplemented sing the rules:

1. Deyakі understand the theory vibirayutsya like major she is accepted without a warrant.

2. Formulated axioms, which are accepted by these theories without proof, which have the power to understand the main ones.

3. The skin understands the theory, so as not to take revenge on the list of the main ones, it is given appointment, for a new one, it is explained yoga for the help of the main ones and the previous understanding.

4. The skin proposition of the theory, which cannot be missed by the list of axioms, can be brought to light. Such propositions are called theorems and bring them on the basis of axioms and theorems, which are to be reworked.

The system of axioms can be:

a) inconsiderate: we are guilty of buti vpevnenі, scho, roblyachi raznі vysnovki z given system of axioms, not come to superechnosti;

b) independent: none of the axioms is guilty of the following of other axioms of the system.

in) again, even within this framework, it is always possible to bring the chi of the firm, which yogo is listed.

The first proof of the axiomatic motivation of the theory is to be taken into account by Euclid's book of geometry in Yogo "Cobs" (3rd century e.). A significant contribution to the development of the axiomatic method inspiring geometry and algebra was developed by N.I. Lobachevsky and E. Galois. For example, 19 st. Italian mathematician Peano broke up a system of axioms for arithmetic.

Basic understanding and understanding of the axiomatic theory of the natural number. Appointment of a natural number.

As the main (non-significant) understanding in the deakіy multiplicity N choose shutter , and navіt vikoristovuyutsya theoretical-multiple understanding, і navіt the rules of logic.

An element that follows the element without interruption a, signify a".

Seemingly, “without intermediary follow for” are satisfied with the upcoming axioms:

Axioms Peano:

Axiom 1. At the faceless N іsnuє element, without middle not offensive there are no multipliers for any element. Let's call yoga loneliness that symbolize 1 .

Axiom 2. For skin element a h N basic single element a" , relentlessly advancing for a .

Axiom 3. For skin element a h Nіsnuє not more than one element, for which it follows without intermediary a .

Axiom 4. Be like a multiplier M faceless N spіvpadє z N , yakscho maє power: 1) 1 take revenge in M ; 2) from what a take revenge in M , next, what i a" take revenge in M.

Appointment 1. Bezlich N , for the elements of which a shutter is installed "Immediately follow", which satisfies axioms 1-4, is called bezlіchchu natural numbers, and yoga elements - natural numbers.

This appointed person has nothing to say about the nature of the elements of the multiplier N . So, you can be there. Vibirayuchi like a faceless N the day is a specific multiplier, on which a specific reference is given “without intermediary follow”, which satisfies the axioms 1-4, we take it model of this system axioms.

The standard model of Peano's system of axioms is a series of numbers, which is the root of the process of historical development of the succession: 1,2,3,4,... The natural series starts from the number 1 (axiom 1); after the natural number of the skin, one natural number immediately follows (axiom 2); a skin natural number follows no more than one natural number (axiom 3); starting from the number 1 and moving in order to the natural numbers advancing one after another, we take all the multipliers of the numbers (axiom 4).

Otzhe, we developed the axiomatic pobudov system of natural numbers with the choice of the main vodnosiny "without intermediary follow for" that axiom, in some descriptions of yoga of power. A little further on pobudov's theory of transferring a look at the powers of natural numbers and operations from them. The stench may be rozkritі at the appointed and theorems, tobto. introduced by the daily logical path of the introduction of “without middle consideration”, and axioms 1-4.

The first thing to understand, as we introduce after the designation of a natural number, is shutter "immediately forward" , yake often vikoristovuyut for an hour to look at the powers of the natural series.

Appointment 2. What is a natural number b follow without intermediary natural number a, that number a called directly ahead(otherwise the front) number b .

Vіdnoshennia "pereduє" maє next to authorities.

Theorem 1. Unity does not have a forward natural number.

Theorem 2. Skin is a natural number a, Vіdmіnne vіd 1, maє one forward number b, so what b"= a.

The axiomatic rationale of the theory of natural numbers is not seen either in the middle school or in the middle school. Prote dominion vіdnosinі "without intermediary following", like it was in Peano's axioms, є the subject of study in the cob course of mathematics. Already at the first class, it’s an hour to look at the numbers of the first ten, it’s clear, as you can get a skin number. At whom the words “slid” and “before” are understood. The skin is a new number as a continuation of the twisted twist of the natural series of numbers. Learn to reconsider at tsiom, scho with a skin number, it’s the same, and more than one, that the natural series of numbers is inexhaustible.

Addition of natural numbers

For the rules of prompting axiomatic theory, designating the addition of natural numbers, it is necessary to carry out, vicarious, "immediately follow", i understand "natural number"і "previous number".

Viperedimo vyznachennya folded by advancing mirkuvannyami. How to any natural number a add 1, then take the number a", relentlessly advancing on a, then. a+ 1= a" And, then, we take the rule of adding 1 to any natural number. Ale yak add to a natural number b, vіdmіnne vіd 1? We are speeding up the coming fact: if we see that 2 + 3 = 5, then the sum is 2 + 4 = 6, which follows the number 5 without intermediary. In this order, 2 + 4 = 2 + 3 " =(2+3)". In the hot look like maybe, .

This fact is the basis for the designation of natural numbers in axiomatic theory.

Appointment 3. Adding natural numbers an algebraic operation is called, which can be powerful:

Number a + b called sum of numbers aі b , and the numbers themselves aі b - dodanki.

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Recommendations are recognized for students of pedagogical universities, as they teach the discipline "Algebra and Number Theory". Within the framework of this discipline, the division "Numbers of the system" is developed in the 6th semester. These recommendations include material about the axiomatic rationale for systems of natural numbers (Peano's system of axioms), systems of integers and rational numbers. Tsya axiomatics allows you to better understand what such a number is, as one of the main ones to understand the school mathematics course. For the shortest assimilation of the material, the introduction of relevant topics is suggested. For example, recommendations and recommendations, statements, tasks.

Reviewer: Ph.D., prof. Dalinger V.A.

Signed to friend - 22.10.98

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1. NATURAL NUMBERS.

With the axiomatic reasoning of the system of natural numbers, it is important to take into account the understanding of the multiplier, the blue, functions and other multi-theoretical understandings.

1.1 Peano's system of axioms and the simplest inferences.

The common understanding in the axiomatic theory of Peano is the impersonal N (as it is called the impersonality of natural numbers), especially the number zero (0) from the new and binary relation "follows" to N, which is denoted by S (a) (or a ().
AXIOM:
1. ((a(N) a"(0 (This is a natural number 0, which does not follow any number.))
2. a=b (a"=b"
3. a "=b" (a=b (Skin natural number follows more than one number.)
4. (axiom of induction) As a multiplier M(N and M satisfies two minds:
A) 0(M;
B) ((a(N) a(M ® a)(M, then M=N).
In functional terminology, ze means that S:N®N is inactive. From axioms 1 it is clear that S:N®N fermentation is not sur'active. Axiom 4 is the basis for proving the hard work "by the method of mathematical induction".
Significantly acts of the power of natural numbers, which without intermediary cry out for axioms.
Power 1. Skin is a natural number a(0 following one and more than one number.
Bringing. Significantly through M impersonal natural numbers, scho vanish zero and all natural numbers, skin of any following for any number. It is sufficient to show that M=N, unity is evident from axioms 3. Let us prove axiom of induction 4:
A) 0(M - by prompt multiplier M;
B) even a(M, those a"(M, more a" follows a.
Mean from axioms 4 M=N.
Power 2. Like a (b, then a "(b").
Power is brought by the method "from the unacceptable", vicorist axiom 3. Similarly, such power is brought 3, vicorist axiom 2.
Power 3. Like a "(b", then a (b.)"
Power 4. ((a(N)a(a". (No natural number follows it).)
Bringing. Let M=(x(x(N, x(x))). ) in such a rank of Umov A) axioms 4 0(M - win. If x(M, then x(x"), then 2 x"((x")" is in power, and tse means that Umov B) x ( M ® x"(M. Alethodically follows axiom 4 M=N."
Let (- the deuce of power of natural numbers. The fact that the number a has power (, write down ((a)).
Task 1.1.1. Let me tell you that axiom 4 of the designation of the impersonal natural numbers is closer to the advancing hardness: for any kind of authority (, like ((0) i, then).
Task 1.1.2. The unary operation (: a(=c, b(=c, c(=a)) is defined in this way on the trielement multiplier A=(a,b,c).)
Task 1.1.3. Let A \u003d (a) - one-element multiplier, a (= a) Yaki with Peano's axioms of truth on the multiplier A with the operation (?)
Task 1.1.4. On a multiplicity of N, a significantly unary operation is significant, no matter who. Explain what will be true of Peano's axioms formulated in terms of the operation.
Task 1.1.5. Come on. Prove that A is closed using the operation (. Reverse the truth of Peano's axioms on the multiplier A with the operation (.).
Task 1.1.6. Come on, . Significantly on A is a unary operation, however. How are Peano's axioms true on the multiplier A of the operation?

