142-442: Georg Ferdinand Ludwig Philipp Cantor ( / ˈ k æ n t ɔːr / KAN -tor ; German: [ˈɡeːɔʁk ˈfɛʁdinant ˈluːtvɪç ˈfiːlɪp ˈkantoːɐ̯] ; 3 March [ O.S. 19 February] 1845 – 6 January 1918) was a mathematician who played a pivotal role in the creation of set theory , which has become a fundamental theory in mathematics. Cantor established the importance of one-to-one correspondence between
284-520: 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. Cantor also introduced the Cantor set during this period. The fifth paper in this series, " Grundlagen einer allgemeinen Mannigfaltigkeitslehre" (" Foundations of
426-453: A "scientific charlatan", a "renegade" and a "corrupter of youth". Kronecker objected to Cantor's proofs that the algebraic numbers are countable, and that the transcendental numbers are uncountable, results now included in a standard mathematics curriculum. 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
568-549: 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 Richard 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 wrote to Dedekind: " Je le vois, mais je ne le crois pas! " ("I see it, but I don't believe it!") The result that he found so astonishing has implications for geometry and
710-460: 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 R 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
852-415: 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
994-543: A Latin commentary on Book 1 of Spinoza's Ethica . Trendelenburg was also the examiner of Cantor's Habilitationsschrift . In 1888, Cantor published his correspondence with several philosophers on the philosophical implications of his set theory. In an extensive attempt to persuade other Christian thinkers and authorities to adopt his views, Cantor had corresponded with Christian philosophers such as Tilman Pesch and Joseph Hontheim , as well as theologians such as Cardinal Johann Baptist Franzelin , who once replied by equating
1136-642: A calendar change, respectively. Usually, they refer to the change from the Julian calendar to the Gregorian calendar as enacted in various European countries between 1582 and 1923. In England , Wales , Ireland and Britain's American colonies , there were two calendar changes, both in 1752. The first adjusted the start of a new year from 25 March ( Lady Day , the Feast of the Annunciation ) to 1 January,
1278-565: A castle, probably from the early Middle Ages, around which the city had developed. While nothing is known of the former castle, remains of it were uncovered during excavations after World War II. The new town hall was built in 1887. A tile mosaic in front of the town hall shows the heraldic eagle of the Kingdom of Prussia (of which Wiesbaden was a part at the time), the coat of arms of the Prussian Province of Hesse-Nassau , and
1420-660: A center for mathematical research. Cantor was a good student, and he received his doctoral degree in 1867. Cantor submitted his dissertation on number theory at the University of Berlin in 1867. After teaching briefly in a Berlin girls' school, he took up a position at the University of Halle , where he spent his entire career. He was awarded the requisite habilitation for his thesis, also on number theory, which he presented in 1869 upon his appointment at Halle. In 1874, Cantor married Vally Guttmann. They had six children,
1562-668: A challenge to the uniqueness of the absolute infinity in the nature of God – on one occasion equating the theory of transfinite numbers with pantheism – a proposition that Cantor vigorously rejected. Not all theologians were against Cantor's theory; prominent neo-scholastic philosopher Constantin Gutberlet was in favor of it and Cardinal Johann Baptist Franzelin accepted it as a valid theory (after Cantor made some important clarifications). The objections to Cantor's work were occasionally fierce: Leopold Kronecker 's public opposition and personal attacks included describing Cantor as
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#17327908154031704-544: A change which Scotland had made in 1600. The second discarded the Julian calendar in favour of the Gregorian calendar, skipping 11 days in the month of September to do so. To accommodate the two calendar changes, writers used dual dating to identify a given day by giving its date according to both styles of dating. For countries such as Russia where no start-of-year adjustment took place, O.S. and N.S. simply indicate
1846-835: A coalition of Germanic tribes from beyond the Limes , captured the fort around 260. Later, in the 370s, when the Romans and Alamanni were allied, the Alemanni gained control of the Wiesbaden area and were in charge of its defense against other Germanic tribes. After the Franks under Clovis I defeated the Alamanni in the Battle of Tolbiac in 496, the Franks eventually displaced the Alamanni in
1988-420: A colleague, perceiving him as a "corrupter of youth" for teaching his ideas to a younger generation of mathematicians. Worse yet, Kronecker, a well-established figure within the mathematical community and Cantor's former professor, disagreed fundamentally with the thrust of Cantor's work ever since he had intentionally delayed the publication of Cantor's first major publication in 1874. Kronecker, now seen as one of
2130-508: A construction that proves the set of transcendental numbers has the same "power" (see below) as the set of real numbers. 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 Leopold Kronecker, who admitted mathematical concepts only if they could be constructed in
2272-512: A fatal heart attack on 6 January 1918, in the sanatorium where he had spent the last year of his life. Cantor's work between 1874 and 1884 is the origin of set theory . Prior to this work, the concept of a set was a rather elementary one that had been used implicitly since the beginning of mathematics, dating back to the ideas of Aristotle . 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
2414-459: A letter dated "12/22 Dec. 1635". In his biography of John Dee , The Queen's Conjurer , Benjamin Woolley surmises that because Dee fought unsuccessfully for England to embrace the 1583/84 date set for the change, "England remained outside the Gregorian system for a further 170 years, communications during that period customarily carrying two dates". In contrast, Thomas Jefferson , who lived while
2556-586: 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 milder winters 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 August 1862, he then graduated from
2698-678: A mountain basin at the southern foot of the Taunus , protected by the mountains in the north and west, gives Wiesbaden a mild climate. It has been called the " Nice of the North" because of its climate and architecture. The city of Wiesbaden is one of the wealthiest cities in Germany and one of those with above-average purchasing power. The United States Army Europe and Africa headquarters are located in Wiesbaden-Erbenheim . Wiesbaden
