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Encyclopedia > Georg Cantor
Born March 3, 1845 Saint Petersburg, Russia January 6, 1918 (aged 72) Halle, Germany Russia (1845–1856), Germany (1856–1918) Mathematics University of Halle ETH Zurich, University of Berlin Set theory

Georg Ferdinand Ludwig Philipp Cantor (March 3, 1845[1]January 6, 1918) was a German mathematician. He is best known as the creator of set theory, which has become a foundational theory in mathematics. Cantor established the importance of one-to-one correspondence between sets, defined infinite and well-ordered sets, and proved that the real numbers are "more numerous" than the natural numbers. In fact, Cantor's theorem implies the existence of an "infinity of infinities". He defined the cardinal and ordinal numbers, and their arithmetic. Cantor's work is of great philosophical interest, a fact of which he was well aware.[2] Georg Cantor Cantor died in 1918; therefore, all photographs of him are in the public domain. ... is the 62nd day of the year (63rd in leap years) in the Gregorian calendar. ... 1845 was a common year starting on Wednesday (see link for calendar). ... Saint Petersburg (Russian: &#1057;&#1072;&#1085;&#1082;&#1090;-&#1055;&#1077;&#1090;&#1077;&#1088;&#1073;&#1091;&#769;&#1088;&#1075;, English transliteration: Sankt-Peterburg), colloquially known as &#1055;&#1080;&#1090;&#1077;&#1088; (transliterated Piter), formerly known as Leningrad (&#1051;&#1077;&#1085;&#1080;&#1085;&#1075;&#1088;&#1072;&#769;&#1076;, 1924&#8211;1991) and... is the 6th day of the year in the Gregorian calendar. ... 1918 (MCMXVIII) was a common year starting on Tuesday of the Gregorian calendar (see link for calendar) or a common year starting on Wednesday of the Julian calendar. ... , Halle (also called Halle an der Saale (literally Halle on the Saale, and in some historic references is not uncommonly called Saale after the river) in order to distinguish it from Halle in North Rhine-Westphalia) is the largest city in the German State of Saxony-Anhalt. ... Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... The Martin-Luther-University of Halle-Wittenberg is located in the German cities of Halle, Saxony-Anhalt and Wittenberg. ... The ETH Zurich, often called Swiss Federal Institute of Technology, is a science and technology university in the city of Zurich, Switzerland. ... There is no institution called the University of Berlin, but there are four universities in Berlin, Germany: Humboldt University of Berlin (Humboldt-Universität zu Berlin) Technical University of Berlin (Technische Universität Berlin) Free University of Berlin (Freie Universität Berlin) Berlin University of the Arts (Universität der... Set theory is the mathematical theory of sets, which represent collections of abstract objects. ... is the 62nd day of the year (63rd in leap years) in the Gregorian calendar. ... 1845 was a common year starting on Wednesday (see link for calendar). ... is the 6th day of the year in the Gregorian calendar. ... 1918 (MCMXVIII) was a common year starting on Tuesday of the Gregorian calendar (see link for calendar) or a common year starting on Wednesday of the Julian calendar. ... Leonhard Euler, considered one of the greatest mathematicians of all time A mathematician is a person whose primary area of study and research is the field of mathematics. ... Set theory is the mathematical theory of sets, which represent collections of abstract objects. ... Foundations of mathematics is a term sometimes used for certain fields of mathematics itself, namely for mathematical logic, axiomatic set theory, proof theory, model theory, and recursion theory. ... In mathematics, a bijection, bijective function, or one-to-one correspondence is a function that is both injective (one-to-one) and surjective (onto), and therefore bijections are also called one-to-one and onto. ... In set theory, an infinite set is a set that is not a finite set. ... In mathematics, a well-order (or well-ordering) on a set S is a total order on S with the property that every non-empty subset of S has a least element in this ordering. ... In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ... In mathematics, a natural number can mean either an element of the set {1, 2, 3, ...} (i. ... In Zermelo-FrÃ¤nkel set theory, Cantors theorem states that the power set (set of all subsets) of any set A has a strictly greater cardinality than that of A. Cantors theorem is obvious for finite sets, but surprisingly it holds true for infinite sets as well. ... The infinity symbol âˆž in several typefaces. ... Aleph-0, the smallest infinite cardinal In mathematics, cardinal numbers, or cardinals for short, are a generalized kind of number used to denote the size of a set, known as its cardinality. ... In set theory, ordinal, ordinal number, and transfinite ordinal number refer to a type of number introduced by Georg Cantor in 1897, to accommodate infinite sequences and to classify sets with certain kinds of order structures on them. ...

The harsh criticism has been matched by international accolades. In 1904, the Royal Society of London awarded Cantor its Sylvester Medal, the highest honor it can confer.[11] Today, the vast majority of mathematicians who are neither constructivists nor finitists accept Cantor's work on transfinite sets and arithmetic, recognizing it as a major paradigm shift.[citation needed] Cantor believed his theory of transfinite numbers had been communicated to him by God.[12] In the words of David Hilbert: "No one shall expel us from the Paradise that Cantor has created."[13] ... The Sylvester Medal is a bronze medal awarded every three years by the Royal Society for the encouragement of mathematical research. ... In the philosophy of mathematics, constructivism asserts that it is necessary to find (or construct) a mathematical object to prove that it exists. ... In the philosophy of mathematics, finitism is an extreme form of constructivism, according to which a mathematical object does not exist unless it can be constructed from natural numbers in a finite number of steps. ... Paradigm shift is the term first used by Thomas Kuhn in his 1962 book The Structure of Scientific Revolutions to describe a change in basic assumptions within the ruling theory of science. ... David Hilbert (January 23, 1862, KÃ¶nigsberg, East Prussia â€“ February 14, 1943, GÃ¶ttingen, Germany) was a German mathematician, recognized as one of the most influential and universal mathematicians of the 19th and early 20th centuries. ...

### Youth and studies

Cantor was born in 1845 in the Western merchant colony in Saint Petersburg, Russia, and brought up in the city until he was eleven. Georg, the eldest of six children, was an outstanding violinist, having inherited his parents' considerable musical and artistic talents. Cantor's father had been a member of the Saint Petersburg stock exchange; when he became ill, the family moved to Germany in 1856, first to Wiesbaden then to Frankfurt, seeking winters milder than those of Saint Petersburg. In 1860, Cantor graduated with distinction from the Realschule in Darmstadt; his exceptional skills in mathematics, trigonometry in particular, were noted. In 1862, Cantor entered the Federal Polytechnic Institute in Zurich, today the ETH Zurich, and began studying mathematics. Saint Petersburg (Russian: &#1057;&#1072;&#1085;&#1082;&#1090;-&#1055;&#1077;&#1090;&#1077;&#1088;&#1073;&#1091;&#769;&#1088;&#1075;, English transliteration: Sankt-Peterburg), colloquially known as &#1055;&#1080;&#1090;&#1077;&#1088; (transliterated Piter), formerly known as Leningrad (&#1051;&#1077;&#1085;&#1080;&#1085;&#1075;&#1088;&#1072;&#769;&#1076;, 1924&#8211;1991) and... The violin is a bowed string instrument with four strings tuned in perfect fifths. ... The Old Bourse seen from the Neva River The old Saint Petersburg Bourse is the one of the most important monuments of the Greek Revival not only in the capital of Imperial Russia but in the whole of the Russian Empire and the world. ... Wiesbaden is a city in central Germany. ... For other uses, see Frankfurt (disambiguation). ... For other uses, see Darmstadt (disambiguation). ... Wikibooks has a book on the topic of Trigonometry All of the trigonometric functions of an angle Î¸ can be constructed geometrically in terms of a unit circle centered at O. Trigonometry (from Greek trigÅnon triangle + metron measure[1]), informally called trig, is a branch of mathematics that deals with... The ETH Zurich, often called Swiss Federal Institute of Technology, is a science and technology university in the city of Zurich, Switzerland. ... Location within Switzerland   ZÃ¼rich[?] (German pronunciation IPA: ; usually spelled Zurich in English) is the largest city in Switzerland (population: 366,145 in 2004; population of urban area: 1,091,732) and capital of the canton of ZÃ¼rich. ... The ETH Zurich, often called Swiss Federal Institute of Technology, is a science and technology university in the city of Zurich, Switzerland. ...

