Georg Cantor

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Georg Cantor

Born: March 3, 1845 Died: January 6, 1918 Georg Cantor lived at the end of the 19th century and early 20th century. This is a time period in both mathematics and the world that is referred to as "the age of abstraction". Ideas and philosophies were changing from the concrete to the abstract. This could be seen in many fields along with mathematics. In economics abstract notions of different types of economies such as communism were described Marx And Engle and capitalism was described by Adam Smith. The world of art was changing to a more abstract form. Artists moved from being a "camera" that could reproduce what the human eye could see to having an abstract eye. For example the works of Cezanne, Van Gogh and Gauguin differed greatly from the works of Monet. Mathematicians began to cross the gap of what visual or physical reality would dictate, such as the innovation of Bolyai and Lobachevski concerning non- Euclidean geometries.[1 p 246]

Georg Cantor was born in Denmark and grew up with a deep appreciation for culture and the arts which was instilled in him by his mother who had considerable musical talent as a violinist. In terms of religion Georg's mother and father were a mixed marriage, his father was a Protestant who had converted from Judaism and his mother Roman Catholic. Georg was raised as a Protestant. Georg Cantor's father was a successful merchant and stock broker in St. Petersburg whose wealth enabled him to afford a private tutor for Georg's early education. The family moved to Germany because his father's health required a warmer climate. When the family first moved to Germany young Cantor lived at home and studied at the Gymnasium in Wiesbaden. Later the family moved to Frankfurt where he went away to boarding school at the Realschule in Darmstadt. In 1860 he graduated with an excellent academic record with exceptional skills in mathematics, and in particular trigonometry. He attended the Hoheren Gewerbeschule in Darmstadt for two years to study engineering and then entered the Polytechnic of Zurich in 1862. In that year, Cantor sought his father's permission to study mathematics and was overjoyed when his father gave his consent. His studies were cut short by the death of his father in the next year. [2] Cantor moved to the University of Berlin where he had instructors such as Weierstrass, Kummer and Kronecker. Cantor would occasionally travel to Gttingen to study. He would complete his dissertation at the University of Berlin in the area of number theory in 1867. In Berlin he was involved with the Mathematical Society and would become its president in 1864-1865.[2] Cantor accepted a position teaching at girl's school in Berlin. He joined the Schellbach Seminar for mathematics teachers while completing his habilitation degree in 1869. Cantor received an appointment at Halle and the focus of his research changed from number theory to analysis. This is because one of his colleagues challenged him to prove an open problem concerning the representation of functions as trigonometric functions, in particular sines and cosines. This was a famous problem that had been attempted by Dirchlet, Lipschitz and Riemann. Cantor solved the problem in April 1870 and his solution clearly reflected the teaching of Weierstrass.[2] In 1872 Cantor was promoted to Extraordinary Professor at Halle. He published a paper that year in which he characterized irrational numbers as convergent series of rational numbers. Cantor had started a friendship with Dedekind (they met on holiday in Switzerland) who referred to Cantor's result in famous characterization of the real numbers consisting as "Dedekind Cuts".[2] Cantor's work with series of trigonometric functions and the convergence of series led him to consider intrinsic differences among various sets of numbers. In particular this meant devising a means for comparing the size of sets that did not rely on the concept of counting. Cantor used the idea of "equinumerosity" to characterize if two set contain the same number of elements. This is a notion where sets are compared by an element by element pairing to see if allelements in one set could be paired with all elements in a second set. If such a pairing exists we call the sets equinumerous or in modern terms we say the sets are in one-to-one correspondence.[1 p. 253]

In Cantor's own words: Two sets M and N are equivalent...if it is possible to put them , by some law, in such a relation to one another that to every element of each one of them corresponds one and only one element of the other.[1 p. 253]

Cantor was moving to a completely new concept of characterizing the infinite. The mathematicians who preceded Cantor objected to the idea that a process that considered an infinite set as being able to be "completed" or a process that referenced them as finished was not sound reasoning. Gauss once commented: ...I protest above all against the use of an infinite quantity as a completed one, which in mathematics is never allowed. The infinite is only a manner of speaking...[1 p 254]