2840-592: A paper presented by Julius König at the Third International Congress of Mathematicians . The paper attempted to prove that the basic tenets of transfinite set theory were false. Since the paper had been read in front of his daughters and colleagues, Cantor perceived himself as having been publicly humiliated. Although Ernst Zermelo demonstrated less than a day later that König's proof had failed, Cantor remained shaken, and momentarily questioning God. Cantor suffered from chronic depression for
2982-852: A planned " Fluxus " publication with concerts of antique musical instruments, the “Fluxus Internationale Festspiele Neuester Music” (Fluxus International Festival of Newest Music) at the Museum Wiesbaden. Fourteen concerts were performed on four weekends between 1 and 23 September which marked the beginning of the Fluxus movement. Work by musicians such as John Cage , György Ligeti , Krzysztof Penderecki , Terry Riley , Brion Gysin and others were performed alongside new performance pieces written by Higgins, Knowles, George Brecht , Nam June Paik , Ben Patterson , Robert Filliou , Emmett Williams , and others. One performance in particular, "Piano Activities" by Philip Corner , became notorious by challenging
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#17327908154033124-497: A population of 86,100, hosted 126,000 visitors annually. Famous visitors to the springs included Johann Wolfgang von Goethe , Fyodor Dostoevsky , Richard Wagner , Johannes Brahms , and Henrik Pontoppidan . In those years, more millionaires were living in Wiesbaden than in any other city in Germany. Gambling followed bathing en suite , and in the 19th century, Wiesbaden was famous for both. Its casino ( Spielbank ) rivalled those of Bad Homburg , Baden-Baden , and Monaco . In 1872,
3266-540: A reconciliation with Kronecker. Nevertheless, the philosophical disagreements and difficulties dividing them persisted. In 1889, Cantor was instrumental in founding the German Mathematical Society , and he chaired its first meeting in Halle in 1891, where he first introduced his diagonal argument; his reputation was strong enough, despite Kronecker's opposition to his work, to ensure he was elected as
3408-517: 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 . Cantor wrote on the Goldbach conjecture in 1894. In 1895 and 1897, Cantor published
3550-459: A set of limit points S ω+1 , and so on. He had examples that went on forever, and so here was a naturally occurring infinite sequence of infinite numbers ω , ω + 1, ω + 2, ... Between 1870 and 1872, Cantor published more papers on trigonometric series, and also a paper defining irrational numbers as convergent sequences of rational numbers . Dedekind, whom Cantor befriended in 1872, cited this paper later that year, in
3692-415: A set, the resulting contradiction implies only that the ordinals form an inconsistent multiplicity. In contrast, Bertrand Russell treated all collections as sets, which leads to paradoxes. In Russell's set theory, the ordinals form a set, so the resulting contradiction implies that the theory is inconsistent . From 1901 to 1903, Russell discovered three paradoxes implying that his set theory is inconsistent:
3834-859: A start-of-year adjustment works well with little confusion for events before the introduction of the Gregorian calendar. For example, the Battle of Agincourt is well known to have been fought on 25 October 1415, which is Saint Crispin's Day . However, for the period between the first introduction of the Gregorian calendar on 15 October 1582 and its introduction in Britain on 14 September 1752, there can be considerable confusion between events in Continental Western Europe and in British domains. Events in Continental Western Europe are usually reported in English-language histories by using
3976-467: 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. Felix Bernstein supplied a correct proof in his 1898 PhD thesis; hence the name Cantor–Bernstein–Schröder theorem . 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
4118-531: A two-part paper in Mathematische Annalen under Felix Klein 's editorship; these were his last significant papers on set theory. 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
4260-511: A warehouse full of 4,000 cases of champagne. After the war's end, American rock artist Elvis Presley was stationed in Friedberg and often visited Wiesbaden. After World War II, the state of Hesse was established (see Greater Hesse ), and Wiesbaden became its capital, though nearby Frankfurt am Main is much larger and works as Hesse's economic and financial centre. Wiesbaden however suffered much less than Frankfurt from air bombing. There
4402-536: Is Germany's 24th-largest city . Wiesbaden forms a conurbation with a population of around 500,000 with the neighbouring city of Mainz . This conurbation is in turn embedded in the Rhine-Main Metropolitan Region —Germany's second-largest metropolitan region after Rhine-Ruhr —which also includes the nearby cities of Frankfurt am Main , Darmstadt , Offenbach am Main , and Hanau , and has a combined population exceeding 5.8 million. The city
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4544-597: Is "laughable" and "wrong". Cantor's recurring bouts of depression from 1884 to the end of his life have been blamed on the hostile attitude of many of his contemporaries, though some have explained these episodes as probable manifestations of a bipolar disorder . The harsh criticism has been matched by later accolades. In 1904, the Royal Society awarded Cantor its Sylvester Medal , the highest honor it can confer for work in mathematics. David Hilbert defended it from its critics by declaring, "No one shall expel us from
4686-536: Is 9 February 1649, the date by which his contemporaries in some parts of continental Europe would have recorded his execution. The O.S./N.S. designation is particularly relevant for dates which fall between the start of the "historical year" (1 January) and the legal start date, where different. This was 25 March in England, Wales, Ireland and the colonies until 1752, and until 1600 in Scotland. In Britain, 1 January
4828-489: Is a "law of thought". Cantor extended his work on the absolute infinite by using it in a proof. Around 1895, he began to regard his well-ordering principle as a theorem and attempted to prove it. In 1899, he sent Dedekind a proof of the equivalent aleph theorem: the cardinality of every infinite set is an aleph . First, he defined two types of multiplicities: consistent multiplicities (sets) and inconsistent multiplicities (absolutely infinite multiplicities). Next he assumed that
4970-553: Is a persistent rumour that the U.S. Army Air Force spared the town with the intention of turning it into a postwar HQ, but USAAF sources claim this to be a myth, arguing that Wiesbaden's economic and strategic importance simply did not justify more bombing. Wiesbaden was host to the Headquarters, U.S. Air Forces, Europe based at the former Lindsey Air Station from 1953 to 1973. American armed forces have been present in Wiesbaden since World War II. The U.S. 1st Armored Division
