Set theory forms the foundation of modern mathematics. Explain the basic axioms of Zermelo-Fraenkel set theory with the Axiom of Choice (ZFC). Discuss how mathematical concepts like numbers, functions, and relations can be constructed within set theory. Explore historical paradoxes in set theory (like Russell's paradox) and how they were addressed through axiomatic approaches. Consider alternative set theories and their potential advantages or limitations compared to ZFC.