The family of Georg Kantor (1845-1918) moved from Russia to Germany when he was still a child. It was there that he began to study mathematics. In 1868 he defended his dissertation on number theory and received his doctorate from the University of Berlin. At the age of 27, Kantor published an article containing a general solution to a very complex mathematical problem - and ideas that later grew into his famous theory - set theory. In 1878, he introduced and formulated a significant number of new concepts, gave the definition of a set and the first definition of a continuum, and developed the principles of comparing sets. He gave a systematic presentation of the principles of his doctrine of infinity in 1879-1884.

Cantor's insistence on considering infinity as something actually given was big news for that time. Kantor thought of his theory as a completely new calculus of the infinite, "transfinite" (that is, "superfinite") mathematics. According to his idea, the creation of such a calculus was supposed to revolutionize not only mathematics, but also metaphysics and theology, which interested Cantor almost more than scientific research itself. He was the only mathematician and philosopher who believed that the actual infinity not only exists, but is also comprehensible by man in the full sense, and this comprehension will raise mathematicians, and after them theologians, higher and closer to God. He devoted his life to this task. The scientist firmly believed that he was chosen by God to make a great revolution in science, and this belief was supported by mystical visions. However, Georg Cantor's titanic attempt ended strangely: insurmountable paradoxes were discovered in the theory, which cast doubt on the meaning of Cantor's favorite idea - the "ladder of alephs", a sequential series of transfinite numbers. (These numbers are widely known in the designation he adopted: in the form of the letter aleph - the first letter of the Hebrew alphabet.)

The unexpectedness and originality of his point of view, despite all the advantages of the approach, led to a sharp rejection of his work by most scientists. For decades, he waged a stubborn struggle with almost all his contemporaries, philosophers and mathematicians, who denied the legitimacy of building mathematics on the foundation of the actual-infinite. This was taken as a challenge by some, since Cantor assumed the existence of sets or sequences of numbers that have infinitely many elements. The famous mathematician Poincaré called the theory of transfinite numbers a "disease" from which mathematics must someday be cured. L. Kronecker - Cantor's teacher and one of the most respected mathematicians in Germany - even attacked Cantor, calling him a "charlatan", "renegade" and "molester of youth"! Only by 1890, when applications of set theory to analysis and geometry were obtained, Cantor's theory was recognized as an independent branch of mathematics.

It is important to note that Kantor contributed to the creation of a professional association - the German Mathematical Society, which contributed to the development of mathematics in Germany. He believed that his scientific career had suffered from prejudice against his work, and he hoped that an independent organization would allow young mathematicians to independently judge new ideas and develop them. He was also the initiator of the convening of the first International Mathematical Congress in Zurich.

Kantor had a hard time with the contradictions of his theory and the difficulty of accepting it. Since 1884 he suffered from a deep depression and after a few years he retired from scientific activity. Kantor died of heart failure in a psychiatric hospital in Halle.

Kantor proved the existence of a hierarchy of infinities, each of which is "greater" than the previous one. His theory of transfinite sets, having survived years of doubt and attack, eventually grew into a grandiose revolutionary force in 20th-century mathematics. and became its cornerstone.

The beginning of the 19th century was marked by the discovery of non-Euclidean geometry. In 1825 - Nikolai Vasilyevich Lobachevsky, a little later, in 1831 - Janos Bolyai. And the fate of these discoveries was very tragic. Neither one nor the second discovery was recognized. Until the 1860s, before the discoveries of other non-Euclidean geometries - Riemann and others. And the discoverers of non-Euclidean geometry have already died! And now - the theory of sets, which is also not recognized, scolded ... Oh, this strange 19th century ...

