Term Paper: Contributions of Georg Cantor in Mathematics

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This is a term paper on Georg Cantor’s contribution in the field of mathematics. Cantor was the first to show that there was more than one kind of infinity. In doing so, he was the first to cite the concept of a 1-to-1 correspondence, even though not calling it such.

Cantor’s 1874 paper, “On a Characteristic Property of All Real Algebraic Numbers”, was the beginning of set theory. It was published in Crelle’s Journal. Previously, all infinite collections had been thought of being the same size, Cantor was the first to show that there was more than one kind of infinity. In doing so, he was the first to cite the concept of a 1-to-1 correspondence, even though not calling it such. He then proved that the real numbers were not denumerable, employing a proof more complex than the diagonal argument he first set out in 1891. (O’Connor and Robertson, Wikipaedia)

What is now known as the Cantor’s theorem was as follows: He first showed that given any set A, the set of all possible subsets of A, called the power set of A, exists. He then established that the power set of an infinite set A has a size greater than the size of A. consequently there is an infinite ladder of sizes of infinite sets.

Cantor was the first to recognize the value of one-to-one correspondences for set theory. He distinct finite and infinite sets, breaking down the latter into denumerable and nondenumerable sets. There exists a 1-to-1 correspondence between any denumerable set and the set of all natural numbers; all other infinite sets are nondenumerable. From these come the transfinite cardinal and ordinal numbers, and their strange arithmetic. His notation for the cardinal numbers was the Hebrew letter aleph with a natural number subscript; for the ordinals he engaged the Greek letter omega. He proved that the set of all rational numbers is denumerable, but that the set of all real numbers is not and therefore is strictly bigger. The cardinality of the natural numbers is aleph-null; that of the real is larger, and is at least aleph-one. (Wikipaedia)

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