1.2. Non-superelectiveness and categoricalness of Peano's system of axioms.

The system of axioms is called non-superable, as with її axioms it is impossible to bring the theorem T and її transverse (T. It was understood that super-efficient systems of axioms cannot have the same value in mathematics, because in such a theory it is possible to bring everything that Therefore, the lack of superbness of the system of axioms is absolutely essential.
Yakshcho in the Aksіomatic Theoret did not stream theorem t і ї ї ї ї ї ї ї not mean, the system of aksi is not overwhelmed; to the fact that the interpretation of the system of axioms in an obviously non-superequalable theory S, then the system of axioms itself is nonsuperequal.
For the system of axioms of Peano, one can zbuduvat rich different interpretations. Especially rich in interpretation of the theory of multiplicity. One of such interpretations is significant. By natural numbers, we can take multiples (, ((), ((())), (((())),..., we will distinguish zero by the number (. (M), the only element of such and such M. In this order, ("=((), (()"=((()) and so on)). is small: it shows that the system of Peano's axioms is even though the theory of multiples is not superlative, but the proof of the nonsupercity of the system of axioms of the theory of multiples is even more important.
A system of axioms that is not superlative is called independent, because the skin axiom of this system cannot be proved as a theorem on the basis of other axioms. To bring to light that the axiom
(1, (2, ..., (n, ((1))
enough to prove that the system of axioms is non-superable
(1, (2, ..., (n, (((2))
It’s true, yakby (it was possible to vary from the other axioms of system (1), then system (2) was super-smart, the shards of it would be true of the theorem (and axiom ((.)).
Also, in order to bring the independence of the axioms (from other axioms of the system (1), it is enough to encourage the interpretation of the system of axioms (2).
The independence of the system of axioms is a great neobov'yazkova. Sometimes, in order to avoid the proof of "important" theorems, we will create a supra-world (deposit) system of axioms. However, "zayv" axioms make it easier to see the role of axioms in theory, as well as internal logical links between different divisions of theory. In addition, pobudova іnterpretatsіy for fallow systems of axioms is significantly folded, lower for independent ones; even if you have to reconsider the validity of "zayvih" axioms. Of the reasons for the nutrition of the fallow land, among the axioms of long ago, the first importance was given. Try to bring it to your time that the 5th postulate in Euclid's axiomatics "It is no more than one straight line that passes through the point A parallel to the straight line" (", є by the theorem (to lie in the other axioms) and brought to the conclusion of Lobachevsky's geometry).
A non-superscriptive system is called deductively new, as if the proposition A of a given theory can either be brought, or be proclaimed, then either A, or (A is a theorem of the given theory. an axiom is called Deductive povnota - tezh not obov'yazkova vimoga, for example, a system of axioms of the theory of groups, the theory of territory, the theory of watering - not true, shards are based on and kіntsevі and neskіnchennі groups, kіltsya, fields, then in these theories you can’t ask, you can’t bring proposition.: "Group (kіltse, field) to avenge kіltse kіlkіst elements".
It should be noted that in rich axiomatic theories (themselves, in non-formalized ones) impersonal propositions cannot be taken exactly into account, and it is impossible to bring the deductive completeness of the system of axioms of such a theory. The second change is often called categorical. The system of axioms is called categorical, so be it two interpretations isomorphic, so that there is such a mutually unambiguous distinction between multiple cob objects and other interpretations Categoricalness - tezh neobov'yazkova mind. For example, the axiom system of group theory is not categorical. The reason is that the Kintsev group cannot be an isomorphic unskinned group. However, with the axiomatization of the theory of a numerical system, the categorical nature of obov'yazkova; For example, the categorical nature of the system of axioms, which signifies the natural numbers, means that, up to isomorphism, there is only one natural series.
Let us bring the categoricity of Peano's system of axioms. Let (N1, s1, 01) and (N2, s2, 02) be two interpretations of Peano's system of axioms. It is necessary to indicate such a biektivne (mutually unambiguous) expression f: N1®N2, for which you should think:
a) f(s1(x)=s2(f(x)) for any x N1;
b) f(01) = 02
If the unary operations s1 and s2 are offended by the same stroke, then umova a) rewrite
a) f(x()=f(x)(.
Significantly on the multiplier N1(N2)
1) 01f02;
2) how xfy, x(fy(.
Let's change, what is the use of fermentation N1 to N2, then for dermal x s N1
(((y(N2)xfy(1)
Significantly through M1 impersonal elements x N1, for some minds (1) win. Todi
A) 01 (M1 z 1);
B) x(M1 ® x((M1 by virtue of 2) and the power of 1 point 1).
Therefore, according to axiom 4, it is possible that M1=N1, and tse i means that the introduction of f є fermentation of N1 N2. At tsimu z 1) it is obvious that f (01) = 02. Umov 2) is written like this: f(x)=y, then f(x()=y(. It sounds like f(x()=f(x)(. Also, for the reflection of f, think a)) and b.
Significantly through M2, impersonal quiet elements of N2, skin of any of them in the form of one and only one element of N1 when f is displayed.
Shards f(01)=02, then 02 є. If so x(N2 і x(01), then for power 1 item 1 x follows the current element c z N1 і then f(x)=f(c()=f(c)((02. Mean, 02 f) the rank of a single element 01, then 02 (M2.
Go ahead y(M2 і y=f(x), where x is the single preimage of the element y. Then, by virtue of a) y(=f(x)(=f(x()), then y(є the image of the element x ) (. Let c be a pre-image of the element y(, then f(c)=y(. Skіlki y((02, then c(01 і c) is the forward element, which is meaningful through d.)) Then y( =f( c)=f(d()=f(d)(, due to axiom 3 y=f(d)). M2 ® y
All pre-Greek mathematics has little empirical character. All the elements of the theory were drowning in the mass of empirical approaches to the development of practical tasks. The Greeks gave this empirical material of logical analysis, tried to find the connection between different empirical data. For whom the whole sense of geometry has a great role played by Pythagoras and the school (5th century AD). The ideas of the axiomatic method were clearly voiced in the works of Aristotle (4th century AD). Prote, a practical development of these ideas was carried out by Euclid at yoga "Cobs" (3 centuries AD).
Three forms of axiomatic theories can be named.
one). Zmistovna axiomatics, as if it was one until the middle of the last century.
2). Napіvformal axiomatics, scho vinyl in the last quarter of the past century.
3). Formal (otherwise formalized) is axiomatics, the date of birth of which can be taken as 1904, if D. Hilbert published his famous program about the basic principles of formalized mathematics.
The new skin form is not blocked in front, but with a development and clarification, the same is true for the development of the new skin form, lower in the front.
Zmistovna axiomatics are characterized by the fact that they can be understood intuitively clearly before formulating axioms. So, in the "Cobs" of Euclid, under the point of understanding, those who are intuitively self-evident under these understandings. At the same time, there is a great language, and a great intuitive logic, which is more like Aristotle.
The formal axiomatic theories also have a strong language and intuitive logic. However, the first understanders do not rely on the same intuitive sense, they are characterized only by axioms. Tim himself moves strictness, shards of intuition with a singing world conquer strictness. In addition, sleepiness is growing, because the skin theorem, brought in such a theory, will be fair in any interpretation. Clearly in the form of a formal axiomatic theory - Hilbert's theory, included in the book "Imagine Geometry" (1899). The butts of the nap_vformalnyh theories are also the theory of the kіlets and other theories, presented in the course of algebra.
The butt of the formalized theory is the calculation of the number of words, which is developed in the course of mathematical logic. On the vіdmіnu vіd zmіstovnoї and napіvformalії axiomatics, the formalization of the theory victorious especially symbolic mova. The alphabet of the theory is assigned to itself, so that it is a deuce of impersonal symbols, which play the same role as the letters in the original language. Be it a kіntseva sequence of symbols is called a viraz or a word. Among the viruses, there is a class of formulas, and the exact criterion that allows the skin virus to be recognized is indicated by the formula. The formulas play the same role as the speech of the great language. Deyakі formulas goloshuyutsya axioms. In addition, logical rules of vision are set; Such a rule means that in the course of the totality of formulas, the whole formula is without middle. The proof of the theorem itself is the end of the lantz of formulas, the rest of the formula is the theorem itself and the skin formula is either an axiom, or the theorem was brought earlier, otherwise it sings out of the middle of the forward formulas of the lance on one of the rules of observation. In this rank, we should not stand for the evidence about the validity of evidence: otherwise Danish lanciugє proof, or є, there are no conclusive proofs. At the link with the cim, the axiomatics is formalized to get used to the especially subtle principles of priming mathematical theories, if the obvious intuitive logic can lead to pardons, which are the main rank through the inaccuracies and ambiguity of our great movement.
So, as in the formalization of the theory about skin viraz, one can say that it is a formula, then the impersonal propositions of the formalized theory can be taken into account. In connection with this, it is possible, in principle, to break down the argument about the proof of the deductive reason, as well as about the proof of the non-superficiality, without going into interpretation. In a number of the simplest ways, you can see the difference. For example, the lack of superficiality of the calculation is carried out without interpretation.
In non-formalized theories, impersonal propositions are not clearly defined; therefore, the reason for proving non-superficiality, without going to interpretation, is put stupidly. Those same worth and food about the proof of deductive povnoti. However, as such a proposition of an unformalized theory was heard, as it is impossible to bring it or to ask, then the theory, obviously, is deductively inaccurate.
The axiomatic method has long been established not only in mathematics, but also in physics. First, try it directly, Aristotle tried to do it, but he also corrected his own axiomatic method in physics, excluding Newton's robots from mechanics.
At the link with the turbulent process of mathematization of sciences, there is also the process of axiomatization. None of the axiomatic methods are found in various divisions of biology, for example, in genetics.
The possibilities of the axiomatic method are not endless.
It is significant that we should not forget about formalizing theories without ignoring the intuition. The theory itself is formalized without any interpretation of the desired meaning. The blame for this is low on the connection between the formalized theory and the interpretation. In addition, as in the formalization of theories, there is a question about the non-superity, independence and completeness of the system of axioms. The totality of all such food becomes the essence of another theory, as it is called a metatheory of a formalized theory. On the basis of the formalized theory, the language metatheory is the most important everyday language, and the logical mirroring is carried out by the rules of natural intuitive logic. In this manner, the intuition, which is again taken from the formalized theory, reappears in the metatheory.
But the main weakness of the axiomatic method is not in tsoma. Previously, it was already thought about the program of D. Hilbert, as it laid the foundation for a formalized axiomatic method. Hilbert's main idea is to make classical mathematics as a formalized axiomatic theory, to bring non-superability. However, the program in its main points appeared to be utopian. In 1931, the famous Austrian mathematician K. Gödel developed his famous theorems, which made it clear that offending the main tasks set by Hilbert were not published. Yomu went beyond the help of his method of coding to learn for the help of the formulas of formalized arithmetic, and the help of a metatheory to bring that these formulas are not visible in the formalization of arithmetic. In this manner, formalized arithmetic appeared deductively inaccurate. From Gödel's results, it was obvious that even if an unprovable formula is included to the number of axioms, then there is another unprovable formula that expresses the same correct proposition. All this meant that not only all mathematics, but to learn arithmetic - the simplest part, it is impossible to formalize. Zokrema, Gödel, having inspired a formula that confirms the propositions "Formalized arithmetic is non-superable", and showing that the formula cannot be shown either. This fact means that the imperfection of formalized arithmetic cannot be brought to the middle of arithmetic itself. Zrozumіlo, you can encourage a strong formalized theory and її by means of bringing the non-superity of formalized arithmetic, and at the same time blame more importantly for the non-superity of the new theory.
Gödel's results indicate the validity of the axiomatic method. And, more importantly, podstav for pessimistic visnovkіv in the theory of knowledge of the one who does not know the truth, - no. The fact that arithmetic truths are established, which cannot be brought to the formalization of arithmetic, does not mean the manifestation of ignorance of truths and does not mean the obscurity of human thought. Vin means only that the possibilities of our mind will not be reduced to procedures, that they will be more formalized, and that people still need to test and find new principles of proof.

