Born March 3, 1845 in St. Petersburg and grew up there until the age of 11. The father of the family was a member of the St. Petersburg Stock Exchange. When he fell ill, the family, counting on a milder climate, moved to Germany in 1856: first to Wiesbaden, and then to Frankfurt. In 1860, Georg graduated with honors from the real school in Darmstadt; teachers noted his exceptional ability in mathematics, in particular, in trigonometry. He continued his education at the Federal Polytechnic Institute in Zurich. A year later, after the death of his father, Georg received an inheritance and transferred to the University of Berlin. There he attends lectures by Kronecker, Weierstrass, Kummer. Kantor spent the summer of 1866 at the University of Göttingen, an important center of mathematical thought. In 1967 he received his doctorate in Berlin for his work on number theory "De aequationibus secundi gradus indeterminatis".

After a brief stint as a teacher at a girls' school in Berlin, Kantor took a position at the Martin Luther University of Galle, where he would spend his entire career. In 1872, he became an adjunct professor, at the same time, during his vacation, he struck up a friendship with Richard Dedekind. At 34, Kantor became a professor of mathematics. In 1879-84 he systematically expounds his doctrine of infinity; “introduced the concepts of a limit point, a derived set, built an example of a perfect set, developed one of the theories irrational numbers, formulated one of the axioms of continuity. Despite such a successful career, he dreams of a position at a more prestigious university, such as Berlin. However, dreams fail to come true: many contemporaries, including Kronecker, who is now regarded as one of the founders of constructive mathematics, are hostile to Cantor's set theory, since it asserts the existence of sets that satisfy certain properties - without providing specific examples of sets whose elements would actually satisfy these properties.

In 1984, Kantor experienced a bout of deep depression and moved away from mathematics for a while, shifting his interests towards philosophy. Then he returns to work. In 1897, he stops scientific work. Kantor died in Halle on January 6, 1918.

One of the urgent problems of the 19th century was the problem of the infinite division of segments and the existence of a point that belonged to all such contracting segments. This problem required the concept of a real number.

Cantor's construction of the theory of the real number was published in 1872, almost simultaneously with the theory of Weierstrass and Dedekind. In his construction, Cantor proceeds from the existence of rational numbers. Then he introduces fundamental Cauchy sequences and assigns a formal limit to them. Next, he considers splitting all sequences into equivalence classes. Sequences belong to the same class if and only if their difference tends to zero, that is. Further, formal limits are equal to each other if they have two such fundamental sequences that are equivalent to each other or. The order relation is defined as follows.

Thus, equivalence classes describe some real numbers. Let's call them real numbers of the first order. If we try to form a real number of a higher order by composing the fundamental Cauchy sequences, we will again obtain a set of real numbers of the first order. In other words, the set of real numbers is closed.

Kantor draws attention to the fact that in the definition of a real number there is actually an infinite set of rational numbers: "... a certain strictly defined set of the first cardinality of rational numbers always belongs to the definition of some irrational number."

Note that Cantor's construction can be generalized to other objects, which was done by Cantor and his followers, "the development of real number theories was a fairly essential prerequisite for the creation of set theory." For example, on the basis of his construction of the real number, Cantor subsequently developed his theory of transfinite numbers.

In addition, Cantor introduced the concept of cardinality of sets and proved the non-equivalence of irrational and rational numbers.

Georg Ferdinand Ludwig Philipp Kantor was born on March 4, 1845 in St. Petersburg. His parents were Georg-Voldemar Kantor and Maria Anna Boym. Kantor was raised as a staunch Protestant, and his love for art was passed on to him from his parents. It is believed that he was an outstanding violinist. His father was German and his mother was a Russian who attended the Roman Catholic Church. WITH early years Kantor had a private teacher and also attended a school in St. Petersburg. In 1856, when Kantor was eleven years old, his family moved to Germany, which Kantor never fell in love with.

Kantor's father's health began to deteriorate, causing the family to move again, this time to Frankfurt, due to the warmer climate. In Frankfurt, Kantor studied at the gymnasium, from which he graduated with honors in 1960. His teachers noted that he was good at mathematics, especially trigonometry. After high school in 1962, Kantor entered federal university Zurich, where he studied mathematics. Having received the approval of his parents, he studied there for a couple of years, until the death of his father put an end to his studies. After his father's death, Kantor moved to the University of Berlin, where he became friends with Hermann Schwartz and attended lectures by Kronecker, Weierstrass, and Kummer. During the summer he also studied at the University of Göttingen, and in 1867 completed his first dissertation on numbers with the title "De aequationibus secondi gradus indeterminatis".

In the same year he received his doctorate in mathematics.

Career

Early in his career, Kantor was an active member of mathematical unions and communities. He became president of one of the communities in 1865 and 1868. He also took part in the Schellbach conference on mathematics. In 1869 he was appointed professor at the University of Halle. He continued to work on various dissertations on number theory and analysis. At the same time, Kantor decided to continue studying trigonometry and began to reflect on the uniqueness of the geometric representation of the functions of the trigonometric series, which were presented to him by his senior colleague, Heine.

By 1870, Cantor was up to the task, proving the uniqueness of the geometric image, much to Heine's amazement. In 1873, he proved that the rational numbers are countable and can be brought into line with the natural numbers. By the end of 1873, Cantor had proved that both real and relative numbers are also countable. He was promoted to professor extraordinary in 1872, and in 1879 he was appointed professor of the highest category. He was grateful for the appointment, but still wanted a position at a more prestigious university.

In 1882, Kantor began corresponding with Gösta Mittag-Leffler, and soon began publishing his work in Leffler's journal, Acta Mathematica. Kronecker, a contemporary of Kant, constantly ridiculed and oppressed Cantor's theories.

Kantor continued to publish his work, but in 1884 he suffered a nervous breakdown from which he soon recovered and decided to teach philosophy. Soon he began to study the literature of the Elizabethan period.

In 1890 he founded the German Mathematical Society, in which he first published drawings of the diagonal section, thus establishing some relationship with Kronecker. But, despite the fact that the scientists began to communicate, they never reconciled, because of which the tension in their relationship was present until the end of Kantor's life.

Personal life

In 1874 Kantor married Wally Guttman; the couple had six children. It is believed that Kantor, despite the status of a famous mathematician, could not support his family. In the presence of free time, he played the violin and immersed himself in art and literature. He was awarded the Sylvester medal for his work in mathematics. In 1913, Kantor retired because he was mentally unstable, suffered from constant mental disorders, and in the end he ended up in a health resort, where he stayed until his death.