Cantor), Georg (March 3, 1845 - January 6, 1918) - mathematician and thinker, creator of set theory, which has its own basis. object of infinite sets. Genus. In Petersburg. From 1872 - prof. university in Halle. He died in Halle in a psychiatric hospital. clinic. To the creation of the theory of sets (1870), he was led by studies of trigonometric. rows. The creative period in the life of K., which lasted until 1897 (interrupted by a spiritual crisis in 1885), is marked by Op. "On infinite linear point manifolds" ("?ber unendliche, lineare Punktmannigfaltigkeiten", 1879–84), "On the justification of the theory of transfinite sets" ("Beitr?ge zur Begr?ndung der transfiniten Mengenlehre", 1895–97), etc. K. laid the foundations as an abstract theory of sets [studying sets only from the point of view. their "numbers" (cardinality of the set) and order relations between their elements (order types of sets)], and the theory of point sets (i.e., sets consisting of points of the number line and, in general, the number n-dimensional space). K. was one of the first to construct the theory of real numbers, which is still (along with the theories of the German scientists R. Dedekind and K. Weierstrass) usually used as the basis for the construction of mathematical. analysis. Cantor's set theory marked an important step forward in the study of the concept of infinity; its creation was a revolution in everything mathematical. knowledge. In the beginning. 20th century all mathematics was restructured on the basis of set theory; its development and penetration into various areas of mathematics led to the emergence of new scientific. disciplines, for example. topology, abstract algebra, etc. Later, paradoxes were discovered in set theory, which gave a new impetus to the study of logical. foundations of mathematics and led to the emergence of new trends in its philosophy. interpretation (eg, intuitionism). One of the first paradoxes of this kind (associated with the concept of the power of the set of all sets) was discovered by K. himself in 1899. Mathematics, based on the unconditional application of K.'s set theory, in present. time is often called classical. See Mathematics, Set Theory, Mathematical Infinity. Philos. aspect of the ideas of K. consisted in the recognition of the full legitimacy of the concept of actually infinite. K. distinguished two types of mathematical. infinity: the improperly infinite (potential, or syncategorematic, infinite) and the proper infinite (actually infinite), understood by K. as something complete, as a strictly limited whole. In connection with the question of reality, the mathematical concepts K. distinguished: their intrasubjective, or immanent, reality (their internal logical. consistency) and their transsubjective, or transient, reality, by which he understood the correspondence between mathematical. concepts and processes of the real world. In contrast to Kronecker, who rejected those methods of proving the existence of mathematical. objects, to-rye not associated with their construction or calculation, K. put forward the thesis: "the essence of mathematics - in its freedom," DOS. meaning to-rogo was reduced to the assumption of the construction of any logically consistent abstract mathematical. systems, the question of "transient reality" to-rykh is solved by comparing them with the processes of reality. The fruitfulness of this thought of K. was confirmed by the development of mathematics in the 20th century, which brought many examples of the application of newly emerging abstract mathematical concepts. and logical. theories in physics, technology, linguistics, and other fields. By their philosophy. views K. was an objective idealist. He considered the actual infinite in mathematics to be only one of the forms of the existence of the actually infinite in general; the latter acquires the "highest completeness" in a completely independent, out-of-the-world existence - in God; god is absolutely infinite or absolute; in addition, the actual infinite, according to K., objectively exists in the external world. K. criticized Hegel, rejecting his dialectic on the grounds that its core is a contradiction. Hence, attention, especially in the last period of his life, K. paid to theology. His religious philosophy. views took shape under the influence of Aristotle, Plato and the scholastics. Op.: Gesammelte Abhandlungen..., V., 1932. Lit.: Fraenkel?., Georg Cantor, Lpz., 1930. A. Konoplyankin. Moscow.

Great Definition

Incomplete definition ↓

KANTOR Georg (1845-1918)

