Cantor and the Creation of Set Theory
Introduction
The history of set theory is unusual in mathematics, in that it can largely be traced back to the work of one man; Georg Cantor. Cantor was born in St Petersburg in 1845, moving to Germany in early childhood. He studied in Berlin, where he attended lectures by Weierstrass and other leading mathematicians of the day. His early research was in Number Theory, but following his later appointment to the University of Halle, he began the work in analysis which inspired his first development of the theory of point sets. Subsequently, he became interested in infinite sets, with the result that in 1874 he published the astonishing proof that the real numbers are nondenumerable, or uncountable.
This marks the beginning of transfinite set theory, a discipline that Cantor was to develop extensively during the next decade. His research was motivated to a large extent by his desire to prove the Continuum Hypothesis; that there are only two infinite sizes represented on the real line. His attack on this problem required him to extend the known number system to include transfinite numbers, and both his definition and justification of these numbers were heavily reliant on set-theoretic properties.
During Cantor’s lifetime, there was much opposition to his revolutionary ideas. His strong religious beliefs meant he sometimes used theology to defend his theory from criticism, with mixed results. It is particularly interesting to note that whilst Cantor gave several definitions of sethood, none were mathematically precise. The lack of rigour here soon led to paradoxes being discovered within his new theory, but Cantor himself never lost faith in his ideas. It is the intention of this essay to examine his substantial contributions to the creation of set theory and thereby to mathematics.
Chapter One - The Origins of Set Theory in Cantor’s Work
Chapter Two - The Development of Set Theory
Chapter Three - Cantor’s Perception of a Set
