Antiquity

In the ancient period, some ideas appeared that later led to integral calculus, but in that era these ideas were not developed in a strict, systematic way. Calculations of volumes and areas, which are one of the goals of the integral calculus, can be found in the Moscow Mathematical Papyrus from Egypt (c. 1820 BC), but the formulas are more instructions, without any indication of the method, and some are simply erroneous. In the era of Greek mathematics, Eudoxus (c. 408-355 BC) used the exhaustion method to calculate areas and volumes, which anticipates the concept of limit, and later this idea was further developed by Archimedes (c. 287-212 BC) by inventing heuristics that resemble the methods of integral calculus. The exhaustion method was later invented in China by Liu Hui in the 3rd century AD, which he used to calculate the area of ​​a circle. In the 5th AD, Zu Chongzhi developed a method for calculating the volume of a ball, which would later be called Cavalieri's principle.

Middle Ages

In the 14th century, the Indian mathematician Madhava Sangamagrama and the Kerala astronomical-mathematical school introduced many components of calculus such as Taylor series, infinite series approximation, integral convergence test, early forms of differentiation, term-by-term integration, iterative methods for solving non-linear equations, and determining what area under the curve is its integral. Some consider Yuktibhaza (Yuktibhāṣā) to be the first work on calculus.

Modern era

In Europe, the treatise of Bonaventure Cavalieri became a fundamental work, in which he argued that volumes and areas can be calculated as the sum of volumes and areas of an infinitely thin section. The ideas were similar to those set forth by Archimedes in Method, but this treatise by Archimedes was lost until the first half of the 20th century. Cavalieri's work was not recognized, as his methods could lead to erroneous results, and he created a dubious reputation for infinitesimal values.

The formal study of the infinitesimal calculus, which Cavalieri combined with the calculus of finite differences, was being carried out in Europe at about the same time. Pierre Fermat, claiming that he borrowed this from Diophantus, introduced the concept of "quasi-equality" (eng. adequality), which was equality up to an infinitesimal error. Major contributions were also made by John Wallis, Isaac Barrow and James Gregory. The last two around 1675 proved the second fundamental theorem of calculus.

Foundations

In mathematics, foundations refer to a strict definition of a subject, starting from precise axioms and definitions. At the initial stage of the development of calculus, the use of infinitesimal quantities was considered non-strict, it was subjected to harsh criticism by a number of authors, primarily Michel Rolle and Bishop Berkeley. Berkeley famously described infinitesimals as "ghosts of dead quantities" in his book The Analyst in 1734. The development of rigorous foundations for calculus occupied mathematicians for over a century after Newton and Leibniz, and is still somewhat of an active area of ​​research today.

Several mathematicians, including Maclaurin, tried to prove the validity of using infinitesimals, but this was only done 150 years later by the works of Cauchy and Weierstrass, who finally found means of how to avoid simple "little things" of infinitesimals, and the beginnings were laid differential and integral calculus. In Cauchy's writings we find a universal range of underlying approaches, including the definition of continuity in terms of infinitesimals and the (somewhat imprecise) prototype of the (ε, δ)-limit definition in the definition of differentiation. In his work, Weierstrass formalizes the concept of limit and eliminates infinitesimal quantities. After this work by Weierstrass, limits, and not infinitesimal quantities, became the general basis for calculus. Bernhard Riemann used these ideas to give a precise definition of the integral. Also, during this period, the ideas of calculus were generalized to Euclidean space and to the complex plane.

In modern mathematics, the foundations of calculus are included in the section of real analysis, which contains complete definitions and proofs of theorems in calculus. The scope of calculus research has become much wider. Henri Lebesgue developed the theory of set measures and used it to define integrals of all but the most exotic functions. Laurent Schwartz introduced generalized functions, which can be used to calculate the derivatives of any function at all.

The introduction of limits determined not the only rigorous approach to the basis of the calculus. An alternative would be, for example, Abraham Robinson's non-standard analysis. Robinson's approach, developed in the 1960s, uses technical tools from mathematical logic to extend the system of real numbers to infinitesimals and infinites, as was the original Newton-Leibniz concept. These numbers, called hyperreals, can be used in the usual rules of calculus, similar to what Leibniz did.

