Provide a detailed examination of Zermelo-Fraenkel set theory with the Axiom of Choice (ZFC). Discuss the historical development of set theory, from Cantor's naive set theory to the formalization of axioms. Explain each axiom of ZFC in detail and discuss their necessity. Address Russell's paradox and how ZFC avoids it. Explore Gödel's constructible universe L and the independence of the Continuum Hypothesis from ZFC. Discuss the implications of these independence results for the foundations of mathematics.