How did Georg Cantor develop set theory?

Georg Cantor developed set theory between 1874 and 1897, transforming the study of infinite sets into a rigorous mathematical discipline. His work originated from investigations into trigonometric series and the nature of real numbers.12

Cantor’s approach was revolutionary because he treated infinite sets as completed mathematical objects, rather than merely potential infinities. By utilizing one-to-one correspondence—a technique also employed by Richard Dedekind and influenced by Bernhard Bolzano—Cantor demonstrated that different "sizes" of infinity exist. For instance, he proved that while the set of integers and rational numbers are countable, the set of real numbers is strictly larger.13425

Using his diagonalization argument, Cantor established that the power set of any set is always larger than the original, leading to a hierarchy of transfinite numbers. Although his work initially faced significant resistance from critics like Leopold Kronecker, it eventually became a foundational pillar of modern mathematics.531