Importance

Although some ideas of calculus had previously been developed in Egypt, Greece, China, India, Iraq, Persia and Japan, the modern use of calculus began in Europe in the 17th century, when Isaac Newton and Gottfried Wilhelm Leibniz built on the work of previous mathematicians its basic principles. The development of calculus was based on the earlier concepts of instantaneous motion and area under a curve.

Differential calculus is used in calculations related to speed and acceleration, curve angle and optimization. Applications of integral calculus include calculations involving areas, volumes, arc lengths, centers of mass, work, and pressure. More complex applications include calculations of power series and Fourier series.

Calculus [ ] is also used to gain a more accurate understanding of the nature of space, time, and motion. For centuries, mathematicians and philosophers have struggled with the paradoxes associated with dividing by zero or finding the sum of an infinite series of numbers. These questions arise in the study of motion and the calculation of areas. The ancient Greek philosopher Zeno of Elea gave several famous examples of such paradoxes. Calculus provides tools for resolving these paradoxes, in particular limits and infinite series.

Limits and infinitesimals

Notes

1. morris kline, Mathematical thought from ancient to modern times, Vol. I
2. archimedes, method, in The Works of Archimedes ISBN 978-0-521-66160-7
3. Dun, Liu; Fan, Dainian; Cohen, Robert Sonne. A comparison of Archimdes" and Liu Hui"s studies of circles (English): journal. - Springer, 1966. - Vol. 130 . - P. 279 . - ISBN 0-792-33463-9., Chapter, p. 279
4. Zill, Dennis G.; Wright, Scott; Wright, Warren S. Calculus: Early Transcendentals (indefinite). - 3. - Jones & Bartlett Learning (English)Russian, 2009. - S. xxvii. - ISBN 0-763-75995-3., Extract of page 27
5. Indian mathematics
6. von Neumann, J., "The Mathematician", in Heywood, R. B., ed., The Works of the Mind, University of Chicago Press, 1947, pp. 180-196. Reprinted in Bródy, F., Vámos, T., eds., The Neumann Compedium, World Scientific Publishing Co. Pte. Ltd., 1995, ISBN 9810222017, pp. 618-626.
7. André Weil: Number theory. An approach through history. From Hammurapi to Legendre. Birkhauser Boston, Inc., Boston, MA, 1984, ISBN 0-8176-4565-9, p. 28.
8. Leibniz, Gottfried Wilhelm. The Early Mathematical Manuscripts of Leibniz. Cosimo, Inc., 2008. Page 228. Copy
9. Unlu, Elif Maria Gaetana Agnesi (indefinite) . Agnes Scott College (April 1995). Archived from the original on September 5, 2012.

Links

• Ron Larson, Bruce H. Edwards (2010). "Calculus", 9th ed., Brooks Cole Cengage Learning. ISBN 978-0-547-16702-2
• McQuarrie, Donald A. (2003). Mathematical Methods for Scientists and Engineers, University Science Books. ISBN 978-1-891389-24-5
• James Stewart (2008). Calculus: Early Transcendentals, 6th ed., Brooks Cole Cengage Learning.

Introduction

L. Euler is the most productive mathematician in history, the author of more than 800 works on mathematical analysis, differential geometry, number theory, approximate calculations, celestial mechanics, mathematical physics, optics, ballistics, shipbuilding, music theory, etc. Many of his works have had a significant influence on the development of science.

Euler spent almost half his life in Russia, where he energetically helped create Russian science. In 1726 he was invited to work in St. Petersburg. In 1731-1741 and starting from 1766 he was an academician of the St. Petersburg Academy of Sciences (in 1741-1766 he worked in Berlin, remaining an honorary member of the St. Petersburg Academy). He knew Russian well, he published part of his works (especially textbooks) in Russian. The first Russian academicians in mathematics (S.K. Kotelnikov) and in astronomy (S.Ya. Rumovsky) were students of Euler. Some of his descendants still live in Russia.

