Georg Cantor astonished the mathematical world with his discovery that the integers and the real numbers cannot be placed into one-to-one onto correspondance, showing that there is not just one infinity but at least two infinities: countable infinity and uncountable infinity. It was later discovered that there is actually an infinite tower of infinities, a result of the theorem that the power set of any set has larger cardinality than the set itself.
Georg Cantor was German, but he spent most of the first eleven years of his life in St. Petersburg, Russia, and then his family moved back to Germany where he spent the rest of his life. He got his doctorate from the University of Berlin in 1867, with a dissertation on number theory. In his years as a mathematician and professor, he corresponded frequently with Dedekind, who published one of best rigorous definitions of real numbers (now called Dedekind cuts), and he inspired Cantor to work on this and other problems related to Analysis. In 1878 Cantor published his paper that made the concept of a 1-1 correspondence precise, and this began a long period where he was looked upon with disfavor by many important mathematicians of the time, because his ideas were so controversial and so contrary to the accepted mathematical understanding.
Unfortunately Cantor suffered through many periods of depression, which was certainly aggravated by the fact that many mathematicians disliked his ideas. When not working on math he worked on philosophy and also his pet literary project, which was his theory that Francis Bacon actually wrote all of Shakespeare's plays. From 1905 onward he was ill much of the time, and spent most of the rest of his life in a hospital, dying in 1918 of a heart attack.
Although Cantor's theorems were about pure mathematics, his proof techniques have led to many important results in computer science. In fact, the halting problem is often proved to be undecidable using the exact same diagonalization technique that Cantor invented to show that the real numbers are not countable.
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