Georg Ferdinand Ludwig Cantor
Born	March 3, 1845(1845-03-03) Saint Petersburg, Russia
Died	January 6, 1918 (aged 72) Halle, Germany
Residence	Russia (1845–1856), Germany (1856–1918)
Fields	Mathematics
Institutions	University of Halle
Alma mater	ETH Zurich, University of Berlin
Doctoral advisor	Ernst Kummer Karl Weierstrass
Doctoral students	Alfred Barneck
Known for	Set theory

Georg Ferdinand Ludwig Philipp Cantor (March 3, 1845^[1] – January 6, 1918) was a German mathematician. Events 1284 - Statute of Rhuddlan incorporated the Principality of Wales into England 1575 - Indian Year 1845 ( MDCCCXLV) was a Common year starting on Wednesday (link will display the full calendar of the Gregorian calendar (or a Common Saint Petersburg ( tr: Sankt-Peterburg,) is a city and a federal subject of Russia located on the Neva River Russia (Россия Rossiya) or the Russian Federation ( Rossiyskaya Federatsiya) is a transcontinental Country extending Events 1066 - Harold Godwinson is crowned King of England. 1205 - Philip of Swabia becomes King Year 1918 ( MCMXVIII) was a Common year starting on Tuesday (link will display the full calendar of the Gregorian calendar (or a Common Halle is the largest city in the German State of Saxony-Anhalt. Germany, officially the Federal Republic of Germany ( ˈbʊndəsʁepuˌbliːk ˈdɔʏtʃlant is a Country in Central Europe. Russia (Россия Rossiya) or the Russian Federation ( Rossiyskaya Federatsiya) is a transcontinental Country extending Germany, officially the Federal Republic of Germany ( ˈbʊndəsʁepuˌbliːk ˈdɔʏtʃlant is a Country in Central Europe. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and The Martin Luther University of Halle-Wittenberg (Martin-Luther-Universität Halle-Wittenberg also referred to as MLU, is a public University in the cities of Alma mater is Latin for "nourishing mother" It was used in Ancient Rome as a title for the mother Goddess, and in Medieval For other universities in Berlin see List of Universities in Berlin. A doctorate is an Academic degree that indicates the highest level of academic achievement Ernst Eduard Kummer ( 29 January 1810 - 14 May 1893) was a German Mathematician. Karl Theodor Wilhelm Weierstrass ( Weierstraß) ( October 31, 1815 &ndash February 19, 1897) was a German mathematician Events 1284 - Statute of Rhuddlan incorporated the Principality of Wales into England 1575 - Indian Year 1845 ( MDCCCXLV) was a Common year starting on Wednesday (link will display the full calendar of the Gregorian calendar (or a Common Events 1066 - Harold Godwinson is crowned King of England. 1205 - Philip of Swabia becomes King Year 1918 ( MCMXVIII) was a Common year starting on Tuesday (link will display the full calendar of the Gregorian calendar (or a Common Germany, officially the Federal Republic of Germany ( ˈbʊndəsʁepuˌbliːk ˈdɔʏtʃlant is a Country in Central Europe. A mathematician is a person whose primary area of study and research is the field of Mathematics. He is best known as the creator of set theory, which has become a fundamental theory in mathematics. Foundations of mathematics is a term sometimes used for certain fields of Mathematics, such as Mathematical logic, Axiomatic set theory, Proof theory 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 Mathematics, a bijection, or a bijective function is a function f from a set X to a set Y with the property In Set theory, an infinite set is a set that is not a Finite set. In Mathematics, a well-order relation (or well-ordering) on a set S is a Total order on S with the property that every In Mathematics, the real numbers may be described informally in several different ways In Mathematics, a natural number (also called counting number) can mean either an element of the set (the positive Integers or an In fact, Cantor's theorem implies the existence of an "infinity of infinities". Note in order to fully understand this article you may want to refer to the Set theory portion of the Table of mathematical symbols. Infinity (symbolically represented with ∞) comes from the Latin infinitas or "unboundedness He defined the cardinal and ordinal numbers, and their arithmetic. This article describes cardinal numbers in mathematics For cardinals in linguistics see Names of numbers in English. In Set theory, an ordinal number, or just ordinal, is the Order type of a Well-ordered set. Cantor's work is of great philosophical interest, a fact of which he was well aware. ^[2]