German mathematician, logician, theologian, creator of the theory of transfinite (infinite) sets, which had a decisive influence on the development of mathematical sciences at the turn of the 19th and 20th centuries. Graduated from the University of Berlin (1867), professor at the University of Halle (1879-1913). Main work: "Fundamentals of the General Doctrine of Varieties" (1902). K.'s research, initiated by the need to solve pressing problems in the theory of infinite Fourier series, became the basis for further fundamental research in the direction of the theory of numerical sets, where he introduced: the general definition of a set, transfinite numbers, the general concept of "power of a set" (as the number of elements of a set) , cardinalities of various transfinite sets. Under the set, K. understood "... in general, any many things that can be thought of as a single unit, that is, any set of certain elements that can be connected into one whole with the help of some law ...". Fundamental to the concept of a set is the act of combining different objects into a single whole, defined as a set. Elements of sets can be any objects of real reality, human intuition or intellect. The presence in the definition of K. of the phrase "... a set of certain elements that can be connected into one whole with the help of a certain law ..." completely determines the set of its elements or law (characteristic features, properties), according to which the act of combining various objects takes place into a single whole - a multitude. Therefore, the fundamental concept of set theory is not the concept of a set itself, but the relation of belonging of objects to a set. The tradition of dividing infinity into actual and potential goes back to Aristotle: "The alternative remains, according to which the infinite has a potential existence ... Actually the infinite does not exist" (Aristotle, "Physics"). This tradition was continued by Descartes (“Infinity is recognizable, but not cognizable”) and even in the time of K. Gauss (“In mathematics, an infinite value can never be used as something final; infinity is nothing more than facon de parle / manner of expression - С.С / , meaning the limit to which some quantities tend, when others decrease indefinitely"). K., as M. Kline wrote, departed from a long tradition "already by the fact that he considered infinite sets as single entities, moreover, entities accessible to the human mind." Sharply disagreeing with his fellow mathematicians in his views on mathematical infinity, K. motivated the need to introduce actually infinite sets by the fact that "potential infinity actually depends on the actual infinity logically preceding it." A classic example of an actually infinite set according to K. are the decimal expansions of irrational numbers, since each "finite segment of such a decomposition gives only a finite approximation to an irrational number." By 1873, K. began research on the classification of actually infinite sets. A little later, K. defined an infinite set as a set for which there is a one-to-one correspondence with its own subset (that is, different from the entire set). One of the consequences of this approach was, for example, the possibility of establishing a one-to-one correspondence between points of a straight line and points of a manifold of any dimension. Based on his own definition of infinite sets, K. was able to establish for each pair of them the relation of equivalence (equal power). In 1874 K. proved the uncountability of the set of all real numbers, establishing the existence of pairs of infinite sets with different cardinalities (nonequivalent sets). Systematically the foundations of his theory of mathematical infinity K. outlined in 1879-1884. The basis of the hierarchy of infinities K. was proved in the first half of the 1890s by the well-known theorem of K.-Bernstein: "if two sets A and B are such that there is a one-to-one correspondence between the set A and a subset of the set B and between the set B and the subset of the set A , then it is also possible to establish a one-to-one correspondence between the set A and the set B", i.e. establish the equivalence (equivalence) of the sets A and B. At the same time, K. determined that if the set A can be put in one-to-one correspondence with its own subset B, and the set B cannot be put in one-to-one correspondence with its own subset A, then the set B according to definition is greater than the set A. According to M. Klein, such a definition generalizes to the case of infinite sets what is "immediately obvious in the case of finite sets." Following this approach, K. proved that for any "given set there is always a set greater than the original" (for example, the set of all subsets of a given set is greater than the original set). The fact that between two powers it is possible to establish relations "equality", "more" and "less", gave K. there is reason to call "numbers" the symbols for designating the cardinalities of infinite sets (for finite sets, the symbols for designating their cardinalities are the numbers of the natural series that determine the number of elements in each of the equivalent finite sets). In contrast to the numbers of the natural series [ordinal numbers / from him. Die Ordinalzahl (Ordnungzahl) - ordinal numerals - C.C.I, K. called cardinal numbers (from German Die Kardinalzahl - quantitative numbers)] "numbers" designating the power of infinite sets. K. believed that the area of ​​certain values ​​is not limited to finite values, tk. about "the actual infinite is also possible demonstrative knowledge". If the concept of cardinality was an extended concept of "quantity" for infinite sets, then the concept of a cardinal number became an extended generalization of the concept of "numbers in general". K. expansion of the concept of "number" in the realm of the Infinite marked the transition of mathematics to a qualitatively new level of thinking. In fact, the power of sets according to K. reflects in the mind of a human researcher certain relations of sets, i.e. the cardinality of sets in K. is the most general characteristic of equivalent infinite sets. Bolzano in the early 19th century. came to the concept of a one-to-one correspondence between sets (and, consequently, to the concept of cardinalities of sets and their expression by cardinal numbers). However, under the "quantity" until the middle of the 19th century. size was understood. And since each quantity can be expressed as a number by means of the chosen unit of measurement, the idea of ​​quantity was associated with the concept of number. The poet Bolzano was forced to retreat before the serious difficulties arising from the concept of "quantity". Mathematics of that time was generally defined as a science that studies the relationships between quantities and the numbers expressing them. However, as VA Volkov writes, "no matter how important different types of quantities and relationships between them are for practical applications of mathematics, they do not cover all the richness of various quantitative relations and spatial forms of the real world." K. also introduced the concept of "limit point of a derived set" into mathematics, constructed an example of a perfect set ("set K."), and formulated one of the axioms of continuity ("axiom K."). Consequences from the theory of K. revealed contradictions in quite seriously studied areas of the foundations of mathematics. The leaders of mathematics of that time called these contradictions paradoxes (antinomies) for the sole reason that the paradox "can be explained, and mathematicians did not leave the hope that they would eventually be able to resolve all the difficulties they encountered." The theory of mathematical infinity of K., unlike most of the leading mathematicians of that time, was supported by Russell and Hilbert. Russell, considering K. one of the great thinkers of the 19th century, wrote in 1910 that the solution of K. problems, "which have long shrouded the mystery of mathematical infinity, is probably the greatest achievement that our century / 20th century should be proud of - S.S ./". Hilbert in 1926 thought that the theory of K. - is "the most delightful flower of mathematical thought and one of the greatest achievements of human activity in the field of pure thinking." And E. Borel and A. Lebesgue already at the very beginning of the 20th century. generalized the concept of the integral and developed the theory of measure and measurement, which was based on the theory of K. By 1897, K. was forced to stop active mathematical research due to sharp resistance to his ideas (in particular, from L. Kronecker, who called K. a charlatan), putting forward called the "law of conservation of ignorance": "it is not easy to refute any wrong conclusion, once it has been arrived at and it has become sufficiently widespread, and the less it is understood, the more stubbornly it is adhered to." K. always shared the philosophical ideas of Plato and believed that in the world around us "ideas exist independently of man. And in order to realize the reality of these ideas, you only need to think about them." K., being a zealous Lutheran in accordance with the long-standing religious tradition of his family, often used theological argument in his statements. This was especially evident after his departure from mathematics.