1.3. Storing natural numbers

Operations of folding and multiplication of natural numbers by the system of Peano's axes are not postulated, but instead of operations.
Appointment. The addition of natural numbers is called a binary algebraic operation + on the multiplier N, which can be powerful:
1s. ((a(N)a+0=a);
2c. ((a, b (N) a + b (= (a + b)).
Blaming nutrition - what is such an operation, but if it is, then what is it?
Theorem. Addition of natural numbers is necessary and only one.
Bringing. The binary operation of algebra on the multiplicity N is the fermentation (:N(N®N. It is necessary to bring that there is only one fermentation (:N(N®N with powers: 1)) ((x(N) ((x,0)= x ; 2) ((x,y(N) ((x,y()=((x,y))). 0) )=x; ).
Significantly on the multiplier N, binary expression fx by the minds:
a) 0fxx;
b) how yfxz, y(fxz(.
Let's change, what is the use of N to N, then for the skin y z N
(((z(N) yfxz (1)
Significantly, through M, the multiplier of natural numbers y, for which minds (1) is victorious. So think a) vyplyaє, scho 0 (M, a z um b) and power 1 p. and means that the fx is the fermentation of N to N. For which fermentation, think:
1() fx(0)=x - s a);
2() fx((y)=fx(y() - through b).
Tim himself brought the reasoning for folding.
We bring unity. Let + i (- be like two binary operations of algebra on sets N with powers 1c and 2c. It is necessary to bring that
((x, y(N) x + y = x(y)
It is fixed enough number x i is significant through S of impersonal natural numbers y, for which equanimity
x+y=x(y(2)
win. Skіlki zgіdno 1с x+0=x і x(0=x, then
A) 0(S
Now let y(S, so that equality (2) wins. So x+y(=(x+y)(, x(y(=(x(y))(і x+y=x(y, then) ) axioms 2 x+y(=x(y(, so that the mind will win)
B) y(S ® y((S.)
So, by axiom 4 S=N, which completes the proof of the theorem.
Let's bring the authorities to the dodavannya.
1. The number 0 is the neutral element of addition, so a+0=0+a=a for the skin natural number a.
Bringing. Equanimity a+0=a screams from the mind 1s. We bring equality 0+a=a.
Significantly through M impersonal numbers, which won't win. Obviously, 0+0=0 and 0(M. Let a(M, then 0+a=a.) Then 0+a(=(0+a)(=a(i, aka, a((M) ) Otzhe, M=N, how and it is necessary to bring.
Give us a lema.
Lemma. a(+b=(a+b)(.
Bringing. Let M be an impersonal number of all natural numbers b, for which the equality is a(+b=(a+b)(true for any value of a.):
A) 0(M, shards a(+0=(a+0)(;);
C) b(M ® b((M. Definitely, since b(M and 2c) is possible)
a(+b(=(a(+b))(=((a+b)()(=(a+b()(,
so b ((M. Mean, M = N, what i need to bring).
2. The addition of natural numbers is commutative.
Bringing. Let M=(a(a(N(((b(N)a+b=b+a))) Tell me that M=N. Maybe:
A) 0(M - cost 1.
C) a(M ® a((M)
a(+b=(a+b)(=(b+a)(=b+a(.)).
Mean a((M, i from axiom 4 M=N).
3. Adding associatively.
Bringing. Come on
M=(c(c(N(((a,b(N))(a+b)+c=a+(b+c))
It is necessary to bring that M=N. So (a+b)+0=a+b and a+(b+0)=a+b, then 0(M. Let s(M, then (a+b)+c=a+(b+c) ).
(a+b)+c(=[(a+b)+c](=a+(b+c)(=a+(b+c())).
Mean c((M i by axiom 4 M=N).
4. a+1=a(, de 1=0(.
Bringing. a+1=a+0(=(a+0)(=a(.
5. If b(0), then ((a(N)a+b(a)).
Bringing. Let M=(a(a(N(a+b(a)) 0+b=b(0, then 0(M)). 2 p.1 (a+b)((a(otherwise a( +b(a)) means a((M і M=N)).
6. If b(0, then ((a(N)a+b(0))
Bringing. If a=0, then 0+b=b(0, if a(0 і a=c(, then a+b=c(+b=(c+b))((0. So, y be- which time a) + b (0.
7. (The law of trichotomy folding). For any natural numbers a and b, only one and only one of the three similitudes is true:
1) a = b;
2) b=a+u de u(0;
3) a=b+v de v(0.
Bringing. We fix a certain number a and it is significant through M the multiplier of all natural numbers b, for which one of the connotations 1), 2), 3) is victorious. It is necessary to bring that M=N. Let b = 0. If a=0, then 1), and if a(0, just 3), then a=0+a. Otzhe, 0(M.
It is now acceptable that b(M, so that the inverse of a is one of the converses of 1), 2), 3). If a=b, then b(=a(=a+1, then for b(the offset 2 is counted).) If b=a+u, then b(=a+u(, then for b(the offset is counted) 2 ) If a=b+v, then two declinations are possible: v=1 and v(1. If v=1, then a=b+v=b", then for b" the converse ratio 1 is taken. and v(1 , then v=c", de c(0 and then a=b+v=b+c"=(b+c)"=b"+c, de c(0, so for b" we have a converse 3). Later, we brought that b (M ® b "(M, i, also M = N, so for whether a and b, one wants to use one of the consonances 1), 2), 3). they cannot be defeated at once. spіvvіdnoshennia 2) and 3), then small b a = (a + u) + v = a + + (u + v), but it is impossible through the power of 5 and 6. The power of 7 is brought to a close.
Task 1.3.1. Let 1(=2, 2(=3, 3(=4, 4(=5, 5(=6, 6(=7, 7(=8, 8(=9))). Tell me 3+5 =8, 2+4=6.

