ABSTRACT
In mathematics, it is often the case that there is a long period in which ideas are conceived and developed by several different individuals and then at a later time a significant breakthrough is made by more than one individual. Such was the case with the development of calculus and non-Euclidean geometry. However, on the contrary, set theory is primarily the creation of one individual, Georg Ferdinand Ludwig Philipp Cantor (1845-1918). Cantor was born in St. Petersburg, Russia. His father, Georg Waldemar Cantor, was born in Copenhagen, but moved to St. Petersburg as a young man. Cantor’s mother, Maria Anna Bo¨hm, was Russian. In 1856, when young Georg was eleven years old, the family, consisting of Georg, a brother, a sister, and his parents, moved to Wiesbaden, Germany, and later to Frankfurt due to the poor health of his father. Georg’s father wanted him to become an engineer, so he could make a good living. After receiving the proper technical training in high school at Darmstadt from 1860 to 1862, Georg entered the Polytechnic of Zurich in the fall of 1862. Later in 1862, Georg requested his father’s permission to study mathematics instead of engineering and his father consented. After his father’s death in June 1863, Georg transfered to the University of Berlin, where he completed his doctorate in December 1867. In the spring of 1869, Cantor joined the faculty at the University of Halle as a lecturer. In 1872, he was promoted to assistant professor and in 1879 to professor. Cantor spent the remainder of his life in Halle. Cantor’s early publications were in the area of number theory. However, Heinrich Eduard Heine, a senior colleague of Cantor at Halle, challenged Cantor to prove that a function can be represented uniquely as a trigonometric series. Cantor succeeded in doing so in 1870. In 1872, Cantor published a paper on trigonometric series in which he defined the irrational numbers in terms of convergent sequences of rational numbers. Then Cantor began his lifelong work on set theory and the concept of transfinite numbers. In 1873, he proved the rational numbers are countable-that is, that there is a one-to-one correspondence between the rational numbers and the natural numbers. In 1874, Cantor showed that the real numbers are not countable. After Cantor initiated research in the area of set theory, others made significant contributions. Conjectures which Cantor made opened fertile areas of research for others and the paradoxes which arose later because of his work resulted in important study with respect to the foundations of mathematics.