Cantor's theory of transfinite numbers was originally regarded as so counter-intuitive—even shocking—that it encountered resistance from mathematical contemporaries such as Leopold Kronecker and Henri Poincaré^[3] and later from Hermann Weyl and L. E. J. Brouwer, while Ludwig Wittgenstein raised philosophical objections. Transfinite numbers are Cardinal numbers or Ordinal numbers that are larger than all finite numbers yet not necessarily absolutely infinite. In Mathematical logic, the theory of infinite sets was first developed by Georg Cantor. Leopold Kronecker ( December 7, 1823 – December 29, 1891) was a German Mathematician and Logician who argued Jules Henri Poincaré ( 29 April 1854 &ndash 17 July 1912) (ˈʒyl ɑ̃ˈʁi pwɛ̃kaˈʁe was a French Mathematician Hermann Klaus Hugo Weyl ( 9 November 1885 – 8 December 1955) was a German Mathematician. Luitzen Egbertus Jan Brouwer ɛxˈbɛʁtəs jɑn ˈbʁʌuəʁ ( February 27 1881, Overschie – December 2 1966, Blaricum In Mathematical logic, the theory of infinite sets was first developed by Georg Cantor. Some Christian theologians (particularly neo-Scholastics) saw Cantor's work as a challenge to the uniqueness of the absolute infinity in the nature of God,^[4] on one occasion equating the theory of transfinite numbers with pantheism. Christian Theology is discourse concerning Christian faith Christian theologians use biblical Exegesis, rational analysis and argument Neo-Scholasticism is the revival and development from the second half of the Nineteenth century of medieval Scholastic philosophy. God is the principal or sole Deity in Religions and other belief systems that worship one deity. Pantheism ( Greek: πάν ( 'pan') = all and θεός ( 'theos') = God it literally means " God is All ^[5] The objections to his work were occasionally fierce: Poincaré referred to Cantor's ideas as a "grave disease" infecting the discipline of mathematics,^[6] and Kronecker's public opposition and personal attacks included describing Cantor as a "scientific charlatan", a "renegade" and a "corrupter of youth. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and "^[7] Writing decades after Cantor's death, Wittgenstein lamented that mathematics is "ridden through and through with the pernicious idioms of set theory," which he dismissed as "utter nonsense" that is "laughable" and "wrong". ^[8] Cantor's recurring bouts of depression from 1884 to the end of his life were once blamed on the hostile attitude of many of his contemporaries,^[9] but these episodes can now be seen as probable manifestations of a bipolar disorder. Major depressive disorder, also known as major depression, unipolar depression, unipolar disorder, clinical depression, or simply depression ^[10]

The harsh criticism has been matched by international accolades. In 1904, the Royal Society awarded Cantor its Sylvester Medal, the highest honor it can confer. The Royal Society of London for the Improvement of Natural Knowledge, known simply as The Royal Society, is a Learned society for science that was founded in 1660 The Sylvester Medal is a bronze medal awarded every three years by the Royal Society for the encouragement of mathematical research ^[11] Cantor believed his theory of transfinite numbers had been communicated to him by God. ^[12] David Hilbert defended it from its critics by famously declaring: "No one shall expel us from the Paradise that Cantor has created. David Hilbert ( January 23, 1862 &ndash February 14, 1943) was a German Mathematician, recognized as one of the most "^[13]

Life

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. Saint Petersburg ( tr: Sankt-Peterburg,) is a city and a federal subject of Russia located on the Neva River Russia (Россия Rossiya) or the Russian Federation ( Rossiyskaya Federatsiya) is a transcontinental Country extending Georg, the eldest of six children, was an outstanding violinist, having inherited his parents' considerable musical and artistic talents. The violin is a bowed String instrument with four strings usually tuned in Perfect fifths It is the smallest and highest-pitched member 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. For the active stock exchange in Saint Petersburg see Saint Petersburg Stock Exchange. Wiesbaden, a city in southwest Germany, is the capital of the state of Hesse. In 1860, Cantor graduated with distinction from the Realschule in Darmstadt; his exceptional skills in mathematics, trigonometry in particular, were noted. Darmstadt is a city in the Bundesland Circle-trig6svg|300px|thumb|right|All of the Trigonometric functions of an angle θ can be constructed geometrically in terms of a unit circle centered at O. In 1862, Cantor entered the Federal Polytechnic Institute in Zürich, today the ETH Zurich. Zürich (, Zürich German: Züri, Zurich, Zurigo; in English generally Zurich) is the largest city in Switzerland and capital of the After receiving a substantial inheritance upon his father's death in 1863, Cantor shifted his studies to the University of Berlin, attending lectures by Kronecker, Karl Weierstrass and Ernst Kummer. For other universities in Berlin see List of Universities in Berlin. Karl Theodor Wilhelm Weierstrass ( Weierstraß) ( October 31, 1815 &ndash February 19, 1897) was a German mathematician Ernst Eduard Kummer ( 29 January 1810 - 14 May 1893) was a German Mathematician. He spent the summer of 1866 at the University of Göttingen, then and later a very important center for mathematical research. The University of Göttingen ( German: Georg-August-Universität Göttingen) is a University in the city of Göttingen, Germany. In 1867, Berlin granted him the PhD for a thesis on number theory, De aequationibus secundi gradus indeterminatis. "PhD" redirects here for other uses see PhD (disambiguation. A dissertation (also called thesis or disquisition) is a document that presents the author's Research and findings and is submitted in support of candidature 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

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. The Martin Luther University of Halle-Wittenberg (Martin-Luther-Universität Halle-Wittenberg also referred to as MLU, is a public University in the cities of He was awarded the requisite habilitation for his thesis on number theory. Habilitation is the highest academic qualification a person can achieve by their own pursuit in certain European and Asian 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 central Germany It is the highest mountain chain in northern Germany occupying parts of the German states of Lower Saxony, Saxony-Anhalt Julius Wilhelm Richard Dedekind ( October 6, 1831 &ndash February 12, 1916) was a German mathematician who did important Switzerland (English pronunciation; Schweiz Swiss German: Schwyz or Schwiiz Suisse Svizzera Svizra officially the Swiss Confederation

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. The meaning of the word professor ( Latin: professor, person who professes to be an expert in some art or science teacher of highest rank) varies 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. In the Philosophy of mathematics Cantor came to believe that Kronecker's stance would make it impossible for Cantor to ever leave Halle.