After his father's death in 1863, leaving a substantial inheritance, Cantor shifted his studies to the University of Berlin, attending lectures by Kronecker, Karl Weierstrass and Ernst Kummer. He spent the summer of 1866 at the University of Göttingen, then and later a very important center for mathematical research. In 1867, Berlin granted him the PhD for a thesis on number theory, De aequationibus secundi gradus indeterminatis. There is no institution called the University of Berlin, but there are four universities in Berlin, Germany: Humboldt University of Berlin (Humboldt-Universität zu Berlin) Technical University of Berlin (Technische Universität Berlin) Free University of Berlin (Freie Universität Berlin) Berlin University of the Arts (Universität der... Karl Theodor Wilhelm Weierstrass (WeierstraÃŸ) (October 31, 1815 â€“ February 19, 1897) was a German mathematician who is often cited as the father of modern analysis. // Karl Weierstrass was born in Ostenfelde, Westphalia (today Germany). ... Ernst Eduard Kummer (29 January 1810 in Sorau, Brandenburg, Prussia - 14 May 1893 in Berlin, Germany) was a German mathematician. ... The Georg-August University of GÃ¶ttingen (Georg-August-UniversitÃ¤t GÃ¶ttingen, often called the Georgia Augusta) was founded in 1734 by George II, King of Great Britain and Elector of Hanover, and opened in 1737. ... Doctor of Philosophy, abbreviated Ph. ... It has been suggested that Thesis statement be merged into this article or section. ... Number theory is the branch of pure mathematics concerned with the properties of numbers in general, and integers in particular, as well as the wider classes of problems that arise from their study. ...

### Teacher and researcher

After teaching briefly in a Berlin girls' school, Cantor took up a position at the University of Halle, where he spent his entire career. He was awarded the requisite habilitation for his thesis on number theory. The Martin-Luther-University of Halle-Wittenberg is located in the German cities of Halle, Saxony-Anhalt and Wittenberg. ... Habilitation is the highest academic qualification a person can achieve by his/her own pursuit in certain European countries. ...

In 1874, Cantor married Vally Guttmann. They had six children, the last born in 1886. Cantor was able to support a family despite modest academic pay, thanks to his inheritance from his father. During his honeymoon in the Harz mountains, Cantor spent much time in mathematical discussions with Richard Dedekind, whom he befriended two years earlier while on Swiss holiday. The Harz is a mountain range in northern Germany. ... Richard Dedekind Julius Wilhelm Richard Dedekind (October 6, 1831 â€“ February 12, 1916) was a German mathematician who did important work in abstract algebra and the foundations of the real numbers. ...

Cantor was promoted to Extraordinary Professor in 1872, and made full Professor in 1879. To attain the latter rank at the age of 34 was a notable accomplishment, but Cantor desired a chair at a more prestigious university, in particular at Berlin, then the leading German university. However, his work encountered too much opposition for that to be possible.[14] Kronecker, who headed mathematics at Berlin until his death in 1891, became increasingly uncomfortable with the prospect of having Cantor as a colleague,[15] perceiving him as a "corrupter of youth" for teaching his ideas to a younger generation of mathematicians.[16] Worse yet, Kronecker, a well-established figure within the mathematical community and Cantor's former professor, fundamentally disagreed with the thrust of Cantor's work. Kronecker, now seen as one of the founders of the constructive viewpoint in mathematics, disliked much of Cantor's set theory because it asserted the existence of sets satisfying certain properties, without giving specific examples of sets whose members did indeed satisfy those properties. Cantor came to believe that Kronecker's stance would make it impossible for Cantor to ever leave Halle. The meaning of the word professor (Latin: one who claims publicly to be an expert) varies. ... In the philosophy of mathematics, constructivism asserts that it is necessary to find (or construct) a mathematical object to prove that it exists. ...

In 1881, Cantor's Halle colleague Eduard Heine died, creating a vacant chair. Halle accepted Cantor's suggestion that it be offered to Dedekind, Heinrich Weber and Franz Mertens, in that order, but each declined the chair after being offered it. Friedrich Wangerin was eventually appointed, but he was never close to Cantor. Heinrich Eduard Heine (March 15, 1821â€“October 21, 1881) was a German mathematician. ... Richard Dedekind Julius Wilhelm Richard Dedekind (October 6, 1831 â€“ February 12, 1916) was a German mathematician who did important work in abstract algebra and the foundations of the real numbers. ... Heinrich Martin Weber (1842â€“1913) was a German mathematician who specialized in algebra and number theory. ... Franz Mertens (March 20, 1840 - March 5, 1927) was a German mathematician. ...

In 1882 the mathematical correspondence between Cantor and Dedekind came to an end, apparently as a result of Dedekind's refusal to accept the chair at Halle.[17] Cantor also began another important correspondence, with Gösta Mittag-Leffler in Sweden, and soon began to publish in Mittag-Leffler's journal Acta Mathematica. But in 1885, Mittag-Leffler was concerned about the philosophical nature and new terminology in a paper Cantor had submitted to Acta.[18] He asked Cantor to withdraw the paper from Acta while it was in proof, writing that it was "… about one hundred years too soon." Cantor complied, but wrote to a third party: Magnus Gustaf (GÃ¶sta) Mittag-Leffler (16 March 1846â€“7 July 1927) was a Swedish mathematician. ...

 “ "Had Mittag-Leffler had his way, I should have to wait until the year 1984, which to me seemed too great a demand! … But of course I never want to know anything again about Acta Mathematica."[19] ”

Cantor then sharply curtailed his relationship and correspondence with Mittag-Leffler, displaying a tendency to interpret well-intentioned criticism as a deeply personal affront.

Cantor suffered his first known bout of depression in 1884.[20] Criticism of his work weighed on his mind: every one of the fifty-two letters he wrote to Gösta Mittag-Leffler in 1884 attacked Kronecker. A passage from one of these letters is revealing of the damage to Cantor's self-confidence: Magnus Gustaf (GÃ¶sta) Mittag-Leffler (16 March 1846â€“7 July 1927) was a Swedish mathematician. ...

 “ "…I don't know when I shall return to the continuation of my scientific work. At the moment I can do absolutely nothing with it, and limit myself to the most necessary duty of my lectures; how much happier I would be to be scientifically active, if only I had the necessary mental freshness."[21] ”

This emotional crisis led him to apply to lecture on philosophy rather than mathematics. He also began an intense study of Elizabethan literature in an attempt to prove that Francis Bacon wrote the plays attributed to Shakespeare (see Shakespearean authorship question); this ultimately resulted in two pamphlets, published in 1896 and 1897.[22] The philosopher Socrates about to take poison hemlock as ordered by the court. ... The term Elizabethan literature refers to the literature produced in England during the reign of Queen Elizabeth I (1558 - 1603). ... It has been suggested that Idols of the mind be merged into this article or section. ... Wikipedia does not yet have an article with this exact name. ... The frontispiece of the First Folio (1623), the first collected edition of Shakespeares plays. ...

Cantor recovered soon thereafter, and subsequently made further important contributions, including his famous diagonal argument and theorem. However, he never again attained the high level of his remarkable papers of 1874–1884. He eventually sought a reconciliation with Kronecker, which Kronecker graciously accepted. Nevertheless, the philosophical disagreements and difficulties dividing them persisted. It was once thought that Cantor's recurring bouts of depression were triggered by the opposition his work met at the hands of Kronecker.[9] While Cantor's mathematical worries and his difficulties dealing with certain people were greatly magnified by his depression, it is doubtful that they were its cause. Rather, his posthumous diagnosis of bipolarity has been accepted as the root cause of his erratic mood.[10] Cantors diagonal argument, also called the diagonalisation argument, the diagonal slash argument or the diagonal method, was published in 1891 by Georg Cantor as a proof that there are infinite sets which cannot be put into one-to-one correspondence with the infinite set of natural numbers. ... In Zermelo-FrÃ¤nkel set theory, Cantors theorem states that the power set (set of all subsets) of any set A has a strictly greater cardinality than that of A. Cantors theorem is obvious for finite sets, but surprisingly it holds true for infinite sets as well. ... For other uses, see Bipolar. ... A root cause is a cause that is at a root of an effect. ...

In 1890, Cantor was instrumental in founding the Deutsche Mathematiker-Vereinigung and chaired its first meeting in Halle in 1891; his reputation was strong enough, despite Kronecker's opposition to his work, to ensure he was elected as the first president of this society. Setting aside the animosity he felt towards Kronecker, Cantor invited him to address the meeting, but Kronecker was unable to do so because his spouse was dying at the time. The German mathematical society (german:Deutsche Mathematiker-Vereinigung - DMV) is the main professional society of german mathematicians. ...

### Late years

Cantor retired in 1913, and suffered from poverty, even malnourishment, during World War I.[26] The public celebration of his 70th birthday was canceled because of the war. He died on January 6, 1918 in the sanatorium where he had spent the final year of his life. â€œThe Great War â€ redirects here. ... is the 6th day of the year in the Gregorian calendar. ... 1918 (MCMXVIII) was a common year starting on Tuesday of the Gregorian calendar (see link for calendar) or a common year starting on Wednesday of the Julian calendar. ...