At this time (1874) Cantor was engaged to Vally Guttmann a friend of his sisters, who introduced her to Cantor because she feared he was spending to much time on his professional activities. They married on August 9, 1874 they spent their honeymoon in Interlaken in Switzerland where Cantor knew Dedekind was on vacation. Cantor spent much of his honeymoon in mathematical discussions with Dedekind.[2] Cantor's research led him to a discovery that many different types of infinity exist. In fact thereare an infinite number of different types of infinity. Along the way to this discovery he was able to find sets of points that were eqinumerous that he did not expect. He was able to prove that the set of points in the unit interval was equinumerous with the points in the unit solid in n-dimensional space. This surprised and amazed Cantor so much he is attributed with a famous quote: I see it, but I don't believe it! [2]

In 1881 Heine died leaving open a very important position at Halle. Cantor was asked to recommend a replacement for Heine. Cantor asked his good friend Dedekind to fill the place and he was turned down. Weber and Mertens, his second and third choices also turned him down. Cantor realized that his work was not widely accepted and people (even his best friend) did not want there work associated with him. Cantor states clearly the opposition to his ideas: ...I realize that in this undertaking I place myself in a certain opposition to views widely held concerning the mathematical infinite and to opinions frequently defended on the nature of numbers. [2]

Cantor became very depressed. In 1884 he had his first recorded attack of depression that he recovered from after a few weeks. His experience made him lose confidence and fear the treatment he was subject to. The treatment for mental health disorders at this time was confinement in a sanatoria which was very unpleasant. Cantor began to work on the Continuum Hypothesis, but was not able to make much progress. The Continuum Hypothesis was a theory that stated that the cardinality of the real numbers was next in order after the natural numbers. The inability to resolve this worsened his mental state. It was improved and his depression kept in check with his family and personal life. In 1886 he bought a new house and his wife gave birth to the last of his six children. Cantor published a strange paper in 1894 that showed Golbach's conjecture was true for all even numbers up to 1000. This had already been done for all numbers up to 10000 forty years before. This gives more evidence of his state of mind and wanting to be accepted again by the mathematical community. Cantor suffered from periods of depression from 1899 on, following the death of his youngest son. He was in and out of the sanatoria several times between 1902 and 1908. He had to take leave from his teaching responsibilities during many winter semesters. When Cantor suffered from periods of depression he turned away from mathematics and toward his family, philosophy and Shakespeare. He proported a theory that Francis Bacon had wrote Shakespeare's plays. In 1911 Cantor was invited to the University of St. Andrews as a distinguished foreign scholar. Cantor had hoped to meet with Bertrand Russell who had just published Principia Mathematica to discuss his theory on sets, but news his son was ill made him return to Germany. He retired in 1913 and spent his final years ill and starving because of the small amount of food available to German citizens due to the war. A major celebration of his 70th birthday was planned at Halle but was canceled due to the war. In 1917 he entered the sanitarium for the last time were he would continually write his wife asking to go home. He died mentally ill, scared, starving and penniless of a heart attack.[2] Hilbert said the following of Cantor's work: ...the finest product of mathematical genius and one of the supreme achievements of purely intellectual human activity.[2]

[1] Dunham, William. Journey Through Genius The Great Theorems of Mathematics. New York: John Wiley & Sons 1990.

[2] Web site: MacTutor History. (9/2000).

Cantor's work begins with his notion of "equinumerosity". In modern times we would say that two sets are "equinumerous" if there exists a one-to-one correspondence between the two sets. A one-to-one correspondence is a function from one set to another that is both one-to-one and onto. A cardinal number is what is used to denote the class of all sets that can be put into one-to-one correspondence with each other.

That is to say a if there is a function f:AB that is both one-to-one and onto, then we say the set A is in one-to-one correspondence with the set B. Sets and One-to-One CorrespondencesAn important tool