L. Euler made a very great contribution to the development of mathematical analysis.

The purpose of the abstract is to study the history of the development of mathematical analysis in the 18th century.

The concept of mathematical analysis. Historical outline

Mathematical analysis is a set of branches of mathematics devoted to the study of functions and their generalizations using the methods of differential and integral calculus. With such a general interpretation, analysis should also include functional analysis, together with the theory of the Lebesgue integral, complex analysis (TFKP), which studies functions defined on the complex plane, non-standard analysis, which studies infinitely small and infinitely large numbers, as well as the calculus of variations.

In the educational process, analysis includes

differential and integral calculus

The theory of series (functional, power and Fourier) and multidimensional integrals

vector analysis.

At the same time, elements of functional analysis and the theory of the Lebesgue integral are given optionally, and the TFKP, the calculus of variations, the theory of differential equations are taught in separate courses. The rigor of the exposition follows the patterns of the late 19th century and in particular uses naive set theory.

The forerunners of mathematical analysis were the ancient method of exhaustion and the method of indivisibles. All three directions, including analysis, have a common initial idea: decomposition into infinitesimal elements, the nature of which, however, seemed rather vague to the authors of the idea. The algebraic approach (infinitesimal calculus) begins to appear in Wallis, James Gregory and Barrow. The new calculus as a system was created in full measure by Newton, who, however, did not publish his discoveries for a long time. Newton I. Mathematical works. M, 1937.

The official date of birth of differential calculus can be considered May 1684, when Leibniz published the first article "A new method of maxima and minima ..." Leibniz // Acta Eroditorum, 1684. L.M.S., vol. V, p. 220-226. Rus. per.: Success Mat. Nauk, vol. 3, c. 1 (23), p. 166--173.. This article, in a concise and inaccessible form, outlined the principles of a new method called differential calculus.

At the end of the 17th century, a circle arose around Leibniz, the most prominent representatives of which were the Bernoulli brothers, Jacob and Johann, and Lopital. In 1696, using the lectures of I. Bernoulli, Lopital wrote the first textbook L'pital. Analysis of infinitesimals. M.-L.: GTTI, 1935., who presented a new method as applied to the theory of plane curves. He called it "Analysis of infinitesimals", thus giving one of the names to the new branch of mathematics. The presentation is based on the concept of variables, between which there is some connection, due to which a change in one entails a change in the other. In Lopital, this connection is given using plane curves: if M is a moving point of a plane curve, then its Cartesian coordinates x and y, called the diameter and ordinate of the curve, are variables, and a change in x entails a change in y. The concept of a function is absent: wishing to say that the dependence of the variables is given, Lopital says that "the nature of the curve is known." The concept of differential is introduced as follows:

“The infinitely small part by which a variable value is continuously increasing or decreasing is called its differential ... To designate the differential of a variable quantity, which itself is expressed by one letter, we will use the sign or symbol d. There. Ch.1, def.2 %D1%87%D0%B5%D1%81%D0%BA%D0%B8%D0%B9_%D0%B0%D0%BD%D0%B0%D0%BB%D0%B8%D0%B7 - cite_note -4#cite_note-4 ... The infinitesimal part by which the differential of a variable increases or decreases continuously is called ... the second differential. There. Ch.4, def.1.

These definitions are explained geometrically, with infinitesimal increments shown as finite in the figure. Consideration is based on two requirements (axioms). First:

It is required that two quantities differing from each other only by an infinitesimal amount can be taken indifferently one instead of the other. Lopital. Analysis of infinitesimals. M.-L.: GTTI, 1935. ch.1, requirement 1.

dxy = (x + dx)(y + dy) ? xy = xdy + ydx + dxdy = (x + dx)dy + ydx = xdy + ydx

and so on. differentiation rules. The second requirement is:

It is required that one can consider a curved line as a collection of an infinite set of infinitely small straight lines.

