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Abstract

This paper provides an explication of mathematician Georg Cantor’s 1883 proof of the nondenumerability of perfect sets of real numbers. A set of real numbers is denumerable if it has the same (infinite) cardinality as the set of natural numbers {1, 2, 3, …}, and it is perfect if it consists only of so-called limit points (none of its points are isolated from the rest of the set). Directly from this proof, Cantor deduced that every infinite closed set of real numbers has only two choices for cardinality: the cardinality of the set of natural numbers, or the cardinality of the set of real numbers. This result strengthened his belief in his famous continuum hypothesis that every infinite subset of real numbers had one of those two cardinalities and no other. This paper also traces Cantor’s realization that understanding perfect sets was key to understanding the structure of the continuum (the set of real numbers) back through some of his results from the 1874–1883 period: his 1874 proof that the set of real numbers is nondenumerable, which confirmed Cantor’s intuitive belief in the richness of the continuum compared to discrete subsets (such as the set of natural numbers) and proved that there was more than one “size” of infinite cardinal; his 1878 proof that continuous domains of different dimensions (such as a one-dimensional line and a two-dimensional surface) surprisingly have the same cardinality; and his 1883 definition of a continuum as a set that is connected (all of one piece) and perfect.

Author Bio

Laila Awadalla wrote this paper when she was an undergraduate for a History of Mathematics course, taught by Dr. Richard Delaware at the University of Missouri–Kansas City. She will start working toward a Ph.D. in mathematics at the University of Nebraska–Lincoln in the fall.

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