## Mathematical work

In one of his earliest papers, Cantor proved that the set of real numbers is "more numerous" than the set of natural numbers; this showed, for the first time, that there exist infinite sets of different sizes. He was also the first to appreciate the importance of one-to-one correspondences (hereinafter denoted "1-to-1") in set theory. He used this concept to define finite and infinite sets, subdividing the latter into denumerable (or countably infinite) sets and uncountable sets (nondenumerable infinite sets).[29] Please refer to Real vs. ... Natural number can mean either a positive integer (1, 2, 3, 4, ...) or a non-negative integer (0, 1, 2, 3, 4, ...). Natural numbers have two main purposes: they can be used for counting (there are 3 apples on the table), or they can be used for ordering (this is... In mathematics, the cardinality of a set is a measure of the number of elements of the set. There are two approaches to cardinality â€“ one which compares sets directly using bijections and injections, and another which uses cardinal numbers. ... In mathematics, a bijection, bijective function, or one-to-one correspondence is a function that is both injective (one-to-one) and surjective (onto), and therefore bijections are also called one-to-one and onto. ... In mathematics, a set is called finite if there is a bijection between the set and some set of the form {1, 2, ..., n} where is a natural number. ... In set theory, an infinite set is a set that is not a finite set. ... In mathematics, a countable set is a set with the same cardinality (i. ... In mathematics, an uncountable or nondenumerable set is a set which is not countable. ...

Cantor introduced fundamental constructions in set theory, such as the power set of a set A, which is the set of all possible subsets of A. He later proved that the size of the power set of A is strictly larger than the size of A, even when A is an infinite set; this result soon became known as Cantor's theorem. Cantor developed an entire theory and arithmetic of infinite sets, called cardinals and ordinals, which extended the arithmetic of the natural numbers. His notation for the cardinal numbers was the Hebrew letter $aleph$ (aleph) with a natural number subscript; for the ordinals he employed the Greek letter ω (omega). This notation is still in use today. In mathematics, given a set S, the power set (or powerset) of S, written or 2S, is the set of all subsets of S. In axiomatic set theory (as developed e. ... A is a subset of B, and B is a superset of A. In mathematics, especially in set theory, the terms, subset, superset and proper (or strict) subset or superset are used to describe the relation, called inclusion, of one set being contained inside another set. ... In Zermelo-FrÃ¤nkel set theory, Cantors theorem states that the power set (set of all subsets) of any set A has a strictly greater cardinality than that of A. Cantors theorem is obvious for finite sets, but surprisingly it holds true for infinite sets as well. ... In the mathematical field of set theory, there are three usual operations on ordinals: addition, multiplication, and (ordinal) exponentiation. ... Aleph-0, the smallest infinite cardinal In mathematics, cardinal numbers, or cardinals for short, are a generalized kind of number used to denote the size of a set, known as its cardinality. ... Ordinal numbers, or ordinals for short, are numbers used to denote the position in an ordered sequence: first, second, third, fourth, etc. ... In the branch of mathematics known as set theory, the aleph numbers are a sequence of numbers used to represent the cardinality (or size) of infinite sets. ... Look up Î©, Ï‰ in Wiktionary, the free dictionary. ...

### Number theory and function theory

Cantor's first ten papers were on number theory, his thesis topic. At the suggestion of Eduard Heine, the Professor at Halle, Cantor turned to analysis. Heine proposed that Cantor solve an open problem that had eluded Dirichlet, Lipschitz, Bernhard Riemann, and Heine himself: the uniqueness of the representation of a function by trigonometric series. Cantor solved this difficult problem in 1869. Between 1870 and 1872, Cantor published more papers on trigonometric series, including one defining irrational numbers as convergent sequences of rational numbers. Dedekind, whom Cantor befriended in 1872, cited this paper later that year, in the paper where he first set out his celebrated definition of real numbers by Dedekind cuts. Number theory is the branch of pure mathematics concerned with the properties of numbers in general, and integers in particular, as well as the wider classes of problems that arise from their study. ... Heinrich Eduard Heine (March 15, 1821â€“October 21, 1881) was a German mathematician. ... Analysis has its beginnings in the rigorous formulation of calculus. ... An open problem is a problem that can be formally stated and for which a solution is known to exist but which has not yet been solved. ... Johann Peter Gustav Lejeune Dirichlet (February 13, 1805 â€“ May 5, 1859) was a German mathematician credited with the modern formal definition of a function. ... Rudolf Otto Sigismund Lipschitz (May 14, 1832 â€“ October 7, 1903) was a German mathematician and Professor at the University of Bonn from 1864. ... Bernhard Riemann. ... Graph of example function, The mathematical concept of a function expresses the intuitive idea of deterministic dependence between two quantities, one of which is viewed as primary (the independent variable, argument of the function, or its input) and the other as secondary (the value of the function, or output). A... In mathematics, a Fourier series of a periodic function, named in honor of Joseph Fourier (1768-1830), represents the function as a sum of periodic functions of the form where e is Eulers number and i the imaginary unit. ... In mathematics, an irrational number is any real number that is not a rational number â€” that is, it is a number which cannot be expressed as a fraction m/n, where m and n are integers. ... In functional analysis and related areas of mathematics, a sequence space is an important class of function space. ... In mathematics, a rational number is a number which can be expressed as a ratio of two integers. ... In mathematics, a Dedekind cut, named after Richard Dedekind, in a totally ordered set S is a partition of it, (A, B), such that A is closed downwards (meaning that for all a in A, x â‰¤ a implies that x is in A as well) and B is closed upwards...

### Set theory

An illustration of Cantor's diagonal argument for the existence of uncountable sets.[30] The sequence at the bottom cannot occur anywhere in the infinite list of sequences above.

The beginning of set theory as a branch of mathematics is often marked by the publication of Cantor's 1874 paper, "Über eine Eigenschaft des Inbegriffes aller reelen algebraischen Zahlen" ("On a Characteristic Property of All Real Algebraic Numbers").[27] The paper, published in Crelle's Journal thanks to Dedekind's support (and despite Kronecker's opposition), was the first to formulate a mathematically rigorous proof that there was more than one kind of infinity. This demonstration is a centerpiece of his legacy as a mathematician, helping lay the groundwork for both calculus and the analysis of the continuum of real numbers.[31] Previously, all infinite collections had been implicitly assumed to be equinumerous (that is, of "the same size" or having the same number of elements).[32] He then proved that the real numbers were not countable, albeit employing a proof more complex than the remarkably elegant and justly celebrated diagonal argument he set out in 1891.[33] Prior to this, he had already proven that the set of rational numbers is denumerable. Image File history File links This is a lossless scalable vector image. ... Image File history File links This is a lossless scalable vector image. ... Cantors diagonal argument, also called the diagonalisation argument, the diagonal slash argument or the diagonal method, was published in 1891 by Georg Cantor as a proof that there are infinite sets which cannot be put into one-to-one correspondence with the infinite set of natural numbers. ... In mathematics, an uncountable or nondenumerable set is a set which is not countable. ... Crelles Journal, or just Crelle, is the common name for the Journal für die reine und angewandte Mathematik founded by August Leopold Crelle. ... Contrary to what most mathematicians believe, Georg Cantors first proof that the set of all real numbers is uncountable was not his famous diagonal argument, and did not mention decimal expansions or any other numeral system. ... Cantors diagonal argument, also called the diagonalisation argument, the diagonal slash argument or the diagonal method, was published in 1891 by Georg Cantor as a proof that there are infinite sets which cannot be put into one-to-one correspondence with the infinite set of natural numbers. ... In mathematics, a rational number is a number which can be expressed as a ratio of two integers. ...

Joseph Liouville had established the existence of transcendental numbers in 1851, and Cantor's paper established that the set of transcendental numbers is nondenumerable. The logic is as follows: Cantor had shown that the union of two denumerable sets must be denumerable. The set of all real numbers is equal to the union of the set of algebraic numbers with the set of transcendental numbers (that is, every real number must be either algebraic or transcendental). The 1874 paper showed that the algebraic numbers (that is, the roots of polynomial equations with integer coefficients), were denumerable. In contrast, Cantor had also just shown that the real numbers were not denumerable. If transcendental numbers were denumerable, then the result of their union with algebraic numbers would also be denumerable. Since their union (which equals the set of all real numbers) is nondenumerable, it logically follows that the transcendentals must be nondenumerable. The transcendentals have the same "power" (see below) as the reals, and "almost all" real numbers must be transcendental. Cantor remarked that he had effectively reproved a theorem, due to Liouville, to the effect that there are infinitely many transcendental numbers in each interval. Joseph Liouville (born March 24, 1809, died September 8, 1882) was a French mathematician. ... In mathematics, a transcendental number is any complex number that is not algebraic, that is, not the solution of a non-zero polynomial equation with integer (or, equivalently, rational) coefficients. ... In mathematics, an algebraic number is any number that is a root of an algebraic equation, a non-zero polynomial with integer (or equivalently, rational) coefficients. ... ROOT is an object-oriented software package developed by CERN. It was originally designed for particle physics data analysis and contains several features specific to this field, but it is also commonly used in other applications such as astronomy and data mining. ... In mathematics, a polynomial is an expression that is constructed from one or more variables and constants, using only the operations of addition, subtraction, multiplication, and constant positive whole number exponents. ... The integers are commonly denoted by the above symbol. ... In mathematics, a coefficient is a constant multiplicative factor of a certain object. ... In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ... Joseph Liouville (born March 24, 1809, died September 8, 1882) was a French mathematician. ...

