How did Georg Cantor’s set theory change the study of mathematical foundations?

Georg Cantor’s set theory fundamentally transformed the study of mathematical foundations by introducing the revolutionary concept of the actual infinite as a legitimate mathematical object. By treating infinite sets as entities on equal footing with finite ones, Cantor challenged long-standing mathematical traditions that had avoided the "actual" infinite since antiquity.1

His work, which utilized one-to-one correspondence to measure and compare infinite sets, provided a new, precise language for defining complex mathematical concepts. However, because Cantor did not establish rigorous axioms for his theory, it initially lacked the formal structure required for a foundational basis.2134

The discovery of paradoxes—such as those identified by Bertrand Russell—revealed that "naive" set theory was susceptible to contradictions. This necessitated the development of axiomatic set theory, specifically the Zermelo-Fraenkel axioms, to provide a rigorous foundation. Ultimately, Cantor’s work became a cornerstone of modern mathematics, enabling the derivation of nearly all known mathematical systems from the theory of sets.3156