The continuation of each such line is called a tangent to the curve. There. Chapter 2. def. Investigating the tangent passing through the point M = (x, y), L'Hopital attaches great importance to the quantity

reaching extreme values ​​at the inflection points of the curve, while the ratio of dy to dx is not given any special significance.

Finding extremum points is noteworthy. If, with a continuous increase in the diameter x, the ordinate of y first increases and then decreases, then the differential dy is first positive compared to dx, and then negative.

But any continuously increasing or decreasing quantity cannot turn from positive to negative without passing through infinity or zero ... It follows that the differential of the largest and smallest magnitude must equal zero or infinity.

This formulation is probably not perfect, if we recall the first requirement: let, say, y = x2, then by virtue of the first requirement

2xdx + dx2 = 2xdx;

at zero, the right side is zero, but the left side is not. Apparently it should have been said that dy can be transformed in accordance with the first requirement so that at the maximum point dy = 0. In the examples, everything is self-evident, and only in the theory of inflection points does Lopital write that dy is equal to zero at the maximum point, being divided by dx Lopital. Analysis of infinitesimals. M.-L.: GTTI, 1935 § 46.

Further, with the help of differentials alone, conditions for an extremum are formulated and a large number of complex problems are considered, mainly related to differential geometry on the plane. At the end of the book, in ch. 10, what is now called L'Hopital's rule is stated, although in a not quite ordinary form. Let the value of the ordinate y of the curve be expressed as a fraction, the numerator and denominator of which vanish at x = a. Then the point of the curve with x = a has an ordinate y equal to the ratio of the numerator differential to the denominator differential taken at x = a.

According to L'Hopital's idea, what he wrote was the first part of "Analysis", while the second was supposed to contain integral calculus, that is, a way to find the connection of variables by the known connection of their differentials. Its first exposition was given by Johann Bernoulli in his Mathematical Lectures on the Integral Method by Bernulli, Johann. Die erste Integrelrechnunug. Leipzig-Berlin, 1914. Here a method for taking most elementary integrals is given and methods for solving many first-order differential equations are given.

History of calculus

The 18th century is often called the century of the scientific revolution, which determined the development of society up to the present day. This revolution was based on the remarkable mathematical discoveries made in the 17th century and founded in the next century. “There is not a single object in the material world and not a single thought in the realm of the spirit, which would not be affected by the influence of the scientific revolution of the 18th century. None of the elements of modern civilization could exist without the principles of mechanics, without analytical geometry and differential calculus. There is not a single branch of human activity that has not experienced the strong influence of the genius of Galileo, Descartes, Newton and Leibniz. These words of the French mathematician E. Borel (1871 - 1956), uttered by him in 1914, remain relevant in our time. Many great scientists contributed to the development of mathematical analysis: I. Kepler (1571 -1630), R. Descartes (1596 -1650), P. Fermat (1601 -1665), B. Pascal (1623 -1662), H. Huygens (1629 -1695), I. Barrow (1630 -1677), brothers J. Bernoulli (1654 -1705) and I. Bernoulli (1667 -1748) and others.

The innovation of these celebrities in understanding and describing the world around us:

movement, change and variability (life entered with its dynamics and development);

statistical casts and snapshots of her condition.

Mathematical discoveries of the 17th-17th centuries were defined using such concepts as variable and function, coordinates, graph, vector, derivative, integral, series and differential equation.

Pascal, Descartes and Leibniz were not so much mathematicians as philosophers. It is the universal human and philosophical meaning of their mathematical discoveries that is now the main value and is a necessary element of a common culture.

Both serious philosophy and serious mathematics cannot be understood without mastering the appropriate language. Newton, in a letter to Leibniz on solving differential equations, outlines his method as follows: 5accdae10effh 12i…rrrssssttuu.

The founders of modern science - Copernicus, Kepler, Galileo and Newton - approached the study of nature as mathematics. While studying motion, mathematicians developed such a fundamental concept as a function, or a relationship between variables, for example d = kt 2 , where d is the distance traveled by a freely falling body, and t is the number of seconds the body is in free fall. The concept of a function immediately became central in determining the speed at a given time and the acceleration of a moving body. The mathematical difficulty of this problem was that at any moment the body travels zero distance in zero time. Therefore, determining the value of the speed at a moment of time by dividing the path by the time, we will come to the mathematically meaningless expression 0/0.