1.4. MULTIPLYING NATURAL NUMBERS.

Appointment 1. The multiplication of natural numbers is called such a binary operation (on the multiplier N, for which the mind is counted:
1u. ((x(N)x(0=0);
2y. ((x, y(N)x(y)=x(y+x).
I’m vindicating nutrition again - why is such an operation and how is it, then what is the only thing?
Theorem. The operation of multiplying natural numbers is only one.
The proof may be carried out in the same way, as for additional proof. It is necessary to know such an expression (:N(N®N), as
1) ((x(N)) ((x,0)=0;
2) ((x, y (N) ((x, y")) = ((x, y) + x).
We fix quite a number x. It is also possible for skin x(N іsnuvannya vіrazhennya fx: N®N s authority
1") fx(0)=0;
2") ((y(N) fx(y")=fx(y)+x,
then the function ((x,y), which is equal to ((x,y)=fx(y) and satisfies minds 1) and 2).
Later, the proof of the theorem goes up to the proof of the basis of that unity for skin x of the function fx(y) with powers 1") and 2"). Let's set the number of N values according to the following rule:
a) the number zero is set to the number 0,
b) since the number y is given the number c, then the number y (the number c + x is equal).
Let's reconsider that in such a setting the skin number y can be a single image: and it is significant that it is possible to convert N into N. Significantly, through M the impersonality of all natural numbers y, a single image can be formed. Think a) that axiom 1 is correct, so 0(M. Let y(M. Think b) and axiom 2 are clear that y((M. So, M=N, so our proof is N) in N , is signifi- cantly in terms of fx, then fx(0)=0 by reason of a) and fx(y()=fx(y)+x - by reason of b).
Later, the reason for the multiplication operation was confirmed. Let me now (i (- be two binary operations on the multiplier N with powers 1y and 2y. It is left to say that ((x,y(N) x(y=x(y) We fix quite a number x and don’t))
S=(y?y(N(x(y=x(y))
Skip through 1y x(0=0 і x(0=0, then 0(S. Let y(S), then x(y=x(y))
x(y(=x(y+x=x(y+x=x(y(
i, then, y((S. So, S=N, lower i, the proof of the theorem ends).
Significantly many deacons of power.
1. The neutral element is usually the number 1=0(, so ((a(N) a(1=1(a=a))).
Bringing. a(1=a(0(=a(0+a=0+a=a)) In this way, the equality of a(1=a has been completed. N) (1(a=a). So 1(0=0, then 0(M. Let a(M, then 1(a=a)). Then 1(a(=1(a+1=a +1=) a(, i, otzhe, a((M. So, from the axioms 4 M=N, which was necessary to bring).
2. For a set of fairs, a right distributive law, then
((a,b,c(N) (a+b)c=ac+bc).
Bringing. Let M=(c(c(N(((a,b(N))(a+b)c=ac+bc))). , then 0(M. So c(M, then (a+b) c=ac+bc), then (a + b)(c(= (a + b)c +(a + b) = ac + bc +a+b=(ac+a)+(bc+b)= ac(+bc(.) So, c((M і M=N).
3. The multiplication of natural numbers is commutative, that is ((a,b(N) ab=ba).
Bringing. Let's get it right for b (N equal 0 (b = b (0 = 0. Equal b (0 = 0) is clear 1y. Let M = (b (b (N (0 (b = 0))) ) 0=0, then 0(M. So b(M, then 0(b=0, then 0(b(=0(b+0=0)) i, also, b((M. So, M= N, then equality 0(b=b(0 brought to all b(N. Let's go further) S=(a (a(N(ab=ba))). a) (S, then ab = ba. Then a (b = (a + 1) b = ab + b = ba + b = ba (, then a ((S. So S = N), which is necessary to bring) .
4. Multiple distributive folding. Tsya dominion viplivaє z dominion 3 and 4.
5. The plural is associative, that is ((a, b, c (N) (ab) c = a (bc)).
The proof is carried out, like th at the warehouse, induction on s.
6. If a(b=0, then a=0 and b=0, then N has no zero divisors.
Bringing. Let b(0 і b=c(. If ab=0, then ac(=ac+a=0, the signs follow the power of 6 item 3, so a=0).
Task 1.4.1. Let 1(=2, 2(=3, 3(=4, 4(=5, 5(=6, 6(=7, 7(=8, 8(=9))). Tell me what 2(4 =8, 3(3=9.
Let n, a1, a2, ..., an be natural numbers. The sum of the numbers a1, a2,...,an is called the number, as it is denoted through it by the minds; for any natural number k
A subset of the numbers a1, a2,...,an is a natural number, as it is denoted by i and is denoted by minds: ; for any natural number k
How that number is indicated through an.
Task 1.4.2. Bring what
a);
b);
in);
G);
e);
e);
and);
h);
і) .

