100 years of Zermelo's axiom of choice: What was the problem with it? (2006)

A century ago, mathematician Ernst Zermelo introduced the "axiom of choice," a seemingly simple idea that became one of the most debated concepts in math. It sounds harmless—given a bunch of sets, you can always pick one item from each—but it led to bizarre, counterintuitive results (like the infamous "Banach-Tarski paradox," where you could theoretically split a ball into pieces and reassemble them into two identical balls). This article revisits the controversy: Was Zermelo’s axiom a necessary tool or a dangerous loophole in logic?