The problem of determining and calculating the instantaneous rates of change of various quantities attracted the attention of almost all mathematicians of the 17th century, including Barrow, Fermat, Descartes, and Wallis. The disparate ideas and methods proposed by them were combined into a systematic, universally applicable formal method by Newton and G. Leibniz (1646-1716), the creators of the differential calculus. There was a heated debate between them over the priority in developing this calculus, with Newton accusing Leibniz of plagiarism. However, as studies of historians of science have shown, Leibniz created mathematical analysis independently of Newton. As a result of the conflict, the exchange of ideas between the mathematicians of continental Europe and England was interrupted for many years, to the detriment of the British side. English mathematicians continued to develop the ideas of analysis in a geometric direction, while the mathematicians of continental Europe, including I. Bernoulli (1667-1748), Euler and Lagrange, achieved incomparably greater success, following the algebraic, or analytical, approach.

The basis of all mathematical analysis is the concept of a limit. Speed ​​at a point in time is defined as the limit towards which the average speed tends d/t when the value t getting closer to zero. The differential calculus provides a convenient general method for finding the rate of change of a function f (x) for any value X. This speed is called the derivative. From the generality of the record f (x) it is clear that the concept of a derivative is applicable not only in tasks related to the need to find speed or acceleration, but also in relation to any functional dependence, for example, to some ratio from economic theory. One of the main applications of differential calculus is the so-called. tasks for maximum and minimum; Another important range of problems is finding the tangent to a given curve.

It turned out that with the help of the derivative, specially invented for working with problems of motion, it is also possible to find areas and volumes bounded by curves and surfaces, respectively. The methods of Euclidean geometry did not have the proper generality and did not allow obtaining the required quantitative results. Through the efforts of mathematicians of the 17th century. Numerous private methods were created that made it possible to find the areas of figures bounded by curves of one kind or another, and in some cases a connection was noted between these problems and problems of finding the rate of change of functions. But, as in the case of differential calculus, it was Newton and Leibniz who realized the generality of the method and thus laid the foundations of integral calculus.

The Newton-Leibniz method begins by replacing the curve limiting the area to be determined by a sequence of broken lines approaching it, similar to the method of exhaustion invented by the Greeks. Exact area is equal to the sum of areas limit n rectangles when n turns to infinity. Newton showed that this limit could be found by reversing the process of finding the rate of change of a function. The inverse operation of differentiation is called integration. The statement that summation can be carried out by reversing differentiation is called the fundamental theorem of mathematical analysis. Just as differentiation is applicable to a much wider class of problems than the search for velocities and accelerations, integration is applicable to any summation problem, for example, to physical problems involving the addition of forces.

The 19th century is the beginning of a new, fourth period in the history of mathematics - the period of modern mathematics.

We already know that one of the main directions of development of mathematics in the fourth period is the strengthening of the rigor of proofs in all mathematics, especially the restructuring of mathematical analysis on a logical basis. In the second half of the XVIII century. numerous attempts were made to restructure mathematical analysis: the introduction of the definition of the limit (D'Alembert and others), the definition of the derivative as the limit of the ratio (Euler and others), the results of Lagrange and Carnot, etc., but these works lacked a system, and sometimes they were unsuccessful. However, they prepared the ground on which perestroika in the 19th century. could be carried out. In the 19th century this direction of development of mathematical analysis became the leading one. They were taken up by O. Koshi, B. Bolzano, K. Weierstrass and others.

1. Augustin Louis Cauchy (1789−1857) graduated from the Polytechnic School and the Institute of Communications in Paris. Since 1816, a member of the Paris Academy and a professor at the Polytechnic School. In 1830−1838. during the years of the republic, he was in exile because of his monarchist convictions. Since 1848, Cauchy became a professor at the Sorbonne - the University of Paris. He published more than 800 papers on calculus, differential equations, the theory of functions of a complex variable, algebra, number theory, geometry, mechanics, optics, etc. His main areas of scientific interest were mathematical analysis and the theory of functions of a complex variable.