Death and legacy

Georg Kantor died on January 6, 1918 in Halle, after a long mental breakdown. Numerous publications have been published about Cantor, one of which was a publication in the book "Creators of Mathematics" and an article in the "History of Mathematics". He founded the German Mathematical Society, and most of his scientific works is still in use.

Main works

"Infinite sets"
"Uncountable sets"
"cantor set"
"Cardinals and Ordinals"
"The Continuum Hypothesis"
"Number theory and function theories"
"Infinitesimals"
Convergent Series
"transcendental numbers"
Diagonal Argument
"Cantor-Bernstein-Schroeder theorem"
"Continuum Hypothesis"

Publications

"On a Property of the Collection of All Real Algebraic Numbers"
"Foundations of a General Theory of Aggregates"
"Mathematische Annalen"
"Grundlagen einer allgemeinen Mannigfaltigkeitslehre"
"De aequationibus secondi gradus indeterminatis"

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Georg Cantor (photo is given later in the article) is a German mathematician who created set theory and introduced the concept of transfinite numbers, infinitely large, but different from each other. He also defined ordinal and cardinal numbers and created their arithmetic.

Georg Kantor: a short biography

Born in St. Petersburg on 03/03/1845. His father was a Dane of the Protestant faith, Georg-Valdemar Kantor, who was engaged in trade, including on the stock exchange. His mother Maria Bem was a Catholic and came from a family of prominent musicians. When Georg's father fell ill in 1856, the family moved first to Wiesbaden and then to Frankfurt in search of a milder climate. The boy's mathematical talents showed up even before his 15th birthday while studying at private schools and gymnasiums in Darmstadt and Wiesbaden. In the end, Georg Cantor convinced his father of his firm intention to become a mathematician, not an engineer.

After a short study at the University of Zurich, in 1863 Kantor transferred to the University of Berlin to study physics, philosophy and mathematics. There he was taught:

• Karl Theodor Weierstrass, whose specialization in analysis was probably Georg's greatest influence;
• Ernst Eduard Kummer, who taught higher arithmetic;
• Leopold Kronecker, number theorist who later opposed Cantor.

After spending one semester at the University of Göttingen in 1866, the following year Georg wrote a doctoral dissertation entitled "In mathematics the art of asking questions is more valuable than solving problems", concerning a problem that Carl Friedrich Gauss had left unsolved in his Disquisitiones Arithmeticae (1801) . After briefly teaching at the Berlin School for Girls, Kantor began working at the University of Halle, where he remained until the end of his life, first as a teacher, from 1872 as an assistant professor, and from 1879 as a professor.

Research

At the beginning of a series of 10 papers from 1869 to 1873, Georg Cantor considered number theory. The work reflected his passion for the subject, his studies of Gauss and the influence of Kronecker. At the suggestion of Heinrich Eduard Heine, Cantor's colleague in Halle, who recognized his mathematical talent, he turned to the theory of trigonometric series, in which he expanded the concept of real numbers.

Based on the work on the function of a complex variable by the German mathematician Bernhard Riemann in 1854, in 1870 Kantor showed that such a function can be represented in only one way - by trigonometric series. The consideration of a set of numbers (points) that would not contradict such a representation led him, firstly, in 1872 to a definition in terms of rational numbers (fractions of integers) and then to the beginning of work on his life's work, set theory and the concept transfinite numbers.

set theory

Georg Cantor, whose set theory originated in correspondence with the mathematician of the Technical Institute of Braunschweig Richard Dedekind, was friends with him since childhood. They came to the conclusion that sets, whether finite or infinite, are collections of elements (for example, numbers, (0, ±1, ±2 . . .)) that have a certain property while retaining their individuality. But when Georg Cantor used a one-to-one correspondence (for example, (A, B, C) to (1, 2, 3)) to study their characteristics, he quickly realized that they differ in the degree of their membership, even if they were infinite sets , i.e. sets, a part or subset of which includes as many objects as it itself. His method soon gave amazing results.

In 1873, Georg Cantor (mathematician) showed that rational numbers, although infinite, are countable because they can be put in one-to-one correspondence with natural numbers (i.e. 1, 2, 3, etc.). He showed that the set of real numbers, consisting of irrational and rational ones, is infinite and uncountable. More paradoxically, Cantor proved that the set of all algebraic numbers contains as many elements as the set of all integers, and that the non-algebraic transcendental numbers, which are a subset of the irrational numbers, are uncountable and, therefore, their number is greater than the number of integers, and must be considered as infinite.

Opponents and supporters

But Kantor's paper, in which he first put forward these results, was not published in the journal Krell, since one of the reviewers, Kronecker, was categorically against it. But after the intervention of Dedekind, it was published in 1874 under the title On the Characteristic Properties of All Real Algebraic Numbers.

Science and personal life

In the same year, during his honeymoon with his wife Valli Gutman, Kantor met Dedekind, who spoke favorably of his new theory. George's salary was small, but with the money of his father, who died in 1863, he built a house for his wife and five children. Many of his papers were published in Sweden in the new journal Acta Mathematica, edited and founded by Gesta Mittag-Leffler, who was among the first to recognize the talent of the German mathematician.

Connection with metaphysics

Cantor's theory became an entirely new subject of study concerning the mathematics of the infinite (eg series 1, 2, 3, etc., and more complex sets), which depended heavily on one-to-one correspondence. The development by Cantor of new methods for posing questions concerning continuity and infinity gave his research an ambiguous character.

When he argued that infinite numbers really exist, he turned to ancient and medieval philosophy regarding actual and potential infinity, as well as to the early religious education that his parents gave him. In 1883, in his book Foundations of General Set Theory, Cantor combined his concept with Plato's metaphysics.

Kronecker, who claimed that only integers “exist” (“God created the integers, the rest is the work of man”), for many years ardently rejected his reasoning and prevented his appointment at the University of Berlin.

transfinite numbers

In 1895-97. Georg Cantor fully formed his notion of continuity and infinity, including infinite ordinal and cardinal numbers, in his most famous work, published as Contributions to the Establishment of the Theory of Transfinite Numbers (1915). This essay contains his concept, to which he was led by demonstrating that an infinite set can be put in a one-to-one correspondence with one of its subsets.