In 1881, Cantor's Halle colleague Eduard Heine died, creating a vacant chair. Heinrich Eduard Heine ( March 15 1821 &ndash October 21, 1881) was a German mathematician. Halle accepted Cantor's suggestion that it be offered to Dedekind, Heinrich M. Weber and Franz Mertens, in that order, but each declined the chair after being offered it. Julius Wilhelm Richard Dedekind ( October 6, 1831 &ndash February 12, 1916) was a German mathematician who did important Heinrich Martin Weber ( 5 March 1842 &ndash 17 May 1913) was a German Mathematician who specialized in Algebra Franz Mertens ( March 20, 1840 - March 5, 1927) was a German Mathematician. Friedrich Wangerin was eventually appointed, but he was never close to Cantor.

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. Magnus Gustaf (Gösta Mittag-Leffler ( 16 March 1846 – 7 July 1927) was a Swedish Mathematician. 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:

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 Mittag-Leffler in 1884 attacked Kronecker. A passage from one of these letters is revealing of the damage to Cantor's self-confidence:

This emotional crisis led him to apply to lecture on philosophy rather than mathematics. Philosophy is the study of general problems concerning matters such as existence knowledge truth beauty justice validity mind and language 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. The term Elizabethan literature refers to the English literature produced during the reign of Queen Elizabeth I (1558 - 1603 Francis Bacon 1st Viscount St Alban KC QC (22 January 1561 – 9 April 1626 was an English Philosopher, Statesman, and author William Shakespeare ( baptised The Shakespeare authorship question is the debate dating back to the early 18th century about whether the works attributed to William Shakespeare of Stratford-upon-Avon ^[22]

Cantor recovered soon thereafter, and subsequently made further important contributions, including his famous diagonal argument and theorem. Cantor's diagonal argument, also called the diagonalisation argument, the diagonal slash argument or the diagonal method, was published in 1891 Note in order to fully understand this article you may want to refer to the Set theory portion of the Table of mathematical symbols. 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. A root cause is an initiating Cause of a Causal chain which leads to an outcome or effect of interest ^[10]

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. The German Mathematical Society ( German:Deutsche Mathematiker-Vereinigung - DMV is the main professional society of German Mathematicians. 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.

Late years

After Cantor's 1884 hospitalization, there is no record that he was in any sanatorium again until 1899. A sanatorium (also sanitorium, sanitarium) is a medical facility for long-term illness typically Tuberculosis. ^[20] Soon after that second hospitalization, Cantor's youngest son died suddenly (while Cantor was delivering a lecture on his views on Baconian theory and William Shakespeare), and this tragedy drained Cantor of much of his passion for mathematics. The Baconian theory of Shakespearean authorship holds that Sir Francis Bacon wrote the plays conventionally attributed to William Shakespeare William Shakespeare ( baptised ^[23] Cantor was again hospitalized in 1903. One year later, he was outraged and agitated by a paper presented by Julius König at the Third International Congress of Mathematicians. Julius König ( 16 Dec 1849 – 8 Apr 1913 was a Hungarian Mathematician. The International Congress of Mathematicians (ICM is the largest congress in the Mathematics community The paper attempted to prove that the basic tenets of transfinite set theory were false. Since it had been read in front of his daughters and colleagues, Cantor perceived himself as having been publicly humiliated. ^[24] Although Ernst Zermelo demonstrated less than a day later that König's proof had failed, Cantor remained shaken, even momentarily questioning God. Ernst Friedrich Ferdinand Zermelo ( July 27 1871, Berlin, German Empire – May 21 1953, Freiburg im Breisgau ^[11] Cantor suffered from chronic depression for the rest of his life, for which he was excused from teaching on several occasions and repeatedly confined in various sanatoria. The events of 1904 preceded a series of hospitalizations at intervals of two or three years. ^[25] He did not abandon mathematics completely, however, lecturing on the paradoxes of set theory (Burali-Forti paradox, Cantor's paradox, and Russell's paradox) to a meeting of the Deutsche Mathematiker–Vereinigung in 1903, and attending the International Congress of Mathematicians at Heidelberg in 1904. In Set theory, a field of Mathematics, the Burali-Forti paradox demonstrates that naively constructing "the set of all Ordinal numbers quot leads to In Set theory, Cantor's paradox is the Theorem that there is no greatest Cardinal number, so that the collection of "infinite sizes" is itself Part of the Foundations of mathematics, Russell's paradox (also known as Russell's antinomy) discovered by Bertrand Russell in 1901 showed that the

In 1911, Cantor was one of the distinguished foreign scholars invited to attend the 500th anniversary of the founding of the University of St. Andrews in Scotland. The University of St Andrews is the oldest University in Scotland and third oldest in the English-speaking world, having been founded between Scotland ( Gaelic: Alba) is a Country in northwest Europethat occupies the northern third of the island of Great Britain. Cantor attended, hoping to meet Bertrand Russell, whose newly published Principia Mathematica repeatedly cited Cantor's work, but this did not come about. Bertrand Arthur William Russell 3rd Earl Russell, OM, FRS (18 May 1872 – 2 February 1970 was a British Philosopher, Historian The Principia Mathematica is a 3-volume work on the Foundations of mathematics, written by Alfred North Whitehead and Bertrand Russell The following year, St. Andrews awarded Cantor an honorary doctorate, but illness precluded his receiving the degree in person. An honorary degree or a degree honoris causa ( Latin: 'for the sake of the honour' is an Academic degree for which a university (or other degree-awarding