5112-451: Is at an elevation of 115 metres (377 ft). Wiesbaden covers an area of 204 km (79 sq mi). It is 17.6 kilometres (10.9 mi) from north to south and 19.7 kilometres (12.2 mi) from west to east. In the north are vast forest areas, which cover 27.4% of the urban area. In the west and east are vineyards and agricultural land, which cover 31.1% of the area. Of the municipality's 79 kilometres (49.1 mi)-long border,
5254-400: 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. The difficulty Cantor had in proving
5396-740: Is located on the Rhine ( Upper Rhine ), at the foothills of the Taunus , opposite the Rhineland-Palatine capital of Mainz , and the city centre is located in the wide valley of the small Salzbach stream. Wiesbaden lies in the Rheingau wine-growing region , one of Germany's 13 wine regions . Three of Wiesbaden's boroughs were part of the city of Mainz until 1945, and still bear the designation "Mainz" in their names—the so-called AKK-boroughs of Mainz-Amöneburg , Mainz-Kastel , and Mainz-Kostheim . This so-called AKK-Konflikt ( de:AKK-Konflikt )
5538-415: Is no record that he was in any sanatorium again until 1899. Soon after that second hospitalization, Cantor's youngest son Rudolph died suddenly on 16 December (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. Cantor was again hospitalized in 1903. One year later, he was outraged and agitated by
5680-598: Is situated on the right (northern) bank of the Rhine , above the confluence of the Main , where the Rhine's main direction changes from north to west. The city is across the Rhine from Mainz, the capital of the state of Rhineland-Palatinate . Frankfurt am Main is located about 38 kilometres (23.6 mi) east. To the north of the city are the Taunus Mountains, which trend in a northeasterly direction. The city center,
5822-684: Is the main cause for the rivalry between Mainz and Wiesbaden. Wiesbaden Main Station is connected to Frankfurt am Main by the Rhine-Main S-Bahn rapid transit system. Historically, Wiesbaden was a Nassauian city. From 1170 to 1629, it lay in the County of Nassau , and from 1629 to 1721, it was in the county and later principality of Nassau-Idstein, all of which were territories within the Holy Roman Empire ruled by branches of
Georg Cantor - Misplaced Pages Continue
5964-464: Is the set of limit points of S . If S k+1 is the set of limit points of S k , then he could construct a trigonometric series whose zeros are S k+1 . Because the sets S k were closed, they contained their limit points, and the intersection of the infinite decreasing sequence of sets S , S 1 , S 2 , S 3 ,... formed a limit set, which we would now call S ω , and then he noticed that S ω would also have to have
6106-593: The Deutsche Mathematiker-Vereinigung in 1903, and attending the International Congress of Mathematicians at Heidelberg in 1904. In 1911, Cantor was one of the distinguished foreign scholars invited to the 500th anniversary of the founding of the University of St. Andrews in Scotland. Cantor attended, hoping to meet Bertrand Russell , whose newly published Principia Mathematica repeatedly cited Cantor's work, but
6248-462: The Burali-Forti paradox (which was just mentioned), Cantor's paradox , and Russell's paradox . Russell named paradoxes after Cesare Burali-Forti and Cantor even though neither of them believed that they had found paradoxes. In 1908, Zermelo published his axiom system for set theory . He had two motivations for developing the axiom system: eliminating the paradoxes and securing his proof of
6390-401: The Cantor set , discovered by Henry John Stephen Smith in 1875, is nowhere dense , but has the same cardinality as the set of all real numbers, whereas the rationals are everywhere dense, but countable. He also showed that all countable dense linear orders without end points are order-isomorphic to the rational numbers . Cantor introduced fundamental constructions in set theory, such as
6532-687: The Confessing Church resistance movement against the Nazis, is an Honorary Citizen of Wiesbaden. He presented his last sermon before his arrest in Wiesbaden's Market Church. In World War II, Wiesbaden was the headquarters for Germany's Wehrkreis XII . This military district included the Eifel , part of Hesse , the Palatinate , and the Saarland . After the Battle of France , this Wehrkreis
6674-548: The Holy Roman Empire ). The town was part of Franconia , the heartland of East Francia. In the 1170s, the Count of Nassau , Walram I, received the area around Wiesbaden as a fiefdom . When Franconia fragmented in the early 13th century, Nassau emerged as an independent state as part of the Holy Roman Empire. In 1232 Wiesbaden became a Reichsstadt , an imperial city, of the Holy Roman Empire. However, in 1242, during
6816-662: The House of Nassau . In 1728, the city found itself in the principality of Nassau-Usingen , and in 1744, Biebrich Palace became the main residence of the House of Nassau-Usingen. In 1806, the city became the capital of the Duchy of Nassau . Since 1841, the newly built Wiesbaden City Palace was the principal Nassauian residence. From 1868 to 1944, the city lay in the Prussian Province of Hesse-Nassau , and from 1944 to 1945, it
6958-546: The Limes Germanicus , which was a line of Roman frontier fortifications in the Taunus . Wiesbaden is just south of the Taunus. The capital of the province of Germania Superior , Mogontiacum (present-day Mainz ), base of 2 (at times 3) Roman legions, was just over the Rhine and connected by a bridge at the present-day borough of Mainz-Kastel (Roman " castellum "), a strongly fortified bridgehead. The Alamanni ,
7100-605: The Luftwaffe was stationed here. In the Kristallnacht pogrom on 10 November 1938, Wiesbaden's large synagogue on Michelsberg was destroyed. The synagogue had been designed by Phillip Hoffmann and built in 1869. Another synagogue in Wiesbaden-Bierstadt was also destroyed. When the Nazis came to power in Germany, there were 2,700 Jews living in Wiesbaden. By June 1942 nearly all of them had been deported to
7242-609: The Russian Empire and the very beginning of Soviet Russia . For example, in the article "The October (November) Revolution", the Encyclopædia Britannica uses the format of "25 October (7 November, New Style)" to describe the date of the start of the revolution. The Latin equivalents, which are used in many languages, are, on the one hand, stili veteris (genitive) or stilo vetere (ablative), abbreviated st.v. , and meaning "(of/in) old style" ; and, on
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#17327908154037384-801: The Stadtmitte , is located in the north-easternmost part of the Upper Rhine Valley at the spurs of the Taunus mountains, about 5 kilometres (3.1 mi) from the Rhine. The landscape is formed by a wide lowland between the Taunus heights in the north, the Bierstadter Höhe and the Hainerberg in the east, the Mosbacher Mountain in the south, and the Schiersteiner Mountain in the west, an offshoot of
7526-459: The absolute . The transfinite is increasable in magnitude, while the absolute is unincreasable. For example, an ordinal α is transfinite because it can be increased to α + 1. On the other hand, the ordinals form an absolutely infinite sequence that cannot be increased in magnitude because there are no larger ordinals to add to it. In 1883, Cantor also introduced the well-ordering principle "every set can be well-ordered" and stated that it
