Cantor was bornRussiaSt.Petersburg。Father isjudeaBloodlineDenmarkA businessman whose mother was born in an art family.The family moved to Germany in 1856Frankfurt。First in a middle school, then inWiesbadenA university ofprep schoolstudy.[1]
In 1862, he entered the University of Zurich to study engineering, and transferred toUniversity of BerlinStudy mathematics and theology, and learn fromKummer （Kummer，Ernst Eduard，1810.1.29-1893.5.14）、Weierstrass (Karl Theodor Wilhelm, 1815.10.31-1897.2.19) and Kronecker (Leopold, 1823.12.7-1891.12.29), who went there in 1866GettingenStudy for one semester.In 1867, under the guidance of Kumer, general integer coefficients were solvedIndefinite equationaxtwo+bytwo+cztwo=The dissertation on solving the problem was awarded a doctor's degree.After graduation, under the direct influence of Weierstrass, he turned from number theory to rigorous analytical theory and soon came to the fore.At the beginning of his teaching at Harley University (1869-1913), he proved the uniqueness of trigonometric series expansion of functions with compound variables, and then used rationalsequence limit Define irrational numbers.He became an associate professor of the university in 1872 and a professor in 1879.Due to the heavy blow to his academic views, Cantor once suffered fromSchizophreniaAlthough he recovered in 1887 and continued to work, he was plagued by illness in his later years.
Cantor has a wide range of hobbies and personality. He believes in religion all his life.His early interest in mathematics was number theory. In 1870, he began to study trigonometric series, which led to the establishment of the greatest mathematical achievements at the end of the 19th century and the beginning of the 20th century - set theory and super finite number theory.In addition, he also tried to exploreNew theoryThe mathematical philosophy involved in the founding process.From 1888 to 1893, Cantor was appointedBerlinHe was the first president of the Mathematician Association and led the establishment of the German Federation of Mathematicians in 1890.
Main contributions
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Cantor's contribution to mathematics is set theory and super finite number theory.
For more than 2000 years, scientists have been exposed to infinity, but they are unable to grasp and understand it. This is indeed a sharp challenge to mankind.Cantor, with his unique thinking, rich imagination and novel methods, drew a masterpiece of human intelligence - set theory and ultrapoor number theory, which shocked the entire mathematical and even philosophical circles at the turn of the 19th and 20th centuries.It is no exaggeration to say that "the revolution about mathematical infinity was almost completed by him alone."
The establishment of set theory
In the 19th century, due to the strictness of analysis andFunction theoryMathematicians raised a series of important questions andIrrational numberTheory, nocontinuous function The research results in this field laid the necessary ideological foundation for Cantor's later work.
Cantor is looking for a function to expand intoTrigonometric seriesRepresentedUniquenessDiscriminant criterionIn his work, he realized the importance of infinite sets and began to engage in general theoretical research on infinite sets.As early as 1870 and 1871, Cantor《Journal of Mathematics》A paper was published on thef(x)The uniqueness theorem of trigonometric series representation is proved, and even if it does not converge at a finite number of discontinuities, the theorem still holds.In 1872, he published a paper entitled "Generalization of a Theorem in Trigonometric Series" in the "Mathematical Yearbook", which generalized the uniqueness result to allowException valueIs some kind of infinite set case.To describe this set, he first definedpoint setOfLimit point, and then introduced theDerived setAnd derived set of the derived set.This is the beginning from the exploration of uniqueness to the study of point set theory, and has laid a theoretical foundation for point set theory.Later, he published many articles in Mathematics Yearbook and Mathematics Magazine.He called a set a total of certain and different things that people can realize and judge whether a given thing belongs to the total.He also pointed out that if a set can form a part of itOne-to-one correspondenceIt is infinite.He also gave open set, closed set andComplete setThe union and intersection of sets are defined.