Cantor retired in 1913, and suffered from poverty, even malnourishment, during World War I. World War I (abbreviated WWI; also known as the First World War, the Great War, and the War to End All ^[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. Events 1066 - Harold Godwinson is crowned King of England. 1205 - Philip of Swabia becomes King Year 1918 ( MCMXVIII) was a Common year starting on Tuesday (link will display the full calendar of the Gregorian calendar (or a Common

Mathematical work

Cantor's work between 1874 and 1884 is the origin of set theory. ^[27] Prior to this work, the concept of a set was a rather elementary one that had been used implicitly since the beginnings of mathematics, dating back to the ideas of Aristotle. Aristotle (Greek Aristotélēs) (384 BC – 322 BC was a Greek philosopher a student of Plato and teacher of Alexander the Great. ^[28] No one had realized that set theory had any nontrivial content: Before Cantor, there were only finite sets (which are easy to understand) and "the infinite" (which was considered a topic for philosophical, rather than mathematical, discussion). By proving that there are (infinitely) many possible sizes for infinite sets, Cantor established that set theory was not trivial, and it needed to be studied. Set theory has come to play the role of a foundational theory in modern mathematics, in the sense that it interprets propositions about mathematical objects (for example, numbers and functions) from all the traditional areas of mathematics (such as algebra, analysis and topology) in a single theory, and provides a standard set of axioms to prove or disprove them. Foundations of mathematics is a term sometimes used for certain fields of Mathematics, such as Mathematical logic, Axiomatic set theory, Proof theory Algebra is a branch of Mathematics concerning the study of structure, relation, and Quantity. Analysis has its beginnings in the rigorous formulation of Calculus. Topology ( Greek topos, "place" and logos, "study" is the branch of Mathematics that studies the properties of The basic concepts of set theory are now used throughout mathematics.

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. In Mathematics, the real numbers may be described informally in several different ways In Mathematics, a natural number (also called counting number) can mean either an element of the set (the positive Integers or an In Mathematics, the cardinality of a set is a measure of the "number of elements of the set" He was also the first to appreciate the importance of one-to-one correspondences (hereinafter denoted "1-to-1") in set theory. In Mathematics, a bijection, or a bijective function is a function f from a set X to a set Y with the property 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). In Mathematics, a set is called finite if there is a Bijection between the set and some set of the form {1 2. In Set theory, an infinite set is a set that is not a Finite set. ^[29]

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. In Mathematics, given a set S, the power set (or powerset) of S, written \mathcal{P}(S P ( S) 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. Note in order to fully understand this article you may want to refer to the Set theory portion of the Table of mathematical symbols. Cantor developed an entire theory and arithmetic of infinite sets, called cardinals and ordinals, which extended the arithmetic of the natural numbers. In the mathematical field of Set theory, ordinal arithmetic describes the three usual operations on Ordinal numbers addition multiplication and exponentiation This article describes cardinal numbers in mathematics For cardinals in linguistics see Names of numbers in English. In Set theory, an ordinal number, or just ordinal, is the Order type of a Well-ordered set. His notation for the cardinal numbers was the Hebrew letter (aleph) with a natural number subscript; for the ordinals he employed the Greek letter ω (omega). OMEGA is the premier Counter-terrorism unit of Latvia. Founded in 1992 OMEGA cooperates with many other counter-terrorism units over the world This notation is still in use today.

The Continuum hypothesis, introduced by Cantor, was presented by David Hilbert as the first of his twenty-three open problems in his famous address at the 1900 International Congress of Mathematicians in Paris. In Mathematics, the continuum hypothesis (abbreviated CH) is a Hypothesis, advanced by Georg Cantor, about the possible sizes of Infinite David Hilbert ( January 23, 1862 &ndash February 14, 1943) was a German Mathematician, recognized as one of the most Hilbert's problems are a list of twenty-three problems in Mathematics put forth by German Mathematician David Hilbert at the Paris The International Congress of Mathematicians (ICM is the largest congress in the Mathematics community Paris (ˈpærɨs in English; in French) is the Capital of France and the country's largest city Cantor's work also attracted favorable notice beyond Hilbert's celebrated encomium. ^[13] The US philosopher Charles Peirce praised Cantor's set theory, and, following public lectures delivered by Cantor at the first International Congress of Mathematicians, held in Zurich in 1897, Hurwitz and Hadamard also both expressed their admiration. Charles Sanders Peirce (pronounced purse) (September 10 1839 &ndash April 19 1914 was an American Logician mathematician, philosopher Adolf Hurwitz ( 26 March 1859 - 18 November 1919) (ˈadɒlf ˈhurvits was a German mathematician and was described by Jean-Pierre Jacques Salomon Hadamard ( December 8, 1865 – October 17, 1963) was a French Mathematician best known for his proof of At that Congress, Cantor renewed his friendship and correspondence with Dedekind. From 1905, Cantor corresponded with his British admirer and translator Philip Jourdain on the history of set theory and on Cantor's religious ideas. Philip Edward Bertrand Jourdain ( 16 October 1879 - 1 October 1919) was a British logician and follower of Bertrand Russell. This was later published, as were several of his expository works.