Between 1879 and 1884, Cantor published a series of six articles in Mathematische Annalen that together formed an introduction to his set theory. At the same time, there was growing opposition to Cantor's ideas, led by Kronecker, who admitted mathematical concepts only if they could be constructed in a finite number of steps from the natural numbers, which he took as intuitively given. For Kronecker, Cantor's hierarchy of infinities was inadmissible, since accepting the concept of actual infinity would open the door to paradoxes which would challenge the validity of mathematics as a whole.[34] Cantor also discovered the Cantor set during this period. The Mathematische Annalen is a German mathematical research journal published by Springer-Verlag. ... In the philosophy of mathematics, finitism is an extreme form of constructivism, according to which a mathematical object does not exist unless it can be constructed from natural numbers in a finite number of steps. ... Infinity is a word carrying a number of different meanings in mathematics, philosophy, theology and everyday life. ... The Cantor set, introduced by German mathematician Georg Cantor, is a remarkable construction involving only the real numbers between zero and one. ...

The fifth paper in this series, "Grundlagen einer allgemeinen Mannigfaltigkeitslehre" ("Foundations of a General Theory of Aggregates"), published in 1883, was the most important of the six and was also published as a separate monograph. It contained Cantor's reply to his critics and showed how the transfinite numbers were a systematic extension of the natural numbers. It begins by defining well-ordered sets. Ordinal numbers are then introduced as the order types of well-ordered sets. Cantor then defines the addition and multiplication of the cardinal and ordinal numbers. In 1885, Cantor extended his theory of order types so that the ordinal numbers simply became a special case of order types. A monograph is a scholarly book or a treatise on a single subject or a group of related subjects. ... Transfinite numbers, also known as infinite numbers, are numbers that are not finite. ... In mathematics, a well-order (or well-ordering) on a set S is a total order on S with the property that every non-empty subset of S has a least element in this ordering. ... Ordinal numbers, or ordinals for short, are numbers used to denote the position in an ordered sequence: first, second, third, fourth, etc. ... Aleph-0, the smallest infinite cardinal In mathematics, cardinal numbers, or cardinals for short, are a generalized kind of number used to denote the size of a set, known as its cardinality. ...

In 1891, he published a paper containing his elegant "diagonal argument" for the existence of an uncountable set. He applied the same idea to prove Cantor's theorem: the cardinality of the power set of a set A is strictly larger than the cardinality of A. This established the richness of the hierarchy of infinite sets, and of the cardinal and ordinal arithmetic that Cantor had defined. His argument is fundamental in the solution of the Halting problem and the proof of Gödel's first incompleteness theorem. In Zermelo-FrÃ¤nkel set theory, Cantors theorem states that the power set (set of all subsets) of any set A has a strictly greater cardinality than that of A. Cantors theorem is obvious for finite sets, but surprisingly it holds true for infinite sets as well. ... In mathematics, the cardinality of a set is a measure of the number of elements of the set. There are two approaches to cardinality â€“ one which compares sets directly using bijections and injections, and another which uses cardinal numbers. ... Aleph-0, the smallest infinite cardinal In mathematics, cardinal numbers, or cardinals for short, are a generalized kind of number used to denote the size of a set. ... In the mathematical field of set theory, there are three usual operations on ordinals: addition, multiplication, and (ordinal) exponentiation. ... In computability theory the halting problem is a decision problem which can be informally stated as follows: Given a description of a program and a finite input, decide whether the program finishes running or will run forever, given that input. ... GÃ¶dels incompleteness theorem - Wikipedia /**/ @import /skins-1. ...

In 1895 and 1897, Cantor published a two-part paper in Mathematische Annalen under Felix Klein's editorship; these were his last significant papers on set theory.[35] The first paper begins by defining set, subset, etc., in ways that would be largely acceptable now. The cardinal and ordinal arithmetic are reviewed. Cantor wanted the second paper to include a proof of the continuum hypothesis, but had to settle for expositing his theory of well-ordered sets and ordinal numbers. Cantor attempts to prove that if A and B are sets with A equivalent to a subset of B and B equivalent to a subset of A, then A and B are equivalent. Ernst Schroeder had stated this theorem a bit earlier, but his proof, as well as Cantor's, was flawed. Felix Bernstein supplied a correct proof in his 1898 PhD thesis; hence the name Cantor-Schroeder-Bernstein theorem. The Mathematische Annalen is a German mathematical research journal published by Springer-Verlag. ... Felix Christian Klein (April 25, 1849, DÃ¼sseldorf, Germany â€“ June 22, 1925, GÃ¶ttingen) was a German mathematician, known for his work in group theory, function theory, non-Euclidean geometry, and on the connections between geometry and group theory. ... A is a subset of B, and B is a superset of A. In mathematics, especially in set theory, the terms, subset, superset and proper (or strict) subset or superset are used to describe the relation, called inclusion, of one set being contained inside another set. ... In mathematics, a well-order (or well-ordering) on a set S is a total order on S with the property that every non-empty subset of S has a least element in this ordering. ... Ernst Schröder (25 November 1841 - 16 June 1902) was the most significant representative of the algebraic logic school in Germany in the second half of the nineteenth century. ... Felix Bernstein (February 24, 1878, Halle, Germany â€“ December 3, 1956, Zurich, Switzerland) was a German mathematician known for developing a theorem of the equivalence of sets in 1897, and less well known for demonstrating the correct blood group inheritance pattern of multiple alleles at one locus in 1924 through statistical... In set theory, the Cantorâ€“Bernsteinâ€“Schroeder theorem, named after Georg Cantor, Felix Bernstein, and Ernst SchrÃ¶der, states that, if there exist injective functions f : A â†’ B and g : B â†’ A between the sets A and B, then there exists a bijective function h : A â†’ B. In terms of...

#### One-to-one correspondence

Main article: Bijection
A bijective function.

Cantor's 1874 Crelle paper was the first to invoke the notion of a 1-to-1 correspondence, though he did not use that phrase. He then began looking for a 1-to-1 correspondence between the points of the unit square and the points of a unit line segment. In an 1877 letter to Dedekind, Cantor proved a far stronger result: for any positive integer n, there exists a 1-to-1 correspondence between the points on the unit line segment and all of the points in an n-dimensional space. About this discovery Cantor famously wrote to Dedekind: "Je le vois, mais je ne le crois pas!" ("I see it, but I don't believe it!")[36] The result that he found so astonishing has implications for geometry and the notion of dimension. A bijective function. ... Image File history File links Bijection. ... Image File history File links Bijection. ... A bijective function. ... The unit square in a Cartesian coordinate system with coordinates (x,y) is defined as the square consisting of the points where both x and y lie in the unit interval from 0 to 1. ... The geometric definition of a line segment In geometry, a line segment is a part of a line that is bounded by two end points, and contains every point on the line between its end points. ... In mathematics, an n-dimensional space is a vector space or manifold whose dimension is n, or in some cases, a point in, or subset of, such a space. ... 2-dimensional renderings (ie. ...

In 1878, Cantor submitted another paper to Crelle's Journal, in which he defined precisely the concept of a 1-to-1 correspondence, and introduced the notion of "power" (a term he took from Jakob Steiner) or "equivalence" of sets: two sets are equivalent (have the same power) if there exists a 1-to-1 correspondence between them. Cantor defined denumerable (or countably infinite) sets as sets which can be put into a 1-to-1 correspondence with the natural numbers, and proved that the rational numbers are denumerable. He also proved that n-dimensional Euclidean space Rn has the same power as the real numbers R, as does a countably infinite product of copies of R. While he made free use of countability as a concept, he did not write the word "countable" until 1883. Cantor also discussed his thinking about dimension, stressing that his mapping between the unit interval and the unit square was not a continuous one. In mathematics, the cardinality of a set is a measure of the number of elements of the set. There are two approaches to cardinality â€“ one which compares sets directly using bijections and injections, and another which uses cardinal numbers. ... Jakob Steiner (18 March 1796 â€“ April 1, 1863) was a Swiss mathematician. ... In mathematics, an equivalence relation is a binary relation between two elements of a set which groups them together as being equivalent in some way. ... In mathematics, a countable set is a set with the same cardinality (i. ... Natural number can mean either a positive integer (1, 2, 3, 4, ...) or a non-negative integer (0, 1, 2, 3, 4, ...). Natural numbers have two main purposes: they can be used for counting (there are 3 apples on the table), or they can be used for ordering (this is... Around 300 BC, the Greek mathematician Euclid laid down the rules of what has now come to be called Euclidean geometry, which is the study of the relationships between angles and distances in space. ... Please refer to Real vs. ... 2-dimensional renderings (ie. ... In mathematics and related technical fields, the term map or mapping is often a synonym for function. ... In mathematics, the unit interval is the interval [0,1], that is the set of all real numbers x such that zero is less than or equal to x and x is less than or equal to one. ... In mathematics, a continuous function is a function for which, intuitively, small changes in the input result in small changes in the output. ...