7668-455: The absolute infinite with God, 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. He was a devout Lutheran whose explicit Christian beliefs shaped his philosophy of science. Joseph Dauben has traced the effect Cantor's Christian convictions had on the development of transfinite set theory. Debate among mathematicians grew out of opposing views in
7810-699: The extermination camps in German-occupied Poland . General Ludwig Beck from Wiesbaden was one of the planners of the 20 July 1944 assassination attempt of Adolf Hitler . Beck was designated by his fellow conspirators to be future Head of State (Regent) after elimination of Hitler. The plot failed, however, and Beck was forced to commit suicide. Today, the city annually awards the Ludwig Beck prize for civil courage in his honor. Lutheran pastor and theologian Martin Niemöller , founder of
7952-501: 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. Some Christian theologians saw Cantor's work as a challenge to the uniqueness of the absolute infinity in the nature of God. 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". Cantor strongly believed that this view
8094-409: The paradise that Cantor has created ." Georg Cantor, born in 1845 in Saint Petersburg , Russian Empire, was brought up in that city until the age of eleven. The oldest of six children, he was regarded as an outstanding violinist. His grandfather Franz Böhm (1788–1846) (the violinist Joseph Böhm 's brother) was a well-known musician and soloist in a Russian imperial orchestra. Cantor's father had been
8236-473: 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. 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
8378-399: 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
8520-426: The unit interval and the unit square was not a continuous one. This paper displeased Kronecker and Cantor wanted to withdraw it; however, Dedekind persuaded him not to do so and Karl Weierstrass supported its publication. Nevertheless, Cantor never again submitted anything to Crelle. Cantor was the first to formulate what later came to be known as the continuum hypothesis or CH: there exists no set whose power
8662-432: The well-ordering theorem . Zermelo had proved this theorem in 1904 using the axiom of choice , but his proof was criticized for a variety of reasons. His response to the criticism included his axiom system and a new proof of the well-ordering theorem. His axioms support this new proof, and they eliminate the paradoxes by restricting the formation of sets. In 1923, John von Neumann developed an axiom system that eliminates
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#17327908154038804-705: The "Höhere Gewerbeschule Darmstadt", now the Technische Universität Darmstadt . In 1862 Cantor entered the Swiss Federal Polytechnic in Zurich. After receiving a substantial inheritance upon his father's death in June 1863, Cantor transferred to the University of Berlin , attending lectures by Leopold Kronecker , Karl Weierstrass and Ernst Kummer . He spent the summer of 1866 at the University of Göttingen , then and later
8946-464: The 19th century, a practice that the author Karen Bellenir considered to reveal a deep emotional resistance to calendar reform. Wiesbaden Wiesbaden ( German pronunciation: [ˈviːsˌbaːdn̩] ; lit. ' meadow baths ' ) is the capital of the German state of Hesse , and the second-largest Hessian city after Frankfurt am Main . With around 283,000 inhabitants, it
9088-539: The 4th century , had drifted from reality . The Gregorian calendar reform also dealt with the accumulated difference between these figures, between the years 325 and 1582, by skipping 10 days to set the ecclesiastical date of the equinox to be 21 March, the median date of its occurrence at the time of the First Council of Nicea in 325. Countries that adopted the Gregorian calendar after 1699 needed to skip an additional day for each subsequent new century that
9230-792: The Allied occupation of the Rhineland and was occupied by the French army in 1918. In 1921, the Wiesbaden Agreement on German reparations to France was signed in the city. In 1925, Wiesbaden became the headquarters of the British Army of the Rhine until the withdrawal of occupying forces from the Rhineland in 1930. In 1929, an airport was constructed in Erbenheim on the site of a horse-racing track. In 1936, Fighter Squadron 53 of
9372-583: The Boyne was commemorated with smaller parades on 1 July. However, both events were combined in the late 18th century, and continue to be celebrated as " The Twelfth ". Because of the differences, British writers and their correspondents often employed two dates, a practice called dual dating , more or less automatically. Letters concerning diplomacy and international trade thus sometimes bore both Julian and Gregorian dates to prevent confusion. For example, Sir William Boswell wrote to Sir John Coke from The Hague
9514-462: The British Isles and colonies converted to the Gregorian calendar, instructed that his tombstone bear his date of birth by using the Julian calendar (notated O.S. for Old Style) and his date of death by using the Gregorian calendar. At Jefferson's birth, the difference was eleven days between the Julian and Gregorian calendars and so his birthday of 2 April in the Julian calendar is 13 April in
9656-410: The British colonies, changed the start of the year from 25 March to 1 January, with effect from "the day after 31 December 1751". (Scotland had already made this aspect of the changes, on 1 January 1600.) The second (in effect ) adopted the Gregorian calendar in place of the Julian calendar. Thus "New Style" can refer to the start-of-year adjustment , to the adoption of the Gregorian calendar , or to
9798-409: The Count of Nassau-Usingen granted a concession for gambling in Wiesbaden. In 1810, the Wiesbaden Casino (German: Spielbank ) was opened in the old Kurhaus. Gambling was later outlawed by Prussian authorities in 1872. As a result of Napoleon 's victory over Austria in the Battle of Austerlitz , the Holy Roman Empire was dissolved in 1805. On 12 July 1806, 16 states in present-day Germany, including
9940-508: The Gregorian calendar. For example, the Battle of Blenheim is always given as 13 August 1704. However, confusion occurs when an event involves both. For example, William III of England arrived at Brixham in England on 5 November (Julian calendar), after he had set sail from the Netherlands on 11 November (Gregorian calendar) 1688. The Battle of the Boyne in Ireland took place a few months later on 1 July 1690 (Julian calendar). That maps to 11 July (Gregorian calendar), conveniently close to
10082-466: The Gregorian calendar. Similarly, George Washington is now officially reported as having been born on 22 February 1732, rather than on 11 February 1731/32 (Julian calendar). The philosopher Jeremy Bentham , born on 4 February 1747/8 (Julian calendar), in later life celebrated his birthday on 15 February. There is some evidence that the calendar change was not easily accepted. Many British people continued to celebrate their holidays "Old Style" well into