By the least transfinite cardinal number, he meant the cardinality of any set that can be put in a one-to-one correspondence with the natural numbers. Cantor called it aleph-null. Large transfinite sets are denoted, etc. He further developed the arithmetic of transfinite numbers, which was analogous to finite arithmetic. Thus, he enriched the concept of infinity.

The opposition he encountered, and the time it took for his ideas to be fully accepted, are explained by the difficulty of re-evaluating the ancient question of what a number is. Cantor showed that the set of points on a line has a higher cardinality than aleph-zero. This led to the well-known problem of the continuum hypothesis - there are no cardinal numbers between aleph-zero and the power of points on the line. This problem in the first and second half of the 20th century aroused great interest and was studied by many mathematicians, including Kurt Gödel and Paul Cohen.

Depression

The biography of Georg Kantor since 1884 was overshadowed by his mental illness, but he continued to work actively. In 1897 he helped hold the first international mathematical congress in Zurich. Partly because he was opposed by Kronecker, he often sympathized with young aspiring mathematicians and sought to find a way to save them from the harassment of teachers who felt threatened by new ideas.

Confession

At the turn of the century, his work was fully recognized as the basis for function theory, analysis, and topology. In addition, the books of Cantor Georg served as an impetus for the further development of the intuitionist and formalist schools of the logical foundations of mathematics. This significantly changed the teaching system and is often associated with the "new mathematics".

In 1911, Kantor was among those invited to the celebration of the 500th anniversary of the University of St. Andrews in Scotland. He went there in the hope of meeting with whom, in his recently published Principia Mathematica, he repeatedly referred to the German mathematician, but this did not happen. The university awarded Kantor an honorary degree, but due to illness, he was unable to accept the award in person.

Kantor retired in 1913, lived in poverty and went hungry during the First World War. Celebrations in honor of his 70th birthday in 1915 were canceled due to the war, but a small ceremony took place at his home. He died on 01/06/1918 in Halle, in a psychiatric hospital, where he spent last years own life.

Georg Kantor: biography. Family

On August 9, 1874, the German mathematician married Wally Gutmann. The couple had 4 sons and 2 daughters. The last child was born in 1886 in a new house purchased by Kantor. His father's inheritance helped him support his family. Kantor's state of health was strongly affected by the death of his youngest son in 1899 - since then depression has not left him.

Cantor's theory of transfinite numbers was initially perceived as so illogical, paradoxical, and even shocking that it ran into sharp criticism from contemporary mathematicians, in particular, Leopold Kronecker and Henri Poincaré; later - Hermann Weyl and Leutzen Brouwer, and Ludwig Wittgenstein raised objections of a philosophical nature (see Disputes about Cantor's theory). Some Christian theologians (especially representatives of neo-Thomism) saw in Cantor's work a challenge to the uniqueness of the absolute infinity of the nature of God, once equating the theory of transfinite numbers with pantheism. Criticism of his works was sometimes very aggressive: for example, Poincaré called his ideas a "serious disease" affecting mathematical science; and in Kronecker's public statements and personal attacks on Kantor, such epithets as "scientific charlatan", "apostate" and "corrupter of the youth" sometimes flashed. Decades after Cantor's death, Wittgenstein noted bitterly that mathematics was "trodden up and down by the destructive idioms of set theory", which he dismissed as "buffoonery", "ludicrous" and "erroneous". Periodic bouts of depression from 1884 until the end of Kantor's days were for some time blamed on his contemporaries for taking an overly aggressive stance, but it is now believed that these bouts may have been a manifestation of bipolar disorder.

Sharp criticism was opposed by worldwide fame and approval. In 1904, the Royal Society of London awarded Cantor the Sylvester Medal, the highest honor it could bestow. Cantor himself believed that the theory of transfinite numbers was communicated to him from above. At one time, defending it from criticism, David Hilbert boldly declared: "No one will drive us out of the paradise that Kantor founded."

Biography

Young years and studies

Kantor was born in 1845 in the Western colony of merchants in St. Petersburg and grew up there until the age of 11. Georg was the eldest of six children. He played the violin virtuoso, having inherited significant artistic and musical talents from his parents. The father of the family was a member of the St. Petersburg Stock Exchange. When he fell ill, the family, counting on a milder climate, moved to Germany in 1856: first to Wiesbaden, and then to Frankfurt. In 1860, Georg graduated with honors from the real school in Darmstadt; teachers noted his exceptional ability in mathematics, in particular, in trigonometry. In 1862, the future famous scientist entered the Federal Polytechnic Institute in Zurich (now the Swiss Higher Technical School of Zurich). A year later his father died; having received a solid inheritance, Georg is transferred to the Humboldt University of Berlin, where he begins to attend lectures by such famous scientists as Leopold Kronecker, Karl Weierstrass and Ernst Kummer. He spent the summer of 1866 at the University of Göttingen, then, and still is, a very important center of mathematical thought. In 1867, the University of Berlin awarded him a Ph.D. for his work on number theory "De aequationibus secundi gradus indeterminatis".

Scientist and researcher

After a brief stint as a teacher at a girls' school in Berlin, Kantor took a position at the Martin Luther University of Galle, where he would spend his entire career. He received the habilitation required for teaching for his dissertation on number theory.

In 1874 Kantor married Vally Guttmann. They had 6 children, the last of which was born in 1886. Despite a modest academic salary, Kantor was able to provide the family with a comfortable living thanks to the inheritance received from his father. During his honeymoon in the Harz mountains, Kantor spent a lot of time in mathematical conversations with Richard Dedekind, with whom he struck up a friendship two years earlier during a holiday in Switzerland.

Kantor received the title of Visiting Professor in 1872 and became Full Professor in 1879. Receiving this title at 34 was a great achievement, but Kantor dreamed of a position at a more prestigious university, such as Berlin - at that time the leading university in Germany. However, his theories are seriously criticized, and dreams fail to materialize. Kronecker, head of the Department of Mathematics at the University of Berlin, became increasingly unenthusiastic about the prospect of having a colleague like Kantor, perceiving him as a "corrupter of the youth" who filled the heads of the younger generation of mathematicians with his ideas. Moreover, Kronecker, being a prominent figure in the mathematical community and a former teacher of Cantor, fundamentally disagreed with the content of the theories of the latter. Kronecker, who is now regarded as one of the founders of constructive mathematics, disliked Cantor's set theory because it asserted the existence of sets satisfying certain properties without providing specific examples of sets whose elements would actually satisfy those properties. Kantor realized that Kronecker's position would not even allow him to leave the University of Gaul.