This paper, like the 1874 paper, displeased Kronecker, and Cantor wanted to withdraw it; however, Dedekind persuaded him not to do so and Weierstrass also supported its publication.[37] Nevertheless, Cantor never again submitted anything to Crelle. Karl Theodor Wilhelm Weierstrass (WeierstraÃŸ) (October 31, 1815 â€“ February 19, 1897) was a German mathematician who is often cited as the father of modern analysis. // Karl Weierstrass was born in Ostenfelde, Westphalia (today Germany). ...

#### Continuum hypothesis

Main article: Continuum hypothesis

Cantor was the first to formulate what later came to be known as the continuum hypothesis or CH: there exists no set whose power is greater than that of the naturals and less than that of the reals (or equivalently, the cardinality of the reals is exactly aleph-one, rather than just at least aleph-one). Cantor believed the continuum hypothesis to be true and tried for many years to prove it, in vain. His inability to prove the continuum hypothesis caused him considerable anxiety.[9] In mathematics, the continuum hypothesis is a hypothesis about the possible sizes of infinite sets. ... In mathematics, the continuum hypothesis is a hypothesis about the possible sizes of infinite sets. ... In mathematics, a proof is a demonstration that, assuming certain axioms, some statement is necessarily true. ...

The difficulty Cantor had in proving the continuum hypothesis has been underscored by later developments in the field of mathematics: a 1940 result by Gödel and a 1963 one by Paul Cohen together imply that the continuum hypothesis can neither be proved nor disproved using standard Zermelo-Fraenkel set theory plus the axiom of choice (the combination referred to as "ZFC").[38] Kurt GÃ¶del (IPA: ) (April 28, 1906 BrÃ¼nn, Austria-Hungary (now Brno, Czech Republic) â€“ January 14, 1978 Princeton, New Jersey) was an Austrian American mathematician and philosopher. ... Paul Joseph Cohen (April 2, 1934 â€“ March 23, 2007[1]) was an American mathematician. ... Zermelo-Fraenkel set theory, commonly abbreviated ZFC, is the most common form of axiomatic set theory, and as such is the most common foundation of mathematics. ... In mathematics, the axiom of choice, or AC, is an axiom of set theory. ...

Discussions of set-theoretic paradoxes began to appear around the end of the nineteenth century. Some of these implied fundamental problems with Cantor's set theory program.[39] In an 1897 paper on an unrelated topic, Cesare Burali-Forti set out the first such paradox, the Burali-Forti paradox: the ordinal number of the set of all ordinals must be an ordinal and this leads to a contradiction. Cantor discovered this paradox in 1895, and described it in an 1896 letter to Hilbert. Criticism mounted to the point where Cantor launched counter-arguments in 1903, intended to defend the basic tenets of his set theory.[11] Look up paradox in Wiktionary, the free dictionary. ... Cesare Burali-Forti (13 August 1861 - 21 January 1931) was an Italian mathematician. ... In set theory, a field of mathematics, the Burali-Forti paradox demonstrates that naÃ¯vely constructing the set of all ordinal numbers leads to a contradiction and therefore shows an antinomy in a system that allows its construction. ... In set theory, ordinal, ordinal number, and transfinite ordinal number refer to a type of number introduced by Georg Cantor in 1897, to accommodate infinite sequences and to classify sets with certain kinds of order structures on them. ... David Hilbert (January 23, 1862, KÃ¶nigsberg, East Prussia â€“ February 14, 1943, GÃ¶ttingen, Germany) was a German mathematician, recognized as one of the most influential and universal mathematicians of the 19th and early 20th centuries. ...

In 1899, Cantor discovered his eponymous paradox: what is the cardinal number of the set of all sets? Clearly it must be the greatest possible cardinal. Yet for any set A, the cardinal number of the power set of A is strictly larger than the cardinal number of A (this fact is now known as Cantor's theorem). This paradox, together with Burali-Forti's, led Cantor to formulate a concept called limitation of size,[40] according to which the collection of all ordinals, or of all sets, was an "inconsistent multiplicity" that was "too large" to be a set. Such collections later became known as proper classes. Cantors paradox, also known as the paradox of the greatest cardinal, demonstrates that there is no cardinal greater than all other cardinalsâ€”that the class of cardinal numbers is infinite. ... In Zermelo-FrÃ¤nkel set theory, Cantors theorem states that the power set (set of all subsets) of any set A has a strictly greater cardinality than that of A. Cantors theorem is obvious for finite sets, but surprisingly it holds true for infinite sets as well. ... This article or section is in need of attention from an expert on the subject. ... In set theory and its applications throughout mathematics, a class is a collection of sets (or sometimes other mathematical objects) that can be unambiguously defined by a property that all its members share. ...

One common view among mathematicians is that these paradoxes, together with Russell's paradox, demonstrate that it is not possible to take a "naive", or non-axiomatic, approach to set theory without risking contradiction, and it is certain that they were among the motivations for Zermelo and others to produce axiomatizations of set theory. Others note, however, that the paradoxes do not obtain in an informal view motivated by the iterative hierarchy, which can be seen as explaining the idea of limitation of size. Some also question whether the Fregean formulation of naive set theory (which was the system directly refuted by the Russell paradox) is really a faithful interpretation of the Cantorian conception.[41] Part of the foundation of mathematics, Russells paradox (also known as Russells antinomy), discovered by Bertrand Russell in 1901, showed that the naive set theory of Frege leads to a contradiction. ... Ernst Friedrich Ferdinand Zermelo (July 27, 1871, Berlin, German Empire â€“ May 21, 1953, Freiburg im Breisgau, West Germany) was a German mathematician, whose work has major implications for the foundations of mathematics and hence on philosophy. ... This article or section is in need of attention from an expert on the subject. ... In axiomatic set theory and related branches of mathematics, the Von Neumann universe, or Von Neumann hierarchy of sets is the class of all sets, divided into a transfinite hierarchy of individual sets. ... Friedrich Ludwig Gottlob Frege (8 November 1848, Wismar â€“ 26 July 1925, IPA: ) was a German mathematician who became a logician and philosopher. ... In abstract mathematics, naive set theory[1] was the first development of set theory, which was later to be framed more carefully as axiomatic set theory. ...

## Philosophy, religion and Cantor's mathematics

The concept of the existence of an actual infinity was an important shared concern within the realms of mathematics, philosophy and religion. Preserving the orthodoxy of the relationship between God and mathematics, although not in the same form as held by his critics, was long a concern of Cantor's.[42] He directly addressed this intersection between these disciplines in the introduction to his Grundlagen einer allgemeinen Mannigfaltigkeitslehre, where he stressed the connection between his view of the infinite and the philosophical one.[43] To Cantor, his mathematical views were intrinsically linked to their philosophical and theological implications—he identified the Absolute Infinite with God,[44] and he considered his work on transfinite numbers to have been directly communicated to him by God, who had chosen Cantor to reveal them to the world.[45] Infinity is a word carrying a number of different meanings in mathematics, philosophy, theology and everyday life. ... â€œOrthodoxâ€ redirects here. ... The Absolute Infinite is Georg Cantors concept of an infinity that transcended the transfinite numbers. ... This article discusses the term God in the context of monotheism and henotheism. ...