Number theory and function theory

Cantor's first ten papers were on number theory, his thesis topic. 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 At the suggestion of Eduard Heine, the Professor at Halle, Cantor turned to analysis. Heinrich Eduard Heine ( March 15 1821 &ndash October 21, 1881) was a German mathematician. Analysis has its beginnings in the rigorous formulation of Calculus. 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. An open problem is a known problem with importance or interest in some scientific area that can be accurately stated and has not yet been solved (no solution for it is known Johann Peter Gustav Lejeune Dirichlet (ləʒœn diʀiçle February 13, 1805 &ndash May 5, 1859) was a German Mathematician Rudolf Otto Sigismund Lipschitz ( May 14, 1832 &ndash October 7, 1903) was a German Mathematician and professor at the The Mathematical concept of a function expresses dependence between two quantities one of which is given (the independent variable, argument of the function In Mathematics, a trigonometric series is any series of the form \frac{1}{2}A_{o}+\displaystyle\sum_{n=1}^{\infty}(A_{n} \cos{nx} + B_{n} 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. In Mathematics, an irrational number is any Real number that is not a Rational number — that is it is a number which cannot be expressed as a fraction In Functional analysis and related areas of Mathematics, a sequence space is a Vector space whose elements are infinite Sequences of Complex In Mathematics, a rational number is a number which can be expressed as a Ratio of two Integers Non-integer rational numbers (commonly called fractions 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. In Mathematics, a Dedekind cut, named after Richard Dedekind, in a Totally ordered set S is a partition of it into two non-empty

Set theory

An illustration of Cantor's diagonal argument for the existence of uncountable sets. Cantor's diagonal argument, also called the diagonalisation argument, the diagonal slash argument or the diagonal method, was published in 1891 ^[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 Imbegriffes aller reellen 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. Crelle's Journal, or just Crelle, is the common name for a leading German -language Mathematical journal, the Journal für 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. Georg Cantor 's first uncountability proof demonstrates that the set of all Real numbers is uncountable. Cantor's diagonal argument, also called the diagonalisation argument, the diagonal slash argument or the diagonal method, was published in 1891 ^[33] Prior to this, he had already proven that the set of rational numbers is countable. In Mathematics, a rational number is a number which can be expressed as a Ratio of two Integers Non-integer rational numbers (commonly called fractions

Joseph Liouville had established the existence of transcendental numbers in 1851, and Cantor's paper established that the set of transcendental numbers is uncountable. Joseph Liouville ( March 24 1809 &ndash September 8 1882) was a French Mathematician. In Mathematics, a transcendental number is a Complex number that is not algebraic, that is not a solution of a non-zero Polynomial equation The logic is as follows: Cantor had shown that the union of two countable sets must be countable. 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 countable. In Mathematics, an algebraic number is a Complex number that is a root of a non-zero Polynomial in one variable with rational (or ROOT is an object-oriented program and library developed by CERN. In Mathematics, a polynomial is an expression constructed from Variables (also known as indeterminates and Constants using the operations The integers (from the Latin integer, literally "untouched" hence "whole" the word entire comes from the same origin but via French In Mathematics, a coefficient is a Constant multiplicative factor of a certain object In contrast, Cantor had also just shown that the real numbers were not countable. In Mathematics, the real numbers may be described informally in several different ways If transcendental numbers were countable, then the result of their union with algebraic numbers would also be countable. Since their union (which equals the set of all real numbers) is uncountable, it logically follows that the transcendentals must be uncountable. 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 ( March 24 1809 &ndash 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. The Mathematische Annalen (abbreviated as Math Ann or Math Annal 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. 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 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. In metaphysics, Aristotle distinguished between actual and potential infinities. ^[34] Cantor also discovered the Cantor set during this period. In Mathematics, the Cantor set, introduced by German Mathematician Georg Cantor in 1883 (but discovered in 1875 by Henry John Stephen Smith

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. A monograph ( Classical Greek, "One Writer" or "Single Writing") is a work of writing upon a single subject usually also by a single It contained Cantor's reply to his critics and showed how the transfinite numbers were a systematic extension of the natural numbers. Transfinite numbers are Cardinal numbers or Ordinal numbers that are larger than all finite numbers yet not necessarily absolutely infinite. It begins by defining well-ordered sets. In Mathematics, a well-order relation (or well-ordering) on a set S is a Total order on S with the property that every Ordinal numbers are then introduced as the order types of well-ordered sets. In Set theory, an ordinal number, or just ordinal, is the Order type of a Well-ordered set. Cantor then defines the addition and multiplication of the cardinal and ordinal numbers. This article describes cardinal numbers in mathematics For cardinals in linguistics see Names of numbers in English. In 1885, Cantor extended his theory of order types so that the ordinal numbers simply became a special case of order types.