10224-660: The House of Nassau and thereby, Wiesbaden, received the right of coinage from Holy Roman Emperor Louis the Bavarian . In 1355, the County of Nassau-Weilburg was divided among the sons of Gerlach. The County of Nassau's holdings would be subdivided many times among heirs, with the parts being brought together again whenever a line died out. Wiesbaden became the seat of the County of Nassau-Wiesbaden under Count Adolf I (1307–1370), eldest son of Gerlach. It would eventually fall back to Nassau-Weilburg in 1605. Due to its participation in
10366-430: The Julian and Gregorian dating systems respectively. The need to correct the calendar arose from the realisation that the correct figure for the number of days in a year is not 365.25 (365 days 6 hours) as assumed by the Julian calendar but slightly less (c. 365.242 days). The Julian calendar therefore has too many leap years . The consequence was that the basis for the calculation of the date of Easter , as decided in
10508-527: The Julian calendar had added since then. When the British Empire did so in 1752, the gap had grown to eleven days; when Russia did so (as its civil calendar ) in 1918, thirteen days needed to be skipped. In the Kingdom of Great Britain and its possessions, the Calendar (New Style) Act 1750 introduced two concurrent changes to the calendar. The first, which applied to England, Wales, Ireland and
10650-517: The Julian date of the subsequent (and more decisive) Battle of Aughrim on 12 July 1691 (Julian). The latter battle was commemorated annually throughout the 18th century on 12 July, following the usual historical convention of commemorating events of that period within Great Britain and Ireland by mapping the Julian date directly onto the modern Gregorian calendar date (as happens, for example, with Guy Fawkes Night on 5 November). The Battle of
10792-547: The Prussian-dominated imperial government closed down all German gambling houses. The Wiesbaden casino was reopened in 1949. The Schloßplatz ("palace square") is situated in the center of the city, surrounded by several outstanding buildings. The ducal palace was begun under William, Duke of Nassau . Its foundations were laid in 1837 and it was completed in November 1841 (two years after William's death). For
10934-518: The Rhine makes up 10.3 kilometres (6.4 mi). Wiesbaden has a temperate - oceanic climate ( Köppen : Cfb ; Trewartha : Dobk ) with relatively cold winters and warm summers. Its average annual temperature is 9.8 °C (49.6 °F), with monthly mean temperatures ranging from 1.2 °C (34.2 °F) in January to 18.9 °C (66.0 °F) in July. The Wiesbaden weather station has recorded
11076-815: The Taunus range. The downtown is drained only by the narrow valley of the Salzbach, a tributary of the Rhine, on the eastern flanks of the Mosbacher Mountain. The city's main railway line and the Mainz road ( Mainzer Straße ) follow this valley. Several other streams drain into the Salzbach within the city center: the Wellritzbach, the Kesselbach, the Schwarzbach, the Dambach, and the Tennelbach, as well as
11218-583: The Wiesbaden area over the course of the 6th century. In the 8th century, Wiesbaden became the site of a royal palace of the Frankish kingdom. The first documented use of the name Wiesbaden is by Einhard , the biographer of Charlemagne , whose writings mention "Wisabada" sometime between 828 and 830. When the Frankish Carolingian Empire broke up in 888, Wiesbaden was in the eastern half, called East Francia (which would evolve into
11360-402: 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. 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
11502-576: The chair after being offered it. 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 declining the chair at Halle. 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
11644-600: The city became the ducal residence. Building activity started to give the city a magnificent appearance. Most of the historical center of Wiesbaden dates back to this time. In the Revolutions of 1848 , 30,000 citizens of Nassau assembled in Wiesbaden on 4 March. They demanded a constitution from the Duke, which they received. In the Austro-Prussian War of 1866, Nassau took Austria's side. This decision led to
11786-462: The city. Many wealthy persons chose Wiesbaden as their retirement seat, as it offered leisure and medical treatment alike. In the latter part of the 19th century, Wiesbaden became the German city with the most millionaires. In 1894, the present Hessian State Theater, designed by the Vienna architects Fellner and Helmer, was built on behalf of Kaiser Wilhelm II. After World War I, Wiesbaden fell under
11928-430: The class of all sets. This axiom implies that these big classes are not sets, which eliminates the paradoxes since they cannot be members of any class. Von Neumann also used his axiom to prove the well-ordering theorem: Like Cantor, he assumed that the ordinals form a set. The resulting contradiction implies that the class of all ordinals is not a set. Then his axiom provides a one-to-one correspondence between this class and
12070-438: The class of all sets. This correspondence well-orders the class of all sets, which implies the well-ordering theorem. In 1930, Zermelo defined models of set theory that satisfy von Neumann's axiom . 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
12212-492: The combination of the two. It was through their use in the Calendar Act that the notations "Old Style" and "New Style" came into common usage. When recording British history, it is usual to quote the date as originally recorded at the time of the event, but with the year number adjusted to start on 1 January. The latter adjustment may be needed because the start of the civil calendar year had not always been 1 January and
12354-456: 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. This crisis led him to apply to lecture on philosophy rather than on mathematics. He also began an intense study of Elizabethan literature , thinking there might be evidence that Francis Bacon wrote
12496-416: The continuum hypothesis has been underscored by later developments in the field of mathematics: a 1940 result by Kurt Gödel and a 1963 one by Paul Cohen together imply that the continuum hypothesis can be neither proved nor disproved using standard Zermelo–Fraenkel set theory plus the axiom of choice (the combination referred to as " ZFC "). In 1883, Cantor divided the infinite into the transfinite and
12638-482: The encounter did not come about. The following year, St. Andrews awarded Cantor an honorary doctorate, but illness precluded his receiving the degree in person. Cantor retired in 1913, and lived in poverty and suffered from malnourishment during World War I . The public celebration of his 70th birthday was canceled because of the war. In June 1917, he entered a sanatorium for the last time and continually wrote to his wife asking to be allowed to go home. Georg Cantor had