In order to extend the concept of the number of elements of a finite set to an infinite set, he proposed the concept of set equivalence based on the principle of one-to-one correspondence.Two sets are equivalent only when one to one correspondence can be established between their elements.This is the first time that various infinite sets are classified according to the "number" of their elements.He also introduced the concept of "can be listed"positive integerAny set that forms a one-to-one correspondence is called a countable set.In 1874, he《Journal of Mathematics》In the paper published onRational numberSets are countable, and later he proved that allAlgebraic numberThe collection of all components of is also countable.As for whether the set of real numbers can be listed, Cantor gaveDedekind (Dedkind, Julins Wilhelm Richard, 1831.10.6-1916.2.12), but soon he himself got the answer that the set of real numbers is unlisted.Since the set of real numbers is not countable, while the set of algebraic numbers is countable, he gets that there must beTranscendental numberThe transcendental number is "much more than" algebraic number.It was constructed in the same yearTheory of real variable functionMedium famous“Cantor set”, giving theUncountable setAn example of.He also skillfully corresponded the points on a straight line with the points on the whole plane one by one, and even could match the straight line with the wholeN-dimensional spaceCarry out one-to-one correspondence of points.From 1879 to 1883, Cantor wrote six series of papers with the general title of "On Infinite Linear Points"manifold”The first four of them are similar to the previous papers, discussing some mathematical achievements of set theory, especially some interesting applications of set theory in analysis.The fifth paper was later published in a separate edition with the title of "Fundamentals of General Set Theory".The sixth paper is a supplement to the fifth.Cantor's creed is: "Mathematics is completely free in its own developmentConceptual limitationsIt only lies in that it must be free from contradiction and in harmony with the concept introduced by the precise definitionThe essence of mathematics lies in its freedom. "
The Establishment of the Theory of Super Finite Numbers
The main achievement of General Set Theory Foundation (hereinafter referred to as Foundation) in mathematics is the introduction of ultrafinite numbers. In the process of developing this theory, Cantor applied the following principles:
First generation principle: starting from the number of any given point, itsSuccessor number。
Second generation principle: Any sequence without the maximum number in it can generate a new number as the limit of the sequence, which is defined as the successor number greater than all the numbers in the sequence.
The third (limitation) principle: ensure that a natural interruption occurs in the above ultrapoor sequence, so that the second number class has a definite limit, thus forming a larger number class.
Repeatedly apply the three principles to obtain the sequence of over finite numbers
ω，ω1，ω2，…
Using the concept of set potential introduced previously, Cantor pointed out that the important difference between the first number class (I) and the second number class (II) is that the potential of (II) is greater than that of (I).In Chapter 13 of the Foundation, Cantor pointed out for the first time that the potential of number class (II) is the potential immediately following the potential of number class (I).
In "Foundation", Cantor also gaveWell ordered setSum infinityGood orderThe concept of set number points out that the entire set of super finite numbers is well ordered, and any infinite well ordered set has a unique number in the second number class as the number representing its order characteristics.Cantor also defined the addition, multiplication andInverse operation。
"Essay on the Basis of Ultra poor Number Theory" is Cantor's last important mathematical work. After 20 years of hard exploration, Cantor hopes to systematically summarize the strictFundamentals of Mathematics。Xianwen is divided into two parts. The first part is“Total orderResearch on Sets ", published in Mathematics Yearbook in May 1895.The second part, published in the Mathematical Yearbook in May 1897, is about "the study of well ordered sets".The publication of Xianwen marks the transition of set theory from point set theory to abstract set theory.However, because it is not yet axiomatic, and some of its logical premises andMethod of proofWithout proper restrictions, paradox will be derived, so Cantor's set theory usually becomesClassical set theoryorNaive set theory。
What happened to Cantor
