How did set theory become connected to the foundations of mathematics?

Set theory became central to the foundations of mathematics through the work of Georg Cantor, who developed it as a mathematical discipline while investigating infinite sets of real numbers. By treating infinite sets as mathematical objects, Cantor provided a framework that allowed for the definition of numbers and complex mathematical concepts in terms of sets and their elements.21

Gottlob Frege furthered this by attempting to reduce all mathematics to logic, defining integers and other concepts using set-theoretic primitives. However, the discovery of Russell’s paradox—which highlighted contradictions in "naive" set theory—necessitated a more rigorous approach. This led to the development of axiomatic set theory, most notably the Zermelo-Fraenkel axioms, which restricted set formation to avoid such paradoxes.1324

Ultimately, set theory became a foundational language for mathematics, with the Nicolas Bourbaki group later noting that nearly all known mathematics could be derived from this single source.2