Cauchy published his lectures on analysis, delivered at the Polytechnic School, in three compositions: "Course of Analysis" (1821), "Summary of Lectures on Infinitesimal Calculus" (1823), "Lecture on Applications of Analysis to Geometry", 2 volumes (1826, 1828). in these books, for the first time, mathematical analysis is based on the theory of limits. they marked the beginning of a radical restructuring of mathematical analysis.

Cauchy gives the following definition of the limit of a variable: “If the values ​​successively assigned to the same variable approach a fixed value indefinitely, so that in the end they differ arbitrarily little from it, then the latter is called the limit of all others.” The essence of the matter is well expressed here, but the words "arbitrarily small" themselves need to be defined, and besides, the definition of the limit of a variable, and not the limit of a function, is formulated here. Further, the author proves various properties of limits.

Then Cauchy gives the following definition of the continuity of a function: a function is called continuous (at a point) if an infinitesimal increment of the argument generates an infinitesimal increment of the function, i.e., in modern language

Then he has various properties of continuous functions.

In the first book, he also considers the theory of series: he defines the sum of a number series as the limit of its partial sum, introduces a number of sufficient criteria for the convergence of number series, as well as power series and the region of their convergence - all this both in the real and in the complex region.

He expounds differential and integral calculus in the second book.

Cauchy defines the derivative of a function as the limit of the ratio of the increment of the function to the increment of the argument when the increment of the argument tends to zero, and the differential as the limit of the ratio From here it follows that. Next, we consider the usual formulas for derivatives; the author often uses Lagrange's mean value theorem.

In integral calculus, Cauchy for the first time puts forward a definite integral as a basic concept. He also introduces it for the first time as the limit of integral sums. Here we prove an important theorem on the integrability of a continuous function. The indefinite integral is defined for him as such a function of the argument that. In addition, expansions of functions in Taylor and Maclaurin series are considered here.

In the second half of the XIX century. a number of scientists: B. Riemann, G. Darboux and others found new conditions for the integrability of a function and even changed the very definition of a definite integral in such a way that it could be applied to the integration of some discontinuous functions.

In the theory of differential equations, Cauchy was mainly engaged in proving fundamentally important existence theorems: the existence of a solution to an ordinary differential equation, first of the first, and then of the th order; the existence of a solution for a linear system of partial differential equations.

In the theory of functions of a complex variable, Cauchy is the founder; many of his articles are devoted to it. In the XVIII century. Euler and d'Alembert only laid the foundation for this theory. In the university course on the theory of functions of a complex variable, we constantly meet the Cauchy name: the Cauchy − Riemann conditions for the existence of a derivative, the Cauchy integral, the Cauchy integral formula, etc.; many theorems on residues of a function are also due to Cauchy. B. Riemann, K. Weierstrass, P. Laurent and others also obtained very important results in this area.

Let us return to the basic concepts of mathematical analysis. In the second half of the century, it became clear that the Czech scientist Bernard Bolzano (1781-1848) had done a lot in the field of substantiating analysis before Cauchy and Weierstrasse. Before Cauchy, he gave definitions of the limit, continuity of a function and convergence of a number series, proved a criterion for the convergence of a number sequence, and also, long before Weierstrass had it, a theorem: if a number set is bounded from above (from below), then it has an exact upper ( exact lower) edge. He considered a number of properties of continuous functions; Recall that in the high school course of mathematical analysis there are Bolzano-Cauchy and Bolzano-Weierstrass theorems on functions that are continuous on a segment. Bolzano also investigated some issues of mathematical analysis, for example, he built the first example of a function that is continuous on a segment, but does not have a derivative at any point on the segment. During his lifetime, Bolzano was able to publish only five small works, so his results became known too late.