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 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 (eg 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 irrational numbers, are uncountable and therefore more numerous than integers. , and should be treated 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, is 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 novice 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 work Principia Mathematica, he repeatedly referred to a 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 the last years of his 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.

Ed., Gesammelte Abhandlungen mathematical und philosophical inhalts, mit erlä uternden anmerkungen sowie mit ergä nzungen aus dem briefwechsel Cantor- Dedekind, Berlin, Verlag von Julius Springer, 1932

1. Development period (1845−1871)

Georg Ferdinand Ludwig Philipp Kantor, creator of set theory, one of the greatest new phenomena in the world of science, was born in St. Petersburg on February 19, o.s. style (March 3, new style) 1845. His father Georg Voldemar Kantor, originally from Copenhagen, arrived in St. Petersburg in his youth; he kept a brokerage there under his own name, sometimes under the name "Kantor and K." A diligent and successful businessman, he achieved great success and left after his death (1863) a very significant fortune; apparently, he enjoyed high respect both in St. Petersburg and later in Germany. Due to lung disease, in 1856 he moved with his family to Germany; there he soon chose to stay in Frankfurt am Main, where he lived in the position of a rentier. Kantor's mother, Maria née Boehm, came from a family many of whose members were gifted in various fields of art; her influence was evident, no doubt, in the rich fantasy of her son. His grandfather, Ludwig Böhm, was a bandmaster; grandfather's brother Joseph, who lived in Vienna, was the teacher of the famous virtuoso cellist Joachim; Maria Kantor's brother was also a musician, and her sister Annette had an artist daughter who taught at the Munich School of Artistic Crafts. An artistic streak is also noticeable in Georg Kantor's brother, Konstantin, who was a talented pianist, and in his sister Sophia, who was especially fond of drawing.

