Georg Cantor Essay Example For Students

Georg Cantor Essay Georg CantorI. Georg CantorGeorg Cantor founded set theory and introduced the concept of infinite numberswith his discovery of cardinal numbers. He also advanced the study oftrigonometric series and was the first to prove the nondenumerability of thereal numbers. Georg Ferdinand Ludwig Philipp Cantor was born in St. Petersburg,Russia, on March 3, 1845. His family stayed in Russia for eleven years until thefathers sickly health forced them to move to the more acceptable environment ofFrankfurt, Germany, the place where Georg would spend the rest of his life. Georg excelled in mathematics. His father saw this gift and tried to push hisson into the more profitable but less challenging field of engineering. Georgwas not at all happy about this idea but he lacked the courage to stand up tohis father and relented. However, after several years of training, he became sofed up with the idea that he mustered up the courage to beg his father to becomea mathematician. Finally, just before entering college, his father let Georgstudy mathematics. In 1862, Georg Cantor entered the University of Zurich onlyto transfer the next year to the University of Berlin after his fathers death. At Berlin he studied mathematics, philosophy and physics. There he studied undersome of the greatest mathematicians of the day including Kronecker andWeierstrass. After receiving his doctorate in 1867 from Berlin, he was unable tofind good employment and was forced to accept a position as an unpaid lecturerand later as an assistant professor at the University of Halle in1869. In 1874,he married and had six children. It was in that same year of 1874 that Cantorpublished his first paper on the theory of sets. While studying a problem inanalysis, he had dug deeply into its foundations, especially sets and infinitesets. What he found baffled him. In a series of papers from 1874 to 1897, he wasable to prove that the set of integers had an equal number of members as the setof even numbers, squares, cubes, and roots to equations; that the number ofpoints in a line segment is equal to the number of points in an infinite line, aplane and all mathematical space; and that the number of transcen dental numbers,values such as pi(3.14159) and e(2.71828) that can never be the solution to anyalgebraic equation, were much larger than the number of integers. Before inmathematics, infinity had been a sacred subject. Previously, Gauss had statedthat infinity should only be used as a way of speaking and not as a mathematicalvalue. Most mathematicians followed his advice and stayed away. However, Cantorwould not leave it alone. He considered infinite sets not as merely going onforever but as completed entities, that is having an actual though infinitenumber of members. He called these actual infinite numbers transfinite numbers. By considering the infinite sets with a transfinite number of members, Cantorwas able to come up his amazing discoveries. For his work, he was promoted tofull professorship in 1879. However, his new ideas also gained him numerousenemies. Many mathematicians just would not accept his groundbreaking ideas thatshattered their safe world of mathematics. One of these critics was LeopoldKronecker. Kronecker was a firm believer that the only numbers were integers andthat negatives, fractions, imaginaries and especially irrational numbers had nobusiness in mathematics. He simply could not handle actual infinity. Using hisprestige as a professor at the University of Berlin, he did all he could tosuppress Cantors ideas and ruin his life. Among other things, he delayed orsuppressed completely Cantors and his followers publications, belittled hisideas in front of his students and blocked Cantors life ambition of gaining aposition at the prestigious University of Berlin. Not all mathematicians we rehostile to Cantors ideas. Some greats such as Karl Weierstrass, and long-timefriend Richard Dedekind supported his ideas and attacked Kroneckers actions. However, it was not enough. Cantor simply could not handle it. Stuck in a third-rate institution, stripped of well-deserved recognition for his work and underconstant attack by Kronecker, he suffered the first of many nervous breakdownsin 1884. In 1885 Cantor continued to extend his theory of cardinal numbers andof order types. He extended his theory of order types so that now his previouslydefined ordinal numbers became a special case. In 1895 and 1897 Cantor publishedhis final double treatise on sets theory. Cantor proves that if A and B are setswith A equivalent to a subset of B and B equivalent to a subset of A then A andB are equivalent. This theorem was also proved by Felix Bernstein and by Schrder. The rest of his life was spent in and out of mental institutions and hiswork nearly ceased completely. Much too late for him to really enjoy it, histheory finally began to gain recognition by the turn of the century. In 1904, hewas awarded a medal by the Royal Society of London and was made a member of boththe London Mathematical Society and the Society of Sciences in Gottingen. Hedied in a mental institution on January 6, 1918. Today, Cantors work is widelyused in the many fields of mathematics. His theory on infinite sets reset thefoundation of nearly every mathematical field and brought mathematics to itsmodern form. Software Licensing And Piracy EssayThese one-to-one correspondence sets show that even though we add an unknownvariable, multiply by two, and square a set, the upper and lower sets stillremain equal. Since we will never run out of numbers any correspondence set withtwo infinite values will be equal. All these sets clearly have the samecardinality since its members can be put in a one-to-one correspondence witheach other on and on forever. These sets are said to be countably infinite andtheir cardinality is denoted by the Hebrew letter aleph with a subscript nought,. OTHER INFINITIESCantor thought once you start dealing with infinities, everything is the samesize. This did not turn out to be the case. Cantor developed an entire theory oftransfinite arithmetic, the arithmetic of numbers beyond infinity. Although thesizes of the infinite sets of counting numbers, even numbers, odd numbers,square numbers, etc., are the same, there are other sets, the set of numbersthat can be expressed as decimals, for instance, that are larger. Cantors workrevealed that there are hierarchies of ever-larger infinities. The largest oneis called the Continuum. Some mathematicians who lived at the end of the 19thcentury did not want to accept his work at all. The fact that his results wereso paradoxical was not the problem so much as the fact that he consideredinfinite sets at all. At that time, some mathematicians held that mathematicscould only consider objects that could be constructed directly from the countingnumbers. You cant list all the elements in an infinite set, they said, soanything that you say about infinite sets is not mathematics. The most powerfulof these mathematicians was Leopold Kronecker who even developed a theory ofnumbers that did not include any negative numbers. Although Kronecker did notpersuade very many of his contemporaries to abandon all conclusions that reliedon the existence of negative numbers, Cantors work was so revolutionary thatKroneckers argument that it went too far seemed plausible. Kronecker was amember of the editorial boards of the important mathematical journals of his day,and he used his influence to prevent much of Cantors work from being publishedin his lifetime. Cantor did not know at the time of his death, that not onlywould his ideas prevail, but that they would shape the course of 20th centurymathematics. 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