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The new and epoch-making set theory initiated by Cantor is the first time in the history of human cognition to establish an abstract form for infinity since more than 2000 years in the ancient Greek eraSemiotic systemIt reveals the nature of infinity, makes a revolutionary change in the concept of infinity, penetrates into all mathematical branches, fundamentally transforms the structure of mathematics, promotes the establishment and development of many other new branches of mathematics, and becomes the theory of real variable functionAlgebraic topology, group theory andfunctional analysisThe foundation of such theories also has a profound impact on logic and philosophy.But Cantor's set theory is not perfect. On the one hand, Cantor's“Continuum hypothesis”And the "well ordered theorem";On the other hand, the Braley Forty paradox discovered at the turn of the 19th and 20th centuriesCantor's paradox andRussell paradoxIt makes people seriously doubt the reliability of set theory.In addition, the emergence of set theory has really impacted the traditional concepts and overturned many previous ideas, which was difficult to be accepted by mathematicians at that time. Many people opposed it, the most vehement of which wasBerlin SchoolKronecker, a constructivist, is one of the representative figures of.Kronecker believed that mathematical objects must be constructible, and those that cannot be constructed by finite steps are suspicious and should not be regarded as mathematical objects. He opposed the theory of irrational numbers and continuous functions, and also severely criticized and viciously attacked Cantor's infinite sets andTransfinite numberTheory is not mathematics butMysticism。He said Cantor's set theory was empty and empty.In addition to Kronecker, some famous mathematicians also published on set theoryObjections。French mathematicianPoincare (Poincare, J ules Henri, 1854.4.29 － 1912.7.17) said: "I personally, and I am not alone, think it is important not to introduce something that cannot be completely defined in a limited number of words".He regards set theory as an interesting“PathologyAnd predicted that "the later generation will regard (Cantor) set theory as a disease, and people have recovered from it".Weyl, Claude Hugo Hermann (1885.11.9 － 1955.12.8), a German mathematician, believed that Cantor's hierarchical view of cardinal numbers was "fog"AboveFog ".Klein (Christian Felix, 1849.4.25-1925.6.22) also disagreed with the idea of set theory.Mathematician H.A. Schwartz was a good friend of Cantor, but he broke off with Cantor because he opposed set theory.After the appearance of the paradox of set theory, they began to think that set theory was basically a pathological state, and they developed intoempiricism, semi empiricismIntuitionism, constructivism and other schools formed the anti Cantor camp in the basic war.
In 1884, because the hypothesis of continuity could not be proved for a long time, coupled with the sharp confrontation with Kronecker, the spirit was repeatedly hit. At the end of May, he could not support it. For the first timeMental breakdown。He was depressed and could not concentrate on the study of set theory. Since then, he was deeply involved in theology, philosophy and literature.However, whenever he returned to his normal state, his thoughts always became unusually clear and he continued his work on set theory.
Cantor's set theory has been publicly recognized and warmly praisedSwitzerlandZurichThe first session heldInternational Congress of MathematiciansOn the Internet.Professor, Zurich University of Technology, SwitzerlandHurwitz(Hurwitz, Adolf, 1859.3.26 － 1919.11.18)comprehensive reportIt clearly explains that Cantor's set theory has played a great role in promoting the development of function theoryFor the first timeIt shows to the international mathematical community that Cantor's set theory is not a dispensable philosophy, but a theoretical tool that really plays a role in the development of mathematics.At the group meeting, the French mathematician Hadamard Jacques (1865.12.8-1963.10.17) also reported the important role of Cantor in his work.As time goes by, people gradually realize the importance of set theory.Hilbert David (1862.1.23-1943.2.14) highly praised Cantor's set theory as "the best work of mathematical genius" and "pure humanIntellectual activityIt is one of the highest achievements of the era.At the Second International Congress of Mathematicians in 1900,HilbertSpoke highly of the importance of Cantor's workContinuum hypothesisIt was listed as the first of 23 important mathematical problems to be solved at the beginning of the 20th century.When Cantor'sNaive set theoryWhen a series of paradoxes occur, Kronecker's successorBrower(1881.2.27-1966.12.2) and others took this opportunity to make a big contribution,HilbertWith firm language, he announced to his contemporaries that "no one can drive us out of the Eden created by Cantor".