In 1881 Eduard Heine, Cantor's colleague, died, leaving behind a vacant post. The university management accepted Kantor's offer to invite Richard Dedekind, Heinrich Weber or Franz Mertenz (in that order) to this post, but they all refused. As a result, Friedrich Wangerin took the post, but he was never a friend of Cantor.

In 1882, scientific correspondence with Dedekind broke off, probably as a result of the latter's refusal from his post in Halle. At the same time, Kantor established another important correspondence, with Gösta Mittag-Leffler, who lived in Sweden, and soon began publishing in his journal Acta mathematica. However, in 1885 Mittag-Leffler became alarmed at the philosophical overtones and new terminology in an article sent to him by Kantor for publication. He asked Cantor to withdraw his article while it was still being proofread, writing that the article was "a hundred years ahead of its time." Kantor agreed, but noted in correspondence with another person:

Following this, Kantor abruptly ended his relationship and correspondence with Mittag-Leffler, showing a tendency to take well-intentioned criticism as a deep personal insult.

Kantor experienced his first known bout of depression in 1884. Criticism of his work weighed heavily on his mind: each of the 52 letters he wrote to Mattag-Leffler in 1884 was attacked by Kronecker. An excerpt from one letter shows the extent of the damage done to Kantor's sense of self-confidence:

This emotional crisis forced him to shift his interest from mathematics to philosophy and begin lecturing on it. In addition, Kantor began to intensively study English literature of the Elizabethan era; he tried to prove that those plays attributed to Shakespeare were actually written by Francis Bacon (see Shakespeare's authorship question); the results of this work were eventually published in two prospectuses in 1896 and 1897.

Shortly thereafter, Cantor recovered, and immediately made several important additions to his theory, in particular his famous diagonal argument and theorem. However, he will never be able to reach the high level that was in his works of 1874-1884. In the end, he turned to Kronecker with an offer of peace, which he favorably accepted. However, the philosophical differences and difficulties that separated them remained. For some time, it was believed that Kantor's periodic bouts of depression were associated with Kronecker's harsh rejection of his work. But although his depression had a great influence on Kantor's mathematical anxieties and his problems with some people, it is unlikely that all this was its cause. On the contrary, his posthumous diagnosis of manic-depressive psychosis was approved as the main reason for his unpredictable mood.

In 1890 Kantor contributed to the organization of the German Mathematical Society (Deutsche Mathematiker-Vereinigung) and was chairman of its first meeting in Halle in 1891; at that time his reputation was strong enough, despite Kronecker's opposition, to be chosen as the first president of this society. Closing his eyes to his dislike for Kronecker, Kantor invited him to make a report, but Kronecker could not do this due to the death of his wife.

Objects named after Kantor

• Cantor set- continuum set of zero measure on the segment;
• Cantor function (Cantor ladder);
• Cantor's numbering function - displaying the Cartesian power of a set natural numbers into itself;
• Cantor's theorem (see also Cantor's theorem (disambiguation)) that the cardinality of the set of all subsets of a given set is strictly greater than the cardinality of the set itself;
• The Cantor-Bernstein theorem on the equivalence of sets A and B under the condition that A is equivalent to a subset of B and that B is equivalent to a subset of A;
• The Cantor-Heine theorem on the uniform continuity of a continuous function on a compact;
• Cantor-Bendixon theorem
• The Kantor Medal is a mathematical award given by the German Mathematical Society;
• as well as other mathematical objects.

Compositions

• Cantor G. Gesammelte Abhandlungen und philosophischen Inhalts / Hrsg. von E. Zermelo. B., 1932.

As a manuscript.

Popov N.A., Popov A.N.

NAIVE SET THEORY
AND SOLUTION TO THE CANTOR PARADOX

CONTENTS
page

Preface. . . . . . . . . 5

Chapter I. Introduction. Basic information from the theory of sets. . 8

Chapter II. Is Cantor's naive set theory contradictory?
Solution to Cantor's paradox. . . .19

Chapter III. Axiomatics of Cantor's set theory. . . . . . . .60

Chapter IV. Z-theorem and its two proofs. . . . . . . . . . .72

Chapter V. Difference problem (generalization of the Z-theorem). . . . . . . . .90

Chapter VI. About logical paradoxes. . . . . . . . . . . . . . .87

FOREWORD

To bring the general logical foundations of modern mathematics into such a state that they can be expounded at school to adolescents of 14-15 years old.
Kolmogorov A.N. Simplicity - complex // Izvestia. 1962. 31 Dec.