A gifted boy who attended elementary school in St. Petersburg, already very early showed a passionate desire to begin the study of mathematics. His father, however, did not agree with this, considering the profession of an engineer to be more promising in terms of earnings. The son at first obeyed; for some time he attended the gymnasium in Wiesbaden, as well as private schools in Frankfurt am Main; then he entered, in the spring of 1859, the provincial real school of the Grand Duchy of Hesse in Darmstadt, where they also taught Latin; from there he moved in 1860 to the general course of the Higher Craft School (later the Higher Technical School). His father directed his education with unusually high standards; he attached special importance to the education of energy, firmness of character and religiosity, penetrating all life; in particular, he emphasized the importance of a complete mastery of the main modern languages. His father instructed him (in his confirmation letter in 1860) to stand firm, in spite of all enmity, and always get his way; this call was remembered more than once by the son in the hours of difficult trials, and, perhaps, it is to this paternal upbringing that we owe the fact that his creative spirit was not prematurely broken and its fruits were not lost to posterity.

Over time, the son's deep attraction to mathematics could not but affect his father, whose letters also testify to his respect for science. In a letter from Darmstadt, dated May 25, 1862, representing the first surviving letter from Kantor, the son could already thank his father for his approval of his plans: “Dear dad! You can imagine how glad your letter made me; it determines my future. I have spent the last days in doubt and uncertainty; and could not come to any decision. Duty and attraction were constantly at war. Now I am happy to see that I will not grieve you by following my own inclination in my choice. I hope, dear father, that I will still be able to bring you joy, because my soul, my whole being lives in my calling; a person does what he wants and can, and what his unknown, mysterious voice leads him to! .. "

In the autumn of 1862, Kantor began his studies in Zurich, from which, however, he left after the first semester due to the death of his father. Since the autumn of 1863 he studied mathematics, physics and philosophy in Berlin, where the triumvirate of Kummer, Weierstrass and Kronecker attracted the best talents, exciting the minds of the (then still rather narrow) circle of listeners in the most diverse directions. He spent only the spring semester of 1866 in Göttingen. Weierstrass undoubtedly had the strongest influence on his scientific development. It is remarkable and characteristic of the breadth of Weierstrass's views, for his unprejudiced and insightful judgment, with what sympathetic understanding and how early he appreciated the unconventional ideas of his student, thereby responding to the deep respect that he invariably showed him throughout his life, despite transient quarrels. During his Berlin years, Kantor was not only a member of the Mathematical Society, but also a narrower circle of young colleagues who met weekly at Remel's tavern; this circle included, apart from occasional guests, Henoch (the future publisher of Fortschritte (Successes), Lampe, Mertens, Max Simon, Thoma; the last of them was especially close to Kantor. Further, G A. Schwartz, who was two years older, later, however, he met the ideas of Cantor with the strongest distrust, in contrast to his teacher Weierstrass, and until the very end of his life, like Kronecker, he especially warned his students against them.December 14, 1867 The twenty-two-year-old student completed a thesis at the University of Berlin, which arose from an in-depth study of Legendre's Disquisitiones arithmeticae (Studies in Arithmetic) and Legendre's Theory of Numbers and was rated by the faculty as "dissertatio docta et ingeniosa" (Scholarly and ingenious reasoning) * This work adjoins the Gauss formulas for solving the Diophantine equation ax 2 + a"x" 2 + a"x" 2 = 0; some relation is established in it, which is not given explicitly by Gauss. A detailed discussion of Cantor's work is contained in a detailed biography I wrote of him, published in Jahresbericht der Deutschen Mathematikervereininung, vol. 39 (1930), pp. 189−266, and also in a separate book: Georg Kantor, Leipzig and Berlin, 1930; he dedicated it to his guardians (at the same time the guardians of his brother and sister). In the oral exam, he received "magna cum laude" ("with special distinction"). Of the three theses he proposed to defend, the third one is especially characteristic: “In re mathematica ars propenendi questionem pluris facienda est quam solvendi” (In mathematics, the art of posing questions is more important than the art of solving them.) Perhaps even the results he obtained in set theory are inferior in value to revolutionary formulations issues so far reaching in their influence beyond his own writings.