Debate among mathematicians grew out of opposing views in the philosophy of mathematics regarding the nature of actual infinity. Some held to the view that infinity was an abstraction which was not mathematically legitimate, and denied its existence.[46] Mathematicians from three major schools of thought (constructivism and its two offshoots, intuitionism and finitism) opposed Cantor's theories in this matter. For constructivists such as Kronecker, this rejection of actual infinity stems from fundamental disagreement with the idea that nonconstructive proofs such as Cantor's diagonal argument are sufficient proof that something exists, holding instead that constructive proofs are required. Intuitionism also rejects the idea that actual infinity is an expression of any sort of reality, but arrive at the decision via a different route than constructivism. Firstly, Cantor's argument rests on logic to prove the existence of transfinite numbers as an actual mathematical entity, whereas intuitionists hold that mathematical entities cannot be reduced to logical propositions, originating instead in the intuitions of the mind.[6] Secondly, the notion of infinity as an expression of reality is itself disallowed in intuitionism, since the human mind cannot intuitively construct an infinite set.[47] Mathematicians such as Brouwer and especially Poincaré adopted an intuitionist stance against Cantor's work. Citing the paradoxes of set theory as an example of its fundamentally flawed nature, Poincaré held that "most of the ideas of Cantorian set theory should be banished from mathematics once and for all."[6] Finally, Wittgenstein's attacks were finitist: he believed that Cantor's diagonal argument conflated the intension of a set of cardinal or real numbers with its extension, thus conflating the concept of rules for generating a set with an actual set.[8] // Philosophy of mathematics is the branch of philosophy that studies the philosophical assumptions, foundations, and implications of mathematics. ... In the philosophy of mathematics, constructivism asserts that it is necessary to find (or construct) a mathematical object to prove that it exists. ... In the philosophy of mathematics, intuitionism, or neointuitionism (opposed to preintuitionism), is an approach to mathematics as the constructive mental activity of humans. ... In the philosophy of mathematics, finitism is an extreme form of constructivism, according to which a mathematical object does not exist unless it can be constructed from natural numbers in a finite number of steps. ... In mathematics, a nonconstructive proof, is a mathematical proof that purports to demonstrate the existence of something, but which does not say how to construct it. ... In mathematics, a constructive proof is a method of proof that demonstrates the existence of a mathematical object with certain properties by creating or providing a method for creating such an object. ... Luitzen Egbertus Jan Brouwer (February 27, 1881 - December 2, 1966), usually cited as L. E. J. Brouwer, was a Dutch mathematician, a graduate of the University of Amsterdam, who worked in topology, set theory, measure theory and complex analysis. ... Henri PoincarÃ©, photograph from the frontispiece of the 1913 edition of Last Thoughts Jules Henri PoincarÃ© (April 29, 1854 â€“ July 17, 1912), generally known as Henri PoincarÃ©, was one of Frances greatest mathematicians, theoretical scientists and a philosopher of science. ... In the philosophy of mathematics, intuitionism, or neointuitionism (opposed to preintuitionism), is an approach to mathematics as the constructive mental activity of humans. ... Ludwig Josef Johann Wittgenstein (IPA: ) (April 26, 1889 in Vienna, Austria â€“ April 29, 1951 in Cambridge, England) was an Austrian philosopher who contributed several ground-breaking ideas to philosophy, primarily in the foundations of logic, the philosophy of mathematics, the philosophy of language, and the philosophy of mind. ... Intension refers to the meanings or characteristics encompassed by a given word. ... In any of several studies that treat the use of signs, for example, linguistics, logic, mathematics, semantics, and semiotics, the extension of a concept, idea, or sign consists of the things to which it applies, in contrast with its comprehension or intension, which consists very roughly of the ideas, properties...

Christian theologians saw Cantor's work as a challenge to the uniqueness of the absolute infinity in the nature of God.[48] In particular, Neo-Thomist thinkers saw the existence of an actual infinity that consisted of something other than God as jeopardizing "God's exclusive claim to supreme infinity".[49] Cantor strongly believed that this view was a misinterpretation of infinity, and was convinced that set theory could help correct this mistake:[50] Neo-Scholasticism is the development of the Scholasticism of the Middle Ages during the latter half of the nineteenth century. ...

 “ "…the transfinite species are just as much at the disposal of the intentions of the Creator and His absolute boundless will as are the finite numbers."[51] ”

Cantor also believed that his theory of transfinite numbers ran counter to both materialism and determinism—and was shocked when he realized that he was the only faculty member at Halle who did not hold to deterministic philosophical beliefs.[52] In philosophy, materialism is that form of physicalism which holds that the only thing that can truly be said to exist is matter; that fundamentally, all things are composed of material and all phenomena are the result of material interactions; that matter is the only substance. ... Determinism is the philosophical proposition that every event, including human cognition and behavior, decision and action, is causally determined by an unbroken chain of prior occurrences. ...

In 1888, Cantor published his correspondence with several philosophers on the philosophical implications of his set theory. In an extensive attempt to persuade Christian thinkers and authorities to adopt his views, Cantor had corresponded with Christian philosophers such as Tilman Pesch and Joseph Hontheim,[53] as well as theologians such as Cardinal Johannes Franzelin, who once replied by equating the theory of transfinite numbers with pantheism.[5] Cantor even sent one letter directly to Pope Leo XIII himself, and addressed several pamphlets to him.[50] Tilman Pesch (b. ... Pantheism (Greek: Ï€Î¬Î½ ( pan ) = all and Î¸ÎµÏŒÏ‚ ( theos ) = God) literally means God is All and All is God. It is the view that everything is of an all-encompassing immanent abstract God; or that the universe, or nature, and God are equivalent. ... Pope Leo XIII (March 2, 1810â€”July 20, 1903), born Vincenzo Gioacchino Raffaele Luigi Pecci, was the 256th Pope of the Roman Catholic Church, reigning from 1878 to 1903, succeeding Pope Pius IX. Reigning until the age of 93, he was the oldest pope, and had the third longest pontificate...

Cantor's philosophy on the nature of numbers led him to affirm a belief in the freedom of mathematics to posit and prove concepts apart from the realm of physical phenomena, as expressions within an internal reality. The only restrictions on this metaphysical system are that all mathematical concepts must be devoid of internal contradiction, and that they follow from existing definitions, axioms, and theorems. This belief is summarized in his famous assertion that "the essence of mathematics is its freedom."[54] These ideas parallel those of Edmund Husserl.[55] Plato (Left) and Aristotle (right), by Raphael (Stanza della Segnatura, Rome) Metaphysics is the branch of philosophy concerned with explaining the ultimate nature of reality, being, and the world. ... Edmund Gustav Albrecht Husserl (April 8, 1859, ProstÄ›jov â€“ April 26, 1938, Freiburg) was a German philosopher, known as the father of phenomenology. ...

Cantor's 1883 paper reveals that he was well aware of the opposition his ideas were encountering: In this article, Cantors Theory refers to the pre-formal ideas about set theory introduced by Georg Cantor in the latter part of the nineteenth century. ...

 “ "…I realize that in this undertaking I place myself in a certain opposition to views widely held concerning the mathematical infinite and to opinions frequently defended on the nature of numbers."[56] ”

Hence he devotes much space to justifying his earlier work, asserting that mathematical concepts may be freely introduced as long as they are free of contradiction and defined in terms of previously accepted concepts. He also cites Aristotle, Descartes, Berkeley, Leibniz, and Bolzano on infinity. Broadly speaking, a contradiction is an incompatibility between two or more statements, ideas, or actions. ... Aristotle (Greek: AristotÃ©lÄ“s) (384 BC â€“ 322 BC) was a Greek philosopher, a student of Plato and teacher of Alexander the Great. ... RenÃ© Descartes (French IPA: ) (March 31, 1596 â€“ February 11, 1650), also known as Renatus Cartesius (latinized form), was a highly influential French philosopher, mathematician, scientist, and writer. ... George Berkeley (IPA: , Bark-Lee) (12 March 1685 â€“ 14 January 1753), also known as Bishop Berkeley, was an influential Irish philosopher whose primary philosophical achievement is the advancement of a theory he called immaterialism (later referred to as subjective idealism by others). ... â€œLeibnizâ€ redirects here. ... Bernard Bolzano Bernard (Bernhard) Placidus Johann Nepomuk Bolzano (October 5, 1781 â€“ December 18, 1848) was a Bohemian mathematician, theologian, philosopher, logician and antimilitarist of German mother tongue. ...

## Cantor's ancestry

Cantor's paternal grandparents were from Copenhagen, and fled to Russia from the disruption of the Napoleonic Wars. Cantor himself called them "Israelites". However, there is no direct evidence on whether his grandparents practiced Judaism; there is very little direct information on them of any kind.[57] Jakob Cantor, Cantor's grandfather, gave his children Christian saints' names. Further, several of his grandmother's relatives were in the Czarist civil service, which would not welcome Jews, unless they, or their ancestors, converted to Orthodox Christianity. Cantor's father, Georg Woldemar Cantor, was educated in the Lutheran mission in Saint Petersburg, and his correspondence with his son shows both of them as devout Lutherans. His mother, Maria Anna Böhm, was an Austrian born in Saint Petersburg and baptized Roman Catholic; she converted to Protestantism upon marriage. However, there is a letter from Cantor's brother Louis to their mother, saying "We are the descendants of Jews", which could imply that she was of Jewish ancestry.[58] For other uses, see Copenhagen (disambiguation). ... Combatants Austria[1] Portugal Prussia[1] Russia[2] Sicily  Spain[3]  Sweden United Kingdom[4] French Empire Holland Italy Naples [5] Duchy of Warsaw Bavaria[6] Saxony[7] Denmark-Norway [8] Commanders Archduke Charles Prince Schwarzenberg Karl Mack von Leiberich JoÃ£o Francisco de Saldanha Oliveira e Daun Gebhard von... This article or section does not cite its references or sources. ... Topics in Christianity Movements Â· Denominations Â· Other religions Ecumenism Â· Preaching Â· Prayer Music Â· Liturgy Â· Calendar Symbols Â· Art Â· Criticism Important figures Apostle Paul Â· Church Fathers Constantine Â· Athanasius Â· Augustine Anselm Â· Aquinas Â· Palamas Â· Luther Calvin Â· Wesley Arius Â· Marcion of Sinope Archbishop of Canterbury Â· Catholic Pope Coptic Pope Â· Ecumenical Patriarch Christianity Portal This box:      Christianity is... In traditional Christian iconography, Saints are often depicted as having halos. ... Lutheranism is a major branch of Protestant Christianity that follows the teachings of the sixteenth-century German reformer Martin Luther. ... â€œCatholic Churchâ€ redirects here. ... Protestantism encompasses the forms of Christian faith and practice that originated with the doctrines of the Reformation. ...