Cantor's intuitive so-called "naive" set theory is considered a controversial theory among mathematicians. To substantiate such an assessment, one usually points to Cantor's too vague, "insufficiently mathematical" definition of the concept of a set. Some will recall the paradoxes of naive theory - Russell's paradox and Cantor's paradox. But what these paradoxes consist of, few can explain.
We know of no other grounds for considering the "naive" theory contradictory. All this was the motive for the attempt, presented below, to find out whether it is possible to justify the construction of a naive theory of sets, proceeding only from Cantor's definition of the concept of a set and the principle of volume.
The initial impetus for this work was the strange circumstance that, simultaneously with the Cantor paradox mentioned in some textbooks (for example, ) in the same textbooks, a clearly erroneous, as it seemed to us, proof of the famous Cantor theorem is presented in the same textbooks. But, unfortunately, as it turned out a little later, the obviousness of the logical error of the proof was almost obvious to no one. But the evidence was different: for more than 100 years, no serious mathematician has challenged the proof of Cantor's theorem. So it can't be! The attitude towards those who dispute Cantor's theorem (and these are rare isolated cases) has developed approximately the same as towards the inventors of a perpetual motion machine.
As the practice of discussing this problem has shown, all reasoning thought out and set out on paper is rather difficult to perceive and requires considerable mental effort and, most importantly, time. Therefore, there was no serious criticism of our work. The topic of discussion very rarely met with a serious and conscientious attitude. Not a single opponent (and their number is calculated in units) could not present a single convincing objection to the above considerations.
However, the job is done. Cantor's paradox has been explored and resolved. The results of his research are as follows.
In the main Kantor was right. We managed to prove his famous theorem and find out from which axioms it follows. And all the contradictory examples known to us, examples of sets that contradict his theorem, including the set of all sets, turned out to be untenable. In the sense that these sets turned out to be internally contradictory formations: one of the axioms that define the concept of a set, namely, the axiom of certainty formulated in Chapter III, does not hold for them. However, the generally accepted, standard proof of Cantor's theorem, presented in all textbooks, is erroneous. The fallacy of the proof consists in the fact that a contradiction that follows only from an inconsistent definition of a set is presented in the standard proof by contradiction as evidence that the assumption to the contrary is false.
A small digression about the "crisis in the foundations" of set theory should give the reader an idea of ​​the content of the work and its relation to the current state of set theory.
In modern literature on the foundations of mathematics, in particular, monographs such as "Introduction to Metamathematics", Kleene, "Fundamentals of Set Theory", Frenkel A.A., Bar-Hillel, , the state of this field of knowledge is characterized as still unresolved crisis. An impetus to identify far-reaching differences of opinion and points of view about the most basic mathematical concepts was the discovery at the turn of the 19th and 20th centuries of the so-called antinomies (paradoxes) at the very foundations of the recently emerged theory of sets. In an effort to rid the theory of contradictions that seemed unacceptable and as a result of revising its foundations, so-called axiomatic set theories arose, free from paradoxes known by that time. This success was achieved at the cost of reducing the scope of the main concept of the theory - the concept of a set. The reason for the antinomies was seen in the consideration of "too vast" (???) sets. Some intuitive collections, such as the set of all sets or the set of all cardinalities, have been declared classes rather than sets. Cantor's theory of sets was actually abandoned, declaring it contradictory.
From our point of view, based on the results of the study and the aforementioned paradoxes of set theory, and the so-called diagonal proofs, the correct solution to the problem of paradoxes has not been reached. Paradoxes were eliminated from the theory, but not resolved, that is, the causes of the contradictions were not fully disclosed. As a result, in the now universally recognized set theory (ZF), and even in some theorems of mathematical logic (see Section V.7 of Chapter V on the proof of A. Tarski's theorem), erroneous methods of proof are used. We argue that all the proofs of Cantor's theorem in textbooks on set theory, mathematical logic, and the theory of functions of a real variable (for example, see) are erroneous.
A careful study of set-theoretic paradoxes would reveal the cause of the contradictions in them. These, as shown in Sections II.4 - II.11, are nothing more than conflicting definitions of sets. With a clear understanding of this reason, there would be no talk of a crisis in the foundations of mathematics.
The general work plan is as follows.
Chapter I gives basic information on set theory. The chapter is addressed to readers who are not familiar with set theory, or who wish to refresh their knowledge in this area. Readers with even a superficial knowledge of set theory can skip this chapter (except for Section I.7) without compromising their understanding of the material that follows.
The content of Chapter II is a presentation of the study of the problem of Cantor's paradox by careful thought through the problem, an study based solely on the logic of common sense. This research continued intermittently for many years. The main result of the work is that Cantor's paradox has been investigated and resolved.
In Chapter III an attempt is made to axiomatically construct Cantor's "naive" set theory.
Chapters IV and V present the so-called Z-theorem, which generalizes the family of diagonal paradoxes and explains set-theoretic paradoxes from a unified standpoint. Chapter VI is devoted to the analysis of several of the most famous paradoxes.
To understand the work, no special knowledge is required, even a superficial acquaintance with the basic concepts of set theory (the concepts of "set", "function", "domain of definition" and the like) and some habit of perceiving mathematical reasoning is enough, so the work is quite accessible to students of physics - mathematical faculties and just a person with a university, higher technical or higher teacher education. The authors of the work set themselves the task of presenting the results of their research into the paradoxes of set theory in a language understandable even to a high school student. To what extent they succeeded in solving this problem, let the reader judge.
We thank
N.A. Dmitrieva
for valuable discussions on the topic of the work, as well as VNIIEF staff
M.I.Kaplunova,
G.S. Klinkov, I.V. Kuzmitsky,
V.S. Lebedeva,
B.V. Pevnitsky, V.I. Filatov, V.A. Shcherbakov and I.T. Shmorin, who read fragments of our work in manuscripts and discussed it.
Lists of sources used in this edition are given separately for each chapter.

CHAPTER I
INTRODUCTION BASIC INFORMATION FROM SET THEORY

I.1. On the concept of a set. . . . . . . . . . . . . . . . . . 8
I.2. Methods for describing sets. . . . . . . . . . . . . . . . 10
I.3. Set-theoretic operations. . . . . . . . . . . . . eleven
I.4. Quantitative comparison of sets. . . . . . . . . . . . . eleven
I.5. The concept of a subset. . . . . . . . . . . . . . . . . . 13
I.6. Cantor's theorem (formulation). . . . . . . . . . . . . . 14
I.7. Underdetermined sets. . . . . . . . . . . . . . . . 14
I.8. On uncountable sets. . . . . . . . . . . . . . . . . . 16
List of used sources. . . . . . . . . . . . . . 19

This chapter aims to provide the basics of set theory to the reader who is unfamiliar with this theory, or who wishes to refresh his knowledge in this area. Readers with knowledge of set theory, at least in the scope of the course for the physics and mathematics faculties of pedagogical universities, can skip this chapter (except for Section I.7) without prejudice to understanding the subsequent material.

I.1. On the concept of a set.