2. In mathematical analysis, the absence of a clear definition of the function was more and more clearly felt. A significant contribution to resolving the dispute about what is meant by a function was made by the French scientist Jean Fourier. He was engaged in the mathematical theory of heat conduction in a solid and in connection with this he used trigonometric series (Fourier series)

these series later became widely used in mathematical physics - a science that deals with mathematical methods for studying partial differential equations encountered in physics. Fourier proved that any continuous curve, regardless of what heterogeneous parts it is composed of, can be defined by a single analytical expression - a trigonometric series, and that this can also be done for some curves with discontinuities. The study of such series, carried out by Fourier, again raised the question of what is meant by a function. Can we assume that such a curve defines a function? (This is a renewal of the old 18th century controversy about the relationship between function and formula on a new level.)

In 1837, the German mathematician P. Dierechle for the first time gave a modern definition of a function: “there is a function of a variable (on the segment if, each value (on this segment) corresponds to a completely definite value, and it doesn’t matter how this correspondence is established - by an analytical formula, graph, table or even just in words". The addition is noteworthy: "it makes no difference how this correspondence is established." Direkhlet's definition gained general recognition rather quickly. True, it is now customary to call the correspondence itself a function.

3. The modern standard of rigor in mathematical analysis first appeared in the works of Weierstrass (1815−1897), worked for a long time as a mathematics teacher in gymnasiums, and in 1856 became a professor at the University of Berlin. The listeners of his lectures gradually published them in the form of separate books, thanks to which the content of Weierstrass's lectures became well known in Europe. It was Weierstrass who began to systematically use language in mathematical analysis. He gave the definition of the limit of a sequence, the definition of the limit of a function in the language (which is often incorrectly called the definition of Cauchy), strictly proved theorems on limits and the so-called Weierstrass theorem on the limit of a monotone sequence: an increasing (decreasing) sequence, bounded from above (from below), has a finite limit. He began to use the concepts of the exact upper and lower bounds of a numerical set, the concept of a limit point of a set, proved a theorem (which also has another author - Bolzano): a bounded numerical set has a limit point, considered some properties of continuous functions. Weierstrass devoted many works to the theory of functions of a complex variable, substantiating it with the help of power series. He also worked on the calculus of variations, differential geometry and linear algebra.

4. Let us dwell on the theory of infinite sets. Its creator was the German mathematician Kantor. Georg Kantor (18451918) worked for many years as a professor at the University of Halle. He published works on set theory starting from 1870. He proved the uncountability of the set of real numbers, thus establishing the existence of non-equivalent infinite sets, introduced the general concept of the cardinality of a set, and found out the principles for comparing powers. Kantor built a theory of transfinite, "improper" numbers, attributing the lowest, smallest transfinite number to the cardinality of a countable set (in particular, the set of natural numbers), the cardinality of the set of real numbers - a higher, greater transfinite number, etc.; this enabled him to construct an arithmetic for transfinite numbers similar to ordinary arithmetic for natural numbers. Cantor systematically used actual infinity, for example, the possibility of completely "exhausting" the natural series of numbers, while before him in mathematics of the 19th century. only potential infinity was used.

Cantor's set theory aroused objections from many mathematicians when it first appeared, but recognition gradually came when its great importance for substantiating topology and the theory of functions of a real variable became clear. But logical gaps remained in the theory itself, in particular, the paradoxes of set theory were discovered. Here is one of the most famous paradoxes. Denote by the set all such sets that are not elements of themselves. Does the inclusion also hold and is not an element, since by the condition only such sets are included as elements that are not elements of themselves; if, by condition, the inclusion-contradiction holds in both cases.

These paradoxes were connected with the internal inconsistency of some sets. It became clear that not all sets could be used in mathematics. The existence of paradoxes was overcome by the creation already at the beginning of the 20th century. axiomatic set theory (E. Zermelo, A. Frenkel, D. Neumann, etc.), which, in particular, answered the question: what sets can be used in mathematics? It turns out that one can use the empty set, the union of given sets, the set of all subsets of a given set, etc.