1.5. ORDER OF THE SYSTEM OF NATURAL NUMBERS.

The statement "follows" is antireflexive and antisymmetric, but not transitive and does not follow that order. We are significantly changing the order, relying on the addition of natural numbers.
Appointment 1. a
Destination 2. a(b (((x(N) b=a+x)).
Perekonaєmosya, scho vіdnoshennia Vіdznachimo deyaki vlastnostі natural numbers, povyazanih іz vіdnosinami іnоnostі і nerіvnostі.
1.
1.1 a=b (a+c=b+c).
1.2 a = b (ac = bc).
1.3a
1.4a
1.5 a+c=b+c (a=b).
1.6ac=bc(c(0(a=b).
1.7a+c
1.8ac
1.9a
1.10a
Bringing. Dominance 1.1 and 1.2 exude from the uniqueness of the operations of folding and multiplication. Yakscho a
2. ((a(N) a
Bringing. Oskils a(=a+1, then a
3. The least element N is 0, and the least element N\(0) is the number 1.
Bringing. So ((a(N) a=0+a, then 0(a, i, hence, 0 is the smallest element of N.) Then, like x(N\(0), then x=y(, y(N ) , otherwise x = y + 1. The answer is that ((x (N \ (0)) 1 (x, so 1 is the smallest element in N \ (0)).
4. Suggestion ((a, b (N) ((n (N)) b (0 (nb> a)).
Bringing. Obviously, for any natural a, there is also a natural number n, which
a Such a number є, for example, n = a (. Dahl, if b (N \ (0), then for power 3
1(b(2)
Z (1) and (2) on the basis of powers 1.10 and 1.4 take aa.

1.6. THE REAL ORDER OF THE SYSTEM OF NATURAL NUMBERS.

Appointment 1. As a skin non-empty submultiplier of an ordered multiplier (M; Reconsider that the new order is linear. Let a and b be two elements from a whole ordered multiplier (M; Lema) . 1) a
Bringing.
1) a((b (b=a(+k, k(N)(b=a+k(, k((N\(0)))
2) a(b(b=a+k, k(N)(b(=a+k(, k((N\(0)))
Theorem 1. The natural order on the set of natural numbers is a higher order.
Bringing. Let M be empty of the impersonal natural numbers, and S is the immateriality of the lower inters in N, so S = (x (x (N (((m (M)) x (m)). next, 0(S. Yakby was victorious and other Umov's axioms 4 n(S(n((S, then small b S=N)).
Theorem 2. If there is a non-empty boundary for the beast of impersonal natural numbers, there may be the greatest element.
Bringing. Let M be a non-empty boundary between the beast of the impersonal natural numbers, and S is the impersonality of the upper cordons, so S=(x(x(N((m(M)) m(x)).) Significantly through x0, the smallest element of y S. If m
Task 1.6.1. Bring what
a);
b);
in).
Task 1.6.2. Come on (- deak power of natural numbers and k - more than a natural number. Bring what
a) be-like a natural number may be power (like only 0 may be power for whatever n (0
b) whether it is a natural number, greater than or equal to k, maє power (, if only k maє tsyu power i for whatever n (k (n) s omission, scho n maє power (, next, scho number n + 1 also Volodya tsієyu power). ;
c) whether it is a natural number, greater than or equal to k, may have power (as only k may have power and for whatever n (n>k) is an allowance, that all numbers t, assigned by mental k (t

1.7. INDUCTION PRINCIPLE.

Vikoristovuyuchi povryadkovannost of the system of natural numbers, you can bring such a theorem, one of the foundations of the methods of proof, titles by the method of mathematical induction.
Theorem (principle of induction). Usі vyslovlyuvannya z sequent A1, A2, ..., An, ... є іstnymi, yakshcho vykonuyutsya mind:
1) A1 is true;
2) how to use Ak with k
Bringing. It is admissible not to accept: think 1) and 2) to win, but if the theorem is not true, then we will not allow є impersonal M = (m (N (N \ (0), Am - hibno)). element, which is meaningful in terms of n. mentally 1) A1 is true, and An is bad, then 1(n, i, aka, 1)
For confirmation by the method of induction, two stages can be seen. At the first stage, which is called the basis of induction, the mentality of the mind is overturned 1). From the other side of the stage, called the induction crock, the mind is brought to mind 2). At the most often, vipads are traversed, if to prove the truth of An, it is not possible to use victoriousness of the truth of Ak at k
butt. To bring unevenness Payable = Sk. It is necessary to bring the truth of the derivation Ak=(Sk The sequence of the deduction, as described in Theorem 1, can come from the predicate A(n) assigned to the set N or to the th subset Nk=(x(x(N, x(k)), where k is a fixed natural number.
Sokrema, if k=1, then N1=N(0), and the numbering can be carried out for additional equalities A1=A(1), A2=A(2), ..., An=A(n), .. If k(1, then the sequence of occurrences can be taken from additional evennesses A1=A(k), A2=A(k+1), ..., An=A(k+n-1), .. .Vidpovidno to such values, Theorem 1 can be formulated in a different form.
Theorem 2. The predicate A(m) is also true on the multiplier Nk, so you know:
1) A(k) is true;
2) how to use A(m) for m
Task 1.7.1. Let me tell you that this kind of equality does not make a decision in the gallery of natural numbers:
a) x + y = 1;
b) 3x = 2;
c) x2 = 2;
d) 3x+2=4;
e) x2+y2=6;
f) 2x+1=2y.
Task 1.7.2. Bring, victorious principle of mathematical induction:
a) (n3+(n+1)3+(n+2)3)(9;
b);
in);
G);
e);
e).