12780-581: The end of the duchy. After the Austrian defeat, Nassau was annexed by Prussia and became part of the Prussian province of Hesse-Nassau . The deposed duke Adolph of Nassau in 1890 became the Grand Duke of Luxembourg (see House of Nassau ). This turned out to be a fortunate change for the city, as it then became an international spa town. A rise in construction commenced after the aristocracy followed
12922-543: The end of the following December, 1661/62 , a form of dual dating to indicate that in the following twelve weeks or so, the year was 1661 Old Style but 1662 New Style. Some more modern sources, often more academic ones (e.g. the History of Parliament ) also use the 1661/62 style for the period between 1 January and 24 March for years before the introduction of the New Style calendar in England. The Gregorian calendar
13064-523: The first president of this society. Setting aside the animosity Kronecker had displayed towards him, Cantor invited him to address the meeting, but Kronecker was unable to do so because his wife was dying from injuries sustained in a skiing accident at the time. Georg Cantor was also instrumental in the establishment of the first International Congress of Mathematicians , which took place in Zürich, Switzerland, in 1897. After Cantor's 1884 hospitalization there
13206-603: The following extreme values: While evidence of settlement at present-day Wiesbaden dates back to the Neolithic era, historical records document continuous occupancy after the erection of a Roman fort in 6 AD which housed an auxiliary cavalry unit. The thermal springs of Wiesbaden are first mentioned in Pliny the Elder 's Naturalis Historia . They were famous for their recreation pools for Roman army horses and possibly as
13348-742: 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. Whenever Cantor applied for a post in Berlin, he was declined, and the process usually involved Kronecker, so Cantor came to believe that Kronecker's stance would make it impossible for him ever to leave Halle. In 1881, Cantor's Halle colleague Eduard Heine died. Halle accepted Cantor's suggestion that Heine's vacant chair be offered to Dedekind, Heinrich M. Weber and Franz Mertens , in that order, but each declined
13490-505: The history of set theory and on Cantor's religious ideas. This was later published, as were several of his expository works. 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 Peter Gustav Lejeune Dirichlet , Rudolf Lipschitz , Bernhard Riemann , and Heine himself:
13632-583: 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
13774-475: The important status of the piano in post-war German homes. Wiesbaden has long been famous for its thermal springs and spa. Use of the thermal springs was first documented by the Romans. The business of spring bathing became important for Wiesbaden near the end of the Middle Ages . By 1370, 16 bath houses were in operation. By 1800, the city had 2,239 inhabitants and 23 bath houses. By 1900, Wiesbaden, with
13916-583: The inconsistency of infinitesimals . The beginning of set theory as a branch of mathematics is often marked by the publication of Cantor's 1874 paper , "Ueber eine Eigenschaft des Inbegriffes aller reellen algebraischen Zahlen" ("On a Property of the Collection of All Real Algebraic Numbers"). This paper was the first to provide a rigorous proof that there was more than one kind of infinity. Previously, all infinite collections had been implicitly assumed to be equinumerous (that is, of "the same size" or having
14058-437: The intuitions of the mind. 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. Mathematicians such as L. E. J. Brouwer and especially Henri Poincaré adopted an intuitionist stance against Cantor's work. Finally, Wittgenstein 's attacks were finitist: he believed that Cantor's diagonal argument conflated
14200-557: The last (Rudolph) born in 1886. Cantor was able to support a family despite his 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 had met at Interlaken in Switzerland two years earlier while on holiday. Cantor was promoted to extraordinary professor in 1872 and made full professor in 1879. To attain
14342-410: 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, at that time the leading German university. However, his work encountered too much opposition for that to be possible. Kronecker, who headed mathematics at Berlin until his death in 1891, became increasingly uncomfortable with the prospect of having Cantor as
14484-555: The lead of the Hohenzollern emperors, who began annual trips to Wiesbaden. The period around the turn of the 20th century is regarded as the heyday of the city. Kaiser Wilhelm II visited the city regularly in summer, such that it became an unofficial "summer residence". The city was also popular among the Russian nobility. In the wake of the imperial court, numerous nobles, artists, and wealthy businessmen increasingly settled in
14626-447: The members of two sets, defined infinite and well-ordered sets , and proved that the real numbers are more numerous than the natural numbers . Cantor's method of proof of this 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 he was well aware of. Originally, Cantor's theory of transfinite numbers
14768-507: The natural numbers. His notation for the cardinal numbers was the Hebrew letter ℵ {\displaystyle \aleph } ( ℵ , aleph ) with a natural number subscript; for the ordinals he employed the Greek letter ω {\displaystyle \omega } ( ω , omega ). This notation is still in use today. The Continuum hypothesis , introduced by Cantor,
14910-445: The notion of dimension . 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 countable sets (or denumerable sets) as sets which can be put into
15052-418: The old city hall, was built in 1609 and 1610. No older buildings are preserved due to two fires in 1547 and 1561. In 1648, at the end of the devastating Thirty Years' War , chronicles tell that Wiesbaden had barely 40 residents left. In 1659, the County of Nassau-Weilburg was divided again. Wiesbaden became part of the County of Nassau-Usingen. In 1744, the seat of Nassau-Usingen was moved to Biebrich . In 1771,
15194-467: The ordinals form a set, proved that this leads to a contradiction, and concluded that the ordinals form an inconsistent multiplicity. He used this inconsistent multiplicity to prove the aleph theorem. In 1932, Zermelo criticized the construction in Cantor's proof. Cantor avoided paradoxes by recognizing that there are two types of multiplicities. In his set theory, when it is assumed that the ordinals form
15336-403: The other, stili novi or stilo novo , abbreviated st.n. and meaning "(of/in) new style". The Latin abbreviations may be capitalised differently by different users, e.g., St.n. or St.N. for stili novi . There are equivalents for these terms in other languages as well, such as the German a.St. (" alter Stil " for O.S.). Usually, the mapping of New Style dates onto Old Style dates with