The term "set" in everyday life is used to refer to large quantities of some objects that can be counted. We say: lots of mistakes, lots of pictures, lots of people.
The everyday concept of “set” is rather vague, it is impossible to indicate the number, for example, of cows, which should be called a set of cows. The so-called “heap paradox” is known on this topic: starting from what amount of grains form a heap of grains?
In order to be able to build a theory, the concepts of this theory must be quite clear. To construct a theory of sets, it is necessary to have a clear concept of a set. The ingenious founder of set theory, Georg Cantor (1845 - 1918), gave his famous definition of the concept of a set. Here it is.
“By “set” we mean the union into one whole M of certain completely distinguishable objects m of our perception or thinking (which will be called “elements” of the set M)”.
Whether this definition can be considered sufficiently clear, we will discuss a little later, and now we will note some of its features.
To begin with, we note that nothing is said about the number of combined items. This means that already two elements form a set. This also means that a set remains a set if one element is removed from it. Guided by this principle, we come to the concept of a unit set, which is obtained if one of them is removed from the set of two elements. And here we find that Cantor's definition of a set is not complete: in the case of a single set, we do not see any unification.
Further more. Removing its only element from the identity set, we arrive at the concept of an empty set. Not everyone can digest this abstraction. At the first acquaintance with the concept of a set, not everyone agrees to recognize the empty set as a set. In this regard, the author of the monograph "Introduction to Metamathematics" S. Kleene seemed to find Cantor's definition of the concept of a set insufficiently complete, and he supplemented it as follows:
“Sets are joined by an empty set that has no elements, and unit sets, each of which has one single element.”
Indeed, no "combination into one whole" in an empty and single set is visible at first glance. However, as V.A. Shcherbakov noted, if the “unification” is carried out according to a certain attribute, then for some attributes both single and empty sets will arise, and then the Kleene complement is no longer required.
The need to consider unit sets and the empty set along with the rest is evident from the fact that, defining a set in one way or another, we may not know in advance whether it contains more than one or at least one element.
Here it is necessary to emphasize that the unit set and its only element are essentially different concepts and different things. The difference is that the unit set has all the properties of sets: it has subsets, set-theoretic operations can be applied to it, while an element of the unit set does not have these properties, unless it is a set itself.
Further on, Cantor's definition speaks of "certain and quite distinct objects of our perception or thought." Here we will not discuss this fundamental concept - the concept of an object, postponing its analysis for a while and considering it clear enough for a first acquaintance with the concept of a set. For us now it is much more important to grasp that side of the concept of a set, that inherent property of a set, about which nothing is said in Cantor's definition. This property is expressed as follows:
A set is completely determined by its elements.
In axiomatic, formal theories, this side of the concept of a set is formulated as an axiom, called the axiom of volume, or the axiom of extensionality. But even when presenting the meaningful ("naive") Cantor's theory of sets, this provision is either implied or formulated explicitly, for example, as the "intuitive principle of volume" in R. Stoll's textbook "Sets. Logic. Axiomatic theories".
The axiom of volume states that a set does not depend on the order of enumeration or the order in which its elements are arranged. Only one set can consist of the same elements. For example, different permutations made up of the same characters:

(a, b, c, d), (a, c, d, b), (b, d, c, a), etc.,

They are one and the same set, and as sets do not differ. This means, further, that different sets can differ only by the presence or absence of at least one element in them.
From this it becomes clear that there is only one empty set, since in the absence of elements, sets have no signs of difference. The empty set is denoted by ;.
According to their composition, as can be seen from Kantor's definition, sets can be thought of as consisting of real objects (the set of cats in the city of Sarov, for example) or of conceivable, conceptual entities (the set of natural numbers). Among the latter, a very important kind of sets are infinite sets, that is, those consisting of an infinite number of elements.
Two things must be noted here. On the one hand, it is clear that these are purely mental abstractions, that the sets of real objects cannot be infinite. On the other hand, it is precisely infinite sets that give special value, beauty, and originality to Cantor's theory of sets. Cantor is justly credited for his scientific courage when he began to consider infinite sets as entities accessible to the human mind.
We also note that the very concept of a set is a purely mental concept, in the words of Cantor - the object of our thought.

I.2. Ways to describe sets

If the letter M stands for a certain set, and the letter x is some "definite and completely distinguishable object of our perception or thought", then the expression "x; M" is read as "x belongs to M", or "x enters M", or "x is an element of M", or in some other similar way. Crossed entry sign; means the negation of the entry statement.
If there are not too many elements a, b, c, ... of the set M, then it is possible to describe the set by listing its elements inside curly braces:
M = (a, b, c, ... ).
Otherwise, the set is usually described using some membership condition P(x):
M = (x: P(x)).
This expression reads as follows: the set M consists of all such and only such x for which the proposition P(x) is true. The reader may notice that the second way of designating a set is more general, and the first form of describing a set can be reduced to the second. For example, using a logical formula:
M \u003d (x: x \u003d a, or x \u003d b, or x \u003d c, or ...),

And if a, b, c,... are numbers (whatever they are), then, for example, using the equation:

M \u003d (x: (x-a) (x-b) (x-c) ... \u003d 0).

I.3. Set-theoretic operations.

Operations can be performed on sets. The most commonly used operations are union and intersection.
The union of two sets is a set that combines the elements of both combined sets. This operation is indicated by the symbol ;. For example, if set A=(a,b,c), and set B=(c,d,e), then
A;B=(a,b,c,d,e).
The intersection of two sets is the set consisting of common elements of these sets. This operation is indicated by the symbol ;. For two sets of the previous example A;B=(c).
Other, more complex operations on sets are also used.

I.4. Quantitative comparison of sets.

For finite sets, the question of comparing their numbers is solved simply: for this, it is enough to count the compared sets, and we can already compare natural numbers with elementary school. But how to compare infinite sets? Kantor proposed to compare infinite sets quantitatively according to the principle of one-to-one correspondence.
DEFINITION. We say that a one-to-one correspondence is established between set A and set B if each element of set A is associated with one and only one element of set B so that each element of set B is associated with one and only one element of set A.
A one-to-one correspondence will be denoted by the shorter term “1-1-correspondence”, or, even shorter, by a bijection.
According to this principle, two sets are considered to be of equal number, or, more precisely, of equal power, or equivalent, if a bijection can be established between them. If it is impossible to establish a bijection between them, then the most powerful one is considered to be the one on the part of which the other can be mapped one-to-one.
Obviously, the equivalence relation between sets is symmetric, reflexive, and transitive. It is also clear that finite sets can also be compared by the 1-1 correspondence method, and that this method is a generalization of the usual way of comparing finite sets by recalculating them. In essence, the method of recalculation is the method of comparison by 1-1-correspondence with the standard set - the set of natural numbers.
Examples of comparison of infinite sets.
Even Galileo noticed that the set of all squares of natural numbers can be put in 1-1 correspondence with the set of all natural numbers:

1, 2, 3, 4, 5, …
1, 4, 9, 16, 25, …

And in this sense, there are exactly as many squares of natural numbers as there are numbers themselves. The situation is the same with even numbers: there are exactly the same number of them. We see that with the method of quantitative comparison of sets proposed by Cantor, a part of an infinite set turned out to be quantitatively equivalent to the whole. Kantor proposed to take this property of infinite sets as a defining attribute of an infinite set.
Sets for which it is possible to establish a bijection with the set of natural numbers, in other words, to renumber their elements, are called countable sets. Countable sets are obviously both the set of all squares of integers and the set of all even numbers. The set of all integers (positive and negative) is also countable. This can be seen from the fact that all integers can be arranged in the form of such a chain:
0, 1, -1, 2, -2, 3, -3, 4, -4, 5, -5, . . .