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. Note in order to fully understand this article you may want to refer to the Set theory portion of the Table of mathematical symbols. In Mathematics, the cardinality of a set is a measure of the "number of elements of the set" This established the richness of the hierarchy of infinite sets, and of the cardinal and ordinal arithmetic that Cantor had defined. This article describes cardinal numbers in mathematics For cardinals in linguistics see Names of numbers in English. In the mathematical field of Set theory, ordinal arithmetic describes the three usual operations on Ordinal numbers addition multiplication and exponentiation His argument is fundamental in the solution of the Halting problem and the proof of Gödel's first incompleteness theorem. In computability theory, the halting problem is a Decision problem which can be stated as follows given a description of a program and a finite input In Mathematical logic, Gödel's incompleteness theorems, proved by Kurt Gödel in 1931 are two Theorems stating inherent limitations of all but the most

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. The Mathematische Annalen (abbreviated as Math Ann or Math Annal Felix Christian Klein ( 25 April 1849 &ndash 22 June 1925) was a German Mathematician, known for his work in Group ^[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. In Mathematics, a well-order relation (or well-ordering) on a set S is a Total order on S with the property that every 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 Schröder had stated this theorem a bit earlier, but his proof, as well as Cantor's, was flawed. For the actor see Ernst Schröder (actor. Ernst Schröder ( 25 November, 1841 Mannheim Germany – Felix Bernstein supplied a correct proof in his 1898 PhD thesis; hence the name Cantor–Bernstein–Schroeder theorem. Felix Bernstein ( February 24, 1878, Halle, Germany – December 3[[ 956]] Zurich, Switzerland) was a German In set theory, the Cantor–Bernstein–Schroeder Theorem, named after Georg Cantor, Felix Bernstein, and Ernst Schröder, states

One-to-one correspondence

Main article: Bijection

A bijective function. In Mathematics, a bijection, or a bijective function is a function f from a set X to a set Y with the property

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. In Mathematics, a bijection, or a bijective function is a function f from a set X to a set Y with the property 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. The unit square is a square with all of the side lengths equalling 1 In Geometry, a line segment is a part of a line that is bounded by two distinct end points, and contains every point on the line between its end points 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. In Mathematics, an n -dimensional space is a Topological space whose Dimension is n (where n is a fixed Natural 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. In mathematics the dimension of a Space is roughly defined as the minimum number of Coordinates needed to specify every point within it

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. In Mathematics, the cardinality of a set is a measure of the "number of elements of the set" Jakob Steiner ( 18 March, 1796 &ndash 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" Cantor defined countable sets (or denumerable 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. In Mathematics, a natural number (also called counting number) can mean either an element of the set (the positive Integers or an 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. In Mathematics, the real numbers may be described informally in several different ways Cartesian square redirects here For Cartesian squares in Category theory, see Cartesian square (category theory. 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 dimension of a Space is roughly defined as the minimum number of Coordinates needed to specify every point within it 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, that is the set of all Real numbers x such that zero is less than or equal to x 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. Karl Theodor Wilhelm Weierstrass ( Weierstraß) ( October 31, 1815 &ndash February 19, 1897) was a German mathematician ^[37] Nevertheless, Cantor never again submitted anything to Crelle.

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). In Mathematics, the continuum hypothesis (abbreviated CH) is a Hypothesis, advanced by Georg Cantor, about the possible sizes of Infinite In Mathematics, the continuum hypothesis (abbreviated CH) is a Hypothesis, advanced by Georg Cantor, about the possible sizes of Infinite Cantor believed the continuum hypothesis to be true and tried for many years to prove it, in vain. In Mathematics, a proof is a convincing demonstration (within the accepted standards of the field that some Mathematical statement is necessarily true His inability to prove the continuum hypothesis caused him considerable anxiety. ^[9]

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"). Kurt Gödel (kʊɐ̯t ˈgøːdl̩ (April 28 1906 – January 14 1978 was an Austrian American Logician, Mathematician and Philosopher Paul Joseph Cohen ( April 2, 1934 &ndash March 23, 2007) was an American Mathematician best known for his proof of Zermelo–Fraenkel set theory with the axiom of choice, commonly abbreviated ZFC, is the standard form of Axiomatic set theory and as such is the most common In Mathematics, the axiom of choice, or AC, is an Axiom of Set theory. ^[38]

Paradoxes of set theory

Discussions of set-theoretic paradoxes began to appear around the end of the nineteenth century. A paradox is a true statement or group of statements that leads to a Contradiction or a situation which defies intuition; or inversely 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. Cesare Burali-Forti ( 13 august 1861 Arezzo - 21 january 1931 Turin) was an Italian Mathematician In Set theory, a field of Mathematics, the Burali-Forti paradox demonstrates that naively constructing "the set of all Ordinal numbers quot leads to In Set theory, an ordinal number, or just ordinal, is the Order type of a Well-ordered set. Cantor discovered this paradox in 1895, and described it in an 1896 letter to Hilbert. David Hilbert ( January 23, 1862 &ndash February 14, 1943) was a German Mathematician, recognized as one of the most Criticism mounted to the point where Cantor launched counter-arguments in 1903, intended to defend the basic tenets of his set theory. ^[11]

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. In Set theory, Cantor's paradox is the Theorem that there is no greatest Cardinal number, so that the collection of "infinite sizes" is itself 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). Note in order to fully understand this article you may want to refer to the Set theory portion of the Table of mathematical symbols. 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. In the Philosophy of mathematics, specifically the philosophical foundations of Set theory, limitation of size is a concept developed by Philip Jourdain Such collections later became known as proper classes. In Set theory and its applications throughout Mathematics, a class is a collection of sets (or sometimes other mathematical objects that can be unambiguously

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. Part of the Foundations of mathematics, Russell's paradox (also known as Russell's antinomy) discovered by Bertrand Russell in 1901 showed that the Ernst Friedrich Ferdinand Zermelo ( July 27 1871, Berlin, German Empire – May 21 1953, Freiburg im Breisgau 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. In Set theory and related branches of Mathematics, the von Neumann universe, or von Neumann hierarchy of sets, denoted V, is the class 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. Friedrich Ludwig Gottlob Frege ( 8 November 1848, Wismar, Grand Duchy of Mecklenburg-Schwerin  &ndash 26 July 1925 Naive set theory is one of several theories of sets used in the discussion of the Foundations of mathematics. ^[41]