1.8. VIDCHITANNYA I DELENNYA NATURAL NUMBERS.

Designation 1. The difference between natural numbers a and b is such a natural number x that b+x=a. The difference of natural numbers a and b is denoted through a-b, and the operation of the difference of the difference is called the difference. Vіdnimannya is not an operation of algebra. Tse vyplyvaє iz nastupnoї theorem.
Theorem 1. Retail a-b is the only difference and only one, if b(a. If there is a difference, then only one).
Bringing. If b(a, then for the designation of the reference (if it is a natural number x, then b+x=a. Ale ce i means that x=a-b. that b + x = a. Alece means that b (a.
We bring unity retail a-b. Let a-b=x and a-b=y. The same goes for the appointments 1 b+x=a, b+y=a. Zvіdsi b+x=b+y і, also, x=y.
Destination 2. The fraction of two natural numbers a and b(0) is called a natural number c such that a = bc.
Theorem 2. It's more private than one.
Bringing. Come on = x that = y. The same goes for appointments 2 a=bx and a=by. Zvіdsi bx=by і, also, x=y.
It is worthy of note that the operations carried out on that occasion can be counted literally the same way, as in the case of school assistants. Tse means that in paragraphs 1-7, on the basis of Peano's axioms, the theoretical foundation of the arithmetic of natural numbers was laid, and further developments are subsequently established in the high school course in mathematics and in the university course "Algebra and Number Theory".
Task 1.8.1. Bring the justice of such assertions, admitting that all the differences that are stated in their formulas are clear:
a) (a-b)+c=(a+c)-b;
b) (a-b) (c = a (c-b (c);
c) (a+b)-(c+b)=a-c;
d) a-(b+c)=(a-b)-c;
e) (a-b)+(c-d)=(a+c)-(b+d);
e) (a-b)-(c-d)=a-c;
g) (a+b)-(b-c)=a+c;
h) (a-b)-(c-d)=(a+d)-(b+c);
i) a-(b-c)=(a+c)-b;
to) (a-b)-(c+d)=(a-c)-(b+d);
k) (a-b)(c+d)=(ac+ad)-(bc+bd);
l) (a-b)(c-d)=(ac+bd)-(ad+bc);
m) (a-b)2=(a2+b2)-2ab;
o) a2-b2=(a-b)(a+b).
Task 1.8.2. To bring the justice of the coming hardships, admitting that everything is private, that they are stated in the given formula, it is clear.
a); b); in); G); e); e); and); h); i); to); l); m); n); about); P); R).
Task 1.8.3. To prove that mothers of two different natural solutions cannot be so equal: a) ax2+bx=c (a,b,c(N); b) x2=ax+b (a,b(N); c) 2x=ax2 + b(a,b(N).
Task 1.8.4. Untie natural numbers equal:
a) x2+(x+1)2=(x+2)2; b) x + y = x (y; c); d) x2+2y2=12; e) x2-y2 = 3; e) x + y + z = x (y (z.
Task 1.8.5. To prove that there is no such equal solution in the sphere of natural numbers: a) x2-y2=14; b) x-y = xy; in); G); e) x2=2x+1; f) x2 = 2y2.
Task 1.8.6. Unraveling the natural numbers of the unevenness: a) ; b); in); d) x+y2 Task 1.8.7. Tell me that in the realm of natural numbers, the onset of spiving is fair: a) 2ab(a2+b2; b) ab+bc+ac(a2+b2+c2; c) c2=a2+b2 (a2+b2+c2 1.9. KILKISNIY DEATH natural numbers.
Really, the natural numbers should be placed as the head rank of the rahunka of the elements, and what is required to establish a quarterly substitute for the natural numbers theoretically by Peano.
Destination 1. Anonymous (x(x(N, 1(x(n)) is called in contrast to the natural series) and is denoted through (1; n ()).
Appointment 2. A kіntsevoj multiplier is called whether it is a multiplier, equal to any counter of the natural series, and also an empty multiplier. Bezlich, like not є kіtsevim, is called unskinned.
Theorem 1 to the wet(Tobto podmnozhini, vіdmіny vіd A).
Bringing. How A=(, the theorem is true, there are no empty shards of empty submultiples. Let us A((і A equally hard (1,n((A((1,n()).)) We can prove the theorem by induction on n. How n= 1 , then A((1,1(, then we use the single submultiplier of the multiplier A is an empty multiplier). It was clear that A(i, also, for n=1, the theorem is true. Assume that the theorem is true for n=m, then all terminals multipliers, equal strengths in the wind (1,m(, do not think of equal strengths in the wind). inverse)) (1,m+1(in A. If ((k) is known by ak, k=1,2,...,m+1, then the impersonal A can be written as A=(a1, a2, ...)) , am, am+1) Our goal is to prove that A does not have equally strong power submultiples.
Let's look at the multipliers A1 = A (am + 1) and B1 = B (am + 1). Since f(am+1)=am+1, then the function f zdіysnyuvatime bioactively displaying the multiplier A1, by the multiplier B1. In this rank, the impersonal A1 will be equal to its powerful submultiple B1. Ale oskіlki A1((1,m(, do not supersede the allowance of induction).
Conclusion 1. The absence of natural numbers is not limited.
Bringing. From Peano's axioms, it is clear that S:N®N\(0), S(x)=x(bjectively) is fermented.
Conclusion 2. If the kіntsev's multiplier A is not empty, it is equal to one and only one counterpart of the natural series.
Bringing. Let A((1,m(і A((1,n(. Todі) (1,m(((1,n(, due to Theorem 1 it is clear), so m=n.)).
Last 2 allows you to enter a designation.
Designation 3. As A((1,n(, then the natural number n is called the number of elements in the multiplier A), and the process of establishing mutually unambiguous similarity between the multipliers A and (1,n (called the number of elements in the multiplier A. The number of natural elements of the multiple of the empty enter) the number zero.
About the greatness of the significance of the rahunka for a practical life, speak zayve.
Respectfully, knowing the calculus of a natural number, it would be possible to calculate the multiplication operation through the addition itself:
.
We didn’t send this way for now, to show that arithmetic itself is not required in the calculus sense: the calculus sense of the natural number is needed only in additions to arithmetic.

1.10. THE SYSTEM OF NATURAL NUMBERS AS A DISCRETE REVERSE IS ORDERLY BAGATO.

We have shown that the impersonal natural numbers are compatible with the natural order and the whole ordering. If so, ((a(N) a
1. for any number a(N іsnuє sudіdnє coming after him 2. for any number a(N \ (0) іsnuє suіdnє yoma in front of you) The whole order of the impersonal (A;()) with powers 1 and 2 is called memo discrete cycle It appears that the ordering with powers 1 and 2 is the characteristic power of the system of natural numbers. element i, also, axiom 1 Peano wins).
So it’s like a linear order, then for any element a there is a single element following it and no more than one forward sudidny element. think:
1) a0(M, where a0 is the smallest element of A;
2) a(M (a((M.))
Let's say that M=N. Admissible is not accepted, then A\M((. Significantly, through b, the smallest element in A\M.
We also brought the possibility of another designation of the system of natural numbers.
Appointment. The system of natural numbers is called whether a multiplicity is ordered as a whole, on which minds are counted:
1. for any element, there is a next advancing element behind it;
2. for any element, the least visible element, the main judicial element.
Іsnuyut іnshі pіdhodi destination of the system of natural numbers, on which we do not here zupinaєmosya.

2. TSILI AND RATIONAL NUMBERS.

2.1. SIGNIFICANCE AND POWER OF THE SYSTEM OF NUMBERS.
Apparently, there are no number of numbers in the mind of an intuitive mind, and the number is how to fold that multiplier, moreover, the number is to avenge all the natural numbers. It was understood that there is no swearing in the kіltsі tsіlih numbers, like it would avenge all natural numbers. The qi of power, it seems, can be laid as the basis for a strict designation of a system of numbers. In paragraphs 2.2 and 2.3, the correctness of such a designation will be brought.
Appointment 1. The system of numbers is called an algebraic system, for which the mind is:
1. Algebraic system є kіltse;
2. Anonymity of natural numbers should be taken into account, moreover, adding that multiplication in kіltsі on submultiples is taken from the additions of those multiplications of natural numbers, tobto
3. (umova minimality). Z is the minimum for the inclusion of the multiplier with power 1 and 2. In other words, in order to avenge the natural numbers, then Z0=Z.
Appointment 1 can be given an axiomatic character. The first concepts in this axiomatic theory will be:
1) Anonymous Z, the elements of which are called whole numbers.
2) A special integer number, as it is called zero and is indicated through 0.
3) Ternary vіdnosini + ta (.
Through N, as usual, the impersonal natural numbers are denoted by folding (and multiplications (. As a matter of fact, up to the designation 1, the system of integers is called such a system of algebra (Z; +, (, N), for which the following axioms are victorious):
1. (Axioms of the kіltsya.)
1.1.
This axiom means that + є is a binary operation of algebra on the set Z.
1.2. ((a,b,c(Z) (a+b)+c=a+(b+c)).
1.3. ((a, b (Z) a + b = b + a).
1.4. ((a(Z) a+0=a, so the number 0 can be added as a neutral element).
1.5. ((a(Z)((a((Z) a+a(=0), so for the skin integer there is the opposite number a()).
1.6. ((a,b(Z))((! d(Z) a(b=d)).
This axiom means that the multiplication is a binary operation of algebra on the multiplier Z.
1.7. ((a, b, c(Z)) (a(b)(c = a((b(c))).
1.8. ((a, b, c (Z) (a + b) (c = a (c + b (c, c ((a + b)) = c (a + c (b))
2. (Axioms of the link between Z and the system of natural numbers.)
2.1. N(Z.
2.2. ((a, b (N) a + b = a (b).
2.3. ((a, b(N)) a(b = a(b).
3. (Axiom of minimality.)
If Z0 is the end of the ring Z and N(Z0, then Z0=Z.
Significantly acts of power of the system of numbers.
1. The number of skins can be represented by looking at the difference between two natural numbers. The appearance is ambiguous, moreover, z=a-b and z=c-d, de a, b, c, d (N, both and only if a+d=b+c).
Bringing. Significantly, through Z0, the absence of all integers, the skin of any of them, looks like two natural numbers. Obviously, ((a(N) a=a-0, i, aka, N(Z0).
Let's go x,y(Z0, then x=a-b, y=c-d, de a,b,c,d(N. Then x-y=(a-b)-(c-d)=(a+d)--(b + c)=(a(d)-(b(c)), x(y=(a-b)(c-d)=(ac+bd)-(ad+bc)=(a(c(b(d))- ( a(d(b(c). It can be seen that x-y, x(y(Z0 i, henceforth, Z0 is a subset of the ring Z, to avenge the impersonal N.)).
2. The ring of integers is a commutative ring with unity, and the zero of the ring is the natural number 0, and the unity of the ring is the natural number 1.
Bringing. Let x,y(Z. Valid to power 1 x=a-b, y=c-d, de a,b,c,d(N.) Then x(y=(a-b)((c-d)=(ac+bd)- (ad) +bc)=(a(c(b(d))-(a(d(b(c)), y(x=(c-d))(a-b)=(ca+db)-(da+ cb)=(c( a(d(b)-(d(a(c(b))). Therefore, due to the commutativity of the multiplication of natural numbers, it fits that xy=yx. The commutativity of the multiplication in the ring Z has been brought. 2 vyplyvayut from the offensive obvious equalities, in which, through 0 and 1, the natural numbers zero and one are known: x+0=(a-b)+0=(a+(-b))+0=(a+0)+(-b) =(a(0)+ (-b) = a-b = x x (1 = (a-b) (1 = a (1-b (1 = a (1-b (1 = a-b = x)))

2.2. ІSNUVANNYA SYSTEM CYLIKH NUMBER.