15478-728: The outflow of many thermal and mineral springs in the Kurhaus (spa) district. Above the city center, the Salzbach is better known as the Rambach. The highest point of the Wiesbaden municipality is located northwest of the city center near the summit of the Hohe Wurzel, with an elevation of 608 metres (1,995 ft) above sea level . The lowest point is the harbour entrance of Schierstein at 83 metres (272 ft) above sea level. The central square (the Schlossplatz , or palace square)
15620-455: The paper where he first set out his celebrated definition of real numbers by Dedekind cuts . While extending the notion of number by means of his revolutionary concept of infinite cardinality, Cantor was paradoxically opposed to theories of infinitesimals of his contemporaries Otto Stolz and Paul du Bois-Reymond , describing them as both "an abomination" and "a cholera bacillus of mathematics". Cantor also published an erroneous "proof" of
15762-440: The paradoxes by using an approach similar to Cantor's—namely, by identifying collections that are not sets and treating them differently. Von Neumann stated that a class is too big to be a set if it can be put into one-to-one correspondence with the class of all sets. He defined a set as a class that is a member of some class and stated the axiom: A class is not a set if and only if there is a one-to-one correspondence between it and
15904-470: The plays attributed to William Shakespeare (see Shakespearean authorship question ); this ultimately resulted in two pamphlets, published in 1896 and 1897. Cantor recovered soon thereafter, and subsequently made further important contributions, including his diagonal argument and theorem . However, he never again attained the high level of his remarkable papers of 1874–84, even after Kronecker's death on 29 December 1891. He eventually sought, and achieved,
16046-410: The real algebraic numbers as a sequence a 1 , a 2 , a 3 , .... In other words, the real algebraic numbers are countable. Cantor starts his second construction with any sequence of real numbers. Using this sequence, he constructs nested intervals whose intersection contains a real number not in the sequence. Since every sequence of real numbers can be used to construct a real not in
16188-574: 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 assertion that "the essence of mathematics is its freedom." These ideas parallel those of Edmund Husserl , whom Cantor had met in Halle. Old Style and New Style dates Old Style ( O.S. ) and New Style ( N.S. ) indicate dating systems before and after
16330-615: The remaining counties of Nassau-Usingen and Nassau-Weilburg, formally left the Holy Roman Empire and joined in the Confederation of the Rhine . Napoleon was its "protector". Under pressure from Napoleon, both counties merged to form the Duchy of Nassau on 30 August 1806. At the 1815 Congress of Vienna , the Duchy of Nassau joined the German Confederation . The capital of Nassau was moved from Weilburg to Wiesbaden, and
16472-399: The rest of his life, for which he was excused from teaching on several occasions and repeatedly confined to various sanatoria. The events of 1904 preceded a series of hospitalizations at intervals of two or three years. 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
16614-443: The same form as held by his critics, was long a concern of Cantor's. 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. To Cantor, his mathematical views were intrinsically linked to their philosophical and theological implications – he identified
16756-531: The same number of elements). Cantor proved that the collection of real numbers and the collection of positive integers are not equinumerous. In other words, the real numbers are not countable . His proof differs from the diagonal argument that he gave in 1891. Cantor's article also contains a new method of constructing transcendental numbers . Transcendental numbers were first constructed by Joseph Liouville in 1844. Cantor established these results using two constructions. His first construction shows how to write
16898-439: The sequence, the real numbers cannot be written as a sequence – that is, the real numbers are not countable. By applying his construction to the sequence of real algebraic numbers, Cantor produces a transcendental number. Cantor points out that his constructions prove more – namely, they provide a new proof of Liouville's theorem: Every interval contains infinitely many transcendental numbers. Cantor's next article contains
17040-627: The source of a mineral used for red hair dye (which was very fashionable around the turn of BC/AD among women in Rome). The Roman settlement is first mentioned using the name Aquae Mattiacorum ( Latin for "Waters of the Mattiaci") in 121. The Mattiaci were a Germanic tribe , possibly a branch of the neighboring Chatti , who lived in the vicinity at that time. The town also appears as Mattiacum in Ptolemy 's Geographia (2.10). The Roman Empire built
17182-400: The theory of transfinite numbers with pantheism . Although later this Cardinal accepted the theory as valid, due to some clarifications from Cantor's. Cantor even sent one letter directly to Pope Leo XIII himself, and addressed several pamphlets to him. 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
17324-460: 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. 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 . He
17466-477: The twenty-six remaining years of ducal authority it was the residence of the ruling family. It later served as a secondary residence for the King of Prussia 1866 to 1918. It was later used as a headquarters for French and British occupying forces after World War I, then as a museum. Since 1945, the building has served as Landtag (parliamentary building) for the state of Hesse. The site of the palace had been that of
17608-504: The uniqueness of the representation of a function by trigonometric series . Cantor solved this problem in 1869. It was while working on this problem that he discovered transfinite ordinals, which occurred as indices n in the n th derived set S n of a set S of zeros of a trigonometric series. Given a trigonometric series f(x) with S as its set of zeros, Cantor had discovered a procedure that produced another trigonometric series that had S 1 as its set of zeros, where S 1
17750-552: The uprisings of the German Peasants' War of 1525, Wiesbaden lost all its privileges for over 40 years. During this time, Wiesbaden became Protestant with the nomination of Wolf Denthener as first Lutheran pastor on 1 January 1543. The same day, the first Latin school was opened, preparing pupils for the gymnasium in Idstein . In 1566, the privileges of the city were restored. The oldest remaining building of Wiesbaden,
17892-608: The war of Emperor Frederick II against the Pope, the Archbishop of Mainz , Siegfried III , ordered the city's destruction. Wiesbaden returned to the control of the House of Nassau in 1270 under Count Walram II, Count of Nassau . However, Wiesbaden and the castle at Sonnenberg were again destroyed in 1283 in conflict with Eppstein . Walram's son and successor Adolf would later become king of Germany from 1292 until 1298. In 1329, under Adolf's son Gerlach I of Nassau-Weilburg