Thus Cantor was not himself Jewish by faith, but has nevertheless been called variously German, Jewish,[59] Russian, and Danish. This article or section does not cite its references or sources. ...

## Historiography

Until the 1970s, the chief academic publications on Cantor were two short monographs by Schönflies (1927)—largely the correspondence with Mittag-Leffler—and Fraenkel (1930). Both were at second and third hand; neither had much on his personal life. The gap was largely filled by Eric Temple Bell's Men of Mathematics (1937), which one of Cantor's modern biographers describes as "perhaps the most widely read modern book on the history of mathematics"; and as "one of the worst".[60] Bell presents Cantor's relationship with his father as Oedipal, Cantor's differences with Kronecker as a quarrel between two Jews, and Cantor's madness as Romantic despair over his failure to win acceptance for his mathematics, and fills the picture with stereotypes. Grattan-Guinness (1971) found that none of these claims were true, but they may be found in many books of the intervening period, owing to the absence of any other narrative. There are other legends, independent of Bell—including one that labels Cantor's father a foundling, shipped to St Petersburg by unknown parents.[61] Arthur Moritz SchÃ¶nflies (April 17, 1853 Landsberg an der Warthe(GorzÃ³w) â€“ May 27, 1928) was a German mathematician, known for his contributions to the application of group theory to crystallography, and for work in topology. ... For other persons named Eric Bell, see Eric Bell (disambiguation). ... Men of Mathematics by E.T. Bell Men of Mathematics is a well-known book on the history of mathematics written by the mathematician E.T. Bell. ... For a list of biographies of mathematicians, see list of mathematicians. ...

• Cantor's back-and-forth method
• Cantor function
• Heine–Cantor theorem
• Cantor medal—award by the Deutsche Mathematiker-Vereinigung in honor of Georg Cantor.

In mathematical logic, especially set theory and model theory, Cantors back-and-forth method, named after Georg Cantor, is a method for showing isomorphism between countably infinite structures satisfying specified conditions. ... In mathematics, the Cantor function is a function c : [0,1] &#8594; [0,1] defined as follows: Express x in base 3. ... In mathematics, the Heineâ€“Cantor theorem, named after Eduard Heine and Georg Cantor, states that if M is a compact metric space, then every continuous function f : M â†’ N where N is a metric space is uniformly continuous. ... The Cantor medal of the Deutsche Mathematiker-Vereinigung is named in honor of Georg Cantor. ... The German mathematical society (german:Deutsche Mathematiker-Vereinigung - DMV) is the main professional society of german mathematicians. ...

## Notes

1. ^ In the Gregorian calendar (Grattan-Guinness 2000, p. 351). Some modern Russian sources give February 19, 1845, the equivalent date according to the Julian calendar, which was in use in Saint Petersburg at the time.
3. ^ Dauben 2004, p. 1.
4. ^ Dauben, 1977, p. 86; Dauben, 1979, pp. 120 & 143.
5. ^ a b Dauben, 1977, p. 102.
6. ^ a b c Dauben 1979, p. 266.
7. ^ Dauben 2004, p. 1. See also Dauben 1977, p. 89 15n.
8. ^ a b Rodych 2007
9. ^ a b c Dauben 1979, p. 280:"…the tradition made popular by [Arthur Moritz Schönflies] blamed Kronecker's persistent criticism and Cantor's inability to confirm his continuum hypothesis" for Cantor's recurring bouts of depression.
10. ^ a b Dauben 2004, p. 1. Text includes a 1964 quote from psychiatrist Karl Pollitt, one of Cantor's examining physicians at Halle Nervenklinik, referring to Cantor's mental illness as "cyclic manic-depression".
11. ^ a b c Dauben 1979, p. 248.
12. ^ Dauben 2004, pp. 8, 11 & 12–13.
13. ^ a b Reid 1996, p. 177.
14. ^ Dauben 1979, p. 163.
15. ^ Dauben 1979, p. 34.
16. ^ Dauben 1977, p. 89 15n.
17. ^ Dauben 1979, pp. 2–3; Grattan-Guinness 1971, pp. 354–355.
18. ^ Dauben 1979, p. 138.
19. ^ Dauben 1979, p. 139.
20. ^ a b Dauben 1979, p. 282.
21. ^ Dauben 1979, p. 136; Grattan-Guinness 1971, pp. 376–377. Letter dated June 21, 1884.
22. ^ Dauben 1979, pp. 281–283.
23. ^ Dauben 1979, p. 283.
24. ^ For a discussion of König's paper see Dauben 1979, 248–250. For Cantor's reaction, see Dauben 1979, p. 248; 283.
25. ^ Dauben 1979, p. 283–284.
26. ^ Dauben 1979, p. 284.
27. ^ a b Johnson 1972, p. 55.
28. ^ This paragraph is a highly abbreviated summary of the impact of Cantor's lifetime of work. More details and references can be found later.
29. ^ A countable set is a set which is either finite or denumerable; the denumerable sets are therefore the infinite countable sets. However, this terminology is not universally followed, and sometimes "denumerable" is used as a synonym for "countable".
30. ^ This follows closely the first part of Cantor's 1891 paper.
31. ^ Moore 1995, pp. 112 & 114; Dauben 2004, p. 1.
32. ^ For example, geometric problems posed by Galileo and John Duns Scotus suggested that all infinite sets were equinumerous — see Moore 1995, p. 114.
33. ^ For this, and more information on the mathematical importance of Cartan's work on set theory, see e.g., Suppes 1972.
34. ^ Dauben 1977, p. 89.
35. ^ The English translation is Cantor 1955.
36. ^ Wallace 2003, p. 259.
37. ^ Dauben 1979, p. 69; 324 63n. The paper had been submitted in July 1877. Dedekind supported it, but delayed its publication due to Kronecker's opposition. Weierstrass actively supported it.
38. ^ Some mathematicians consider these results to have settled the issue, and, at most, allow that it is possible to examine the formal consequences of CH or of its negation, or of axioms that imply one of those. Others continue to look for "natural" or "plausible" axioms that, when added to ZFC, will permit either a proof or refutation of CH, or even for direct evidence for or against CH itself; among the most prominent of these is W. Hugh Woodin. One of Gödel's last papers argues that the CH is false, and the continuum has cardinality Aleph-2.
39. ^ Dauben 1979, pp. 240–270; see especially pp. 241 & 259.
40. ^ Hallett 1986.
41. ^ Weir 1998, p. 766: "…it may well be seriously mistaken to think of Cantor's Mengenlehre [set theory] as naive…"
42. ^ Dauben 1979, p. 295.
43. ^ Dauben, 1979, p. 120.
44. ^ Hallett 1986, p. 13. Compare to the writings of Thomas Aquinas.
45. ^ Dauben 2004, pp. 8, 11 & 12–13.
46. ^ Dauben 1979, p. 225
47. ^ Snapper 1979, p. 3
48. ^ Dauben, 1977, p. 86; Dauben, 1979, pp. 120 & 143.
49. ^ Davenport 1997, p.3
50. ^ a b Dauben, 1977, p. 85.
51. ^ Cantor 1932, p. 404. Translation in Dauben 1977, p. 95.
52. ^ Dauben 1979, p. 296.
53. ^ Dauben, 1979, p. 144.
54. ^ Dauben 1977 pp. 91–93.
55. ^ On Cantor, Husserl, and Frege, see Hill and Rosado Haddock (2000).
56. ^ Dauben 1979, p. 96.
57. ^ E.g., Grattan-Guinness's only evidence on the grandfather's date of death is that he signed papers at his son's engagement.
58. ^ For more information, see: Dauben 1979, p. 1 and notes; Grattan-Guinness 1971, pp. 350–352 and notes; Purkert and Ilgauds 1985; the letter is from Aczel 2000, pp. 93–94, from Louis' trip to Chicago in 1863. It is ambiguous in German, as in English, whether the recipient is included.
59. ^ Cantor was frequently described as Jewish in his lifetime. For example, Jewish Encyclopedia, art. "Cantor, Georg"; Jewish Year Book 1896–1897, "List of Jewish Celebrities of the Nineteenth Century", p.119; this list has a star against people with one Jewish parent, but Cantor is not starred.
60. ^ Grattan-Guinness 1971, p. 350.
61. ^ Grattan-Guinness 1971 (quotation from p. 350, note), Dauben 1979, p.1 and notes. (Bell's Jewish stereotypes appear to have been removed from some postwar editions.)