It is clear that all integers will fall into this chain, and we can renumber the numbers of this entire chain.
But here's a more complicated example. Is it possible to renumber all positive rational numbers? Cantor proposed the following way of numbering the set of all positive rational numbers. Let's arrange this set in the form of an infinite table - an infinite number of infinite rows. In the first line we place all fractions with a denominator 1, that is, natural numbers in ascending order. In the second line we will arrange all the fractions with a denominator 2 in ascending order of the numerator, in the third line - in the same order all the fractions with a denominator 3, and so on. After that, we first number all the fractions with the sum of the numerator and denominator equal to 2 (this is just one fraction 1/1), then all the fractions with the sum of the denominator and numerator equal to 3 (this; and 2/1), then - with the sum of the denominator and a numerator equal to 4 (these are 1/3, 2/2 and 3/1), and so on. In this case, we will skip the reducible fractions, since they have already been numbered earlier. It is clear that with this method of numbering, the number will get any positive rational number. On fig. I.1 shows the numbering scheme for the set of all rational numbers proposed by Cantor; the arrows indicate the numbering order.
1/1, ; 2/1, 3/1, ; 4/1, 5/1, ; …
; ; ; ;
1/2, 2/2, 3/2, 4/2, 5/2, … .
; ; ; ;
1/3, 2/3, 3/3, 4/3, 5/3 …
; ;
1/4, 2/4, 3/4, 4/4, 5/4 …
; ;
1/5, 2/5, 3/5, 4/5, 5/5, …

This numbering scheme is carved on a monument at Kantor's grave.
Using the same numbering scheme, one can also renumber the set of all ordered pairs of natural numbers (since each positive rational number corresponds to an ordered pair of natural numbers - the numerator and denominator). Further, placing the numbered set of ordered pairs on one line, we can apply the same trick to numbering the set of all ordered triples of natural numbers, then fours, and generally ordered n-tuples, where n is any natural number.

I.5. The concept of a subset.

A set M is called a subset of a set N if there are no elements in M ​​that are not included in N (in particular, M can coincide with N).
In other words, a subset should not contain “outside” elements, if this term characterizes all elements outside the wider (generally speaking) set N.
This definition is good in that it also covers the empty set: the empty set does not contain any, and therefore “foreign” elements. It is thus a subset of any set. If we define the concept of a subset in a more understandable way, as a set consisting only of elements of the main set, then the empty set will have to be counted among the subsets “in a separate line”. The need for such an enumeration is evident from the same considerations as the need to supplement with an empty set general concept sets (see above).
If the set M is a subset of the set N, then this circumstance can be briefly noted in the notation of the set M:

M = (x;N: P(x))

(it reads: the set M consists of all such and only such x from N for which the proposition P(x) is true).

I.5.1. Own and improper subsets.
The empty set, as already mentioned, is a subset of any set. In this sense, it stands apart, and therefore it is called an improper subset.
In addition to the empty set, an improper subset is also called a subset that coincides with the entire set. The remaining subsets are called proper. They constitute the "correct" parts of the main set, while the improper subsets are the "wrong" parts: this is the part equal to the whole, or the zero part.

I.5.2. How many subsets does the simplest set have?
The least numerous empty set has 0 elements. How many subsets does it have? Despite the absence of elements, the empty set still has one subset. This is itself, this is its doubly improper subset: firstly, because it is empty, and, secondly, because it coincides with the whole set. (Note that 20=1.)
The unit set, in which there is only one element, already has two subsets, both of which are improper: this is an empty set and a subset that coincides with the entire set. (Note again that 21 = 2.)
For a set consisting of two elements, two proper subsets are added to two improper subsets - single subsets containing one of the elements of the set each. The total is 4. (Note again that 22 = 4.)
By induction or otherwise, the reader can easily prove that a finite set of n elements has 2n subsets.

I.6. Cantor's theorem (formulation)

We see that for any n 2n > n, that is, the number of subsets of a finite set is always more number elements. This obvious property of finite sets was generalized by Cantor to infinite sets, proving his famous theorem, which states:
the cardinality of the set of all subsets is greater than the cardinality of the original set.
At first glance, this generalization is so natural that there is no doubt about the validity of Cantor's theorem. We will, however, give an example of the opposite property. The number of all possible ordered pairs of elements of a finite set of n elements is given by the formula n2, and we see that for n>1 n2>n. However, we have seen (see Section I.3) that the cardinality of the set of ordered pairs of an infinite set of natural numbers is not greater than the cardinality of the original set.
A general objection to both examples of ratios between the numbers of finite sets is that the analogy is not a proof.