18034-517: The year 1984, which to me seemed too great a demand! ... But of course I never want to know anything again about Acta Mathematica ." Cantor suffered his first known bout of depression in May 1884. Criticism of his work weighed on his mind: every one of the fifty-two letters he wrote to Mittag-Leffler in 1884 mentioned Kronecker. A passage from one of these letters is revealing of the damage to Cantor's self-confidence: ... I don't know when I shall return to
18176-582: Was a misinterpretation of infinity, and was convinced that set theory could help correct this mistake: "... 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.". Prominent neo-scholastic German philosopher Constantin Gutberlet was in favor of such theory, holding that it didn't oppose the nature of God. Cantor also believed that his theory of transfinite numbers ran counter to both materialism and determinism – and
18318-444: Was also the first to appreciate the importance of one-to-one correspondences (hereinafter denoted "1-to-1 correspondence") in set theory. He used this concept to define finite and infinite sets , subdividing the latter into denumerable (or countably infinite) sets and nondenumerable sets (uncountably infinite sets). Cantor developed important concepts in topology and their relation to cardinality . For example, he showed that
18460-480: Was altered at different times in different countries. From 1155 to 1752, the civil or legal year in England began on 25 March ( Lady Day ); so for example, the execution of Charles I was recorded at the time in Parliament as happening on 30 January 164 8 (Old Style). In newer English-language texts, this date is usually shown as "30 January 164 9 " (New Style). The corresponding date in the Gregorian calendar
18602-498: Was attacked by both RAF and United States Air Force bombers on 66 days. In the attacks, about 18% of the city's homes were destroyed. During the war, more than 25% of the city's buildings were damaged or worse and 1,700 people were killed. Wiesbaden was the location of a camp for Sinti and Romani people (see Romani Holocaust ), and two subcamps of the Hinzert concentration camp , mostly for Luxembourgish prisoners. Wiesbaden
18744-434: Was captured by U.S. Army forces on 28 March 1945. The U.S. 317th Infantry Regiment attacked in assault boats across the Rhine from Mainz while the 319th Infantry attacked across the river Main near Hochheim am Main . The attack started at 01:00 and by early afternoon the two forces of the 80th U.S. Infantry Division had linked up with the loss of only three dead and three missing. The Americans captured 900 German soldiers and
18886-596: Was celebrated as the New Year festival from as early as the 13th century, despite the recorded (civil) year not incrementing until 25 March, but the "year starting 25th March was called the Civil or Legal Year, although the phrase Old Style was more commonly used". To reduce misunderstandings about the date, it was normal even in semi-official documents such as parish registers to place a statutory new-year heading after 24 March (for example "1661") and another heading from
19028-425: Was concerned about the philosophical nature and new terminology in a paper Cantor had submitted to Acta . 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 then curtailed his relationship and correspondence with Mittag-Leffler, writing to a third party, "Had Mittag-Leffler had his way, I should have to wait until
19170-446: 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
19312-535: Was extended to include Lorraine , including Nancy , and the Grand Duchy of Luxembourg . The commander was General der Infanterie Walther Schroth . Wehrkreis XII was made up of three subordinate regions: Bereich Hauptsitze Koblenz , Mannheim and Metz . During the war, Wiesbaden was, between August 1940 and the end of 1942, bombed by the Royal Air Force and from 1943 through to March 1945,
19454-702: Was headquartered at the Wiesbaden Army Airfield , just off the autobahn toward Frankfurt, until the Division completed relocation to Fort Bliss, Texas, in 2011. Wiesbaden is now home to the U.S. Army Europe Headquarters and the General John Shalikashvili Mission Command Center. In 1962, the American artists George Maciunas , Dick Higgins and Alison Knowles traveled to Europe to promote
19596-549: Was implemented in Russia on 14 February 1918 by dropping the Julian dates of 1–13 February 1918 , pursuant to a Sovnarkom decree signed 24 January 1918 (Julian) by Vladimir Lenin . The decree required that the Julian date was to be written in parentheses after the Gregorian date, until 1 July 1918. It is common in English-language publications to use the familiar Old Style or New Style terms to discuss events and personalities in other countries, especially with reference to
19738-750: Was presented by David Hilbert as the first of his twenty-three open problems in his address at the 1900 International Congress of Mathematicians in Paris. Cantor's work also attracted favorable notice beyond Hilbert's celebrated encomium. The US philosopher Charles Sanders Peirce praised Cantor's set theory and, following public lectures delivered by Cantor at the first International Congress of Mathematicians, held in Zürich in 1897, Adolf Hurwitz and Jacques Hadamard also both expressed their admiration. At that Congress, Cantor renewed his friendship and correspondence with Dedekind. From 1905, Cantor corresponded with his British admirer and translator Philip Jourdain on
19880-522: Was regarded as counter-intuitive – even shocking. This caused it to encounter resistance from mathematical contemporaries such as Leopold Kronecker and Henri Poincaré and later from Hermann Weyl and L. E. J. Brouwer , while Ludwig Wittgenstein raised philosophical objections ; see Controversy over Cantor's theory . Cantor, a devout Lutheran Christian , believed the theory had been communicated to him by God. Some Christian theologians (particularly neo-Scholastics ) saw Cantor's work as
20022-536: Was shocked when he realized that he was the only faculty member at Halle who did not hold to deterministic philosophical beliefs. It was important to Cantor that his philosophy provided an "organic explanation" of nature, and in his 1883 Grundlagen , he said that such an explanation could only come about by drawing on the resources of the philosophy of Spinoza and Leibniz. In making these claims, Cantor may have been influenced by F. A. Trendelenburg , whose lecture courses he attended at Berlin, and in turn Cantor produced
20164-552: Was the capital of the Province of Nassau . In 1945, it became the capital of Greater Hesse and subsequently, in 1946, of Hesse . Wiesbaden is one of the oldest spa towns in Europe. Its name translates to "meadow baths", and there are 15 mineral springs —14 of which are hot springs —in the city centre. With a yield of around 2 million liters daily, Wiesbaden is the second-most productive German spa after Aachen . Its location in
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