The Gregorian calendar is the most widely used calendar in the world. ... [[Media:Italic text]]{| style=float:right; |- | |- | |} is the 50th day of the year in the Gregorian calendar. ... 1845 was a common year starting on Wednesday (see link for calendar). ... The Julian calendar was introduced in 46 BC by Julius Caesar and came into force in 45 BC (709 ab urbe condita). ... Arthur Moritz SchÃ¶nflies (April 17, 1853 Landsberg an der Warthe(GorzÃ³w) â€“ May 27, 1928) was a German mathematician, known for his contributions to the application of group theory to crystallography, and for work in topology. ... is the 172nd day of the year (173rd in leap years) in the Gregorian calendar. ... Year 1884 (MDCCCLXXXIV) was a leap year starting on Tuesday (link will display the full calendar) of the Gregorian calendar (or a leap year starting on Sunday of the 12-day-slower Julian calendar). ... In mathematics, a countable set is a set with the same cardinality (i. ... Galileo can refer to: Galileo Galilei, astronomer, philosopher, and physicist (1564 - 1642) the Galileo spacecraft, a NASA space probe that visited Jupiter and its moons the Galileo positioning system Life of Galileo, a play by Bertolt Brecht Galileo (1975) - screen adaptation of the play Life of Galileo by Bertolt Brecht... John Duns Scotus (c. ... W. Hugh Woodin is a set theorist at University of California, Berkeley. ... Saint Thomas Aquinas, O.P.(also Thomas of Aquin, or Aquino; c. ... Friedrich Ludwig Gottlob Frege Friedrich Ludwig Gottlob Frege (November 8, 1848 - July 26, 1925) was a German mathematician, logician, and philosopher who is regarded as a founder of both modern mathematical logic and analytic philosophy. ... The Jewish Encyclopedia was an encyclopedia originally published between 1901 and 1906 by Funk and Wagnalls. ... The Jewish Year Book is an almanac targetted at the Jewish community in the United Kingdom. ...

## References

Older sources on Cantor's life should be treated with some caution. See the section on historiography above.
Primary literature in English
• Cantor, Georg (1955, 1915). Contributions to the Founding of the Theory of Transfinite Numbers. New York: Dover. ISBN 978-0486600451
• Ewald, William B. (ed.) (1996). From Kant to Hilbert: A Source Book in the Foundations of Mathematics. ISBN 978-0198532712
Primary literature in German
• Cantor, Georg (1932). Gesammelte Abhandlungen mathematischen und philosophischen inhalts. (PDF) Almost everything that Cantor wrote.
Secondary literature
• Aczel, Amir D. (2000). The mystery of the Aleph: Mathematics, the Kabbala, and the Human Mind. New York: Four Walls Eight Windows Publishing. ISBN 0760777780. A popular treatment of infinity, in which Cantor is frequently mentioned.
• Dauben, Joseph W. (1977). Georg Cantor and Pope Leo XIII: Mathematics, Theology, and the Infinite. Journal of the History of Ideas 38.1.
• Dauben, Joseph W. (1979). Georg Cantor: his mathematics and philosophy of the infinite. Boston: Harvard University Press. The definitive biography to date. ISBN 978-0-691-02447-9
• Dauben, Joseph (1993, 2004). "Georg Cantor and the Battle for Transfinite Set Theory" in Proceedings of the 9th ACMS Conference (Westmont College, Santa Barbara, CA) (pp. 1–22). Internet version published in Journal of the ACMS 2004.
• Davenport, Anne A. (1997). The Catholics, the Cathars, and the Concept of Infinity in the Thirteenth Century. Isis 88.2:263–295.
• Grattan-Guinness, Ivor (1971). Towards a Biography of Georg Cantor. Annals of Science 27:345–391.
• Grattan-Guinness, Ivor (2000). The Search for Mathematical Roots: 1870–1940. Princeton University Press. ISBN 978-0691058580
• Hallett, Michael (1986). Cantorian Set Theory and Limitation of Size. New York: Oxford University Press. ISBN 0-19-853283-0
• Halmos, Paul (1998, 1960). Naive Set Theory. New York & Berlin: Springer. ISBN 3540900926
• Hill, C. O. & Rosado Haddock, G. E. (2000). Husserl or Frege? Meaning, Objectivity, and Mathematics. Chicago: Open Court. ISBN 0812695380 Three chapters and 18 index entries on Cantor.
• Johnson, Phillip E. (1972). The Genesis and Development of Set Theory. The Two-Year College Mathematics Journal 3.1:55–62.
• Moore, A.W. (1995, April). A brief history of infinity. Scientific American.4:112–116.
• Penrose, Roger (2004). The Road to Reality. Alfred A. Knopf. ISBN 0679776311 Chapter 16 illustrates how Cantorian thinking intrigues a leading contemporary theoretical physicist.
• Purkert, Walter & Ilgauds, Hans Joachim (1985). Georg Cantor: 1845–1918. Birkhäuser. ISBN 0-8176-1770-1
• Reid, Constance (1996). Hilbert. New York: Springer-Verlag. ISBN 0387049991
• Rucker, Rudy (2005, 1982). Infinity and the Mind. Princeton University Press. ISBN 0553255312 Deals with similar topics to Aczel, but in more depth.
• Rodych, Victor (2007). "Wittgenstein's Philosophy of Mathematics" in Edward N. Zalta (Ed.) The Stanford Encyclopedia of Philosophy.
• Snapper, Ernst (1979). The Three Crises in Mathematics: Logicism, Intuitionism and Formalism. Mathematics Magazine 524:207–216.
• Suppes, Patrick (1972, 1960). Axiomatic Set Theory. New York: Dover. ISBN 0486616304 Although the presentation is axiomatic rather than naive, Suppes proves and discusses many of Cantor's results, which demonstrates Cantor's continued importance for the edifice of foundational mathematics.
• Wallace, David Foster (2003). Everything and More: A Compact History of Infinity. New York: W.W. Norton and Company. ISBN 0393003388
• Weir, Alan (1998). Naive Set Theory is Innocent!. Mind 107.428:763–798.

Immanuel Kant Immanuel Kant (April 22, 1724 &#8211; February 12, 1804) was a Prussian philosopher, generally regarded as one of Europes most influential thinkers and the last major philosopher of the Enlightenment. ... David Hilbert David Hilbert (January 23, 1862 &#8211; February 14, 1943) was a German mathematician born in Wehlau, near Königsberg, Prussia (now Znamensk, near Kaliningrad, Russia) who is recognized as one of the most influential mathematicians of the 19th and early 20th centuries. ... Ivor Grattan-Guinness (Born 23 June 1941, in Bakewell, England) is a prolific historian of mathematics and logic, at Middlesex University. ... Ivor Grattan-Guinness (Born 23 June 1941, in Bakewell, England) is a prolific historian of mathematics and logic, at Middlesex University. ... Ivor Grattan-Guinness (Born 23 June 1941, in Bakewell, England) is a prolific historian of mathematics and logic, at Middlesex University. ... Ivor Grattan-Guinness (Born 23 June 1941, in Bakewell, England) is a prolific historian of mathematics and logic, at Middlesex University. ... Paul Halmos Paul Richard Halmos (March 3, 1916 â€” October 2, 2006) was a Hungarian-born American mathematician who wrote on probability theory, statistics, operator theory, ergodic theory, functional analysis (in particular, Hilbert spaces), and mathematical logic. ... Paul Halmos Paul Richard Halmos (March 3, 1916 â€” October 2, 2006) was a Hungarian-born American mathematician who wrote on probability theory, statistics, operator theory, ergodic theory, functional analysis (in particular, Hilbert spaces), and mathematical logic. ... Sir Roger Penrose, OM, FRS (born 8 August 1931) is an English mathematical physicist and Emeritus Rouse Ball Professor of Mathematics at the Mathematical Institute, University of Oxford and Emeritus Fellow of Wadham College. ... Sir Roger Penrose, OM, FRS (born 8 August 1931) is an English mathematical physicist and Emeritus Rouse Ball Professor of Mathematics at the Mathematical Institute, University of Oxford and Emeritus Fellow of Wadham College. ... Theoretical physics employs mathematical models and abstractions of physics, as opposed to experimental processes, in an attempt to understand nature. ... Rudy Rucker, Fall 2004, photo by Georgia Rucker. ... Rudy Rucker, Fall 2004, photo by Georgia Rucker. ...

Results from FactBites:

 Cantor, Georg Ferdinand Ludwig Philipp (1845-1918) (679 words) Cantor realized that irrational numbers can be represented as infinite sequences of rational numbers, so that they can be understood as geometric points on the real-number line, just as rational numbers can. In 1873-74 Cantor proved that the rational numbers could be paired off, one by one, with the natural numbers and were therefore countable, but that there was no such one-to-one correspondence with the real numbers. Henri Poincaré said that Cantors theory of infinite sets would be regarded by future generations as a disease from which one has recovered. Kronecker went further and did all he could to ridicule Cantors ideas, suppress publication of his results, and block Cantors ambition of gaining a position at the prestigious University of Berlin.
 Georg Cantor and Cantor's Theorem (514 words) Cantor not only found a way to make sense out an actual, as opposed to a potential, infinity but showed that their are different orders of infinity. Georg Cantor was born March 3, 1845 in Saint Petersburg, Russia. Georg Cantor's academic career was at the University of Halle, a lesser level university.
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