I.7. Underdetermined sets

The existence of underdetermined sets follows from the existence of paradoxical, namely, contradictory judgments. Let's show how it works.
Let us recall the second way of describing sets (see section I.2). Here is how this method is described in R. Stoll's textbook
Intuitive principle of abstraction. Any form P(x) defines some set A by the condition that the elements of the set A are exactly such objects a that P(a) is a true proposition
The expression "form P(x)" means some statement about some object in which the name of this object is replaced by a variable x running through a given range of values. Another term for the notion "form P(x)" is a one-place predicate. Section I.2 uses the expression "membership condition" in the same sense.
But what if, for some values ​​of x (for some objects a), the judgment P(x) turns out to be contradictory?
A specific example of a set with such a membership condition makes the question more understandable.
We will consider the names of some objects, but only unambiguous names, that is, relating to only one specific object. The name contained in the object with this name (the object can be a set, or, for example, a book) will be called the internal name. A name that is not internal will be called external. The set E is the set of external names of the set of objects S, if it is included in the set S and has a name, it gives us an example of an underdetermined set.
Indeed, the name of the set E has, it is expressed by the letter E. To which of the two categories should the name of the set E be assigned? If we recognize it as an external name, that is, one of the elements of the set E, then it will turn out to be an internal name, and vice versa. The judgment about the belonging of the name of the set E to this set does not have the meaning of truth.
The answer to the above question is obvious. For values ​​x that turn P(x) into an inconsistent proposition, it is impossible to establish whether the corresponding object a is an element of the set A. The set A is underdetermined with respect to this object.
But the peculiarity of an underdetermined set is not only and not so much in its underdetermination. More importantly, its underdetermination is the result of the inconsistency of its definition. Such inconsistency, which you will not immediately notice. After all, it manifests itself only in relation to its single element (in our example, to the proper name of a set of external names). Consideration of the condition of belonging to such a set leads to a contradiction. And since we are accustomed to the fact that a contradiction is the result of either an error or the falsity of one of the initial premises of the argument, this is where the temptation arises to prove something.
Meanwhile, the contradiction that follows from the contradictory, or rather, from the unsatisfiable definition, proves absolutely nothing (except the unsatisfiability of this definition). Failure to understand this not very complicated circumstance leads to the appearance of false theorems.
How should we treat sets with conflicting definitions? We see here two possible forms of this relationship (with the same content).
1) It is possible to continue to consider inconsistent sets of the type of the set E described above as sets, allowing for the possibility of inconsistent sets, which was already pointed out by Cantor (he considered the set of all sets to be inconsistent), but then the possibility of the occurrence of such sets cannot be ignored when proving theorems.
Taking into account this possibility, it is not always possible to conclude from the contradiction that is obtained by proving by contradiction that some premise is false: for a contradictory set, the contradiction is its legitimate attribute and does not say anything.
2) It seems more correct to formulate our attitude to contradictory sets (more precisely, to sets with a contradictory definition) by clarifying Cantor's concept of a set in the sense that the question of belonging to a set of any object must have an unambiguous and consistent answer. Collections that do not satisfy this requirement, that do not allow, like the set E, to give such an answer to this question for at least one single element, should not be considered complete sets. These are underdetermined sets.
The possibility of the appearance of underdetermined sets must be taken into account in the proofs of theorems, as already mentioned.
The property of definiteness of a set in the sense indicated above is, of course, implied in the Cantor concept of a set, although, apparently, it was not stated explicitly by Cantor. True, one of the commentators on Cantor's definition of the concept of a set (see section I.1), Robert R. Stoll, interprets the words "certain ... objects" in this definition exactly in this way.
A refinement of the concept of a set in this sense can be formulated in the form of the axiom of the excluded middle, which sets must obey.
The axiom of the excluded middle is a special case of the law of the excluded middle, which states that every proposition is either true or false, and there is no third. But we know that quite meaningful contradictory judgments are also possible, neither true nor false, thus violating the law of the excluded middle, examples of which are judgments from all kinds of paradoxes. Therefore, in order to exclude contradictory sets from the number of admissible ones, we cannot limit ourselves to referring to this law, and must provide for the possibility of its violation by a special axiom.
AXIOM OF THE EXCLUDED THIRD. For any set, the judgment that any object belongs to it is either true or false.
In existing (and present in curricula mathematical faculties of universities) in set theories, underdetermined sets do not arise only because the possibility of paradoxical judgments in these theories is not taken into account.

I.8. On uncountable sets.

The method of quantitative comparison of sets proposed by Cantor by establishing a bijection between the compared sets (see Section I.3.) implicitly assumes that there are (can be encountered) such infinite sets between which it is impossible to establish a bijection. If this were not so, then all infinite sets would turn out to be of equal power, and Cantor's method of comparing sets would be meaningless.
Infinite sets that are equivalent to the set of natural numbers, which means that all their elements can be renumbered, are called countable sets. It follows that uncountable sets (that is, sets that are not countable) are such (so numerous) that it is impossible to renumber all of their elements.
As Cantor showed, the uncountable is the set of all real numbers between 0 and 1, usually called the continuum. The cardinality of a continuum is usually denoted by the letter C. Let us note the following remarkable properties of sets with cardinality of a continuum.
First, the set of real numbers x of a unit segment is equivalent to the set of real numbers y of any segment of the real line. The bijection between these sets is established by the formula:

Y \u003d a + x (b - a),

Where the numbers a and b correspond to the ends of an arbitrary segment.
Secondly, the formula y=tg(x-0.5;) establishes a bijection between a single segment (more precisely, a half-interval) and the entire real line. This means that the cardinality of the set of all real numbers has the same cardinality as the set of numbers of a unit segment (a segment, unlike an interval, includes numbers corresponding to its ends, but this difference does not lead to a difference in cardinalities).
The next important fact of set theory is that the set C (continuum) is equivalent to the set of all subsets of the natural series. Indeed, every real number less than one can be represented one-to-one by a proper infinite binary fraction. To do this, we agree that binary-rational numbers that have two binary representations, one of which ends with an infinite sequence of ones, are represented in exactly the way in which the binary fraction is infinite. And each such fraction is one-to-one determined by a subset of the natural series - the set of numbers of those digits of the binary fraction in which there are ones.
And, finally, another completely unexpected result, which surprised Kantor himself, follows from Cantor's definition of the equality of sets and the possibility of a unique representation of a real number by an infinite binary (or decimal) fraction. Equally powerful set C turned out to be the set of pairs of the same numbers, that is, numbers in the interval from zero to one. Translated into the language of analytic geometry, this means that the set of points of the unit segment turned out to be equivalent to the set of points of the unit square.
Indeed, each real number of a unit segment, represented by an infinite sequence of values ​​​​of decimal (for example) digits of this number, can be one-to-one associated with a pair of the same numbers, one of which is formed from even, and the other from odd digits of the original number.
But this means that the power C - the power of the set of real numbers of any segment - has the set of all points of the plane (the bijection between the unit square and the entire plane is established in the same way as between the unit interval and the entire number line).
In a similar way, the equality of sets of points of a segment and points of a three-dimensional figure - a cube, and hence the set of all points of the entire infinite 3-dimensional and even n-dimensional space is established.
This surprising result, with an unfavorable attitude towards Cantor's theory of sets, can be reproached with this theory: these are the absurd results that Cantor's method of quantitatively comparing sets according to the criterion of one-to-one correspondence leads to.

LIST OF USED SOURCES
(to the introduction and ch. I)

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3. Aleksandrov P.S. Introduction to the general theory of sets and functions. Moscow,
Leningrad. Gostekhizdat. 1948. 412c.
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5. Hausdorff F. Theory of Sets. Moscow, Leningrad. ONTI. 1937.

6. Natanson I.P. The theory of functions of a real variable. M: Gostekhizdat.
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7. Kolmogorov A.N., Dragalin A.G. mathematical logic. Additional chapters
You. Moscow University Press. 1984. 120s.
8. Arkhangelsky A.V. Cantor's theory of sets. Publishing house of Moscow
University. 1988. 112p.
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10. Yashchenko I.V. Paradoxes of the theory of sets. Moscow Publishing House
Center for Continuous Mathematical Education. 2002. 40s.
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A.P. Yushkevich. M: Science. 1985. 432p.
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