Part 4/7:
Building upon Cantor's work, Ernst Zermelo formalized the Axiom of Choice. Recognizing a gap in Cantor's logical structure, Zermelo proved that every set can indeed be well-ordered, given the Axiom of Choice. By conceptually selecting numbers from sets in one fell swoop, Zermelo's proofs demonstrated that understanding mathematics required more than just numbers—it enveloped the inherent logic behind them.
Despite the simplicity of Zermelo's idea, challenges ensued, revolving around whether it was conceivable to declare that an order exists if we couldn't produce it explicitly. This ambiguity incited divisive arguments within the mathematical community.