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. In metaphysics, Aristotle distinguished between actual and potential infinities. 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. The word orthodox, from Greek orthodoxos "having the right opinion" from orthos ("right true straight" + doxa ("opinion ^[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. The Absolute Infinite is Mathematician Georg Cantor 's concept of an " Infinity " that transcended the Transfinite numbers Cantor God is the principal or sole Deity in Religions and other belief systems that worship one deity. ^[12]

Debate among mathematicians grew out of opposing views in the philosophy of mathematics regarding the nature of actual infinity. The philosophy of mathematics is the branch of Philosophy that studies the philosophical assumptions foundations and implications of Mathematics. Some held to the view that infinity was an abstraction which was not mathematically legitimate, and denied its existence. ^[45] Mathematicians from three major schools of thought (constructivism and its two offshoots, intuitionism and finitism) opposed Cantor's theories in this matter. In the Philosophy of mathematics In the Philosophy of mathematics, intuitionism, or neointuitionism (opposed to Preintuitionism) is an approach to Mathematics as the constructive 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 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. In Mathematics, a constructive proof is a method of proof that demonstrates the existence of a Mathematical object with certain properties by creating 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. ^[46] Mathematicians such as Brouwer and especially Poincaré adopted an intuitionist stance against Cantor's work. Luitzen Egbertus Jan Brouwer ɛxˈbɛʁtəs jɑn ˈbʁʌuəʁ ( February 27 1881, Overschie – December 2 1966, Blaricum Jules Henri Poincaré ( 29 April 1854 &ndash 17 July 1912) (ˈʒyl ɑ̃ˈʁi pwɛ̃kaˈʁe was a French Mathematician In the Philosophy of mathematics, intuitionism, or neointuitionism (opposed to Preintuitionism) is an approach to Mathematics as the constructive 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. Not to be confused with the homophone Intention; or the related concept of Intentionality. In any of several studies that treat the use of signs for example in Linguistics, Logic, Mathematics, Semantics, and Semiotics, the ^[8]

Christian theologians saw Cantor's work as a challenge to the uniqueness of the absolute infinity in the nature of God. ^[4] 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". Neo-Scholasticism is the revival and development from the second half of the Nineteenth century of medieval Scholastic philosophy. ^[47] Cantor strongly believed that this view was a misinterpretation of infinity, and was convinced that set theory could help correct this mistake:^[48]

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. The Philosophy of materialism holds that the only thing that can be truly proven to exist is Matter, and is considered a form of Physicalism. Determinism is the philosophical Proposition that every event including human cognition and behaviour decision and action is causally determined ^[50]

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,^[51] as well as theologians such as Cardinal Johannes Franzelin, who once replied by equating the theory of transfinite numbers with pantheism. Tilman Pesch (b at Cologne, 1 February[[ 836]] d at Valkenburg Limburg the Netherlands 18 October[[ 899]] was a German Jesuit philosopher Johann Baptist Franzelin (b at Aldein, in the Tyrol, 15 April[[ 816]] d Pantheism ( Greek: πάν ( 'pan') = all and θεός ( 'theos') = God it literally means " God is All ^[5] Cantor even sent one letter directly to Pope Leo XIII himself, and addressed several pamphlets to him. Pope Leo XIII ( March 2, 1810 – July 20, 1903) born Count Vincenzo Gioacchino Raffaele Luigi Pecci, was the 256th Pope ^[48]

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. Metaphysics is the branch of Philosophy investigating principles of reality transcending those of any particular science This belief is summarized in his famous assertion that "the essence of mathematics is its freedom. "^[52] These ideas parallel those of Edmund Husserl. Edmund Gustav Albrecht Husserl (ˈhʊsɛrl April 8 1859 – April 26 1938) was a philosopher, known as the father of ^[53]

Cantor's 1883 paper reveals that he was well aware of the opposition his ideas were encountering:

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. In Classical logic, a contradiction consists of a logical incompatibility between two or more Propositions It occurs when the propositions taken together yield He also cites Aristotle, Descartes, Berkeley, Leibniz, and Bolzano on infinity. Aristotle (Greek Aristotélēs) (384 BC – 322 BC was a Greek philosopher a student of Plato and teacher of Alexander the Great. George Berkeley (ˈbɑrkli (12 March 1685 14 January 1753 also known as Bishop Berkeley, was a Philosopher. Bernard (Bernhard Placidus Johann Nepomuk Bolzano ( &ndash December 18, 1848) was a Bohemian Mathematician, theologian,