The system of numbers is assigned to 2.1 as the minimum for the inclusion of the ring, which avenges the natural numbers. Vikaє pitanya - what is the same kіltse? In other words, the system of axioms s 2.1 is super-simplistic. In order to bring the non-superity of the system of axioms, it is necessary to induce an interpretation in a clearly non-supervisible theory. Such a theory is taken into account by the arithmetic of natural numbers.
Again, it is necessary to explain the interpretation of the system of axioms 2.1. Let's leave for the impersonal. For whom the impersonal are significantly two binary operations, and a binary setting. If the addition of that multiplication of pairs is reduced to the addition of that multiplication of natural numbers, then for natural numbers, the addition of that multiplication of pairs is commutative, associative, and the multiplication is distributively similar to addition. Let's reconsider, for example, the commutativity of adding pairs: +===+.
Let's take a look at the power of vіdnoshennia ~. Oskіlki a + b = b + a, then ~, then setting ~ reflexively. If ~, then a+b1=b+a1, then a1+b=b1+a, then ~. Otzhe, setting ~ symmetrically. Go ahead ~ i ~. Equalities a+b1=b+a1 and a1+b2=b1+a2 are also valid. Adding up the numbers of equalities, we take away a + b2 = b + a2, then ~. Otzhe, setting ~ also transitively і, otzhe, є equivalent. The class of equivalence that avenges a couple will be determined through. In this rank, the class of equivalence can be assigned to be your own couple and with it
(1)
Anonymity of all classes of equivalence is significant through. Our task is to show that the multiplier in case of a specified operation of folding and multiplication will be the interpretation of the system of axioms from 2.1. Operations on the faceless are significant by equalities:
(2)
(3)
If i is, then on the multiplier N the equality a+b(=b+a(, c+d(=a+c(,)) is valid, the equality (a+c)+(b(+d()=(b ) +d)+(a(+c(), which, by virtue of (1), is acceptable, which. Tse means that equivalence (2) signifies a unique operation of adding on a multiplier, so as not to fall due to the choice of pairs, which denotes additions) and uniqueness of the multiplication of classes In this way, equalities (2) and (3) are assigned to the multiplicity of binary operations of algebra.
Oskіlki adding and multiplying classes can be built up to folding and multiplying pairs, these operations are commutative, associative and multiplying classes are distributively easy folding. From the equalities, it is laid down that the class is a neutral element of the way of folding and the skin class is the proliferative one class. So, the multiplier is a circle, so the axioms of the group 1 from 2.1 are counted.
Let's take a look at the kіl'tsі podmnozhina. If a(b, then via (1) , and if a
On the impersonal, the binary is significant (following (; itself, following the class, following the class, de x (є natural number, coming after x. Class, coming after naturally signified through). the class follows the class i before it is only one.
Let's look at the image. It is obvious that the purpose of the fermentation is biactive and the mind f(0)= , f(x()==(=f(x)(.)). ;, () In other words, algebra (;,() is an interpretation of Peano's system of axioms. Deriving from isomorphic algebras, so you can respectfully consider that the impersonal N itself is submultiplied. ) \u003d a + c, a (c \u003d ac, which means that the addition of that multiplication in the kіltsi on the submultiple N zbіgayutsya zі folded and multiplications of natural numbers. Thus, the addition of the axioms of group 2 is installed.
Come on Z0 - be like a kіltse pіdkіltse, scho to avenge the impersonal N i. Respectfully, scho th, otzhe,. Ale oskіlki Z0 - a kіlce, then the difference between these classes can also lie with a kіltsu Z0. З equalities -= (= fit, sho (Z0 і, aka, Z0=. Non-superity of the system of axioms of item 2.1 is brought).

2.3. UNITY OF THE SYSTEM OF NUMBERS.

I have only one system of numbers for my intuitional mind. Tse means that the system of axioms, that signifies the numbers of numbers, can be categorical, so be the interpretation of the system of axioms isomorphic. Categorical and means that, up to isomorphism, there is only one system of numbers. Perekonayemosya, scho tse true so.
Let (Z1;+,(,N) and (Z2;(,(,N)) be two interpretations of the system of axioms of paragraph 2.1.) are filled with unruly and cream for whatever elements x and y from the ring Z1 fairness
(1)
. (2)
Respectfully, the shards N(Z1 and N(Z2, then
, a(b=a(b. (3)
Let x(Z1 і x=a-b, de a,b(N. Set the element x=a-b to the element u=a(b, de) , stars z (3) a(d=b(c і, otzhe, a(b=c(d)) tse means that our capacity to fall as a representative of the element x as a difference between two natural numbers and cim is shown in f: Z1® Z2, f(a-b)=a(b. Understanding that v(Z2 і v=c(d), then v=f(c-d).) the expression f is sur'jective.
If x = a-b, y = c-d, de a, b, c, d (N і f (x) = f (y), then a (b = c (d). Alethodі a (d = b (d, c) force (3) a+d=b+c, so a-b=c-d We have brought, that the equality of x=y is evident from the equality of f(x)=f(y), then the expression of f is in'active.
If a(N, then a=a-0 і f(a)=f(a-0)=a(0=a.) So, natural numbers are non-violent when f is exaggerated. Far, like x=a-b, y=c-d , de a, b, c, d (N, then x + y = (a + c) - i f (x + y) = (a + c) ((b + d) = (a (c) (( b (d)=(a(b)((c(d)=f(x)+f(y)). Fairness of equality (1) has been proved. Reversible equality (2). Scales f(xy)=(ac+ bd) )((ad+bc)=(a(c(b(d))((a(d(b(c))), and on the other side f(x)(f(y))=(a (b)((c (d)=(a(c(b(d))((a(d(b(c))). So, f(xy)=f(x)(f(y)) , which completes the proof of the categoricity of the system of axioms n.) 2.1.

2.4. VALUE AND POWER OF THE SYSTEM OF RATIONAL NUMBERS.

Anonymous Q rational numbers in the given intuitive rozumіnnі field, for some impersonal Z integer numbers є pіdkіltsem. When it is obvious that Q0 is the subfield of the field Q, to take revenge on the numbers, then Q0 = Q.
Appointment 1. A system of rational numbers is such a system of algebra (Q; +, (; Z), for which the mind is used:
1. algebraic system (Q; +, () є field;
2. ring Z integer numbers є pіdkіltsem field Q;
3. (minimum) if the subfield Q0 of the field Q avenges the subfield Z, then Q0=Q.
In short, the system of rational numbers is the minimum for the included field to avenge the number of numbers. You can give more reports on the axiomatic definition of the system of rational numbers.
Theorem. A skin rational number x can be represented as a private two integers, so
, de a, b (Z, b (0. (1)
The appearance is ambiguous, moreover, de a, b, c, d (Z, b (0, d (0)).
Bringing. Significantly in terms of Q0, there are impersonal rational numbers, as seen in (1). To finish the reconciliation, so Q0 = Q. Come on, de a, b, c, d (Z, b (0, d (0). Then, for the power of the field, it is possible: , and for c (0) Mean Q0 is closed on a non-zero number, i, then, є subfield of the field Q. So if the number a is representable in sight, then Z (Q0. Due to the fact that it is minimal and obvious, Q0 = Q. The proof of the other part of the obvious theorem.