It seems that Kantor taught for a short time in a girls' school in Berlin; in any case, in 1868, having passed the state examination, he entered the well-known Schelbach Seminary, which trained teachers of mathematics.

The doctoral thesis, which gave Kantor the opportunity to become Privatdozent at the University of Halle in the spring of 1869, belongs, together with a few short notes published in 1868-72, to his first, arithmetical circle of interests, to which he rarely returned later. These studies number theory under the direction and with the approval of Kronecker, were not, however, for Cantor just an accidental episode. On the contrary, he experienced the deep inner impact of this discipline, with its special purity and grace. This is evidenced, along with the first, by the third thesis presented by him for defense: “Numeris integros simili modo atque corpora coelestia totum quoddam legibus et relationibus compositum efficere” (“Integer numbers, like celestial bodies, should be interpreted as a single whole, bound by laws and relations "). The establishment of connections between various number-theoretic functions and the Riemann zeta function (adjacent to Riemann's work on prime numbers) also belongs to an early time, possibly already to this period; this work was published by Kantor only in 1880, under the influence of Lipschitz's note in the Paris Comptes Rendus ("Reports"). Cantor's further number-theoretic interests are, in addition to his numerical table, also preserved until 1884, but not implemented, the plan to publish in Acta Mathematica, a work on quadratic forms.

E. Heine, who was an ordinary professor at Halle at the time when Kantor defended his dissertation there, immediately realized that in his young colleague an extraordinary sharpness of mind was happily combined with the richest imagination. Of decisive importance was the fact that soon after Cantor's move to Halle, Heine prompted him to study the theory of trigonometric series. Zealous work on this subject not only resulted in a number of significant achievements, but also led Cantor on the path to the theory of point sets and transfinite ordinal numbers. The works , , and are devoted to the refinement of one of Riemann's assertions about trigonometric series (and the accompanying controversy with Appel, in which the concept of uniform convergence was considered in detail); in his work, Kantor proves a theorem on the uniqueness of the trigonometric representation * It is surprising that Kronecker, who at first had a positive attitude towards Cantor's uniqueness theorem (cf. ), subsequently completely ignores this result; for example, in "Vorlesungen über die Theorie der einfachen und mehrfachen Inegrale" ("Lectures on the Theory of Simple and Multiple Integrals") (1894) he presents the question of uniqueness as still open!. He seeks to generalize this result by giving up any assumptions about the behavior of the series on some exceptional set; this compels him to present in the work a brief outline of ideas “that may be useful for clarifying the relations that arise in all cases when numerical quantities are given in a finite or infinite number. Here, for point sets, limit points and derivatives (of finite order) are introduced. To this end, Cantor, on the one hand, develops his theory of irrational numbers * . In Heine's Elements of the Theory of Functions (J. Math., 74, pp. 172-188, 1872), irrational numbers are introduced in a manner exactly following Cantor's ideas; cf. an introduction to Heine's article, as well as Kantor's work "Mitteilungen zur Lehre vom Transfiniten" ("Toward the Doctrine of the Transfinite"), following the theory of sets immortalized his name, where irrational numbers are considered as fundamental series. On the other hand, for the transition to geometry, he introduces a special axiom (Cantor's axiom), which simultaneously and independently appeared in a slightly different formulation in Dedekind's book Continuity and Irrational Numbers.