Cantor's ancestry

Cantor's paternal grandparents were from Copenhagen, and fled to Russia from the disruption of the Napoleonic Wars. Copenhagen (ˌkəʊpənˈheɪgən ˌkəʊpənˈhɑːgən ˈkəʊpənˌheɪgən ˈkəʊpənˌhɑːgən kʰøb̥ənˈhɑʊ̯ˀn kʰøb̥m̩ˈhɑʊ̯ˀn is the capital and largest city The Napoleonic Wars (1803-1815 involved Napoleon's French Empire and a shifting set of European allies and opposing coalitions In his letters, Cantor referred to them as "Israelites". However, there is no direct evidence on whether his grandparents practiced Judaism; there is very little direct information on them of any kind. Judaism (from the Greek Ioudaïsmos, derived from the Hebrew יהודה Yehudah, " Judah " in Hebrew יַהֲדוּת Yahedut ^[55] Jakob Cantor, Cantor's grandfather, gave his children Christian saints' names. Christianity ( Greek Χριστιανισμός from the word Xριστός ( Christ)is a monotheistic Religion centered on the life and teachings A saint (from the Latin sanctus) is a human being to whom has been attributed (and who has generally demonstrated a high level of Holiness and Sanctity 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. Lutheranism is a major branch of Western Christianity that identifies with the teachings of the sixteenth-century German reformer Martin Luther His mother, Maria Anna Böhm, was an Austrian born in Saint Petersburg and baptized Roman Catholic; she converted to Protestantism upon marriage. Austria (Österreich ( officially the Republic of Austria (Republik Österreich Protestantism refers to the forms of Christian faith and practice that originated in the 16th century Protestant Reformation. However, there is a letter from Cantor's brother Louis to their mother, saying

which could imply that she was of Jewish ancestry. ^[57]

Thus Cantor was not himself Jewish by faith, but has nevertheless been called variously German, Jewish,^[58] Russian, and Danish. Judaism (from the Greek Ioudaïsmos, derived from the Hebrew יהודה Yehudah, " Judah " in Hebrew יַהֲדוּת Yahedut

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). Arthur Moritz Schönflies ( April 17, 1853 &ndash May 27, 1928) was a German Mathematician, known for his contributions 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". Eric Temple Bell ( February 7 Men of Mathematics is a well-known The area of study known as the history of mathematics is primarily an investigation into the origin of new discoveries in Mathematics and to a lesser extent an investigation ^[59] 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 Saint Petersburg by unknown parents. ^[60]

See also

• 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, the back-and-forth method is a method for showing Isomorphism between Countably In Mathematics, the Cantor function, named after Georg Cantor, is an example of a function that is continuous, but not Absolutely continuous In Mathematics, the Heine–Cantor theorem, named after Eduard Heine and Georg Cantor, states that if M is a compact Metric space 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

References

Older sources on Cantor's life should be treated with caution. See Historiography section 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 Immanuel Kant to David Hilbert: A Source Book in the Foundations of Mathematics. Immanuel Kant (ɪmanuəl kant 22 April 1724 12 February 1804 was an 18th-century German Philosopher from the Prussian city of Königsberg David Hilbert ( January 23, 1862 &ndash February 14, 1943) was a German Mathematician, recognized as one of the most 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). Ivor Grattan-Guinness (Born 23 June 1941 Bakewell, England is a Historian of mathematics and Logic. Ivor Grattan-Guinness (Born 23 June 1941 Bakewell, England is a Historian of mathematics and Logic. Towards a Biography of Georg Cantor. Annals of Science 27:345–391.
• Grattan-Guinness, Ivor (2000). Ivor Grattan-Guinness (Born 23 June 1941 Bakewell, England is a Historian of mathematics and Logic. Ivor Grattan-Guinness (Born 23 June 1941 Bakewell, England is a Historian of mathematics and Logic. 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). Paul Richard Halmos ( March 3 1916 &mdash October 2 2006) was a Hungarian -born Jewish American Mathematician Paul Richard Halmos ( March 3 1916 &mdash October 2 2006) was a Hungarian -born Jewish American Mathematician 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). Sir Roger Penrose, PhD, OM, FRS (born 8 August 1931) is an English Mathematical physicist and Emeritus Sir Roger Penrose, PhD, OM, FRS (born 8 August 1931) is an English Mathematical physicist and Emeritus The Road to Reality. Alfred A. Knopf. ISBN 0679776311 Chapter 16 illustrates how Cantorian thinking intrigues a leading contemporary theoretical physicist. Theoretical physics employs Mathematical models and Abstractions of Physics in an attempt to explain experimental data taken of the natural world
• 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). Rudolf von Bitter Rucker (born March 22, 1946 in Louisville Kentucky) is an American Computer scientist and Science fiction Rudolf von Bitter Rucker (born March 22, 1946 in Louisville Kentucky) is an American Computer scientist and Science fiction 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.

External links

• O'Connor, John J. & Robertson, Edmund F. , “Georg Cantor”, MacTutor History of Mathematics archive 
• O'Connor, John J. The MacTutor History of Mathematics archive is an award-winning website maintained by John J & Robertson, Edmund F. , “A history of set theory”, MacTutor History of Mathematics archive  Mainly devoted to Cantor's accomplishment. The MacTutor History of Mathematics archive is an award-winning website maintained by John J
• Georg Cantor at the Mathematics Genealogy Project
• Selections from Cantor's philosophical writing.
• Text of Cantor's 1891 diagonal argument.
• Stanford Encyclopedia of Philosophy: Set theory by Thomas Jech. The Mathematics Genealogy Project is a web-based Database that gives an Academic genealogy based on Dissertation supervision relations Thomas J Jech (Tomáš Jech ˈtɔmaːʃ ˈjɛx born January 29, 1944 in Prague) is a set theorist who was at Penn State for more
• Grammar school Georg-Cantor Halle (Saale): Georg-Cantor-Gynmasium Halle