2.5. FOUNDATION OF THE SYSTEM OF RATIONAL NUMBERS.

The system of rational numbers is designated as the minimum field to avenge the number of numbers. Zvichayno vinikaє pitanya - chi іsnuє such a field, that chi є є nesuperechlivuyu system of axioms, scho vyznaє rational numbers. To confirm non-superity, it is necessary to induce an interpretation of the system of axioms. At whom it is possible to spiral the basis of the system of whole numbers. Let's take a moment to interpret Z(Z\(0) as an immutable number. Two binary operations of algebra are significant on the multiplier
, (1)
(2)
that binary
(3)
Dotsіlnіst sama such a designation of operations and vіdnosinі ~ vyplyaє z that in іy іyіnpretatsії, as I'm going to be, a couple of words are more private.
It is easy to overthink that the operations (1) and (2) are commutative, associative and multiply distributively. All the powers of power are revered on the basis of the higher powers of adding that multiplication of numbers. Pereverimo, for example, the associativity of multiple pairs: .
Similarly, it is reconsidered that the difference is ~ є equivalent, and, hence, the impersonal Z(Z \ (0)) is divided into classes of equivalence. in pairs i by virtue of mind (3) we take:
. (4)
Our task is to designate the operation of folding that multiplier into multiplier, so that it was a field. Number of operations is significant by equalities:
, (5)
(6)
So, then ab1=ba1 and then cd1=dc1, then multiplying the values of equality, we take (ac)(b1d1)=(bd)(a1c1), and tse means that Tse will change us from the one that is equal (6) ) effectively signifies an unambiguous operation on an impersonal class, such as to lie in the choice of representatives of the skin class. Similarly, the uniqueness of the operation (5) is revised.
Since adding and multiplying classes can be reduced to folding and multiplying pairs, then the operations (5) and (6) are commutative, associative and distributive and can be added.
Of equalities, it is laid down that the class is a neutral element when supplemented and for the skin class, the protella yoma element is used. Similarly, it is obvious that the class is a neutral element of the plurality and for the skin class is the corrective class. Also, є the field of operations (5) and (6); first Umov at the appointed point 2.4 wins.
Let's look at the impersonal distance. Obviously, . The impersonality is closed by seeing that plural and, later, by the field’s pidkil’s. Correct, . Let's take a look at the vision, . The sur'jectivity of this manifestation is obvious. If f(x)=f(y), then x(1=y(1 or x=y. Meaning f and injectively. In addition, isomorphic kіltsya, it is possible to understand that Z kіlce is the field's subkіlcem, so that the mind is beaten 2 at the appointed clause 2.4. fields i, come on. Bo, ah, then. Ale oskіlki - the field, then private tsikh elements tezh lie on the field. Tim himself brought it up, what is it, then, tobto. The basis of the system of rational numbers has been completed.

2.6. UNITY OF THE SYSTEM OF RATIONAL NUMBERS.

If there is only one system of rational numbers in the modern intuitive sense, then the axiomatic theory of rational numbers, as it appears here, can be categorical. Categorical and means that, up to isomorphism, there is only one system of rational numbers. Let's show that it's true.
Let (Q1;+, (; Z) and (Q2; (, (; Z)) - be like two systems of rational numbers.
(1)
(2)
for any elements x and y from the field Q1.
Private elements a and b in field Q1 will be denoted by, and in field Q2 - by a:b. Since Z є pіdkіltse kozhny s polіv Q1 і Q2, then for any number of numbers a і b equivalence
, . (3)
Come on and de, . We assign to the given element x the element y=a:b from the field Q2. Even though equality is true in the field Q1, de, the theorem of item 2.4 in the ring Z the equality ab1=ba1 is victorious, otherwise, due to (3) equality, and similarly for the same theorem, the equality a:b=a1:b1 is valid in the field Q2. Tse means that by assigning to the element of the field Q1 the element y=a:b from the field Q2, we will display it, .
Any element from the field Q2 can be represented as a:b, de, otzhe, є the rank of the element from the field Q1. Otzhe, vodobrazhennya f є sur'єktivnym.
Yes, then in the field Q1 and the same. In this way, the fermentation f є bієktivnym and all tsіlі numbers become unruly. It is necessary to bring justice to equalities (1) and (2). Let's say a,b,c,d(Z, b(0, d(0). Then i, signs due to (3) f(x+y)=f(x)(f(y). Similarly, and stars.
Isomorphism of interpretations of (Q1; +, (; Z) and (Q2; (, (; Z)) advancing.

VІDPOVIDI, VKAZIVKI, RISHENNYA.

1.1.1. Solution. Nehai Umov's axiomy 4 is true (such power of natural numbers that ((0) i. Let's do it. So M satisfies the powers of axiom 4, shards ((0) (0(M i. Otzhe), M=N, so be natural) ).the number is powerful (. Back. It is acceptable that for whether or not there is power (from that ((0) i, next. Let M be a submultiplier of N, that 0(M i.) It will be shown that M = N. Let's introduce power (, respectfully. Todi ((0), oskіlki, i.) Otzhe, M=N.
1.1.2. Verdict: True assertion of the 1st and 4th axioms of Peano. Confirmation of the 2nd axioms of Hibne.
1.1.3. Verdict: truthful assertion of 2,3,4 axioms of Peano. Confirmation of the 1st axioms of Hibne.
1.1.4. True assertions 1, 2, 3 Peano's axioms. Statement of the 4th axioms of Hibne. Vkazіvka: to bring, scho satisfied with the possibilities of axiom 4, formulated in terms of the operation, ale.
1.1.5. Vkazіvka: to prove the truth of axiom 4, take a look at the submultiplier M z A, as it satisfies the minds: a) 1 ((M, b), and impersonal.
1.1.6. True assertion of Peano's 1,2,3 axioms. Statement of the 4th axioms of Peano Hibne.
1.6.1. a) Decision: Please let me know if it's 1am. Back. Come on am
1.6.2. a) Decision: Acceptable. Through M, all numbers are signifi- cantly impersonal, so that they cannot be powerful (. By assumption, M((. By virtue of Theorem 1, M has the least element n(0). Be it the number x
1.8.1. f) Tick p. e) and p. c): (a-c)+(c-b)=(a+c)-(c+b)=a-b, also, (a-b)-(c-b)=a-c.
h) Win power.
l) Tick p. b).
l) Tick the p. b) and p. h).
1.8.2. c) Maєmo, otzhe,. Father, .
d) Maemo. Father, .
and).
1.8.3. a) Like (i (different solution equal ax2+bx=c), then a(2+b(=a(2+b(.)) . Exactly ((. However (2=a(+b>a(, also, (>a.))).
c) Nehai (i (- different roots of equal i (>(. Todі 2((-()=(a(2+b))-(a(2+b))=a((-())(( (+( ) Later, a((+()=2), but (+(>2), later, a((+()>2), which is impossible).
1.8.4. a) x = 3; b) x = y = 2 c) x=y(y+2), y is a natural number; d) x = y = 2; e) x = 2, y = 1; f) Exactly up to permutations x=1, y=2, z=3. Solution: For example, let's say x(y(z. Then xyz=x+y+z(3z, so xy(3.) So xy=1, then x=y=1 і z=2+z, so) Impossible : if xy = 2, then x = 1, y = 2. In which case 2z = 3 + z, then z = 3. If xy = 3, then x = 1, y = 3. Then 3z = 4+z, so z=2, to superimpose the allowance y(z.
1.8.5. b) If x=a, y=b is a split, then ab+b=a, then. a>ab, which is impossible. d) If x=a, y=b is a split, then b
1.8.6. a) x=ky, de k,y - enough natural numbers and y(1. b) x - enough natural number, y=1. c) x is a fairly natural number y=1. d) There is no solution. e) x1 = 1; x2=2; x3=3. f) x>5.
1.8.7. a) If a = b, then 2ab = a2 + b2. Come on, for example, a

LITERATURE

1. Redkov M.I. Numerical systems. /Methodological recommendations to the course "Number systems". Part 1. - Omsk: OmDPІ, 1984. - 46s.
2. Ershova T.I. Numerical systems. / Methodical development for practical take. - Sverdlovsk: SDPI, 1981. - 68s.